Edge Detection Revisited

1500
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 34, NO. 3, JUNE 2004
Edge Detection Revisited
Felice Andrea Pellegrino, Walter Vanzella, and Vincent Torre
Abstract—The present manuscript aims at solving four problems of edge detection: the simultaneous detection of all step edges
from a fine to a coarse scale; the detection of thin bars with a width
of very few pixels; the detection of trihedral junctions; the development of an algorithm with image-independent parameters. The
proposed solution of these problems combines an extensive spatial
filtering with classical methods of computer vision and newly developed algorithms. Step edges are computed by extracting local
maxima from the energy summed over a large bank of directional
odd filters with a different scale. Thin roof edges are computed by
considering maxima of the energy summed over narrow odd and
even filters along the direction providing maximal response. Junctions are precisely detected and recovered using the output of directional filters. The proposed algorithm has a threshold for the minimum contrast of detected edges: for the large number of tested images this threshold was fixed equal to three times the standard deviation of the noise present in usual acquisition system (estimated
to be between 1 and 1.3 gray levels out of 256), therefore, the proposed scheme is in fact parameter free. This scheme for edge detection performs better than the classical Canny edge detector in two
quantitative comparisons: the recovery of the original image from
the edge map and the structure from motion task. As the Canny detector in previous comparisons was shown to be the best or among
the best detectors, the proposed scheme represents a significant improvement over previous approaches.
Index Terms—Edge detection, junction detection, performance
evaluation.
I. INTRODUCTION
S
TEP Edges, lines and junctions usually convey the most relevant information of an image [4], [41], [46], hence it is
important to detect them in a reliable way. Edge detection has
been extensively analyzed in computer vision during the last 20
years [5], [7], [10], [18]–[20], [41] but it has not yet been entirely
solved. In fact, despite considerable efforts, the ideal scheme
able to detect and localize with precision edges at all scales,
irrespective of their shape and configuration, has not yet been
produced. The development of the optimal or ideal edge detection scheme is also difficult for the absence of evident and clear
ground truth data, on which to evaluate performances unambiguously [28]. Several criteria have been proposed for the evaluation of edge detection schemes: for instance, task dependent
criteria have been suggested based on the recovery of three-dimensional (3-D) data on a structure from motion task [36], or the
recovery of the original image from the edge map [28]. Human
evaluation criteria also have been proposed and made quantitative by using statistical and psychophysical tools [14], [27].
Manuscript received April 8, 2003; revised August 19, 2003. This paper was
recommended by Associate Editor D. Goldgof.
F. A. Pellegrino is with the Department of Mathematics and Computer Science (DIMI), University of Udine, Udine 33100, Italy.
W. Vanzella and V. Torre are with the Department of Neurobiology,
International School for Advanced Studies, Trieste 34014, Italy (e-mail:
[email protected]).
Digital Object Identifier 10.1109/TSMCB.2004.824147
In all these comparisons the classical edge detection scheme of
Canny [5] provides very often—but not always—the best performances. This scheme, in substance, identifies an edge in the
as a local maximum of an appropriate
original image
function of the convolution of
with odd directional filters. The success of the edge detection scheme [5], proposed
almost 20 years ago, suggests that its essence is correct and
close to being optimal. It is well-known that the human visual
system is able to perform an ideal edge detection, independently
on recognition and reasoning. Therefore, the observation of the
human visual system may suggest how to modify the classical
edge detection scheme of Canny, in order to obtain an almost
ideal edge detection scheme.
In the visual system of higher vertebrates and mammals, the
first stage of processing occurring in the visual area V1, consists
in the convolution of the retinal image with filters having different
orientation, size and shape [16]. Computer vision, also, uses a
large set of filters for image processing. In fact, the detection of
edges, corners, and other important two-dimensional (2-D) features has been obtained by convolving the original image with directional filters having different orientation, size, and shape [11],
[12], [39]. When 2-D features are detected by computing maxima
of the local energy [18], [24], [31] the image is first convolved
with odd and even filters of different orientations. The need for
analyzing images at different scale, i.e., multiscale analysis [19],
[45], is now widely recognized in computer vision and is a standard technique in image processing [13], [22].
