The attribution of Samson and Delilah: Digital analysis of a

The Attribution
of Samson and Delilah:
Digital Analysis
of a Masterpiece
Erica Gasparini
HAIT Master Thesis series nr. 10-004
THESIS SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF ARTS IN COMMUNICATION AND INFORMATION SCIENCES,
MASTER TRACK HUMAN ASPECTS OF INFORMATION TECHNOLOGY,
AT THE FACULTY OF HUMANITIES
OF TILBURG UNIVERSITY
Thesis committee:
Prof. Dr. E.O. (Eric) Postma
Dr. Ir. P.H.M. (Pieter) Spronck
Dr. M.M. (Menno) van Zaanen
Tilburg University
Faculty of Humanities
Department of Communication and Information Sciences
Tilburg, The Netherlands
August 2010
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Abstract
Digital analysis is an important tool used by art historians in solving artistic issues related to
paintings’ attribution.
This Master thesis employs digital analysis to assess the attribution of the painting Samson
and Delilah, supposedly realized by Rubens in 1609-1610 and now exposed at the National
Gallery in London.
The two research questions (RQ) focus (1) on finding which characteristics properties of
Rubens’ painting style can be object of digital analysis and (2) on understanding to what extent
digital analysis can solve the controversy that since the 80s is centered on this painting. The
starting point is solving the first research question studying Rubens’ style, and identifying those
characteristics that lead to the choice of four digital analysis methods: Gabor analysis, color
analysis, gloss analysis, and entropy analysis. Subsequently, these methods are applied to a
database of pictures including, apart from Samson and Delilah, 17 paintings by Rubens, and 8
works realized by two contemporary artists.
Our study shows that there are some properties, typical of Rubens’ painting style that can
be digitally analyzed, providing a positive answer to the first RQ. Unfortunately, the results
obtained with the application of the four digital analysis methods do not succeed in finding a
clear distinction between Rubens and non-Rubens paintings, and therefore, are not capable of
determining the authenticity of Samson and Delilah. The main reasons that may have given rise
to the disappointing results of the RQ2 are discussed.
It is concluded that given the quality of the data available for this study, digital analysis
could not resolve the Samson and Delilah controversy.
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Table of Contents
Abstract
2
Table of Contents
3
1
5
2
3
4
5
6
Introduction
1.1
The painter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2
The controversy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.3
Research Questions and Research Approach . . . . . . . . . . . . . . . . . . . . . . . .
7
1.4
Thesis Outline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
The ambiguity of Samson and Delilah
9
2.1
The five ambiguous characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
Technical relevancy of the ambiguous characteristics. . . . . . . . . . . . . . . . . .
14
Characteristic properties of Rubens’ painting style
15
3.1
Rubens’ stylistic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.2
Analysis methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Computational analysis methods
19
4.1
Gabor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2
Color Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.3
Gloss Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
4.4
Entropy Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Methodology
24
5.1
Dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
5.2
Pre-processing of the images. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
5.3
Computational methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
5.4
Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
5.5
Evaluation of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
Experimental results
6.1
Results of Gabor analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
29
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6.2
Results of Color analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
6.3
Results of Gloss analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
6.4
Results of Entropy analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
7
General discussion
36
8
Final Conclusions and Future Research
38
References
41
Appendix A
45
Appendix B
51
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Chapter 1
Introduction
Digital analysis is becoming a very important tool for art historians (Johnson et al 2008). Even
though computers will never replace the knowledge and sensibility of an art historian, they can
be used as an outstanding resource for supporting their investigations. In fact, art historians
can rely on the support of computers to solve some of the issues related to paintings. In the
past years, many techniques have been developed for the digital analysis of painted features,
such as patterns, brush strokes, color transitions, and brightness (Johnson et al 2008).
The present research is focused on determining the attribution of the painting Samson
and Delilah, currently exposed at the National Gallery in London and supposedly realized by
Rubens in 1609-1610.
To some extent our analysis will follow the argumentation provided by Ms. Doxiadis, a
Greek art historian specialized in Rubens’ masterpieces (Doxiadis, 2010). Our analysis
constitutes a digital comparison of the painting under consideration with other authentic works
by Rubens and with works realized by two related Flemish artists: Sir Anthony Van Dyck and
Jacob Jordaens.
1.1
The Painter
Peter Paul Rubens (1577 - 1640) is one of the most renowned Flemish painters of the Baroque
period. He started his artistic education as an apprentice at the studio of Otto van Veen, who
turned out to be a very important figure for the artistic development of the young artist. In
fact, thanks to his precise knowledge of anatomy, perspective and in particular of the
principles of chiaroscuro, Van Veen provided Rubens with many valuable hints, not only
related to style, but also regarding composition in general (Noel 2009, p.7). Van Veen
suggested Rubens to complete his education as a painter in Italy.
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During the eight years that Rubens spent in Italy, he had the possibility to study the
masterpieces of many of the most relevant artists of all times: Tintoretto, Titian, Michelangelo
and Caravaggio. Inspired by these artists, Rubens adopted a style in which “he absorbed the
luminosity and dramatic expressiveness of the Renaissance” (“Peter Paul Rubens”, Wikipedia
2010), acquiring the strength of naturalism together with the heroic idealization of forms
(“Peter Paul Rubens”, Wikipedia 2010). The works realized by Rubens in this period are very
different from those realized previously, and contain the soft flesh tones characteristic of his
later works (National Gallery 1997). Upon his return to Antwerp in 1608, the burgomaster of
the city commissioned him to create Samson and Delilah. This painting is peculiar in the sense
that it has some visible stylistic differences compared to other contemporary works of Rubens.
1.2
The controversy
The National Gallery of London acquired the alleged Rubens' painting Samson and Delilah in
1980, for around £2.5 million (Doxiadis 2010). The price paid for this masterpiece makes it the
second most expensive painting ever sold at an auction (Morrison 2002). However, since its
purchase it has been subject of a high-pitched discussion between artists and art historians
(Doxiadis 2010). The artists are convinced that the painting is not an original masterpiece of
Rubens, while the art historians are certain that what can be seen in this painting is the real
touch of the Flemish artist.
The aspects that are considered ambiguous have been brought to the international
attention at first by Kasia Pisarek, a Polish art historian who has been studying Rubens and in
particular the painting of Samson and Delilah (Harlow & Januszczak 1997). These ambiguities
were later reasserted by the Greek art expert Euphrosyne Doxiadis, and her supporters.
The debate focuses on the attribution of the painting, rather than on the fact that a
version of Samson and Delilah was actually created by Rubens around 1609-1610. In fact, Mr.
Nicolaas Rockox, the burgomaster of Antwerp, commissioned to him the realization of Samson
and Delilah exactly in those years, and upon completion, the masterpiece featured in the
residence of the burgomaster, as it is evident from numerous documents (Gomez 2005,
Doxiadis 2010). After Rockox’s death, in 1640, the painting was auctioned off. From then on
almost every track attesting the existence of the masterpiece was lost. The only known track is
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a reference of 1674 in the collection of the prince of Liechtenstein, where a painting with the
same title, but initially known as Caritas Romana, was attributed to one of Rubens' students
(Gomez 2005). Samson and Delilah appeared again in Paris in 1929, where it was initially
assigned to a Dutch artist of the 17th century. However, Ludwig Burchard (Gomez 2005)
recognized it as the famous and lost masterpiece of Rubens. Subsequently, it was sold to a
German tobacco dealer, who kept it in his private collection until 1980 (Harlow & Januszczak
1997) when it was purchased by the National Gallery.
There are two different reproductions of Samson and Delilah created by two different
artists, which, therefore, can be considered witnesses of the existence of the painting by
Rubens. The issue then is not if the painting itself existed, but what it looked like. In fact, these
two depictions may be taken as evidence that the original painting was probably different from
the one displayed in London. Jacob Mathaman (1571-1631) realized the first depiction: he
copied Samson and Delilah by Rubens in an engraving on copper plate only a few years after
the artist painted it. The second depiction is a reproduction of Samson and Delilah in a painting
representing Rockox's sitting room: Supper at the House of Burgomaster Rockox, made by the
painter Frans Francken II (1581-1642) around 1630 (Doxiadis 2010).
The attribution of the painting can also be considered through its comparison with other
contemporary works by Rubens. These comparative works will be presented in the next
chapters. Some differences in style can then be noticed, such as the peculiar use of
brushstrokes, the choice of colors or the technique used to paint shapes.
1.3
Research Question and Research Approach
Considering the different methods of digital analysis that are available for art historians,
namely, those based on the analysis of digital reproductions of paintings, and considering that
every artist has some characteristic properties that identify their own painting style, the first
research question reads as follows:
(RQ1) Which are the characteristic properties of Rubens’ painting style that can be the
object of digital analysis?
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Once provided an answer to the first research question, and keeping in consideration the
debate that since the 80s is centered on the famous painting of Rubens, the second research
question reads as follows:
(RQ2) To what extent can digital analysis techniques resolve the Samson and Delilah
controversy?
The research questions are addressed by (1) reviewing Rubens’ painting style in order to
identify characteristic properties of his paintings, (2) determining the digital analysis methods
for measuring these characteristics, and (3) applying the methods to a digital reproduction of
Samson and Delilah, other works by Rubens and related artists.
1.4
Thesis Outline
This work is organized in eight chapters. The present chapter describes the controversy and
formulates the two research questions, followed by a section concerning a short description of
the approach used in the analysis of the paintings.
The second chapter summarizes the ambiguities of Samson and Delilah, relying mainly on
the studies of Ms. Doxiadis (2010).
Chapter 3 answers the first research question, presenting six properties that are typical of
Rubens’ style, which will underpin the development of the digital analysis. These properties
will then be associated with four digital analysis methods (3.2): Gabor analysis, color analysis,
gloss analysis, and entropy analysis. Chapter 4 explains these methods.
The fifth chapter presents the methodology: the database of digital reproductions of
paintings that have been considered in this study (5.1), the pre-processing of the images (5.2),
the computational methods (5.3), the experiments (5.4), and the evaluation of the results (5.5).
The sixth chapter is about the experimental results and provides an answer to the second
research question. Chapter 7 consists of a general discussion of the results obtained, presenting
the limitations of our approach. The last chapter presents the conclusions and future research.
In the two Appendices, the dataset of paintings (Appendix A) and tables related to the
digital analysis methods (Appendix B) are provided.
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Chapter 2
The ambiguity of Samson and Delilah
It is a unique style, the Rubenesque. It is
based on a systematic glorification of energy, a
passionate cult of movement, and a quest for
the dynamic. (Néret 2004, p.11)
This chapter reviews the painting Samson and Delilah as displayed in the National Gallery,
focusing in particular on those details that are of relevance to the attribution of the painting.
Euphrosyne Doxiadis (2010) identified five characteristics that support her claim that
Rubens did not paint this version of Samson and Delilah: the figure of Samson, the old woman,
the head of the barber, the statue of Venus and Cupid, and the carpet.
2.1
The five ambiguous characteristics
The five ambiguous characteristics used by Ms. Doxiadis to support her theory concerning
Samson and Delilah are (1) the figure of Samson, (2) the old woman, (3) the head of the barber,
(4) the statue of Venus and Cupid, and (5) the carpet.
1.
The first element that is worth considering is the figure of Samson (Figure 2.1), the
biblical hero who is sleeping on the lap of his beloved,
unaware of what is happening around him. While
comparing the actual Samson and Delilah exposed at the
National Gallery with the engraving realized by Jacob
Mathaman or the reproduction by Frans Francken II, it can
Figure 2.1 Samson and Delilah
(1609-10), detail. Redrawn after
Doxiadis 2010.
be immediately noticed that the foot of Samson does not
appear entirely in the picture, it has been truncated (Figure
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2.5). This aspect can be considered controversial if we compare it with the deep attention that
Rubens has always paid for the human body and the importance that he gives to its
completeness (Gomez 2005). Even through some toes Rubens has the ability to show the
strength of the characters; the fact that Samson's foot is cut represents an important “missed
opportunity” (Doxiadis 2010). This is quite unlikely for Rubens, who “strove to perfect his
representation of the human body” (Freedberg 1998, p.32).
The reason why Samson's foot is important does not relate
only to the fact that it has been cut; this detail should be
considered also from a different perspective, which regards the
execution of the technique. It can be compared with some other
Rubens’ depictions of feet, like for example the men's feet in The
Raising of the Cross, realized around 1610-1611 (Figure 2.2).
As far as the figure of Samson is concerned, the analysis of
his foot is not the only noteworthy element: his head, in fact, has
Figure 2.2 Men's feet in
The Raising of the Cross.
Reproduced from Doxiadis
2010.
been realized with a pictorial style that does not recall any of the previous works by Rubens. In
this regard, emphasis will be given in particular to the details concerning his left ear and his
hair. We could say that they seem to be in antithesis with the characteristic way in which
Rubens was used to realize these details. The technique that he was using, according to Jane
Pack, painter and professor specialized in Rubens' painting techniques, consisted in building up
“watery layers of transparent oil paint that can give a painted image a distinct luminosity”. It is
after this consideration that Prof. Pack asserts that the Samson and Delilah has been painted
“with a post-impressionist mindset, although it is meant to be a baroque painting” (Gomez
2005).
Samson's head (Figure 2.3) can be compared with the head of Christ (Figure 2.4) from The
Raising of the Cross (1610-1611). It can be noticed that the first one is represented as a
superficial block, while the second one shows the “impetuous power of Rubens’ brush”
(Doxiadis 2010). The beard of Samson, as well, seems to be shapeless and not realistic: it is too
solid and defined to look like a real beard. On the other hand, Christ's beard is represented
with realistic brushstrokes: we can notice the light going through it, giving rhythm and
likelihood to the whole face, together with a strong expressive power (Doxiadis 2010).
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Figure 2.4 Christ's head from
The Raising of the Cross.
Reproduced from Doxiadis
2010.
Figure 2.3 Samson's head from
Samson and Delilah.
Reproduced from Doxiadis
2010
Samson and Delilah
Mathaman's engraving
F. Francken II's reproduction
Figure 2.5 Comparison between the three different representations of Samson's foot. Reproduced from
Doxiadis 2010.
2.
The second analyzed figure is the old woman, who finds her place in the painting just
behind Delilah. If we consider the two reproductions of the Samson and Delilah realized during
Rubens' time, an extremely interesting detail can be noticed: both in the engraving by
Mathaman and in the scaled reproduction by Frans Francken II, the old woman occupies the
same position in the painting, that differs from the one held in Samson and Delilah. In fact, the
head of the old woman, according to the two reproductions, finds its place right behind the
figure of Delilah, over her; while in the canvas exposed at the National Gallery the woman finds
her place behind the girl, but on the left (1992 Report). As far as the technique is concerned
and in comparison with paintings like The Raising of the Cross, it can be noticed that there are
some differences regarding the brushstrokes and the use of colors that give intensity to the
expression on the woman's face (Figure 2.6). Indeed, once again the figure represented in the
Samson and Delilah is lacking harmony and it seems to be unnatural, an unrealistic depiction of
a person: “broken irregular brush marks and tonal confusion provide only a rudimentary
approximation of form” (Doxiadis 2010).
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Samson and Delilah
The Raising of the Cross
Figure 2.6 Comparison of the old woman's head. Reproduced from Doxiadis 2010.
3.
Together with the figure of Samson and the old woman, also the barber gives a very
important insight into the way Rubens used to represent some details of the human body, in
particular head and hair. The head of the barber, according to Doxiadis' analysis (2010), can be
compared to the head of one of the men who are holding Christ's cross in The Raising of the
Cross (Figure 2.7). The man is represented with a “rhythmic, almost symphonic orchestration of
line and form, which brings the character so powerfully to life” (Doxiadis 2010), while the
barber's head, even if it is depicted in the same position, does not have equal intensity.
Moreover, the representation of the skin itself is very different, and it goes from very realistic
(in The Raising of the Cross) to a waxy and unnatural use of color and brushstrokes (in Samson
and Delilah).
Samson and Delilah
The Raising of the Cross
Figure 2.7 The head of the barber, detail extracted from the painting Samson and Delilah, and the head
of the Man, detail extracted from The Raising of the Cross. Reproduced from Doxiadis 2010.
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4.
The fourth interesting aspect to consider is the statue in the background. It represents
Venus and Cupid. Compared to other examples of statues painted by Rubens, it seems that
this one does not have a proper structure. For example, the roundness of the female body, a
typical distinguishing aspect of the Flemish painter, who had a deep
interest in showing full-figured women (“Peter Paul Rubens”,
Wikipedia 2010), is almost completely missing. The body of Cupid can
be considered in the same way, since it is lacking softness and
completeness of shape, two very important features in all Rubens'
works (Doxiadis 2010). During his period in Italy, Rubens developed
the artistic thread that he would follow through his whole life, that is
“his conviction that the prime and most noble task of the painter was
to bring the great works of ancient sculpture to life by recasting them
in pictorial form” (Freedberg 1998, p.32). In this way, he dedicated
himself “to the study and understanding of ancient Greek sculpture”
Figure 2.6 Venus and
Cupid from Samson
and Delilah.
Reproduced from
Doxiadis 2010.
(Freedberg 1998, p.33), which was highly monumental and realistic.
The style of the brushstrokes is completely different from other paintings by Rubens
representing statues. For example, Venus and Cupid can be compared with Seneca in The Four
Philosophers (1611-1612). As far as the representation of light is concerned, it can be noticed
that Venus and Cupid have only some shadows coming from the left side, which do not give
them any realistic look, any softness of the shape, while Seneca's face is totally depicted
thanks to the way in which light falls (Doxiadis 2010), Figure 2.8.
Samson and Delilah
The Four Philosophers
Figure 2.8 Two different ways of
representing a statue: Samson and
Delilah and The Four Philosophers.
Reproduced from Doxiadis 2010.
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5.
The last aspect considered is the carpet that lies under Samson's legs. If compared with
the carpet represented in The Four Philosophers (1611-1612) (Doxiadis 2010), it is possible to
notice a fundamental difference in the realization of the texture. In fact, in the painting just
named, the carpet that covers the table where the four men are sitting at is represented in a
very natural way: it is possible to recognize the movement of the fabric, which is very well
expressed through the brushstrokes. On the other hand, the carpet depicted in Samson and
Delilah has a different style: the carpet’s pattern does not show any relation with the
movement of the fabric represented. It is then not possible to identify the impact in the
representation of the texture, which should be given by the presence of Samson’s foot.