Both biological and artificial vision agree on the need of analyzing images with a large battery of filters with different properties. The next question, obviously, is how to combine the output
of these filters. This is an important issue both in a biological perspective and in computer vision: how is the output of these filters
optimally integrated for detecting edges at all scales, identifying
features of increasing complexity and recognizing objects? The
answer to this question may help to understand the logic underlying processing in the visual cortex and to develop highly performing algorithms for computer vision.
The major aim of this paper is to use the rather large set of
filters, present and implemented in the visual area V1, to improve the edge detection scheme originally proposed in [5] in
such a way to obtain an almost ideal edge detection scheme,
able to detect: 1) step edges [24] from a fine to a coarse scale;
2) thin bars with a small width of two pixels or so; and 3) trihedral junctions with one side having a low contrast. The present
paper aims also at providing a scheme without any parameter
which has to be tuned for the specific image where edges have
to be detected: in fact, the proposed scheme for edge detection
has only one free parameter, i.e., the threshold for the minimum
contrast of detected edges , set equal to three times the standard deviation of the noise present in usual acquisition system.
This threshold , estimated to correspond to four gray levels
1083-4419/04$20.00 © 2004 IEEE
PELLEGRINO et al.: EDGE DETECTION REVISITED
1501
out of 256, was used to extract edges from a large number of
test images, consistently providing very good results.
The proposed scheme for edge detection combines several
previously suggested approaches [5], [24], [9], [17] with new
algorithms. The information obtained by extensive filtering here
described is also used [42] for a final regularization of the original
image using a modified version of the Mumford and Shah functional [26]. This regularization step uses a locally self-adapting
scale and contrast in order to preserve small details. From this
regularized image it is possible to obtain an image segmentation,
providing a convenient decomposition of the image in a limited
number of regions with the same gray level. This segmentation
is, in our opinion, an almost ideal preprocessing for higher level
functions such as categorization and recognition.
The paper is organized in the following way. Section II reviews the previous work and outlines the proposed approach
to edge detection. Section III summarizes the properties of the
most widely used filters in computer vision. Section IV and V
discuss the detection of step and roof edges, respectively, i.e.,
the detection of edges separating two extended regions of the
image (step edges) and the detection of thin bars or roof edges.
Section VI treats the detection and recovery of trihedral junctions. Section VII presents experimental results on a large set
of images of different kind and a comparison with previous approaches to edge detection. Section VIII provides an overview
of the algorithm, adding further information concerning noise
robustness, parameter sensitivity, and computation time.
II. EARLY APPROACHES
Early approaches to edge detection [5], [41] primarily focused their analysis on the identification of the best differential operator necessary to localize sharp changes of the intensity
at location
. These approaches recognized the necessity of a previous filtering, identified as a regularizing stage,
but did not explicitly realize the convenience of a full multiscale
analysis, useful for a reliable detection of both sharp and smooth
edges [8]. Following some biological suggestions, Morrone and
Burr [24] proposed to detect edges as maxima of the local energy
obtained as
[24], [31] when features to be detected are significantly larger
than the size of used filters. This problem will inevitably occur
with digital images when bars having a width below 2 or 3
pixels have to be detected. The simplest way to overcome it is
to oversample the image, but significantly increasing the computational burden of the analysis.
A rather different method of edge detection is obtained
by considering variational approaches [26], [23]. In these
approaches, the set of edges and a suitable approximation
of a given image
are obtained by minimizing,
over the image domain , the functional
(3)
where
is a piece-wise smooth approximation of
is the total length of the edges, is the gradient
operator and and are appropriate constants. Edges obtained
by minimizing the functional (3) and by computing local
maxima of (2) often coincide. Step edges at a given scale are
usually well detected by minimizing the functional (3), but thin
roof edges are not detected and are smeared out.
Inthispaperweadopttheviewofdetectingandlocalizingedges
as local maxima of the energy (2), investigating, in addition, how
to combine the output of different V1-like filters with different
scales, directions and parity for an “almost ideal” detection of
step edges, of thin roof edges and of trihedral junctions at fine and
coarse scales. It will be shown that step edges are best detected by
considering the energy (2) composed only by the output of odd
filters, but summed over all directions and scales. The observation that better performances are obtained considering only odd
filters, is in complete agreement with the early proposal of Canny
[5]. On the contrary, thin roofedges are bestlocalizedby summing
the energy from narrow even and odd filters only in the direction
of maximal response. Trihedral junctions are localized by computing the local curvature, which can be obtained by combining
even and odd filters appropriately, and their sides are precisely
identified by the output of directional filters.