2.2
Technical relevancy of the ambiguous characteristics
Among all the elements just presented, it is worth considering the fact that not all of them are
amenable to digital analysis. In fact, some of them are relevant only from an artistic point of
view. Two examples can be mentioned: the first one is Samson's foot: the fact that it has been
cut can be considered really important knowing that is not highly common that masterpieces
are cut during centuries. Therefore, this element can be a good starting point for the debate
that has been going on until the 80s among art historians and painters, but it is not a feature
that can be taken into consideration when digital analysis is involved. The second example is
the position of the characters in the scene. This is something that is strictly related to a
subjective analysis of the masterpieces and eventually a comparison between the same and
the two witnesses of Mathaman and Frans Francken II (see section 1.2), and is not yet
amenable to digital analysis.
In contrast, the heads of Samson, the barber and the old woman can be object of digital
study, since they can be used as patterns of comparison with the different representations of
heads realized by Rubens. Moreover, also the last element presented, the carpet, can be
considered a useful pattern: in fact, the texture of the brushstrokes can be extracted in order to
allow a digital analysis of similar details represented in different paintings of the same artist.
Therefore, it can be concluded that everything that concerns brushstrokes, light, contrast
and pattern recognition are potential elements of computer based analysis of paintings
through machine learning.
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Chapter 3
Characteristic properties of Rubens’ painting style
This chapter is aimed to answer the first research question: Which are the characteristic
properties of Rubens’ painting style that can be the object of digital analysis? During the
studies conducted throughout this thesis, some relevant properties that are common in
(almost) all the paintings by Rubens, and that can be considered characteristic of his style,
have been recognized, and their importance consists in the fact that they can be the object of
digital analysis.
The properties that distinguish Rubens’ paintings from the works of other artists, and that
are amenable to digital analysis are six:
1. Use of non-bright colors;
2. Diagonal artistic compositions;
3. Gradual transition at contours;
4. Sparsity of paint;
5. Use of specific colors;
6. Use of highlights;
Each of these stylistic properties is associated with a particular analysis method, shortly
described in section 3.2.
3.1
Rubens’ stylistic properties
As abovementioned, the characteristic properties that are peculiar of Rubens’ painting style
and on which the digital analysis will focus are six:
1. The first property concerns the use made by Rubens of non-bright colors. Vander
Auwera & van Sprang (2007) state in their book “Rubens: A Genius at Work” that the
technique applied by the painter consisted in the employment of pale tones. The purpose was
- 15 -
that of leaving in a less advanced stage of coloring the characters that had a minor relevance in
the painting. This technique was aimed to create an impression of physical and, at the same
time, psychological distance in the observer. Similarly, Rubens used to perfect the modeling
and color nuances applying “layers of opaque paint, scumbles, or glazes” (Vander Auwera
2007, p.162) in order to strengthen the contours of the shapes. Moreover, he often allowed
the grayish ground layer to show through in the areas of shadow and half tones.
2. The second property regards the realization of diagonal artistic compositions. This
peculiar aspect can be considered as a clearly visible influence of Baroque’s dynamism, which
is fully embodied by Rubens, to the point that he has been defined as its greatest exponent
(“Peter Paul Rubens”, Wikipedia 2010). In almost all the representations that depict a nonstatic scene, Rubens expresses the motion through a diagonal arrangement of the figures,
which can be noticed in particular from the direction given to the human bodies (Hoocher
2010).
3. The third property deals with the gradual transition of the contours, in particular in
the representation of female figures. In fact, they are not defined with sharp contours. There is
a total integration of the elements of the painting with their surrounding background and the
shapes are usually suggested by the use of faded colors (Parodos 2009).
4. The fourth property concerns the use of specific colors. According to an analysis1 led
on some pigments retrieved in a trunk from Rubens’ studio, it has been asserted that Rubens
was using only few but very peculiar colors to realize his canvases. Through this limited
palette, he could obtain a wide spectrum of nuances. This specific colors are:
White
Lead white
Blue
Genuine ultramarine [lapis lazuli]
Azur d'Allemagne (cobalt)
Black
Ivory black
Red
Madder
Vermilion
Red ochre
Green
Vert azur (oxide of cobalt)
Terre verte [green earth]
Malachite green
Yellow:
Orpiment
Yellow ochre
Yellow lake
Brown
Burnt sienna
Table 3.1 The colors used by Rubens. The main shades are indicated in bold. The pigments are indicated
in italics.
1
The study has been conducted by Hilaire Hiler, as cited in Natural Pigments (2010).
- 16 -
It is worth mentioning that special attention will be given to the different nuances of red. In
fact, in the Report on the Samson and Delilah for the National Gallery (1992), it is stated that
the skirt of Delilah has been realized with “a harsh alizarin crimson” (Report 1992, p.6) that
contrasts with the “warm red ochre and vermilion” (Report 1992, p.6) used in the Francken
reproductions to depict the same detail.
5. The sparsity of paint represents the fifth main property. It is based on Nico van Hout’s
observation that Rubens used to give shape to the figures with a few brushstrokes only (van
Hout personal communication 2010). During my personal study on Rubens’ masterpieces, I
could notice that his style is not characterized by the use of too many details. In fact,
particularly the representations of eyes, noses, and ears show that in the majority of the cases
they were not depicted in detail. They were composed only by few brushstrokes to suggest the
idea of what was represented. Therefore, it is mainly thanks to the context that the observers
can perceive what has been depicted.
6. The last property that will be considered concerns the use of highlights. This property
can be linked to the previous ones regarding the use of non-bright colors and the use of
multiple layers of paint (point 3 and 4). In fact, when Rubens realized the underpainting he
used to apply “few, thick, dry highlights” (Vander Auwera & van Sprang 2007, p. 161), which
represented a point of reference in the realization of his work and remained visible through
the different layers of paint.
3.2
Analysis Methods
The stylistic properties of Rubens’ paintings presented in the previous section are associated to
four analysis methods: Gabor analysis, color analysis, gloss analysis, and entropy analysis.
The first property, which concerns the use of non-bright colors, will be analyzed through
color analysis in combination with the measure of gloss. The first one will focus in particular on
the bodies, mainly on the brightness of the colors used to represent the skin of the different
characters belonging to the various pictures. The second one will verify the brightness of the
same colors according to their reflectance.
- 17 -
Properties two and three, which regard the realization of diagonal artistic compositions
and the gradual transition of the contours, respectively, will be measured through the Gabor
analysis, which is used to assess the orientation of the elements of an image, and to retrieve
the presence (in this case it is better to say the absence) of edges.
The use of specific colors represents the fourth property, and it will be verified with color
analysis. It differs from the first property because what is going to be considered does not
involve the brightness of the colors but the nuance that is given by the use of different
pigments. Color analysis is not invasive, so it will not be possible to find out which peculiar
pigments have been used by Rubens, but it will be possible to notice any difference in the
nuances belonging to the same class of color.
The fifth property, the sparsity of paint, will be studied through Gabor analysis, which will
assess the distribution of the brushstrokes and through entropy analysis. The relevance of the
use of these methods is given by their possibility to attest the presence of highly detailed areas
in the painting.
The last property concerns the use of highlights. The method that will be used to verify this
property is color analysis, with a particular focus on the presence of light elements surrounded
by darker ones.
A detailed explanation of the analysis methods associated to the abovementioned
different properties of Rubens’ style will be provided in chapter 4.
- 18 -
Chapter 4
Computational analysis methods
This chapter explains the different methods that will be used in the digital analysis of Samson
and Delilah. These methods are Gabor analysis (4.1), color analysis (4.2), gloss analysis (4.3),
and entropy analysis (4.4). The previous chapter has already introduced the methods. The
association of the analysis methods to the six typical properties of Rubens’ style can be
summarized as shown in Table 4.1.:
Rubens’ stylistic properties:
1 Use of non-bright colors
2 Diagonal artistic compositions
3 Gradual transition at contours
4 Use of specific colors
5 Sparsity of paint
6 Use of highlights
Analysis method:
Color and Gloss
Gabor
Gabor
Color
Gabor and Entropy
Color
Table 4.1 Association of Rubens’ stylistic properties to their related analysis methods.
4.1
Gabor Analysis
In image processing Gabor analysis is used for the detection of edges, and it consists in
measuring the grade of spatial frequency in a localized area, showing the points where there
are differences in contrast, indeed, where edges are present.
Edges are “locations in an image where intensities change abruptly or at least rapidly2”
(Mallot 2000, p.73), and it usually happens where there are objects’ boundaries. This particular
analysis is insensitive to illumination, so if there is a change in the illumination of an image, the
contrast remains the same (Mallot 2000, p.66).
2
Emphasis given in the original.
- 19 -
This analysis method is used also to detect changes of frequencies and orientations in
images. The representation of frequencies and orientations through the Gabor analysis can be
considered alike to that of the human visual system. In particular, as Hubel & Wiesel assert, the
reaction produced is similar to the way in which the visual neurons of the visual cortex respond
to any change of intensity or orientation of the intensity (as cited in Mallot 2000, p.73).
As showed by Johnson et al. in “Image Processing for Artist Identification” (2008), Gabor
analysis has already been used with successful results in the recognition of paintings by Van
Gogh. In this case, the method has been based on three main principles: the importance of the
contours, the multiple scales analysis of the images and the fact that similarities between
paintings are reflected by the local patterns (Johnson et al, 2008). The digital analysis of
Rubens’ works developed in this thesis will apply the same principles of the Van Gogh analysis.
A broad literature explaining in details the functioning of the Gabor analysis is already
available. For further information, refer to Mallot (2000), Polikar (1996), and Paukner (2007).
4.2
Color Analysis
The analysis of colors, if based on web reproductions of paintings is not always reliable, since
every original picture can have many different digital equivalents whose colors change
according to the light exposition and the camera used. Anyways, it is worth considering this
method as a possibility to retrieve determining features.
When analyzing colors in a digital way, it is necessary to take into consideration the
functioning of the brain’s mechanisms that are responsible in the interpretation of colors3.
Among the most common methods of color analysis in the processing of images, it is
worth mentioning two methods: RGB and HSI. RGB indicates the three colors red, green and
blue, and it is usually represented with a cube (Figure 4.2.1) where the three named colors are
on three corners; cyan, magenta and yellow are on the other three corners; black is at the
origin and white is in the farthest point from the origin. The grey values are located in the line
that joins black and white (Gonzales & Woods, 2001). This model is used as a filter to identify
3
For further information about this topic, refer to Mallot (2000).
- 20 -
the specific colors that compose an image (red, green, and blue) and it is considered ideal for
image color generation (Gonzales & Woods, 2001).
For image color description, the most common method is HSI: this acronym stands for
hue, saturation and intensity, and it corresponds to the way humans can describe colors
(Gonzales & Woods, 2001). Hue stands for the way an observer perceives a color; saturation
represents the amount of white light that is mixed with a hue; and intensity is determined by
the actual amount of light, where more light corresponds to more intense colors (Gonzales &
Woods, 2001). The HSI method is commonly represented through triangular or circular color
planes, where white and black are at the two edges and the intensity is determined by the
position of the plane in the vertical intensity axis (Gonzales & Woods, 2001). For further
information regarding the analysis of colors, refer to Gonzales & Woods (2001).
a)
b)
Figure 4.2.1 Two illustrations of the RGB model: a) schematic representation of the color cube.
Reproduced from Gonzales & Woods, 2001, p.290; b) example of 24-bit color cube. Reproduced from:
http://www.cs.ru.nl/~ths/rt2/col/h2/colorcube.jpg, retrieved July 11, 2010.
- 21 -
Figure 4.2.2 Representation of the HSI model through circular color planes. Reproduced from Gonzales
& Woods, 2001, p.298.
4.3
Gloss Analysis
Gloss is the result of the reflection of light and the interplay between the geometry of the
surface, the surrounding illumination, and the surface reflectance (or albedo), which together
determine the visible properties of a surface (Motoyoshi, Nishida, Sharan & Adelson, 2007).
The different combination of these characteristics creates the glossy appearance of an image.
As observed by Motoyoshi et al. (2007) changes in albedo and gloss are accompanied by
peculiar changes in the luminance histograms. Furthermore, it can be noticed that when the
mean luminance remains the same, the skewness of the histogram increases, both if the
albedo of the surface decreases, and if the gloss augments, Figure 4.1 (Motoyoshi et al. 2007).
Gloss analysis measures the skewness and standard deviation of each image, which are
the result of the computation of their albedo and gloss. Skewness measures the lack of
symmetry. A histogram is considered symmetric if it has the same inclination on both the left
and right side of the central point (Motoyoshi et al. 2007). Standard deviation measures how a
set of data are distributed in relation to their central point; more they are spread, higher is the
deviation compared to average values.
The interested reader can find a more detailed explanation of this technique in Motoyoshi
et al. (2007) and Francken, Cuypers, Mertens & Bekaert (2009).
- 22 -
Figure 4.3 Even if the mean luminance of both stucco-like surfaces represented is equal, the one on
the right looks darker and glossier than one on the left. The luminance histograms show that the image
on the right has a long positive tail while the one on the left has a negative skewness. Reproduced from
Motoyoshi et al. 2007, p.2.
4.4
Entropy Analysis
In the digital analysis of images, the entropy is used to measure the values of randomness or
disorder. It is based on the computation of the similarities of the neighbors of each pixel. If the
local neighborhood contains identical values, it means that the region is homogeneous and the
entropy is low. If the values are very different, the entropy is high. Entropy is a statistical
measure of the randomness or disorder that is commonly used in texture analysis.
The interested reader is referred to Gonzales & Woods (2001) for additional information.
- 23 -
Chapter 5
Methodology
This chapter explains the methodology that has been applied in the development of the digital
analysis. The first section (5.1) presents the dataset on which this study has been based. In
section 5.2, the pre-processing of the images is explained. This part is followed by section 5.3,
where the computation of the four analysis methods presented in chapter 4 is presented, and
by section 5.4, where information concerning the tools used in the experiments can be found.
To conclude the chapter, a description related to the evaluation of the results is provided in
section 5.5.
5.1
The Dataset
This section presents the dataset that has been used in the analysis; it comprises three groups
of paintings:
1.
The first group is made up of seventeen paintings that are officially recognized as real
Rubens. Seven have been chosen according to the study led by the art historian Ms. Doxiadis
and the remaining ten have been chosen among the online database of the Koninklijk Museum
voor Schone Kunsten (KMSKA) in Antwerpen. The paintings belonging to this batch are:
Deposition in the Sepulchre, The Massacre of the Innocents, Artist and his Wife in a
Honeysuckle Bower, Susanna and the Elders, The Raising of the Cross, The Four Philosophers,
Cimon and Pero, The Portrait of Adriana Perez, The Portrait of Nicolaas Rockox, Venus Shivering,
The Holy Family with the Parrot, Return of the Prodigal Son, Maria with the Child, The Last
Communion of St Francis, Christ on the Cross (Le Coup de Lance), Adoration of the Magi, and
The Education of the Virgin.4 In our analyses, whose results are described in Chapter 6, we
4
See Appendix A for further information related to the just mentioned paintings.
- 24 -
used all the paintings, except for The Raising of the Cross, which was omitted from the Gabor
analysis.
2.
The second group is composed of eight masterpieces realized by two contemporaries
of Rubens: Sir Anthony Van Dyck and Jacob Jordaens. The insertion of paintings realized by
these two artists in the current analysis is relevant because both of them were in touch with
Rubens and were influenced by his artistic production. The paintings by Anthony Van Dyck are:
The crowning with Thorns, Portrait of Charles I, Portrait of Maria de Tassis, and Susanna and
the Elders; the works realized by Jacob Jordaens are: Adoration of the Shepherds, Education of
Jupiter, Portrait of a Young Married Couple, and Holy Family and Saint John the Baptist.5
The third and last group is composed only by one painting: Samson and Delilah6. This
3.
masterpiece represents the main object of the present research, and it is labeled as the target
painting7.
5.2
Pre-processing of the images
Before applying the computational methods, it has been necessary to resize the images, since
for some analysis methods, like for example the Gabor Filter, it is fundamental that the pictures
have the same ratio of pixels per centimeter.
After having calculated the ratio for all the images, the smallest value has been selected
and used to rescale the pictures belonging to the three groups. As you can see in Table 1
(Appendix B), the smallest value belongs to The Last Communion of St. Francis by Peter Paul
Rubens, and its ratio is 1,87 pixels per centimeter.
5
Ibidem.
6
For further information about Samson and Delilah, go back to Chapter 2.
7
The expression target painting is in this case used to identify the main object of the research: it
constitutes the starting point of the research questions. Further studies and experiments will be
developed around it, in order to get a result that answers the main issue of the paper.
- 25 -
5.3
Computational methods
This section describes the computation of the four digital analysis methods (Gabor analysis,
colour analysis, gloss analysis, and entropy analysis), which have been presented beforehand in
Chapter 4, and the two methods used to analyze their results: diffusion distance and Multi
Dimensional Scaling (MDS) plot.
Gabor analysis has been chosen to (1) detect the presence of diagonal artistic
compositions, (2) analyze edges, whose absence would determine Rubens’ use of gradual
transition at contours, (3) verify the sparsity of paint, and (4) detect the lack of details. At first,
the resized images have been processed through a global analysis of the wavelet filtering. It has
been performed at 8 scales, from very fine (s=1) to rather coarse (s=8) and 24 orientations (or
dimensions) with steps of 15° counter clockwise, from horizontal (d=1) to almost horizontal
(d=24). This analysis looks for the presence of edges or sharp transitions in the painting. For
every processed file, the global analysis stores as many convolved images as the number of
orientations per scale (24x8=192 convolution images per each painting). This starting analysis
creates, as a result, some histograms, both regular and normalized. The following step is the
computation of the histograms, which consists in reading the convolved images already created
and stored by the first script and determining the maximum value for each scale-orientation
combination. The histograms obtained are then further analyzed in order to have a
multidimensional plot representing the distance between the paintings. Through this analysis
method, it has been possible to obtain a general overview that summarizes the four stylistic
properties of Rubens mentioned in chapter 3.1.
Colour Analysis has been used to (1) detect the use of non-bright colors, (2) analyze the
specific colors used in the paintings, and (3) verify the presence of highlights. This method
consists in assigning to every pixel belonging to an image a value representing its color. The
result is a histogram representing all the values obtained. Subsequently, the histogram
obtained from each image has been compared with those derived from the other elements of
the database, in order to retrieve any similarity. Their distance has been represented in a twodimensional plot.
Gloss Analysis is the method used to test the use of non-bright colors. It has been based
on Motoyoshi et al. (2007) and it computes skewness and standard deviation of every image,
which result from the measure of their albedo and gloss. Skewness measures the lack of
- 26 -
symmetry. For a histogram to be considered symmetric, it needs to have the same inclination
on both the left and right side of the central point (Motoyoshi et al. 2007). Standard
deviation measures the distribution of a set of data from its central point; the more the data
are spread, higher is the deviation compared to average values. The output obtained consists in
a histogram with two values, representing the skewness and the standard deviation, and
defining the location of every image in the two-dimensional plot.
Entropy Analysis has been selected to assess the sparsity of paint and it consists in
calculating the variation of pixels belonging to the same local neighborhood. Since entropy is
computed on the neighbors of each pixel, the result of this analysis is another image where the
pixel values correspond to the local entropy. From the images obtained, it has then been
computed the normalization of the histograms, in order to show the presence of entropy in
every painting. Consequently, the highest peaks of each histogram have been determined and
the coordinates of entropy and frequency have been identified in a two-dimensional plot.
The histograms produced by Gabor analysis, color analysis and entropy analysis are
compared through diffusion distance, a dissimilarity measure to compare histograms that has
been presented by Ling & Okada (2006). This method has been chosen because it has a better
performance in both accuracy and efficiency than any other distance measure (Ling & Okada
2006).
The outcomes obtained after the application of diffusion distance to the results of Gabor
analysis, color analysis and entropy analysis are visualized in a Multi Dimensional Scaling plot
(MDS), which provides a visual representation of the elements of the dataset, based upon their
proximity8 (Kruskal & Wish 1978). The input of the MDS plot is, in our case, the similarity
matrix that results from the computation of each analysis methods. The visual representation
of the relationships between the elements of the dataset is showed on a two-dimensional
space, where elements that are close to each other are considered similar.
The results acquired with the computation of gloss analysis consist in two values per each
element of the dataset. These two values represent the coordinates to visualize each element
in a two-dimensional plot.
8
Proximity is “a number which indicates how similar or how different two objects are, or are perceived
to be” (Kruskal & Wish 1978 p.7).
- 27 -
5.4
Experiments
All the experiments have been done using MATLABTM. The scripts that have been applied in the
four
different
analysis
methods
can
be
retrieved
online
in
the
website:
www.digitalpaintinganalysis.org/Rubens.
5.5
Evaluation of the results
The results obtained from Gabor analysis, color analysis and entropy analysis are represented
on a 2D Multi Dimensional Scaling plot, and those obtained from gloss analysis are visualized in
a two-dimensional plot. Both kinds of representations show the distances between the
paintings, which are graphically illustrated in the plot as points. In order to recognize to which
group every point (or painting) belongs, three different colors have been used:
-
group 1, composed by Rubens paintings, is associated with red;
-
group 2, comprising Non-Rubens paintings, is associated with green; and
-
group 3, the target painting Samson and Delilah, with blue.
The results obtained from the four digital analysis methods abovementioned are
evaluated through statistical analysis of the values indicated in the distance table. The
statistical analysis has been based on the computation of the center of mass of the first and
second group, respectively Rubens paintings (red dots) and Non-Rubens paintings (green dots).
Subsequently, the distribution of the elements of each group has been taken into consideration
to verify their spreading in relation to the center mass (Owens 1997).
As far as the statistical analysis is concerned, no equation is provided in this work, since
the visual representation in the plots is already self-explanatory.
- 28 -
Chapter 6
Experimental results
This chapter presents the computational results, obtained applying the digital analysis
methods to the database of paintings. It is already possible to state that the outcome of this
research does not fulfill the initial expectations. The presentation of the results will be divided
into four sections, as many as the analysis methods.
6.1
Results of Gabor analysis
The Gabor analysis yields orientation-scale histograms. The distances between these
histograms are computed using diffusion distance (Ling & Okada 2006), yielding a distance
matrix where each entry represents the distance between a pair of paintings. To visualize the
distance matrix in a convenient way, we applied multidimensional scaling (MDS) to obtain a 2D
plot. Figure 6.1.1 shows the MDS plot. Each dot represents a painting. The colors of the dots
represent the three classes of paintings: authentic Rubens paintings (red), non-Rubens
paintings (green), and the Samson and Delilah painting (blue). As is evident from the plot, there
is no clear separation between the Rubens and non-Rubens paintings (red and green dots). This
suggests that Gabor analysis is not able to distinguish the stylistic features characteristic of
Rubens and those of other painters. The Samson and Delilah painting (blue dot) is
predominantly surrounded by Rubens paintings (red dots), which indicates that it shares visual
features with authentic Rubens paintings. However, given the lack of separation between
authors (red and green dots), no conclusions can be drawn from this observation. This is
corroborated by a numerical analysis of the distance table. The matrix showing the numerical
values obtained through Gabor analysis has been inserted in the Appendix B (Table 2).
- 29 -
Figure 6.1.1 MDS plot of the diffusion distances between Gabor histograms.
6.2
Results of Color analysis
Color analysis produces hue histograms. Figure 6.2.1 displays all the hue histograms for each of
the three classes. Table 6.2 presents the distance table: the first 16 rows and columns indicate
the numerical values belonging to the elements of group 1, Rubens paintings; the next 8
indicate the numerical values belonging to the elements of group 2, non-Rubens paintings; and
the last one shows the numerical values of Samson and Delilah, which belongs to group 3. As
the Gabor analysis, the distances between the histograms were computed using diffusion
distances yielding a distance matrix. Figure 6.2.2 shows the MDS plot created from this
distance matrix. Again, no clear separation between the authors is obtained indicating that
color analysis is not capable of distinguishing between the color features characteristic of
Rubens and non-Rubens paintings. Statistical analysis of the distance table does not reveal a
significant separation between paintings of different authors.
- 30 -
- 31 -
0
0
0
0
0
0
0
0
0
0
0
0
0
0,8016
Table 6.2. Distance table of color analysis. The first 16 paintings belong to group 1 (Rubens paintings), the next 8 to group 2 (Non-Rubens), and the last 1 to group 3.
0,5746 0,7691 0,5182 0,6141 0,6933 0,5936 1,2294 1,1077 1,0688 0,5155 1,0134 1,6998 0,8548 1,2521 0,9768 0,6531 1,3176 0,8627 0,8341 1,2225 1,4095 0,9395 1,0193 1,7684 0,8016
0
1,8088 1,7684
1,7968 1,0498 1,0193
0,4554 1,2455 0,5709 1,0030 0,9767 1,1434 1,4811 1,3963 1,2046 0,8914 1,0817 1,8190 1,1642 1,4781 1,0170 1,1658 1,5435 1,1894 1,2200 1,3405 1,5677 1,1724 1,0498 1,8088
1,6809 1,7564 1,8070 1,7914 1,5014 1,6351 1,3380 1,6462 1,7858 1,9077 1,6093 0,9432 1,4651 1,3487 1,6430 1,6862 1,7065 1,7547 1,7625 1,3340 1,3239 1,4761 1,7968
0
0,9860 1,4761 1,1724 0,9395
1,2039 1,3232 1,3239 1,5677 1,4095
0,8200 1,1980 0,7349 1,0569 0,8707 1,1363 1,3294 1,1182 1,0345 1,0236 0,8618 1,6835 1,1688 1,2199 0,6647 1,0221 1,4010 1,0673 0,9927 1,2427 1,3232 0,9860
0,8832 1,0232 0,7630 0,7265 0,5271 0,8582 0,7654 0,8758 0,6116 1,1129 0,3868 1,5121 0,9395 0,5601 0,5335 0,6973 1,4319 1,1246 1,2327 0,6880 1,2039
0
1,2285 0,6880 1,2427 1,3340 1,3405 1,2225
1,5021 1,0930 1,2327 0,9927 1,7625 1,2200 0,8341
1,3867 1,3331 1,4634 1,4219 1,2287 1,2502 1,1633 1,0031 1,4496 1,6360 1,2974 1,1841 1,2398 1,2875 1,2909 1,2305 1,1884 1,1623 1,0930 1,2285
1,1058 1,2494 1,0920 0,9468 0,8162 1,0921 0,9945 1,0072 0,8952 1,3557 0,7035 1,3055 1,2373 0,8901 0,8486 0,9578 1,5071 1,2993 1,5021
0
0,7271 1,2993 1,1623 1,1246 1,0673 1,7547 1,1894 0,8627
0,9979 1,1170 1,5071 1,1884 1,4319 1,4010 1,7065 1,5435 1,3176
1,0920 0,8347 1,0389 1,0800 1,0755 0,9093 1,3756 0,9602 1,3731 1,0382 1,3593 1,4639 1,0119 1,4675 1,2833 0,8073 1,1170 0,7271
1,0143 0,7759 1,0108 1,0096 0,9983 0,9599 1,3083 0,8810 1,2040 1,0960 1,1708 1,5071 1,0228 1,3359 1,1124 0,7346 0,9979
0
1,1501 0,7346 0,8073 0,9578 1,2305 0,6973 1,0221 1,6862 1,1658 0,6531
0,7258 1,4014 1,1124 1,2833 0,8486 1,2909 0,5335 0,6647 1,6430 1,0170 0,9768
1,3991 1,1494 1,4344 1,4171 1,3593 1,2953 1,3625 0,8584 1,5140 1,5425 1,4698 1,5857 1,3016 1,4918 1,4014 1,1501
0,8861 0,6217 0,7355 0,5498 0,5572 0,5470 0,9179 0,7599 0,8305 0,8916 0,7601 1,6016 0,7752 0,8939 0,7258
0
0,7475 0,8939 1,4918 1,3359 1,4675 0,8901 1,2875 0,5601 1,2199 1,3487 1,4781 1,2521
1,0262 1,1345 0,7752 1,3016 1,0228 1,0119 1,2373 1,2398 0,9395 1,1688 1,4651 1,1642 0,8548
0,7487 1,1033 0,6786 0,7239 0,5873 0,9478 0,8677 0,9813 0,6042 1,1012 0,3437 1,6262 1,1345 0,7475
1,2260 1,2283 1,0923 0,9684 0,6483 1,0297 0,4757 0,9954 0,8211 1,4284 0,6311 1,4969 1,0262
0
1,5368 1,4969 1,6262 1,6016 1,5857 1,5071 1,4639 1,3055 1,1841 1,5121 1,6835 0,9432 1,8190 1,6998
1,6478 1,1080 0,6311 0,3437 0,7601 1,4698 1,1708 1,3593 0,7035 1,2974 0,3868 0,8618 1,6093 1,0817 1,0134
0,9421 0,9986 0,9010 0,9156 0,6839 0,5331 1,0326 1,0608 1,2529 0,9297 1,1080 1,5368
1,6668 1,6553 1,7890 1,7961 1,5533 1,5971 1,4448 1,5503 1,7920 1,8572 1,6478
0
1,1216 1,8572 0,9297 1,4284 1,1012 0,8916 1,5425 1,0960 1,0382 1,3557 1,6360 1,1129 1,0236 1,9077 0,8914 0,5155
1,1917 0,5354 1,7920 1,2529 0,8211 0,6042 0,8305 1,5140 1,2040 1,3731 0,8952 1,4496 0,6116 1,0345 1,7858 1,2046 1,0688
0,8371 1,1351 0,7423 0,7002 0,5202 0,9711 0,7897 0,9245 0,5354 1,1216
0,6788 1,1129 0,6855 0,7175 0,8478 0,7500 1,4575 1,3908 1,1917
0
1,0332 1,3908 0,9245 1,5503 1,0608 0,9954 0,9813 0,7599 0,8584 0,8810 0,9602 1,0072 1,0031 0,8758 1,1182 1,6462 1,3963 1,1077
0,9722 0,8733 1,4575 0,7897 1,4448 1,0326 0,4757 0,8677 0,9179 1,3625 1,3083 1,3756 0,9945 1,1633 0,7654 1,3294 1,3380 1,4811 1,2294
0,9968 1,1566 0,9023 0,6547 0,7242 1,0588 0,8733 1,0332
1,1782 0,9445 1,1307 0,9504 0,9278 0,9548 0,9722
0
0,9723 0,9548 1,0588 0,7500 0,9711 1,5971 0,5331 1,0297 0,9478 0,5470 1,2953 0,9599 0,9093 1,0921 1,2502 0,8582 1,1363 1,6351 1,1434 0,5936
0,6447 0,7549 0,9278 0,7242 0,8478 0,5202 1,5533 0,6839 0,6483 0,5873 0,5572 1,3593 0,9983 1,0755 0,8162 1,2287 0,5271 0,8707 1,5014 0,9767 0,6933
1,2435 1,2320 1,2281 0,9357 0,7549 0,9723
0,8692 0,8261 0,7865 0,6462 0,6447
0
0,5762 0,6462 0,9357 0,9504 0,6547 0,7175 0,7002 1,7961 0,9156 0,9684 0,7239 0,5498 1,4171 1,0096 1,0800 0,9468 1,4219 0,7265 1,0569 1,7914 1,0030 0,6141
0,6359 0,5687 0,7865 1,2281 1,1307 0,9023 0,6855 0,7423 1,7890 0,9010 1,0923 0,6786 0,7355 1,4344 1,0108 1,0389 1,0920 1,4634 0,7630 0,7349 1,8070 0,5709 0,5182
0,6761 0,9536 0,5687 0,5762
0,7611 0,9481 0,6359
0
0,9782 0,9481 0,9536 0,8261 1,2320 0,9445 1,1566 1,1129 1,1351 1,6553 0,9986 1,2283 1,1033 0,6217 1,1494 0,7759 0,8347 1,2494 1,3331 1,0232 1,1980 1,7564 1,2455 0,7691
1,0490 0,3349 0,7611 0,6761 0,8692 1,2435 1,1782 0,9968 0,6788 0,8371 1,6668 0,9421 1,2260 0,7487 0,8861 1,3991 1,0143 1,0920 1,1058 1,3867 0,8832 0,8200 1,6809 0,4554 0,5746
0,3349 0,9782
1,0490
0
Figure 6.2.1 Visual representation of the hue histograms, showing the different shades of colors.
Figure 6.2.2. MDS plot of the diffusion distances between hue histograms.
- 32 -
6.3
Results of Gloss analysis
Gloss analysis yields two values per painting: the skewness and the standard deviation of the
distribution of intensity values. In Figure 6.3.1 these two values define the coordinates of the points
in a scatter plot. As in the other analyses, the three classes of paintings are represented by colored
dots. The plot reveals that the Rubens and Non-Rubens paintings form overlapping distributions of
skewness and standard-deviation values. The questionable painting (blue dot) lies in the middle of
the two overlapping distributions and hence, it cannot be attributed to the Rubens or non-Rubens
class with any confidence. Table 6.3 shows the numerical values representing the coordinates of
each point, which have been visualized in the two-dimensional plot (Figure 6.3.1).
Group 1: Rubens paintings
Stand.Dev. 0,9223 1,6256
1,2486 1,8713 1,0506 1,6112 1,0680 1,1228 0,5979 1,6110 0,8657
Skewness 0,2055 0,1678
0,2046 0,1790 0,2431 0,2100 0,2184 0,2210 0,2287 0,2060 0,2541
Group 1: Rubens paintings
Group 2: Non-Rubens paintings
Stand.Dev. 0,6022 2,9363 2,5163 1,1229 1,4453 1,9871 1,1326 1,6145 1,3230 1,6886 1,1773
Skewness
0,2675 0,1469 0,1457 0,1708 0,2162 0,2201 0,2010 0,1994 0,2399 0,2036 0,2026
Group 2: Non-Rubens Group 3:
paintings
S&D
Stand.Dev. 1,7363 1,2813 1,7999 1,6462
Skewness 0,1639 0,2282 0,2098
0,2013
Table 6.3 Numerical representation of standard deviation and skewness per each painting, which are
visualized in the two-dimensional plot (Figure 6.3.1)
Figure 6.3.1 MDS plot of the diffusion distances between gloss histograms.
- 33 -
6.4
Results of Entropy analysis
The results of entropy analysis give rise to entropy histograms, which were processed in the same
way as in Gabor analysis and color analysis. The distance table is shown in Table 6.4: the first 16
rows and columns of the table show the numerical values of the red dots (group 1: Rubens
paintings); the next 8 rows show the numerical values of the green dots (group 2: Non-Rubens
paintings); and the last one shows those of the blue dot (group 3: Samson and Delilah).
The resulting MDS plot is depicted in Figure 6.4.1. The separation between Rubens and nonRubens paintings seems to be somewhat better than for the other analyses, but the location of
Samson and Delilah with respect to the Rubens and non-Rubens paintings does not allow for any
firm conclusion with respect to its authenticity. Statistical analysis of the distance table confirms the
lack of statistical evidence for authorship.
Figure 6.4.1 MDS plot of the diffusion distances between entropy histograms.
- 34 -
- 35 -
0
0,0870
0
0
0
0
0
0
0
0
0
0
0
0
0,0724
Table 6.4 Distance table of entropy analysis. The first 16 paintings belong to group 1 (Rubens paintings), the next 8 to group 2 (Non-Rubens), and the last to
group 3.
0,3918 0,0870 0,1855 0,0850 0,3760 1,0343 0,2268 0,1678 0,3131 0,0804 0,1522 0,4435 0,2808 0,0533 0,1830 0,1836 0,4204 0,1625 0,1870 0,1783 0,3969 0,3638 0,1010 0,4146 0,0724
0
0,3052 0,4146
0,2454 0,0667 0,1010
0,4465 0,0504 0,2079 0,0592 0,4057 1,1223 0,3059 0,1440 0,3671 0,0275 0,1797 0,5289 0,1570 0,0742 0,2306 0,2967 0,3092 0,1746 0,1239 0,2214 0,4293 0,4259 0,0667 0,3052
1,1489 0,2428 0,7205 0,1827 0,4087 0,9502 0,2172 0,1988 0,6250 0,2704 0,4478 0,8247 0,0898 0,3165 0,7144 0,5319 0,2279 0,1221 0,1005 0,3646 0,2194 0,7817 0,2454
0
0,5978 0,7817 0,4259 0,3638
0,7321 0,2954 0,2194 0,4293 0,3969
0,6906 0,1104 0,3934 0,0658 0,5225 1,1162 0,3248 0,1782 0,5157 0,0651 0,3016 0,7190 0,1305 0,0982 0,3819 0,3776 0,2110 0,1710 0,0772 0,2835 0,2954 0,5978
0,1001 0,2934 0,0819 0,3643 0,1151 0,5824 0,5041 0,2497 0,0288 0,3336 0,0958 0,0464 0,6222 0,2947 0,0704 0,1043 0,9491 0,3688 0,6819 0,1638 0,7321
0
0,2660 0,1638 0,2835 0,3646 0,2214 0,1783
0,2733 0,2151 0,6819 0,0772 0,1005 0,1239 0,1870
1,2470 0,3888 0,7911 0,2620 0,4309 0,4756 0,2289 0,1958 0,5680 0,2952 0,5279 0,6866 0,2245 0,3479 0,7217 0,3717 0,1666 0,2463 0,2151 0,2660
0,4149 0,1587 0,1970 0,1553 0,0891 0,5832 0,1723 0,0738 0,0800 0,1296 0,0871 0,1481 0,2848 0,1477 0,1826 0,0671 0,4156 0,1481 0,2733
0
0,1072 0,1481 0,2463 0,3688 0,1710 0,1221 0,1746 0,1625
0,3950 0,1509 0,4156 0,1666 0,9491 0,2110 0,2279 0,3092 0,4204
0,9112 0,1569 0,5379 0,0908 0,4427 1,0065 0,1773 0,1629 0,5383 0,1205 0,3342 0,7286 0,0934 0,1570 0,5223 0,4047 0,1509 0,1072
0,6097 0,1104 0,3250 0,0596 0,1741 0,8044 0,0992 0,1095 0,2729 0,1430 0,1376 0,4046 0,1468 0,1103 0,3195 0,1964 0,3950
0
0,5975 0,1964 0,4047 0,0671 0,3717 0,1043 0,3776 0,5319 0,2967 0,1836
0,1227 0,8116 0,3195 0,5223 0,1826 0,7217 0,0704 0,3819 0,7144 0,2306 0,1830
1,3523 0,3584 0,8775 0,2898 0,6905 0,8006 0,3941 0,2347 0,7882 0,2372 0,6778 0,9727 0,1556 0,4153 0,8116 0,5975
0,2977 0,2053 0,1514 0,2152 0,1060 0,3090 0,2250 0,1328 0,0746 0,2154 0,0884 0,0952 0,4212 0,1597 0,1227
0
0,1502 0,1597 0,4153 0,1103 0,1570 0,1477 0,3479 0,2947 0,0982 0,3165 0,0742 0,0533
0,1769 0,4956 0,4212 0,1556 0,1468 0,0934 0,2848 0,2245 0,6222 0,1305 0,0898 0,1570 0,2808
0,0530 0,1640 0,0217 0,2331 0,2279 0,8267 0,4770 0,2224 0,1036 0,1874 0,0850 0,1747 0,4956 0,1502
0,3670 0,0559 0,1786 0,0426 0,2687 0,7991 0,2322 0,1335 0,2337 0,0475 0,1145 0,3895 0,1769
0
0,7259 0,3895 0,1747 0,0952 0,9727 0,4046 0,7286 0,1481 0,6866 0,0464 0,7190 0,8247 0,5289 0,4435
0,1284 0,3464 0,1145 0,0850 0,0884 0,6778 0,1376 0,3342 0,0871 0,5279 0,0958 0,3016 0,4478 0,1797 0,1522
0,8544 0,1178 0,5087 0,0909 0,3989 0,7140 0,2854 0,1434 0,5059 0,1191 0,3464 0,7259
0,2406 0,4086 0,1736 0,4592 0,0933 0,4481 0,4541 0,2865 0,0335 0,4238 0,1284
0
0,1455 0,4238 0,1191 0,0475 0,1874 0,2154 0,2372 0,1430 0,1205 0,1296 0,2952 0,3336 0,0651 0,2704 0,0275 0,0804
0,2703 0,0655 0,0335 0,5059 0,2337 0,1036 0,0746 0,7882 0,2729 0,5383 0,0800 0,5680 0,0288 0,5157 0,6250 0,3671 0,3131
0,2144 0,1212 0,0655 0,1321 0,0950 0,8793 0,2602 0,1343 0,0655 0,1455
0,4194 0,0414 0,1940 0,0407 0,3043 0,9077 0,2738 0,0753 0,2703
0
0,1890 0,0753 0,1343 0,2865 0,1434 0,1335 0,2224 0,1328 0,2347 0,1095 0,1629 0,0738 0,1958 0,2497 0,1782 0,1988 0,1440 0,1678
0,1938 0,3544 0,2738 0,2602 0,4541 0,2854 0,2322 0,4770 0,2250 0,3941 0,0992 0,1773 0,1723 0,2289 0,5041 0,3248 0,2172 0,3059 0,2268
0,1987 0,2539 0,1158 0,2987 0,0564 0,6259 0,3544 0,1890
0,4710 0,0864 0,2345 0,0828 0,1573 0,4419 0,1938
0
0,7095 0,4419 0,6259 0,9077 0,8793 0,4481 0,7140 0,7991 0,8267 0,3090 0,8006 0,8044 1,0065 0,5832 0,4756 0,5824 1,1162 0,9502 1,1223 1,0343
0,4903 0,2397 0,1573 0,0564 0,3043 0,0950 0,0933 0,3989 0,2687 0,2279 0,1060 0,6905 0,1741 0,4427 0,0891 0,4309 0,1151 0,5225 0,4087 0,4057 0,3760
0,8090 0,2493 0,4900 0,2022 0,2397 0,7095
1,2206 0,8865 1,0191 0,8851 0,4903
0
0,2721 0,8851 0,2022 0,0828 0,2987 0,0407 0,1321 0,4592 0,0909 0,0426 0,2331 0,2152 0,2898 0,0596 0,0908 0,1553 0,2620 0,3643 0,0658 0,1827 0,0592 0,0850
0,2357 0,2377 1,0191 0,4900 0,2345 0,1158 0,1940 0,0655 0,1736 0,5087 0,1786 0,0217 0,1514 0,8775 0,3250 0,5379 0,1970 0,7911 0,0819 0,3934 0,7205 0,2079 0,1855
0,3889 0,2567 0,2377 0,2721
0,4908 0,0384 0,2357
0
0,1568 0,0384 0,2567 0,8865 0,2493 0,0864 0,2539 0,0414 0,1212 0,4086 0,1178 0,0559 0,1640 0,2053 0,3584 0,1104 0,1569 0,1587 0,3888 0,2934 0,1104 0,2428 0,0504
0,3533 0,0612 0,4908 0,3889 1,2206 0,8090 0,4710 0,1987 0,4194 0,2144 0,2406 0,8544 0,3670 0,0530 0,2977 1,3523 0,6097 0,9112 0,4149 1,2470 0,1001 0,6906 1,1489 0,4465 0,3918
0,0612 0,1568
0,3533
0
Chapter 7
General discussion
This chapter provides a general discussion about the analysis methods used and the results
obtained, explaining the main reasons why the first ones were unsuccessful.
After the computation of the four digital analysis methods (Gabor analysis, color analysis, gloss
analysis, and entropy analysis) diffusion distance has been applied to calculate the distance
between the various elements belonging to the database, in order to have a visual representation
of the results.
With any of the four methods, it has been possible to get successful results. Already through
the visual representation of the results and after leading statistical analyses of the numerical values
composing the distance tables, it can be noticed that no clustering is present. Therefore, we can
assert that no clear distinction between the paintings belonging to the first and the second group,
respectively Rubens and Non-Rubens paintings, can be identified. Consequently, the attribution of
Samson and Delilah cannot be determined, so the second research question does not have a
positive answer.
There are mainly three technical reasons that can explain why the digital analysis did not lead
to successful results, and they concern the quality of the images, color analysis, and the size of the
database.
The first and most important argument pertains to the quality of the images composing the
database. The reproductions of the paintings used in the digital analyses have been retrieved
online, and they have a very low resolution. Unfortunately, the majority of the information that
could be retrieved from the images is not available because of their low quality. Therefore, it is not
possible to apply the digital analysis methods in a successful way. Moreover, if the resolution of the
images were higher, it would have been possible to focus the analysis not only on the complete
paintings, but also on some peculiar details, e.g. comparing different representations of eyes, noses,
ears, and hands.
- 36 -
The second argument is related to color analysis. It was not possible to find any similarity in the
use of colors among the paintings belonging to the same group, and consequently it was not
possible to retrieve any dissimilarity among paintings belonging to different groups. The main
reason is that the images belonging to the database have been retrieved online, and they come
from different sources. This means that the devices used to take snapshots of the paintings are
different; consequently, color calibration and exposition to the light change from image to image.
For this reason, color analysis cannot provide rewarding results.
The third argument concerns the size of the database. Probably a database composed by a
larger number of reproductions, possibly retrieved from the same source, would represent a more
reliable and complete tool of comparison in digital analysis.
- 37 -
Chapter 8
Conclusions and Future Research
This work dealt with the digital analysis of the painting Samson and Delilah, in order to determine its
attribution. Two research questions have been formulated. The first one read as follows: “Which are
the characteristic properties of Rubens’ painting style that can be the object of digital analysis?”.
This question has found its answer in Chapter 3, where six properties, characteristics of Rubens’
style and amenable to digital analysis, have been presented:
1. Use of non-bright colors;
2. Diagonal artistic compositions;
3. Gradual transition at contours;
4. Sparsity of paint;
5. Use of specific colors;
6. Use of highlights;
The digital analysis methods that have been associated to them are Gabor analysis (applied to
property 2, 3 and 5), color analysis (applied to property 1, 4 and 6), gloss analysis (applied to
property 1), and entropy analysis (applied to property 5).
Answering the first research question has been fundamental in approaching the second research
question, which was formulated as follows: “To what extent can digital analysis techniques resolve
the Samson and Delilah controversy?”. In order to answer to it, three steps have been followed: at
first, Rubens’ painting style has been reviewed, in order to identify some characteristic properties of
his paintings that can be object of digital analysis (RQ 1); secondly, four digital analysis methods for
measuring these characteristics have been determined. Subsequently they have been applied to the
database of paintings, composed by the digital reproduction of Samson and Delilah, by some
authentic works by Rubens and by other works realized by contemporary artists.
Unfortunately, the results achieved applying the four digital analysis methods abovementioned
do not allow us to determine the attribution of the target painting. Already from the visual
representation and through the numerical outcomes of each technique it can be noticed that no
- 38 -
clear distinction between Rubens’ and Non-Rubens’ paintings can be retrieved and, consequently, it
is not possible to classify Samson and Delilah as belonging to one of the two groups.
Therefore, the answer to the second research question is that digital analysis is not always a
reliable tool for image processing when the objects of the analysis are paintings’ reproductions.
The main technical limitations that have influenced the results of our analysis (see Chapter 7)
could have been foreseen to some extent. In fact, color analysis has been chosen being aware of its
limitations as a further possibility of assessing the attribution of the target painting because it is a
potentially useful resource. The size of the database could have been extended for example
including more oil paintings by Rubens, adding more works realized by different artists or probably
also extending the range of the Non-Rubens painters to a wider period. The choice of limiting the
database to only 17 Rubens’ masterpieces can be explained saying that this number includes the
paintings presented by Ms. Doxiadis as terms of comparisons and low-scale versions of some highquality reproductions provided by the Rubens’ experts of the Koninklijk Museum voor Schone
Kunsten (KMSKA) in Antwerpen. The main idea was that despite the limited database, enough
information for this research would have been provided. On the other hand, the quality of the
images belonging to the database is something that could not be changed. In fact, it is not easy to
access to high-quality digital reproductions of masterpieces, so it has been necessary to lead the
analysis with the available tools: low-resolution web versions of the paintings.
Future Research
This work represents just a first attempt to determine the attribution of the painting Samson and
Delilah, and it opens up various possibilities for future research.
The analysis methods used, in particular Gabor analysis and color analysis, can give potentially
reliable results, however higher-quality reproductions of the paintings and a broader database are
needed.
If a more consistent batch of Rubens’ reproductions is provided, probably it is not necessary to
include in the analysis works by other artists. Anyways, as far as the choice of side painters is
concerned, it could be interesting to differentiate the database comparing the target painting with
works realized by various painters, ranging from Rubens’ contemporaries to post-impressionists
- 39 -
since, according to Jane Pack, Samson and Delilah seems to be painted with this kind of mindset
(see p.10, the remark of Gomez, 2005)
Moreover, a further study of Rubens’ paintings can be necessary in order to determine some
other relevant properties typical of his style, which can probably be associated to different digital
analysis methods.
As far as the color analysis is concerned, it would be challenging to compare the use of tones
related mainly to details. An interesting starting point is represented, for example, by the
comparison between the different tones used to depict parts of the human bodies like ears, noses,
eyes, and hair.
- 40 -
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Peter Paul Rubens (April 21, 2010). In Wikipedia, The Free Encyclopedia. Retrieved April 26, 2010
from: http://en.wikipedia.org/wiki/Peter_Paul_Rubens
Peter Paul Rubens (2010) In Encyclopædia Britannica. Retrieved June 08, 2010, from:
http://www.britannica.com/EBchecked/topic/511894/Peter-Paul-Rubens
Rooses, M. (1977) L'œuvre de P. P. Rubens: histoire et description de ses tableaux et dessins, Vol. I-V.
Holland: Edition Davaco Soest.
Rubens fetches record £49.5m. (2002, July 11) BBC News World Edition – Entertainment. Retrieved
June 07, 2010. Available at: http://news.bbc.co.uk/2/hi/entertainment/2119451.stm
Rubens Online 2010 (2010) Rubens Online. Retrieved June 06, 2010 from:
http://www.rubensonline.be
Sorensen, L. (n. d.) Burchard, Ludwig. In Dictionary of Art Historians. Retrieved June 06, 2010 from:
http://www.dictionaryofarthistorians.org/burchardl.htm
Sutton, P. C. (1993) The age of Rubens. New York: Harry N. Abrams Incorporated.
The Masterpieces of Jacob Jordaens. (1907) The Hague: M. Hols Publisher.
Vander Auwera, J., van Sprang, S. (2007) Rubens: A Genius at Work: the works of Peter Paul Rubens
in the Royal Museums of Fine Arts of Belgium reconsidered, Lannoo International. Retrieved
- 43 -
July 05, 2010 from:
http://books.google.com/books?id=kfurKXZJ8fIC&printsec=frontcover&hl=it&source=gbs_ge_s
ummary_r&cad=0#v=onepage&q&f=false
Widjaja, I., Leow, W. K., & Wu, F. C. 2003. Identifying painters from color profiles of
skin patches in painting images. In Proceedings of the International Conference on
Image Processing. IEEE, 845—848, Los Alamitos, CA.
- 44 -
APPENDIX
A:
Paintings composing the database
In this section are provided information relatively to the retrieval of the images used in the analysis
and here classified according to the artist and the date of realization.
Peter Paul Rubens:
 Deposition in the Sepulchre (1602), painted by Rubens in 1602, during his stay in Rome. It can
be noticed that the Italian art had a very visible impact on his style, specially the works of Titian and
Caravaggio, in particular concerning the use of colors, light, and chiaroscuro (Kren & Marx 2010).
Retrieved March 1, 2010. Source:
http://hoocher.com/Peter_Paul_Rubens/Rubens_The_Deposition_1602.jpg
 Samson and Delilah (1609-10) Retrieved February 25, 2010. Source:
http://larryavisbrown.homestead.com/files/OT_history/unit2/Rubens_samson1609.jp
 Artist and his Wife in a Honeysuckle Bower (1609-10), dated 1609-10; this marriage portrait was
realized entirely from the artist himself immediately after his wedding with Isabella Brant, his first
wife, upon return from Italy (Sutton 1993, p.24 - Rooses 1977, Vol. IV, p.261). The image was
retrieved March 1, 2010. Source: http://en.wikipedia.org/ wiki/File:Peter_Paul_Rubens_105.jpg
 Susanna and the Elders, also realized in the same period of the previous paintings and the target
one, namely in 1609-10. As far as the attribution of this painting is concerned, it is worth mentioning
that in the Corpus Rubenianum9 there are some uncertainties regarding the attribution of the same.
At the beginning the masterpiece, which was bought as an authentic by the Real Academia de Bellas
Artes de San Fernando (in Madrid) in 1778, was identified at first as a work by Jordaens, a Flemish
artist that was active in the same period; then as “a workshop, retouched by Rubens” and finally as
9
The Corpus Rubenianum is a collection of all Rubens' painting, sketches and drawings and is based on all the
material assembled by Ludwig Burchard, one of the scholars of the Flemish artist (Sorensen). It is composed of
“29 parts, and each one deals with a particular commission or group of subjects… The Corpus aims in this way
to embody all present-day knowledge of the works of Rubens” (Corpus Rubenianum).
- 45 -
“Probably Rubens, Italian period” (Corpus Rubenianum). In a letter dated 28th April 1618 sent by the
painter to Lord Dudley Carleton, we can read that he was offering him, among other paintings, “Una
Susanna fatta de un mio discipolo pero ritocca de mia mano tutta”10 (Hookham Carpenter 1844,
p.144). Retrieved April 9, 2010. Source: http://www.lib-art.com/imgpainting/4/3/16334-susanna-andthe-elders-pieter-pauwel-rubens.jpg
 The Massacre of the Innocents (1609-11), created between 1609 and 1611. For a very long time
this work had been attributed to a minor Flemish painter, Jan van den Hoecke, one of Rubens'
scholars. Only in 2001 it has been recognized by George Gordon11 as a real Rubens, thanks to its
similarity with the contemporary Samson and Delilah (Rubens fetches record £49.5m, 2002). Image
retrieved March 1, 2010. Source:
http://upload.wikimedia.org/wikipedia/commons/6/6f/Peter_Paul_Rubens_Massacre_of_the_Inno
cents.jpg
 The Raising of the Cross (1610-11) , is a triptych created by Rubens upon return from Italy and
the influence of Michelangelo, Caravaggio and the Venetian painters are easy to notice (Kren & Marx
2010). We are not going to consider all the three parts of the triptych, but only the central panel.
Retrieved March 1, 2010. Source: http://www.lib-art.com/imgpainting/2/7/16272-raising-of-thecross-pieter-pauwel-rubens.jpg
 The Four Philosophers (1611-12), can be considered as “a mirror of the dominant ethical ideas of
the time”, and in this particular case it suggests the moral principles of Stoic philosophy, embodied
by the bust of Seneca and the three humanists that are represented with the painter himself (Kren &
Marx 2010). Retrieved March 1, 2010. Source: http://www.humanitiesweb.org/gallery/156/1.jpg
 Cimon and Pero has been realized in 1612. This painting is considered relevant because in the
dispute that has been going on since the 80s, the National Gallery present it as very similar to our
target painting “in terms of figure drawing, color, composition and paint handling”, together with
10
“A Susanna done by one of my scholars, the whole, however, retouched by my hand” (Hookham Carptenter
1844, p.145).
11
George Gordon is an expert in Dutch and Flemish paintings. After seeing this masterpiece at Sotheby's in
London he determined it as an authentic work by Rubens, because of some similarities, particularly
concerning the style, with the Samson and Delilah, that had been painted by the artist in the same period.
(Rubens fetches record £49.5m, 2002) The value of the painting, which would be sold at auction few months
later, increased exponentially, signing the record of the highest price ever paid at that time in £ sterling for any
Old Master painting purchased at auction (Bond 2010).
- 46 -
The Raising of the Cross and Susanna and the Elders, (National Gallery 1997). Retrieved May 21,
2010. Source: http://www.wga.hu/art/r/ru-bens/61other/060cimon.jpg
 The Portrait of Adriana Perez, (retrieved March 15, 2010. Source: http://www.vkblog.nl/
pub/mm/tempest/652/Image/thomas/rubens_thomas_grt.jpg) and The Portrait of Nicolaas Rockox
(retrieved
March
15,
2010.
Source:
http://www.vkblog.nl/pub/mm/tempest/652/
Image/thomas/rubens_thomas_grt.jpg), are the two lateral panels of the triptych The Incredulity of
Saint Thomas (1613-15), commissioned by Rockox himself and destined to his funeral chapel in the
cathedral of Antwerp (Lambotte 1927). Rockox was friend and patron of Rubens and he
commissioned many of his works, like our target painting; he was nine times burgomaster of the city
and he has been recognized as “one of the key figures who helped to create the spirit of his times, at
the height of the Baroque period in Antwerp” (Rooses 1977, Vol. II, p.158-159).
 Venus Shivering has been realized entirely by Rubens in 1614, and its initial dimensions were
121x95 cm, but later, probably during the 18th century, it has been enlarged both in width and
height it by an unknown artist, defacing the original masterpiece (Rooses 1977, Vol. III, p.181).
Despite that, as far as the realization of the human figures of Venus and Cupid is concerned, this
mythological composition can still be considered as highly representative of the painter’s “lyricism,
eloquence and sensuality that characterize his entire oeuvre” (Néret 2004, p.13). Retrieved March
15,
2010.
Source:
http://www.friendsofart.net/static/images/art1/
pieter-paul-rubens-venus-
frigida.jpg
 The
Holy
Family
with
the
Parrot
(1614)
Retrieved
April
9,
2010.
Source:
http://www.kmska.be/uploadedImages/Museum/Partners/Vlaamsekunstcollectie/VKCAntwerpen.jp
g
 Return of the Prodigal Son represents one of the earliest landscapes by Rubens and it is dated
back to 1618. The biblical subject occupies a minor part in the painting, leaving the rest of the
canvas to the depiction of the environment surrounding the scene (Kren & Marx 2010). Retrieved
March 15, 2010. Source: http://www.terragroda.com/kunsthandel/ image003.png
 Maria with the Child is the left panel that composes the triptych of Christ on the straw, and was
realized in 1618. The image was retrieved on March 15, 2010.
Source: http://www.peterpaul rubens.org/Lamentation-of-Christ-1617-18.html
 The Last Communion of St Francis, is dated 1619, but is noticeably different from the other
paintings of the same period because of the use of the chiaroscuro and the opalescence of the
subjects depicted (Rooses 1977, Vol. II, p.261). Retrieved March 15, 2010. Source:
- 47 -
http://www.settemuse.it/pittori_scultori_europei/rubens/pieter_paul_rubens_006_ultima_comuni
one_san_francesco_1619.jpg
 Christ on the Cross (Le Coup de Lance), realized in 1620, shows all the typical characteristics of
Ruben’s style of those years for the distribution of the colors, the sharpness of the contours, and
the clearness of the brushstrokes (Rooses 1977, Vol. II, p.96-97). This oeuvre has been painted
entirely by the Flemish artist, but there are some details that show a lower quality, which can
probably be attributed to Van Dyck, one of his pupils12. Retrieved March 15, 2010.
Source:http://www.terminartors.com/artworkprofile/Ru-bens_Peter_PaulChrist_on_the_Cross_between_the_Two_Thieves
 Adoration of the Magi, ordered in 1624 by the abbot Matthew Yrsselius, for the main altar of the
Abbey of St. Michael, in Antwerp (Rubens Online 2010 2010). Retrieved March 15, 2010. Source:
http://www.biocrawler.com/w/images/4/4e/Rubens.adoration.650pix.jpg
 The Education of the Virgin painted in 1625-26. The theme of this work was really popular in the
16th century, and after the Counter-Reformation (Kren & Marx 2010), when Rubens became the
main artistic proponent of the spirituality of this time in the north of Europe (“Peter Paul Rubens”,
E. Britannica 2010). Retrieved March 15, 2010. Source:
http://www.lib-art.com/artgallery/16315-the-education-of-the-virgin-pieter-pauwel-rubens.html
Sir Anthony Van Dyck:
Anthony Van Dyck (1599–1641) was active in Rubens’ workshop since he was very young, and the
master himself defined him the best of his pupils (Brown 1999). Later he worked as his assistant, and
he collaborated with Rubens for the realization of many of his paintings. It is then easy to imagine
that Rubens exerted a great influence on the young artist, who used to imitate the way of working of
his teacher, in fact it can be noticed that many of his drawings reproduce some of Rubens’
masterpieces (Meij, de Haan & Webb 2001, p.206). During this period, he absorbed the style and
12
“L’œuvre n’a pas été peinte sans le secours d’un élève. Rubens l’a presqu’entièrement achevée de sa main;
mais il y a des parties mois réussies comme forme et couleur qu’on ne saurait lui attribuer *…+ Le travail de
l’élève, imparfaitement couvert par le retouches du maître, est d’ailleurs fort habile et cette circonstance,
jointe à la date de l’exécution du tableau, nous fait croire que c’est Van Dyck qui, dans l’occurrence, a
assisté son maître” (Rooses 1977, Vol. II, p. 97).
- 48 -
technique of his mentor, and many of his first authentic works were based on Rubens’ compositions.
Also Van Dyck travelled to Italy during his youth, but unlike Rubens, he focused his attention in
particular on the study of the Venetian painters, namely Titian (Meij, de Haan & Webb 2001, p.192196, p.221).
These paintings of Van Dyck have been included in the digital analysis:
 The crowning with Thorns (1620), this first painting by Van Dyck is relevant because of the style
with which the artist depicts the kneeling soldier: it can be noticed, in fact, a clear influence by
Titian, who also realized a painting with the same subject almost two centuries before.
Nevertheless, Van Dyck did not have the chance to travel to Italy and get personally in touch with
that particular painting and the classical artist. So, it is likely that Titan’s influence has at first been
brought to the young Van Dyck by Rubens, who also realized a version of the same painting around
1601-02, when he was in Rome (Meij, de Haan and Webb 2001, p.210). Retrieved June 6, 2010.
Source:
http://lh4.ggpht.com/_SUBGzd1BG60/SZjqTcR6rGI/AAAAAAACONw/YETeDHGXdAo/VanDyck,+Chris
t+crowned+w+thorns+c1620.jpg
 Susanna and the Elders (1621-22) was painted around 1621-22, and it has been chosen for this
analysis mainly because of the subject: in fact, in the second group we can find another painting
with the same title that has been made by Rubens 13 . Retrieved June 6, 2010. Source:
http://www.lib-art.com/imgpainting/1/3/9931-susanna-and-the-elders-sir-anthony-van-dyck.jpg
 Portrait of Charles I, (retrieved June 6, 2010. Source: http://catherinedelors.com/ blog/wpcontent/uploads/Van-Dyck-portrait_de_Charles_1er -web.jpg) was realized by Van Dyck in 1637,
when he was active in London as official court portraitist. This painting, together with the Portrait of
Maria de Tassis (1629-30) (retrieved June 6, 2010. Source: http://catherinedelors.com/ blog/wpcontent/uploads/Van-Dyck-maria-de-tassis-3.jpg) can be considered relevant in this analysis, since
they both represent an example of portraits, respectively of a man and a woman, that could be
compared to Rubens’ panels for the triptych The Incredulity of Saint Thomas (1613-15), representing
Nicolaas Rockox and Adriana Perez.
13
See pages 46-47.
- 49 -
Jacob Jordaens:
Jacob Jordaens (1593–1678) worked as an apprentice in the studio of Van Noort, who had been one
of the first teachers of Rubens (The Masterpieces of Jacob Jordaens 1907). Later he collaborated on
many projects of the notorious Flemish artist, getting in touch with Van Dyck as well. On his
masterpieces, we can identify Rubens’ marks, but also the influence of Italian painters like
Caravaggio and Titian. Unlike his contemporaries, he did not go abroad to examine the classical
masterpieces of those artists, but he got the chance to study them through his mentor, Rubens, and
his private collection. After the death of Rubens and Van Dyck, he affirmed himself as the most
important painter in the south of the Netherlands. His style is characterized by a very high contrast
between lights and shadows and by the richness of figures (“Jacob Jordaens”, E. Britannica, 2010).
These works of Jordaens have been included in the digital analysis:
 The Adoration of the Shepherds of circa 1617 is just one of the six versions realized by Jordaens
about this subject. The scene, very simple, is emphasized by the use of light, which evidence the
interest of the author for Caravaggio (“Jacob Jordaens”, Wikipedia, 2010). Retrieved June 6, 2010.
Source: http://www.jacobjordaens.org/Adoration-of-the-Shepherds-large.html
 Portrait of a Young Married Couple (1615-20) has been chosen mainly because it is a portrait, so
the attention is focused on the faces and bodies of the characters, making this painting a possible
term of comparison with Samson and Delilah, exactly as far as the representation of the human
figures is concerned. Retrieved June 6, 2010. Source:
http://www.jacobjordaens.org/Portrait-of-a-Young-Married-Couple.html
 Holy Family and Saint John the Baptist, probably painted in between 1620-25, has been selected
for this batch because the subject shows some similarities with the painting Maria with the Child,
realized by Rubens in 161814. Retrieved June 6, 2010.
Source: http://www.jacobjordaens.org/Holy-Family-and-Saint-John-the-Baptist.html
 The last painting to consider is The Education of Jupiter (1652), which has been chosen for the
nude figures of the women that can be connected to the Susanna by Rubens and to the way of
representing Samson’s naked body as well. Retrieved June 6, 2010. Source: http://www.libart.com/artgallery/12549-education-of-jupiter-jacob-jordaens.html
14
See page 48.
- 50 -
APPENDIX B:
Table 1. This table shows the ratio of pixels per centimeter of all the images belonging to the dataset, where H
stands for height and W stands for width. The value applied in the resizing process belongs to the painting The
Last Communion of St. Francis. The average ratio per image is 1,87 pixels per centimeter.
Physical
Digital
(cm)
(pixels)
H
W
Name of the paintings
(pix/cm)
W
Avg
(pixels)
H
W
198
218 1000 1152 5,05
5,28
5,17
362
417
Massacre of the Innocents
142
182
767
960
5,40
5,27
5,34
269
337
The Four Philosophers
167
143
480
395
2,87
2,76
2,82
319
262
The Deposition
180
137
580
446
3,22
3,25
3,24
335
258
Artist and His Wife
178
137 2673 2024 15,01 14,82 14,92
336
254
Samson and Delilah
185
205
931 1030 5,03
5,02
5,03
347
384
Cimon and Pero
141
180
972 1255 6,89
6,97
6,93
263
339
The Last Communion of St Francis
422
266
715
546
1,69
2,05
1,87
715
546
The Education of the Virgin
194
140
480
341
2,47
2,44
2,45
366
260
Christ on the Cross
429
311 1170 850
2,73
2,73
2,73
803
583
Adoration of the Magi
447
336
882
650
1,97
1,93
1,95
846
623
Maria with the Child
136
40
469
135
3,45
3,38
3,41
258
74
Portrait of Nicolaas Rockox
146
55
372
144
2,55
2,62
2,58
270
104
Portrait of Adriana Perez
146
55
372
139
2,55
2,53
2,54
275
103
Venus Shivering
142
184 1000 1285 7,04
6,98
7,01
267
343
Return of the Prodigal Son
107
155
290
375
2,71
2,42
2,56
212
274
The Holy Family with the Parrot
163
189
336
383
2,06
2,03
2,04
308
351
Portrait of Charles I
124
96,5
623
480
5,04
4,97
5,01
233
180
The crowning with Thorns
223
196 1600 1358 7,17
6,93
7,05
425
361
Portrait of Maria de Tassis
129
93
640
455
4,96
4,89
4,93
243
173
Susanna and the Elders
194
144
920
668
4,74
4,64
4,69
367
267
Portrait of a Married Couple
125
93
754
566
6,03
6,09
6,06
233
175
Adoration of the Shepherds
125
95,7
753
564
6,02
5,89
5,96
237
177
Holy Family
123
94
754
566
6,13
6,02
6,08
233
175
Education of Jupiter
61
75
674
800 11,05 10,67 10,86
116
138
RUBENS
Susanna and the Elders
DYCK
H
VAN
W
Rescaled
JORDAENS
H
Ratio
- 51 -
Table 2. Matrix of Gabor coefficients. It provides the coefficients (or values) for 24 orientations and 8 scales.
The first 15 rows show the numerical values related to the elements belonging to group 1 (Rubens paintings),
the next 8 rows show those belonging to group 2 (Non-Rubens paintings), and the last one shows those of
Samson and Delilah. The matrix is composed by 24 tables, as the number of orientations. They are divided in 8
columns, according to the number of scales, and 25 rows, which is the number of elements of the dataset.
0,0025
0,0036
0,0024
0,0013
0,0048
0,0030
0,0026
0,0012
0,0028
0,0026
0,0052
0,0029
0,0030
0,0034
0,0027
0,0017
0,0041
0,0022
0,0027
0,0037
0,0022
0,0028
0,0023
0,0017
0,0036
0,0041
0,0055
0,0036
0,0022
0,0070
0,0045
0,0046
0,0021
0,0043
0,0043
0,0086
0,0049
0,0045
0,0057
0,0044
0,0029
0,0066
0,0038
0,0048
0,0059
0,0037
0,0043
0,0033
0,0026
0,0058
0,0059
0,0081
0,0050
0,0035
0,0094
0,0062
0,0072
0,0034
0,0064
0,0072
0,0139
0,0080
0,0065
0,0089
0,0069
0,0045
0,0105
0,0055
0,0075
0,0097
0,0061
0,0073
0,0047
0,0039
0,0084
val(:,:,1) =
0,0083 0,0120
0,0115 0,0168
0,0074 0,0111
0,0051 0,0084
0,0127 0,0177
0,0091 0,0146
0,0108 0,0149
0,0057 0,0087
0,0096 0,0132
0,0131 0,0212
0,0207 0,0315
0,0114 0,0158
0,0103 0,0156
0,0131 0,0166
0,0109 0,0161
0,0061 0,0101
0,0154 0,0225
0,0085 0,0142
0,0147 0,0217
0,0168 0,0220
0,0084 0,0129
0,0122 0,0166
0,0071 0,0101
0,0061 0,0103
0,0122 0,0172
0,0168
0,0225
0,0174
0,0143
0,0263
0,0199
0,0219
0,0155
0,0218
0,0248
0,0441
0,0220
0,0209
0,0232
0,0226
0,0172
0,0293
0,0204
0,0294
0,0368
0,0176
0,0229
0,0174
0,0154
0,0242
0,0236
0,0305
0,0272
0,0219
0,0403
0,0265
0,0357
0,0237
0,0320
0,0417
0,0734
0,0228
0,0422
0,0323
0,0328
0,0262
0,0399
0,0314
0,0554
0,0541
0,0232
0,0394
0,0230
0,0205
0,0326
0,0325
0,0349
0,0392
0,0287
0,0521
0,0421
0,0585
0,0309
0,0512
0,0991
0,1115
0,0304
0,0705
0,0471
0,0507
0,0352
0,0503
0,0499
0,0761
0,0507
0,0288
0,0625
0,0396
0,0288
0,0428
0,0023
0,0037
0,0023
0,0012
0,0046
0,0028
0,0025
0,0012
0,0024
0,0022
0,0052
0,0028
0,0027
0,0029
0,0026
0,0016
0,0041
0,0020
0,0029
0,0034
0,0020
0,0030
0,0022
0,0017
0,0034
0,0039
0,0053
0,0035
0,0021
0,0065
0,0041
0,0043
0,0022
0,0040
0,0042
0,0088
0,0046
0,0038
0,0048
0,0040
0,0028
0,0065
0,0036
0,0046
0,0052
0,0037
0,0044
0,0032
0,0026
0,0054
0,0056
0,0076
0,0051
0,0033
0,0083
0,0061
0,0066
0,0034
0,0060
0,0083
0,0131
0,0065
0,0055
0,0072
0,0059
0,0038
0,0094
0,0056
0,0073
0,0090
0,0061
0,0074
0,0041
0,0039
0,0079
val(:,:,2) =
0,0075 0,0108
0,0124 0,0174
0,0073 0,0109
0,0050 0,0077
0,0122 0,0154
0,0085 0,0118
0,0098 0,0161
0,0059 0,0086
0,0092 0,0128
0,0126 0,0184
0,0194 0,0313
0,0100 0,0125
0,0093 0,0122
0,0106 0,0148
0,0095 0,0140
0,0051 0,0089
0,0161 0,0230
0,0082 0,0124
0,0145 0,0229
0,0161 0,0242
0,0090 0,0137
0,0113 0,0141
0,0060 0,0089
0,0056 0,0094
0,0123 0,0160
0,0156
0,0221
0,0165
0,0124
0,0212
0,0180
0,0225
0,0134
0,0188
0,0235
0,0427
0,0148
0,0135
0,0183
0,0188
0,0155
0,0287
0,0189
0,0292
0,0417
0,0187
0,0191
0,0163
0,0134
0,0239
0,0206
0,0295
0,0251
0,0186
0,0325
0,0250
0,0312
0,0216
0,0292
0,0488
0,0709
0,0206
0,0285
0,0259
0,0280
0,0289
0,0432
0,0308
0,0497
0,0569
0,0236
0,0329
0,0199
0,0175
0,0337
0,0286
0,0353
0,0389
0,0288
0,0458
0,0314
0,0429
0,0384
0,0420
0,0752
0,1183
0,0380
0,0677
0,0332
0,0429
0,0378
0,0594
0,0434
0,0693
0,0669
0,0410
0,0564
0,0315
0,0300
0,0445
0,0022
0,0038
0,0022
0,0012
0,0046
0,0027
0,0024
0,0011
0,0022
0,0022
0,0051
0,0027
0,0025
0,0027
0,0023
0,0015
0,0039
0,0019
0,0030
0,0035
0,0021
0,0029
0,0021
0,0016
0,0033
0,0039
0,0055
0,0036
0,0020
0,0064
0,0042
0,0042
0,0021
0,0037
0,0039
0,0088
0,0039
0,0037
0,0046
0,0039
0,0027
0,0060
0,0034
0,0047
0,0059
0,0039
0,0049
0,0030
0,0026
0,0053
0,0055
0,0078
0,0054
0,0031
0,0081
0,0060
0,0065
0,0039
0,0056
0,0068
0,0133
0,0062
0,0056
0,0070
0,0058
0,0038
0,0089
0,0055
0,0073
0,0095
0,0062
0,0074
0,0040
0,0040
0,0081
val(:,:,3) =
0,0074 0,0106
0,0119 0,0162
0,0074 0,0111
0,0047 0,0074
0,0121 0,0163
0,0077 0,0117
0,0101 0,0161
0,0059 0,0096
0,0082 0,0120
0,0092 0,0146
0,0185 0,0302
0,0095 0,0143
0,0098 0,0123
0,0097 0,0134
0,0086 0,0124
0,0052 0,0082
0,0137 0,0207
0,0082 0,0119
0,0133 0,0228
0,0159 0,0270
0,0092 0,0161
0,0100 0,0158
0,0056 0,0080
0,0058 0,0088
0,0119 0,0161
0,0145
0,0225
0,0167
0,0103
0,0214
0,0174
0,0215
0,0124
0,0180
0,0189
0,0387
0,0157
0,0139
0,0188
0,0170
0,0143
0,0286
0,0191
0,0264
0,0392
0,0223
0,0245
0,0145
0,0140
0,0243
0,0210
0,0335
0,0249
0,0173
0,0261
0,0248
0,0317
0,0260
0,0289
0,0306
0,0678
0,0283
0,0277
0,0254
0,0219
0,0234
0,0408
0,0284
0,0440
0,0439
0,0258
0,0398
0,0211
0,0185
0,0361
0,0285
0,0443
0,0366
0,0213
0,0392
0,0351
0,0515
0,0454
0,0440
0,0447
0,0981
0,0463
0,0440
0,0319
0,0394
0,0282
0,0580
0,0483
0,0637
0,0599
0,0525
0,0520
0,0271
0,0351
0,0452
0,0021
0,0037
0,0022
0,0012
0,0047
0,0027
0,0023
0,0011
0,0021
0,0022
0,0052
0,0027
0,0021
0,0027
0,0023
0,0015
0,0037
0,0019
0,0029
0,0036
0,0021
0,0028
0,0020
0,0016
0,0034
0,0039
0,0058
0,0035
0,0019
0,0063
0,0040
0,0043
0,0022
0,0035
0,0041
0,0087
0,0043
0,0037
0,0044
0,0037
0,0026
0,0059
0,0036
0,0052
0,0068
0,0038
0,0045
0,0030
0,0027
0,0054
0,0055
0,0087
0,0051
0,0029
0,0078
0,0055
0,0070
0,0037
0,0053
0,0055
0,0131
0,0066
0,0053
0,0066
0,0056
0,0040
0,0089
0,0057
0,0081
0,0104
0,0061
0,0068
0,0045
0,0041
0,0085
val(:,:,4) =
0,0073 0,0105
0,0118 0,0152
0,0073 0,0109
0,0043 0,0071
0,0104 0,0144
0,0078 0,0111
0,0109 0,0156
0,0061 0,0105
0,0079 0,0111
0,0081 0,0127
0,0194 0,0303
0,0102 0,0131
0,0079 0,0119
0,0099 0,0149
0,0082 0,0119
0,0058 0,0087
0,0122 0,0188
0,0084 0,0126
0,0114 0,0204
0,0167 0,0250
0,0088 0,0142
0,0097 0,0168
0,0061 0,0088
0,0062 0,0088
0,0112 0,0170
0,0144
0,0233
0,0167
0,0107
0,0201
0,0172
0,0213
0,0155
0,0175
0,0177
0,0387
0,0172
0,0140
0,0184
0,0165
0,0145
0,0257
0,0178
0,0303
0,0411
0,0211
0,0265
0,0131
0,0142
0,0240
0,0196
0,0323
0,0236
0,0186
0,0298
0,0280
0,0348
0,0276
0,0262
0,0267
0,0603
0,0316
0,0205
0,0252
0,0221
0,0184
0,0335
0,0308
0,0376
0,0533
0,0244
0,0364
0,0206
0,0207
0,0329
0,0282
0,0520
0,0351
0,0256
0,0419
0,0344
0,0585
0,0436
0,0433
0,0449
0,0756
0,0401
0,0432
0,0353
0,0310
0,0232
0,0410
0,0493
0,0558
0,0649
0,0474
0,0486
0,0278
0,0346
0,0448
0,0022
0,0038
0,0022
0,0012
0,0045
0,0027
0,0024
0,0011
0,0039
0,0061
0,0035
0,0020
0,0062
0,0041
0,0041
0,0021
0,0055
0,0088
0,0050
0,0030
0,0073
0,0057
0,0069
0,0037
val(:,:,5) =
0,0075 0,0101
0,0128 0,0167
0,0073 0,0108
0,0044 0,0068
0,0100 0,0153
0,0079 0,0116
0,0105 0,0162
0,0060 0,0096
0,0144
0,0253
0,0155
0,0112
0,0232
0,0177
0,0263
0,0151
0,0202
0,0313
0,0250
0,0185
0,0271
0,0222
0,0371
0,0215
0,0284
0,0474
0,0394
0,0264
0,0401
0,0301
0,0637
0,0394
0,0022
0,0038
0,0021
0,0012
0,0045
0,0027
0,0024
0,0011
0,0039
0,0062
0,0034
0,0023
0,0064
0,0042
0,0043
0,0023
0,0055
0,0092
0,0048
0,0034
0,0079
0,0055
0,0071
0,0039
val(:,:,6) =
0,0075 0,0102
0,0124 0,0170
0,0070 0,0101
0,0052 0,0083
0,0099 0,0149
0,0073 0,0114
0,0108 0,0166
0,0062 0,0091
0,0152
0,0255
0,0146
0,0120
0,0213
0,0177
0,0279
0,0136
0,0216
0,0351
0,0231
0,0179
0,0234
0,0267
0,0343
0,0251
0,0284
0,0406
0,0359
0,0270
0,0426
0,0308
0,0487
0,0354
- 52 -
0,0020
0,0023
0,0052
0,0030
0,0018
0,0025
0,0023
0,0015
0,0039
0,0018
0,0030
0,0033
0,0018
0,0028
0,0020
0,0018
0,0033
0,0034
0,0042
0,0084
0,0049
0,0035
0,0043
0,0036
0,0028
0,0064
0,0034
0,0052
0,0063
0,0033
0,0041
0,0029
0,0029
0,0053
0,0051
0,0055
0,0125
0,0074
0,0051
0,0067
0,0052
0,0042
0,0090
0,0053
0,0088
0,0099
0,0048
0,0061
0,0041
0,0046
0,0083
0,0078
0,0073
0,0189
0,0115
0,0057
0,0097
0,0079
0,0057
0,0119
0,0087
0,0150
0,0136
0,0068
0,0089
0,0059
0,0068
0,0109
0,0120
0,0132
0,0273
0,0119
0,0087
0,0128
0,0113
0,0082
0,0177
0,0128
0,0182
0,0202
0,0115
0,0140
0,0089
0,0107
0,0165
0,0184
0,0217
0,0354
0,0165
0,0136
0,0173
0,0162
0,0131
0,0246
0,0170
0,0243
0,0302
0,0152
0,0220
0,0139
0,0148
0,0239
0,0269
0,0369
0,0519
0,0365
0,0180
0,0251
0,0244
0,0204
0,0319
0,0299
0,0475
0,0395
0,0352
0,0272
0,0167
0,0255
0,0296
0,0427
0,0638
0,0698
0,0383
0,0334
0,0337
0,0336
0,0230
0,0403
0,0476
0,0896
0,0536
0,0504
0,0563
0,0326
0,0348
0,0416
0,0020
0,0020
0,0050
0,0033
0,0019
0,0022
0,0022
0,0016
0,0038
0,0018
0,0032
0,0034
0,0018
0,0026
0,0021
0,0019
0,0033
0,0033
0,0034
0,0082
0,0051
0,0036
0,0036
0,0036
0,0029
0,0064
0,0032
0,0064
0,0057
0,0033
0,0039
0,0030
0,0031
0,0052
0,0052
0,0061
0,0122
0,0073
0,0050
0,0061
0,0052
0,0042
0,0096
0,0053
0,0095
0,0093
0,0048
0,0064
0,0039
0,0047
0,0076
0,0081
0,0081
0,0193
0,0119
0,0052
0,0095
0,0075
0,0065
0,0127
0,0080
0,0161
0,0127
0,0065
0,0088
0,0062
0,0072
0,0111
0,0127
0,0154
0,0270
0,0132
0,0084
0,0121
0,0121
0,0089
0,0195
0,0126
0,0186
0,0183
0,0106
0,0135
0,0091
0,0105
0,0156
0,0187
0,0244
0,0344
0,0236
0,0144
0,0160
0,0178
0,0142
0,0257
0,0170
0,0300
0,0320
0,0169
0,0211
0,0129
0,0163
0,0227
0,0272
0,0375
0,0567
0,0318
0,0177
0,0215
0,0232
0,0222
0,0310
0,0285
0,0600
0,0422
0,0269
0,0270
0,0151
0,0257
0,0310
0,0509
0,0600
0,0696
0,0390
0,0367
0,0301
0,0372
0,0271
0,0375
0,0393
0,0836
0,0587
0,0313
0,0472
0,0245
0,0463
0,0350
0,0023
0,0036
0,0021
0,0012
0,0046
0,0027
0,0024
0,0011
0,0019
0,0023
0,0050
0,0029
0,0018
0,0022
0,0023
0,0016
0,0040
0,0018
0,0031
0,0031
0,0018
0,0024
0,0022
0,0020
0,0033
0,0039
0,0059
0,0033
0,0023
0,0062
0,0042
0,0045
0,0023
0,0032
0,0036
0,0080
0,0049
0,0034
0,0036
0,0037
0,0029
0,0062
0,0033
0,0056
0,0052
0,0029
0,0037
0,0031
0,0032
0,0052
0,0056
0,0085
0,0046
0,0035
0,0082
0,0056
0,0073
0,0039
0,0052
0,0058
0,0118
0,0072
0,0041
0,0056
0,0053
0,0044
0,0087
0,0054
0,0095
0,0088
0,0046
0,0057
0,0038
0,0049
0,0072
val(:,:,7) =
0,0074 0,0104
0,0115 0,0156
0,0067 0,0096
0,0056 0,0092
0,0104 0,0148
0,0080 0,0117
0,0120 0,0165
0,0060 0,0085
0,0079 0,0120
0,0078 0,0150
0,0182 0,0254
0,0119 0,0128
0,0064 0,0099
0,0081 0,0121
0,0075 0,0121
0,0067 0,0094
0,0126 0,0175
0,0083 0,0120
0,0158 0,0257
0,0124 0,0174
0,0078 0,0125
0,0081 0,0137
0,0059 0,0087
0,0074 0,0116
0,0107 0,0153
0,0150
0,0239
0,0150
0,0124
0,0204
0,0177
0,0270
0,0133
0,0182
0,0204
0,0347
0,0173
0,0125
0,0155
0,0199
0,0144
0,0262
0,0185
0,0456
0,0299
0,0196
0,0212
0,0102
0,0172
0,0209
0,0210
0,0336
0,0216
0,0214
0,0238
0,0217
0,0346
0,0217
0,0256
0,0296
0,0575
0,0200
0,0190
0,0225
0,0223
0,0203
0,0306
0,0302
0,0737
0,0454
0,0225
0,0275
0,0138
0,0232
0,0316
0,0326
0,0413
0,0332
0,0330
0,0348
0,0291
0,0560
0,0304
0,0451
0,0442
0,0652
0,0429
0,0398
0,0331
0,0308
0,0293
0,0365
0,0365
0,1229
0,0515
0,0257
0,0480
0,0183
0,0508
0,0393
0,0023
0,0033
0,0020
0,0013
0,0047
0,0027
0,0026
0,0012
0,0018
0,0024
0,0048
0,0029
0,0019
0,0022
0,0023
0,0016
0,0038
0,0018
0,0030
0,0028
0,0016
0,0023
0,0021
0,0022
0,0032
0,0041
0,0053
0,0033
0,0023
0,0064
0,0042
0,0047
0,0024
0,0031
0,0039
0,0078
0,0043
0,0032
0,0039
0,0036
0,0028
0,0060
0,0033
0,0051
0,0048
0,0027
0,0038
0,0030
0,0035
0,0050
0,0057
0,0077
0,0046
0,0036
0,0078
0,0058
0,0074
0,0038
0,0050
0,0056
0,0120
0,0071
0,0045
0,0061
0,0055
0,0046
0,0085
0,0053
0,0081
0,0072
0,0044
0,0056
0,0044
0,0051
0,0070
val(:,:,8) =
0,0075 0,0108
0,0111 0,0155
0,0064 0,0093
0,0059 0,0102
0,0112 0,0148
0,0083 0,0113
0,0114 0,0166
0,0060 0,0091
0,0076 0,0116
0,0097 0,0156
0,0180 0,0251
0,0116 0,0121
0,0071 0,0105
0,0091 0,0140
0,0079 0,0119
0,0064 0,0086
0,0130 0,0167
0,0083 0,0119
0,0156 0,0243
0,0116 0,0160
0,0068 0,0120
0,0083 0,0125
0,0061 0,0094
0,0084 0,0128
0,0101 0,0148
0,0153
0,0226
0,0145
0,0161
0,0192
0,0177
0,0263
0,0142
0,0177
0,0232
0,0376
0,0183
0,0132
0,0182
0,0183
0,0145
0,0238
0,0188
0,0331
0,0221
0,0176
0,0207
0,0113
0,0182
0,0233
0,0214
0,0337
0,0204
0,0231
0,0247
0,0200
0,0345
0,0215
0,0262
0,0277
0,0529
0,0293
0,0163
0,0273
0,0229
0,0212
0,0307
0,0322
0,0682
0,0445
0,0194
0,0289
0,0166
0,0257
0,0313
0,0313
0,0453
0,0301
0,0387
0,0354
0,0281
0,0493
0,0237
0,0444
0,0386
0,0681
0,0440
0,0300
0,0352
0,0302
0,0258
0,0435
0,0416
0,1295
0,0589
0,0301
0,0357
0,0228
0,0461
0,0391
0,0024
0,0033
0,0020
0,0015
0,0046
0,0028
0,0027
0,0012
0,0018
0,0024
0,0050
0,0029
0,0020
0,0022
0,0023
0,0017
0,0037
0,0018
0,0030
0,0030
0,0015
0,0024
0,0021
0,0022
0,0030
0,0042
0,0052
0,0033
0,0024
0,0062
0,0042
0,0046
0,0023
0,0032
0,0041
0,0079
0,0047
0,0030
0,0039
0,0037
0,0029
0,0059
0,0032
0,0050
0,0051
0,0027
0,0039
0,0032
0,0036
0,0048
0,0057
0,0074
0,0045
0,0040
0,0076
0,0059
0,0071
0,0038
0,0050
0,0059
0,0116
0,0067
0,0044
0,0061
0,0056
0,0047
0,0089
0,0053
0,0091
0,0075
0,0042
0,0062
0,0043
0,0054
0,0071
val(:,:,9) =
0,0076 0,0108
0,0104 0,0150
0,0065 0,0097
0,0060 0,0098
0,0110 0,0144
0,0085 0,0113
0,0112 0,0167
0,0057 0,0088
0,0075 0,0115
0,0091 0,0132
0,0173 0,0228
0,0109 0,0131
0,0073 0,0123
0,0086 0,0131
0,0084 0,0122
0,0065 0,0086
0,0132 0,0173
0,0078 0,0118
0,0120 0,0155
0,0103 0,0152
0,0066 0,0119
0,0084 0,0120
0,0066 0,0086
0,0079 0,0111
0,0100 0,0143
0,0154
0,0227
0,0148
0,0157
0,0191
0,0170
0,0237
0,0144
0,0185
0,0222
0,0414
0,0187
0,0171
0,0190
0,0165
0,0129
0,0244
0,0197
0,0196
0,0263
0,0166
0,0209
0,0149
0,0154
0,0224
0,0213
0,0337
0,0208
0,0215
0,0232
0,0249
0,0343
0,0206
0,0267
0,0331
0,0595
0,0315
0,0164
0,0271
0,0222
0,0190
0,0368
0,0297
0,0365
0,0446
0,0192
0,0282
0,0211
0,0270
0,0316
0,0330
0,0479
0,0290
0,0338
0,0318
0,0389
0,0566
0,0287
0,0441
0,0298
0,0660
0,0525
0,0276
0,0371
0,0308
0,0250
0,0522
0,0400
0,0664
0,0564
0,0323
0,0357
0,0324
0,0429
0,0393
0,0025
0,0031
0,0020
0,0016
0,0045
0,0029
0,0028
0,0011
0,0019
0,0026
0,0050
0,0028
0,0020
0,0024
0,0025
0,0017
0,0035
0,0018
0,0029
0,0029
0,0017
0,0023
0,0022
0,0022
0,0029
0,0043
0,0051
0,0032
0,0027
0,0059
0,0044
0,0047
0,0022
0,0032
0,0043
0,0079
0,0046
0,0029
0,0042
0,0038
0,0028
0,0056
0,0032
0,0050
0,0052
0,0028
0,0039
0,0032
0,0035
0,0046
0,0059
0,0071
0,0046
0,0045
0,0079
0,0063
0,0071
0,0037
0,0048
0,0064
0,0117
0,0062
0,0044
0,0058
0,0060
0,0045
0,0085
0,0052
0,0081
0,0081
0,0046
0,0058
0,0047
0,0050
0,0070
val(:,:,10) =
0,0078 0,0110
0,0098 0,0150
0,0063 0,0090
0,0070 0,0097
0,0114 0,0145
0,0089 0,0127
0,0107 0,0185
0,0062 0,0085
0,0072 0,0113
0,0100 0,0141
0,0184 0,0257
0,0095 0,0166
0,0063 0,0094
0,0085 0,0131
0,0091 0,0126
0,0066 0,0088
0,0119 0,0172
0,0079 0,0119
0,0108 0,0152
0,0104 0,0162
0,0076 0,0119
0,0093 0,0132
0,0072 0,0099
0,0077 0,0117
0,0101 0,0147
0,0161
0,0238
0,0135
0,0158
0,0192
0,0169
0,0229
0,0118
0,0171
0,0202
0,0390
0,0212
0,0184
0,0184
0,0174
0,0126
0,0219
0,0169
0,0280
0,0225
0,0156
0,0192
0,0155
0,0162
0,0210
0,0222
0,0331
0,0204
0,0234
0,0260
0,0256
0,0352
0,0204
0,0252
0,0269
0,0503
0,0357
0,0216
0,0245
0,0244
0,0171
0,0312
0,0283
0,0348
0,0366
0,0232
0,0273
0,0201
0,0268
0,0298
0,0298
0,0475
0,0311
0,0306
0,0384
0,0447
0,0811
0,0309
0,0358
0,0354
0,0550
0,0517
0,0303
0,0348
0,0447
0,0282
0,0491
0,0375
0,0530
0,0465
0,0401
0,0319
0,0243
0,0422
0,0361
- 53 -
0,0027
0,0029
0,0022
0,0019
0,0048
0,0033
0,0033
0,0013
0,0023
0,0028
0,0050
0,0029
0,0021
0,0028
0,0027
0,0016
0,0037
0,0020
0,0031
0,0026
0,0018
0,0020
0,0025
0,0021
0,0030
0,0045
0,0047
0,0033
0,0033
0,0063
0,0047
0,0053
0,0023
0,0035
0,0048
0,0082
0,0042
0,0030
0,0050
0,0044
0,0026
0,0057
0,0034
0,0051
0,0044
0,0030
0,0037
0,0036
0,0033
0,0049
0,0061
0,0067
0,0046
0,0054
0,0088
0,0066
0,0080
0,0040
0,0049
0,0073
0,0121
0,0061
0,0046
0,0074
0,0067
0,0045
0,0084
0,0053
0,0073
0,0066
0,0049
0,0053
0,0054
0,0050
0,0079
val(:,:,11) =
0,0083 0,0116
0,0090 0,0134
0,0064 0,0091
0,0080 0,0115
0,0126 0,0159
0,0091 0,0135
0,0114 0,0185
0,0067 0,0089
0,0076 0,0114
0,0112 0,0166
0,0181 0,0261
0,0097 0,0169
0,0067 0,0098
0,0100 0,0161
0,0097 0,0131
0,0067 0,0101
0,0128 0,0182
0,0080 0,0122
0,0108 0,0148
0,0105 0,0169
0,0082 0,0107
0,0092 0,0139
0,0078 0,0138
0,0074 0,0109
0,0109 0,0161
0,0160
0,0208
0,0134
0,0171
0,0222
0,0207
0,0255
0,0126
0,0170
0,0216
0,0397
0,0205
0,0176
0,0217
0,0200
0,0181
0,0237
0,0174
0,0283
0,0242
0,0174
0,0207
0,0170
0,0150
0,0246
0,0237
0,0303
0,0194
0,0206
0,0286
0,0272
0,0361
0,0192
0,0278
0,0301
0,0498
0,0395
0,0286
0,0289
0,0299
0,0227
0,0385
0,0280
0,0346
0,0416
0,0349
0,0282
0,0212
0,0222
0,0345
0,0331
0,0397
0,0295
0,0371
0,0420
0,0435
0,0638
0,0317
0,0405
0,0201
0,0611
0,0536
0,0342
0,0433
0,0431
0,0342
0,0473
0,0369
0,0432
0,0511
0,0546
0,0271
0,0320
0,0363
0,0387
0,0028
0,0029
0,0022
0,0021
0,0051
0,0035
0,0035
0,0013
0,0021
0,0034
0,0051
0,0027
0,0021
0,0034
0,0030
0,0016
0,0038
0,0020
0,0031
0,0026
0,0018
0,0020
0,0028
0,0022
0,0033
0,0047
0,0046
0,0032
0,0036
0,0069
0,0051
0,0056
0,0025
0,0035
0,0054
0,0081
0,0046
0,0034
0,0057
0,0051
0,0028
0,0063
0,0037
0,0056
0,0041
0,0031
0,0036
0,0047
0,0035
0,0054
0,0067
0,0067
0,0043
0,0061
0,0100
0,0072
0,0083
0,0040
0,0051
0,0071
0,0121
0,0063
0,0051
0,0091
0,0076
0,0043
0,0089
0,0058
0,0081
0,0068
0,0051
0,0048
0,0076
0,0049
0,0081
val(:,:,12) =
0,0088 0,0124
0,0088 0,0129
0,0060 0,0088
0,0090 0,0142
0,0135 0,0181
0,0096 0,0141
0,0126 0,0199
0,0062 0,0092
0,0078 0,0117
0,0105 0,0203
0,0190 0,0292
0,0102 0,0135
0,0069 0,0122
0,0123 0,0166
0,0112 0,0152
0,0068 0,0105
0,0153 0,0189
0,0086 0,0138
0,0119 0,0165
0,0108 0,0173
0,0073 0,0100
0,0083 0,0120
0,0097 0,0150
0,0070 0,0109
0,0113 0,0176
0,0175
0,0203
0,0134
0,0208
0,0267
0,0207
0,0252
0,0131
0,0180
0,0307
0,0406
0,0205
0,0200
0,0281
0,0229
0,0180
0,0217
0,0195
0,0244
0,0225
0,0198
0,0177
0,0179
0,0158
0,0260
0,0247
0,0340
0,0192
0,0245
0,0321
0,0295
0,0385
0,0224
0,0274
0,0366
0,0521
0,0352
0,0291
0,0443
0,0378
0,0204
0,0387
0,0295
0,0464
0,0355
0,0345
0,0244
0,0277
0,0232
0,0301
0,0363
0,0426
0,0292
0,0381
0,0504
0,0509
0,0596
0,0324
0,0408
0,0500
0,0854
0,0552
0,0413
0,0559
0,0577
0,0290
0,0423
0,0466
0,0687
0,0525
0,0506
0,0340
0,0526
0,0336
0,0359
0,0026
0,0029
0,0020
0,0019
0,0046
0,0032
0,0030
0,0012
0,0020
0,0032
0,0050
0,0026
0,0020
0,0029
0,0030
0,0017
0,0037
0,0018
0,0030
0,0023
0,0019
0,0020
0,0025
0,0020
0,0031
0,0044
0,0047
0,0030
0,0031
0,0065
0,0047
0,0050
0,0022
0,0032
0,0048
0,0078
0,0038
0,0034
0,0051
0,0049
0,0028
0,0060
0,0032
0,0054
0,0036
0,0030
0,0032
0,0042
0,0032
0,0051
0,0063
0,0069
0,0039
0,0051
0,0093
0,0068
0,0071
0,0034
0,0048
0,0072
0,0116
0,0048
0,0053
0,0081
0,0079
0,0043
0,0089
0,0050
0,0080
0,0059
0,0044
0,0042
0,0070
0,0045
0,0078
val(:,:,13) =
0,0087 0,0119
0,0090 0,0128
0,0053 0,0078
0,0080 0,0122
0,0118 0,0162
0,0095 0,0132
0,0107 0,0169
0,0054 0,0083
0,0072 0,0105
0,0091 0,0147
0,0175 0,0261
0,0084 0,0140
0,0086 0,0135
0,0111 0,0166
0,0119 0,0175
0,0068 0,0103
0,0147 0,0194
0,0073 0,0120
0,0124 0,0144
0,0098 0,0137
0,0062 0,0086
0,0074 0,0108
0,0100 0,0130
0,0066 0,0098
0,0103 0,0145
0,0162
0,0192
0,0116
0,0178
0,0245
0,0192
0,0214
0,0125
0,0154
0,0262
0,0420
0,0235
0,0198
0,0265
0,0231
0,0135
0,0277
0,0172
0,0211
0,0186
0,0182
0,0171
0,0184
0,0136
0,0211
0,0231
0,0301
0,0178
0,0256
0,0317
0,0306
0,0338
0,0173
0,0245
0,0365
0,0539
0,0254
0,0288
0,0346
0,0370
0,0192
0,0354
0,0240
0,0418
0,0306
0,0356
0,0224
0,0293
0,0206
0,0297
0,0353
0,0415
0,0269
0,0438
0,0357
0,0465
0,0557
0,0319
0,0403
0,0350
0,0931
0,0359
0,0374
0,0537
0,0549
0,0320
0,0464
0,0377
0,0469
0,0409
0,0399
0,0378
0,0510
0,0329
0,0447
0,0024
0,0031
0,0019
0,0017
0,0044
0,0029
0,0025
0,0010
0,0018
0,0031
0,0048
0,0025
0,0021
0,0026
0,0027
0,0016
0,0036
0,0017
0,0030
0,0021
0,0019
0,0018
0,0023
0,0019
0,0030
0,0042
0,0047
0,0028
0,0028
0,0059
0,0045
0,0042
0,0020
0,0030
0,0053
0,0076
0,0037
0,0034
0,0043
0,0044
0,0026
0,0061
0,0030
0,0048
0,0037
0,0029
0,0028
0,0039
0,0030
0,0049
0,0059
0,0065
0,0037
0,0051
0,0079
0,0067
0,0061
0,0031
0,0046
0,0085
0,0117
0,0047
0,0051
0,0064
0,0071
0,0042
0,0081
0,0047
0,0071
0,0051
0,0045
0,0039
0,0059
0,0047
0,0074
val(:,:,14) =
0,0082 0,0113
0,0096 0,0126
0,0052 0,0074
0,0080 0,0112
0,0100 0,0151
0,0087 0,0115
0,0090 0,0139
0,0047 0,0074
0,0070 0,0104
0,0104 0,0163
0,0170 0,0265
0,0071 0,0120
0,0089 0,0134
0,0095 0,0162
0,0106 0,0155
0,0063 0,0097
0,0131 0,0167
0,0069 0,0104
0,0091 0,0127
0,0087 0,0127
0,0060 0,0097
0,0065 0,0097
0,0088 0,0117
0,0061 0,0080
0,0105 0,0136
0,0151
0,0192
0,0110
0,0150
0,0236
0,0209
0,0196
0,0107
0,0148
0,0236
0,0411
0,0215
0,0171
0,0182
0,0225
0,0152
0,0223
0,0150
0,0222
0,0161
0,0154
0,0158
0,0175
0,0136
0,0207
0,0212
0,0308
0,0166
0,0239
0,0363
0,0289
0,0308
0,0197
0,0212
0,0416
0,0555
0,0213
0,0218
0,0278
0,0331
0,0210
0,0287
0,0196
0,0457
0,0285
0,0294
0,0252
0,0309
0,0206
0,0282
0,0320
0,0316
0,0234
0,0384
0,0390
0,0328
0,0602
0,0263
0,0362
0,0737
0,0728
0,0365
0,0387
0,0474
0,0494
0,0317
0,0477
0,0331
0,0781
0,0449
0,0383
0,0325
0,0581
0,0270
0,0520
0,0023
0,0031
0,0018
0,0015
0,0044
0,0028
0,0023
0,0010
0,0018
0,0027
0,0046
0,0024
0,0020
0,0025
0,0027
0,0041
0,0048
0,0028
0,0027
0,0057
0,0043
0,0038
0,0018
0,0030
0,0054
0,0074
0,0034
0,0033
0,0041
0,0044
0,0058
0,0065
0,0038
0,0045
0,0077
0,0058
0,0058
0,0028
0,0045
0,0081
0,0109
0,0050
0,0056
0,0060
0,0069
val(:,:,15) =
0,0080 0,0105
0,0091 0,0139
0,0052 0,0075
0,0073 0,0112
0,0107 0,0161
0,0077 0,0117
0,0083 0,0123
0,0041 0,0071
0,0067 0,0102
0,0112 0,0163
0,0176 0,0246
0,0070 0,0099
0,0076 0,0119
0,0089 0,0133
0,0105 0,0153
0,0148
0,0188
0,0109
0,0168
0,0241
0,0158
0,0187
0,0114
0,0144
0,0249
0,0364
0,0172
0,0167
0,0159
0,0226
0,0220
0,0273
0,0164
0,0235
0,0308
0,0269
0,0314
0,0157
0,0216
0,0298
0,0527
0,0195
0,0255
0,0254
0,0347
0,0327
0,0354
0,0208
0,0324
0,0440
0,0319
0,0659
0,0232
0,0327
0,0353
0,0731
0,0350
0,0461
0,0511
0,0487
0,0023
0,0032
0,0018
0,0015
0,0045
0,0028
0,0023
0,0010
0,0018
0,0027
0,0046
0,0022
0,0019
0,0024
0,0026
0,0040
0,0048
0,0027
0,0027
0,0058
0,0043
0,0036
0,0017
0,0029
0,0050
0,0072
0,0030
0,0029
0,0037
0,0043
0,0056
0,0063
0,0038
0,0043
0,0082
0,0057
0,0057
0,0028
0,0043
0,0088
0,0106
0,0044
0,0044
0,0058
0,0067
val(:,:,16) =
0,0078 0,0106
0,0082 0,0131
0,0052 0,0073
0,0071 0,0118
0,0113 0,0163
0,0083 0,0115
0,0081 0,0116
0,0042 0,0063
0,0064 0,0096
0,0108 0,0171
0,0163 0,0246
0,0065 0,0086
0,0066 0,0106
0,0086 0,0128
0,0105 0,0160
0,0148
0,0181
0,0107
0,0167
0,0242
0,0170
0,0164
0,0101
0,0147
0,0296
0,0335
0,0146
0,0165
0,0148
0,0248
0,0206
0,0270
0,0152
0,0227
0,0335
0,0272
0,0281
0,0141
0,0225
0,0424
0,0520
0,0164
0,0281
0,0201
0,0309
0,0302
0,0349
0,0227
0,0277
0,0356
0,0397
0,0560
0,0212
0,0319
0,0551
0,0664
0,0220
0,0433
0,0347
0,0501
- 54 -
0,0015
0,0037
0,0017
0,0028
0,0022
0,0016
0,0017
0,0023
0,0019
0,0031
0,0024
0,0059
0,0031
0,0044
0,0035
0,0026
0,0028
0,0039
0,0030
0,0050
0,0037
0,0089
0,0046
0,0064
0,0054
0,0041
0,0038
0,0058
0,0044
0,0074
0,0056
0,0117
0,0068
0,0092
0,0087
0,0059
0,0056
0,0086
0,0057
0,0105
0,0091
0,0158
0,0113
0,0134
0,0127
0,0098
0,0083
0,0137
0,0089
0,0146
0,0137
0,0250
0,0154
0,0214
0,0172
0,0145
0,0161
0,0196
0,0113
0,0212
0,0209
0,0305
0,0181
0,0450
0,0231
0,0308
0,0203
0,0296
0,0179
0,0273
0,0301
0,0496
0,0302
0,0588
0,0343
0,0293
0,0322
0,0489
0,0262
0,0491
0,0014
0,0036
0,0017
0,0030
0,0022
0,0016
0,0018
0,0025
0,0018
0,0030
0,0023
0,0053
0,0030
0,0052
0,0037
0,0026
0,0028
0,0043
0,0028
0,0049
0,0035
0,0093
0,0048
0,0072
0,0052
0,0040
0,0036
0,0067
0,0043
0,0073
0,0054
0,0122
0,0074
0,0108
0,0085
0,0061
0,0052
0,0099
0,0057
0,0098
0,0080
0,0165
0,0103
0,0144
0,0110
0,0105
0,0065
0,0159
0,0085
0,0140
0,0111
0,0265
0,0156
0,0166
0,0176
0,0173
0,0111
0,0220
0,0116
0,0198
0,0201
0,0335
0,0209
0,0399
0,0234
0,0279
0,0180
0,0355
0,0147
0,0269
0,0260
0,0406
0,0321
0,0429
0,0361
0,0457
0,0317
0,0468
0,0254
0,0476
0,0023
0,0030
0,0018
0,0015
0,0046
0,0027
0,0023
0,0010
0,0018
0,0030
0,0044
0,0021
0,0019
0,0023
0,0025
0,0013
0,0035
0,0017
0,0031
0,0022
0,0016
0,0018
0,0025
0,0018
0,0031
0,0039
0,0045
0,0028
0,0026
0,0061
0,0044
0,0037
0,0017
0,0029
0,0050
0,0066
0,0029
0,0031
0,0036
0,0043
0,0022
0,0056
0,0031
0,0058
0,0039
0,0028
0,0026
0,0042
0,0028
0,0049
0,0057
0,0060
0,0037
0,0041
0,0084
0,0060
0,0055
0,0028
0,0043
0,0077
0,0108
0,0045
0,0047
0,0052
0,0069
0,0034
0,0085
0,0049
0,0072
0,0056
0,0043
0,0039
0,0079
0,0041
0,0076
val(:,:,17) =
0,0078 0,0110
0,0083 0,0119
0,0050 0,0071
0,0062 0,0106
0,0116 0,0162
0,0080 0,0119
0,0080 0,0110
0,0043 0,0061
0,0064 0,0095
0,0096 0,0146
0,0159 0,0241
0,0065 0,0078
0,0062 0,0102
0,0080 0,0129
0,0109 0,0170
0,0056 0,0086
0,0129 0,0194
0,0078 0,0104
0,0106 0,0188
0,0086 0,0136
0,0061 0,0113
0,0056 0,0083
0,0106 0,0152
0,0061 0,0085
0,0100 0,0137
0,0163
0,0189
0,0105
0,0150
0,0261
0,0179
0,0181
0,0096
0,0144
0,0276
0,0355
0,0105
0,0143
0,0155
0,0247
0,0110
0,0275
0,0148
0,0174
0,0181
0,0154
0,0107
0,0239
0,0129
0,0199
0,0223
0,0228
0,0162
0,0227
0,0348
0,0238
0,0273
0,0142
0,0221
0,0541
0,0454
0,0157
0,0232
0,0203
0,0355
0,0172
0,0391
0,0249
0,0377
0,0296
0,0268
0,0187
0,0413
0,0184
0,0274
0,0330
0,0279
0,0220
0,0329
0,0425
0,0300
0,0423
0,0222
0,0308
0,0489
0,0620
0,0268
0,0381
0,0381
0,0569
0,0251
0,0382
0,0378
0,0460
0,0359
0,0349
0,0401
0,0509
0,0295
0,0426
0,0023
0,0031
0,0019
0,0015
0,0048
0,0028
0,0023
0,0010
0,0019
0,0031
0,0043
0,0020
0,0019
0,0022
0,0025
0,0014
0,0035
0,0017
0,0030
0,0024
0,0017
0,0018
0,0026
0,0018
0,0031
0,0040
0,0042
0,0028
0,0025
0,0062
0,0045
0,0037
0,0017
0,0030
0,0047
0,0069
0,0032
0,0034
0,0036
0,0044
0,0022
0,0051
0,0030
0,0050
0,0040
0,0029
0,0025
0,0044
0,0027
0,0048
0,0058
0,0060
0,0038
0,0040
0,0083
0,0060
0,0056
0,0029
0,0043
0,0070
0,0108
0,0048
0,0048
0,0055
0,0069
0,0036
0,0079
0,0047
0,0076
0,0060
0,0046
0,0038
0,0077
0,0041
0,0074
val(:,:,18) =
0,0080 0,0112
0,0077 0,0126
0,0054 0,0072
0,0060 0,0102
0,0107 0,0150
0,0080 0,0114
0,0085 0,0111
0,0046 0,0071
0,0062 0,0095
0,0095 0,0159
0,0165 0,0237
0,0062 0,0089
0,0067 0,0114
0,0083 0,0138
0,0105 0,0162
0,0062 0,0091
0,0121 0,0197
0,0075 0,0103
0,0095 0,0208
0,0081 0,0128
0,0067 0,0121
0,0053 0,0081
0,0107 0,0138
0,0059 0,0084
0,0100 0,0139
0,0167
0,0177
0,0108
0,0155
0,0242
0,0170
0,0177
0,0106
0,0150
0,0195
0,0331
0,0142
0,0152
0,0168
0,0268
0,0121
0,0256
0,0146
0,0256
0,0201
0,0180
0,0126
0,0245
0,0130
0,0200
0,0241
0,0215
0,0171
0,0231
0,0351
0,0259
0,0292
0,0139
0,0212
0,0375
0,0523
0,0178
0,0215
0,0192
0,0386
0,0177
0,0378
0,0213
0,0432
0,0332
0,0273
0,0193
0,0460
0,0184
0,0252
0,0362
0,0256
0,0249
0,0371
0,0484
0,0365
0,0369
0,0222
0,0268
0,0532
0,0618
0,0239
0,0429
0,0347
0,0537
0,0302
0,0496
0,0294
0,0632
0,0357
0,0350
0,0415
0,0484
0,0346
0,0376
0,0022
0,0030
0,0019
0,0015
0,0048
0,0028
0,0020
0,0009
0,0020
0,0032
0,0045
0,0021
0,0019
0,0023
0,0026
0,0014
0,0037
0,0017
0,0026
0,0024
0,0017
0,0019
0,0026
0,0017
0,0029
0,0040
0,0043
0,0028
0,0028
0,0065
0,0044
0,0033
0,0017
0,0031
0,0048
0,0071
0,0035
0,0030
0,0040
0,0046
0,0024
0,0058
0,0031
0,0049
0,0039
0,0028
0,0028
0,0041
0,0028
0,0046
0,0057
0,0061
0,0037
0,0043
0,0084
0,0061
0,0050
0,0029
0,0044
0,0065
0,0107
0,0049
0,0047
0,0062
0,0071
0,0037
0,0087
0,0050
0,0087
0,0066
0,0045
0,0042
0,0061
0,0042
0,0066
val(:,:,19) =
0,0077 0,0109
0,0079 0,0120
0,0053 0,0079
0,0070 0,0104
0,0112 0,0151
0,0082 0,0111
0,0077 0,0115
0,0046 0,0073
0,0061 0,0094
0,0091 0,0152
0,0154 0,0236
0,0072 0,0100
0,0076 0,0098
0,0085 0,0114
0,0112 0,0166
0,0064 0,0100
0,0121 0,0206
0,0080 0,0118
0,0100 0,0155
0,0095 0,0132
0,0071 0,0115
0,0059 0,0084
0,0102 0,0156
0,0056 0,0085
0,0093 0,0123
0,0160
0,0169
0,0119
0,0153
0,0217
0,0157
0,0170
0,0115
0,0146
0,0161
0,0354
0,0159
0,0156
0,0171
0,0261
0,0135
0,0288
0,0173
0,0271
0,0217
0,0168
0,0130
0,0265
0,0125
0,0173
0,0239
0,0196
0,0179
0,0249
0,0348
0,0195
0,0241
0,0164
0,0216
0,0331
0,0566
0,0192
0,0232
0,0206
0,0399
0,0196
0,0442
0,0232
0,0310
0,0304
0,0276
0,0196
0,0419
0,0201
0,0245
0,0336
0,0251
0,0270
0,0363
0,0456
0,0347
0,0304
0,0221
0,0301
0,0417
0,0675
0,0274
0,0197
0,0323
0,0555
0,0311
0,0572
0,0332
0,0481
0,0400
0,0398
0,0259
0,0584
0,0330
0,0345
0,0022
0,0029
0,0019
0,0014
0,0049
0,0030
0,0019
0,0010
0,0021
0,0026
0,0042
0,0026
0,0020
0,0025
0,0027
0,0015
0,0039
0,0018
0,0026
0,0025
0,0017
0,0020
0,0022
0,0016
0,0030
0,0040
0,0043
0,0028
0,0027
0,0071
0,0046
0,0031
0,0018
0,0033
0,0047
0,0070
0,0039
0,0029
0,0039
0,0048
0,0025
0,0063
0,0033
0,0043
0,0041
0,0033
0,0032
0,0039
0,0028
0,0047
0,0058
0,0063
0,0039
0,0043
0,0091
0,0061
0,0049
0,0030
0,0046
0,0073
0,0102
0,0054
0,0045
0,0062
0,0071
0,0038
0,0100
0,0049
0,0066
0,0068
0,0053
0,0046
0,0058
0,0041
0,0069
val(:,:,20) =
0,0079 0,0112
0,0085 0,0119
0,0055 0,0079
0,0071 0,0102
0,0122 0,0174
0,0080 0,0109
0,0073 0,0109
0,0042 0,0071
0,0065 0,0100
0,0109 0,0163
0,0165 0,0238
0,0078 0,0100
0,0071 0,0089
0,0086 0,0122
0,0118 0,0166
0,0063 0,0094
0,0142 0,0213
0,0077 0,0111
0,0092 0,0149
0,0104 0,0137
0,0085 0,0111
0,0067 0,0118
0,0094 0,0121
0,0060 0,0084
0,0095 0,0124
0,0152
0,0155
0,0121
0,0150
0,0235
0,0155
0,0154
0,0116
0,0151
0,0204
0,0358
0,0160
0,0156
0,0170
0,0238
0,0137
0,0315
0,0191
0,0205
0,0226
0,0164
0,0123
0,0216
0,0140
0,0163
0,0222
0,0255
0,0176
0,0274
0,0366
0,0205
0,0235
0,0198
0,0236
0,0278
0,0606
0,0251
0,0223
0,0223
0,0360
0,0208
0,0440
0,0262
0,0278
0,0374
0,0233
0,0257
0,0350
0,0231
0,0223
0,0333
0,0303
0,0264
0,0352
0,0515
0,0315
0,0338
0,0217
0,0337
0,0341
0,0754
0,0396
0,0371
0,0337
0,0530
0,0372
0,0497
0,0363
0,0489
0,0546
0,0458
0,0346
0,0652
0,0312
0,0331
0,0023
0,0032
0,0020
0,0013
0,0051
0,0040
0,0046
0,0030
0,0024
0,0073
0,0057
0,0068
0,0041
0,0038
0,0096
val(:,:,21) =
0,0079 0,0114
0,0095 0,0132
0,0059 0,0086
0,0061 0,0096
0,0131 0,0187
0,0162
0,0169
0,0126
0,0143
0,0252
0,0235
0,0243
0,0201
0,0266
0,0356
0,0321
0,0373
0,0271
0,0377
0,0549
0,0023
0,0034
0,0021
0,0013
0,0054
0,0041
0,0052
0,0031
0,0022
0,0075
0,0058
0,0074
0,0043
0,0033
0,0098
val(:,:,22) =
0,0081 0,0117
0,0107 0,0166
0,0061 0,0090
0,0053 0,0105
0,0142 0,0190
0,0178
0,0205
0,0139
0,0163
0,0277
0,0263
0,0327
0,0214
0,0236
0,0413
0,0355
0,0415
0,0319
0,0300
0,0764
- 55 -
0,0029
0,0021
0,0010
0,0021
0,0029
0,0043
0,0029
0,0020
0,0027
0,0027
0,0014
0,0041
0,0019
0,0028
0,0029
0,0018
0,0024
0,0023
0,0016
0,0030
0,0047
0,0034
0,0018
0,0034
0,0049
0,0067
0,0040
0,0032
0,0044
0,0047
0,0025
0,0068
0,0033
0,0045
0,0046
0,0035
0,0036
0,0037
0,0026
0,0049
0,0061
0,0054
0,0031
0,0048
0,0074
0,0111
0,0057
0,0048
0,0061
0,0074
0,0041
0,0108
0,0051
0,0072
0,0073
0,0061
0,0055
0,0060
0,0042
0,0077
0,0079
0,0075
0,0044
0,0067
0,0114
0,0164
0,0074
0,0065
0,0091
0,0123
0,0061
0,0148
0,0078
0,0101
0,0127
0,0100
0,0088
0,0085
0,0061
0,0105
0,0116
0,0118
0,0070
0,0108
0,0177
0,0226
0,0111
0,0104
0,0130
0,0188
0,0096
0,0216
0,0122
0,0158
0,0168
0,0142
0,0109
0,0132
0,0091
0,0141
0,0169
0,0183
0,0110
0,0167
0,0193
0,0355
0,0175
0,0154
0,0208
0,0243
0,0133
0,0312
0,0190
0,0210
0,0269
0,0198
0,0175
0,0186
0,0139
0,0203
0,0221
0,0255
0,0194
0,0241
0,0306
0,0558
0,0262
0,0246
0,0260
0,0411
0,0222
0,0422
0,0283
0,0418
0,0432
0,0319
0,0230
0,0301
0,0214
0,0287
0,0289
0,0373
0,0295
0,0366
0,0437
0,0765
0,0340
0,0271
0,0294
0,0622
0,0382
0,0567
0,0445
0,0342
0,0727
0,0493
0,0452
0,0532
0,0317
0,0360
0,0029
0,0023
0,0010
0,0023
0,0029
0,0046
0,0026
0,0021
0,0030
0,0029
0,0016
0,0039
0,0019
0,0030
0,0031
0,0021
0,0025
0,0025
0,0016
0,0031
0,0043
0,0038
0,0019
0,0035
0,0048
0,0072
0,0046
0,0037
0,0048
0,0047
0,0027
0,0064
0,0034
0,0050
0,0054
0,0040
0,0042
0,0044
0,0026
0,0051
0,0060
0,0061
0,0032
0,0051
0,0075
0,0109
0,0066
0,0061
0,0072
0,0079
0,0041
0,0110
0,0055
0,0081
0,0083
0,0070
0,0065
0,0063
0,0038
0,0081
0,0088
0,0084
0,0047
0,0075
0,0110
0,0163
0,0074
0,0089
0,0106
0,0129
0,0059
0,0158
0,0085
0,0110
0,0138
0,0109
0,0092
0,0096
0,0057
0,0112
0,0126
0,0127
0,0072
0,0113
0,0170
0,0260
0,0106
0,0128
0,0144
0,0212
0,0092
0,0239
0,0130
0,0153
0,0225
0,0164
0,0121
0,0126
0,0090
0,0155
0,0174
0,0181
0,0111
0,0187
0,0204
0,0347
0,0177
0,0187
0,0221
0,0296
0,0140
0,0351
0,0187
0,0198
0,0349
0,0279
0,0187
0,0196
0,0142
0,0227
0,0247
0,0276
0,0190
0,0259
0,0348
0,0553
0,0349
0,0315
0,0256
0,0451
0,0271
0,0499
0,0293
0,0299
0,0524
0,0383
0,0316
0,0295
0,0214
0,0335
0,0323
0,0473
0,0306
0,0377
0,0482
0,0680
0,0489
0,0359
0,0316
0,0590
0,0383
0,0522
0,0430
0,0446
0,0598
0,0567
0,0403
0,0474
0,0394
0,0368
0,0025
0,0034
0,0023
0,0014
0,0056
0,0031
0,0026
0,0011
0,0027
0,0027
0,0049
0,0031
0,0025
0,0036
0,0029
0,0017
0,0044
0,0021
0,0026
0,0037
0,0023
0,0026
0,0027
0,0016
0,0037
0,0043
0,0053
0,0033
0,0025
0,0080
0,0048
0,0045
0,0021
0,0042
0,0048
0,0080
0,0058
0,0040
0,0061
0,0049
0,0026
0,0073
0,0036
0,0046
0,0061
0,0040
0,0044
0,0045
0,0026
0,0060
0,0061
0,0077
0,0046
0,0037
0,0101
0,0065
0,0068
0,0035
0,0053
0,0079
0,0121
0,0080
0,0068
0,0092
0,0079
0,0043
0,0110
0,0058
0,0070
0,0102
0,0065
0,0070
0,0067
0,0039
0,0095
val(:,:,23) =
0,0086 0,0129
0,0117 0,0144
0,0067 0,0099
0,0059 0,0118
0,0143 0,0185
0,0104 0,0149
0,0099 0,0133
0,0053 0,0085
0,0080 0,0129
0,0112 0,0183
0,0177 0,0289
0,0087 0,0134
0,0104 0,0152
0,0135 0,0193
0,0127 0,0196
0,0064 0,0114
0,0169 0,0225
0,0087 0,0142
0,0109 0,0192
0,0165 0,0233
0,0108 0,0173
0,0097 0,0164
0,0102 0,0135
0,0055 0,0084
0,0127 0,0184
0,0188
0,0190
0,0153
0,0170
0,0309
0,0183
0,0203
0,0137
0,0196
0,0270
0,0456
0,0178
0,0233
0,0271
0,0307
0,0150
0,0287
0,0204
0,0281
0,0379
0,0290
0,0282
0,0199
0,0145
0,0260
0,0276
0,0318
0,0232
0,0261
0,0515
0,0268
0,0350
0,0224
0,0286
0,0368
0,0628
0,0263
0,0336
0,0336
0,0458
0,0227
0,0423
0,0311
0,0384
0,0489
0,0433
0,0355
0,0327
0,0210
0,0348
0,0385
0,0385
0,0365
0,0395
0,0828
0,0364
0,0505
0,0376
0,0396
0,0586
0,0839
0,0485
0,0351
0,0459
0,0591
0,0388
0,0683
0,0455
0,0540
0,0449
0,0478
0,0408
0,0428
0,0394
0,0534
0,0028
0,0036
0,0025
0,0015
0,0054
0,0033
0,0027
0,0012
0,0028
0,0030
0,0053
0,0033
0,0030
0,0044
0,0030
0,0017
0,0044
0,0024
0,0027
0,0041
0,0024
0,0030
0,0028
0,0016
0,0043
0,0047
0,0056
0,0036
0,0026
0,0081
0,0050
0,0050
0,0023
0,0043
0,0049
0,0088
0,0058
0,0050
0,0079
0,0053
0,0029
0,0074
0,0041
0,0044
0,0065
0,0037
0,0050
0,0043
0,0025
0,0066
0,0066
0,0082
0,0050
0,0041
0,0116
0,0065
0,0076
0,0038
0,0061
0,0078
0,0132
0,0085
0,0075
0,0120
0,0081
0,0045
0,0110
0,0065
0,0083
0,0117
0,0062
0,0079
0,0070
0,0037
0,0101
val(:,:,24) =
0,0094 0,0142
0,0118 0,0163
0,0074 0,0114
0,0062 0,0113
0,0159 0,0203
0,0106 0,0160
0,0115 0,0168
0,0067 0,0104
0,0090 0,0142
0,0121 0,0175
0,0201 0,0320
0,0117 0,0166
0,0106 0,0185
0,0158 0,0235
0,0119 0,0190
0,0062 0,0117
0,0147 0,0199
0,0098 0,0164
0,0151 0,0204
0,0190 0,0245
0,0087 0,0130
0,0111 0,0184
0,0105 0,0149
0,0061 0,0091
0,0132 0,0180
0,0209
0,0257
0,0177
0,0196
0,0343
0,0214
0,0224
0,0155
0,0221
0,0288
0,0500
0,0261
0,0316
0,0317
0,0311
0,0166
0,0253
0,0255
0,0310
0,0361
0,0273
0,0291
0,0226
0,0164
0,0247
0,0297
0,0362
0,0276
0,0260
0,0455
0,0298
0,0444
0,0241
0,0356
0,0278
0,0755
0,0289
0,0434
0,0358
0,0422
0,0258
0,0465
0,0333
0,0461
0,0477
0,0449
0,0404
0,0319
0,0252
0,0343
0,0427
0,0424
0,0417
0,0422
0,0838
0,0459
0,0670
0,0339
0,0498
0,0545
0,1121
0,0516
0,0614
0,0773
0,0624
0,0372
0,0709
0,0446
0,0744
0,0504
0,0427
0,0533
0,0448
0,0384
0,0469
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