III. FILTERING
(1)
where
and
are the convolution of the
original image
with a quadrature pair of one-dimensional filters in the direction. A quadrature pair of one-dimensional filters consists of one even and one odd-symmetric filters
with zero mean, same norm and orthogonal to each other. These
two filters are one the Hilbert transform of the other. Subsequent
approaches [18] suggested the local energy to be
(2)
In this section properties of some spatial filters often used
in computer vision are briefly recalled: Gabor, log-Gabor, and
gaussian filters will be considered. These filters can be coupled
in order to form a quadrature pair [24]. A quadrature pair of
one-dimensional (1-D) filters consists of one even and one oddsymmetric filters with zero mean, same norm and orthogonal to
each other. These two filters are one the Hilbert transform of the
other.
A 1-D Gabor filter is a gaussian function modulated by a sinusoidal wave.
(4)
i.e., the energy summed over different directions and scales
. In these approaches, edges are identified as local maxima
of the energy (possibly normalized) of (2). In one dimension,
maxima of local energy are also points of maximal phase congruency [43], i.e., points of particular interest in the image.
Local maxima of the energy (2) correctly identify edges [18],
is the
where is a parameter setting the spatial spread and
frequency of the modulating sinusoidal wave. Gabor filters have
Fourier transform
(5)
1502
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 34, NO. 3, JUNE 2004
Fig. 1. Detection of step edges with odd and even filters. (a) One-dimensional intensity profile with bars of different width having the right edge slightly smoothed.
(b) Energy profile using odd and even 1-D gaussian filters and in (c) using only odd 1-D gaussian filters. (d)–(f) as in (a)–(c) for the second bar of panel (a). In (e)
and (f), dot and dash-dot lines refer to Gabor and log-Gabor filters. Filters had nine different sizes with !
ranging from 3–18 pixel/cycle (see Section III).
has its maximum in
defined by its Fourier transform.
. A log-Gabor filter is
(6)
has its maximum in
and is a conAlso,
stant determining the frequency bandwidth. The odd and even
log-Gabor filters in the spatial domain are defined as the back
transform of a quadrature pair of filters, see for instance [32]
F
F
(7)
The pair of gaussian filters is obtained by back-transforming
a quadrature pair of filters.
(8)
. The size
These filters have their maximum for
of a given filter is taken as the period corresponding to
or
. The size of a gaussian filter is simply
.
Even and odd gaussian filters do not have ripples, but have a
broader spectrum. The Fourier transform of log-Gabor filters is
symmetric when plotted on a semilog scale. Gabor filters have
Fourier transform maximally concentrated around
and in
the space domain they have significant ripples and oscillations
introduced by the sinusoidal modulation.
When edge detection is performed considering the energy
summed over the output of odd and even filters, as in (1), Gabor,
log-Gabor, and gaussian filters provide rather similar results
(see Fig. 1). When only odd filters are used, as it will be done in
the next section, gaussian filters provide best results, due to the
absence of ripples.
One-dimensional filters can be extended to two dimensions in
several ways. Two-dimensional filters with a Fourier transform,
which in polar coordinates can be factorized in the product of
two functions one depending on radius and the other on the
angle , are called separable. These filters can be designed and
implemented using the ring-wedge technique (see for instance
[18]). Two-dimensional Gabor and log-Gabor filters were implemented using this technique.
Two-dimensional gaussian filters were constructed in the
space domain multiplying the 1-D profile by a gaussian function
in an orthogonal direction. The ratio between the variance of
the two gaussian functions was 1.2. Two-dimensional gaussian
Filters constructed in this way do not have ripples, which are
present in 2-D Gabor and log-Gabor filters.
It is useful to recall that the output of a directional odd
gaussian filter is a good approximation for the computation
of a derivative in the same direction. Similarly, the output of
an even, zero-mean gaussian filter, is a good approximation
of a second derivative. These remarks will be useful in a later
section, when the local curvature of the image intensity
has to be computed.
IV. DETECTION OF STEP EDGES
In this section we address the issue of the detection of step
edges at all scales, i.e., sharp and blurred edges. The detection
of edges, composing the contour of large features (spanning at
least five and more pixels), can be efficiently solved by detecting
maxima of (1) or (2) as shown previously [18]. The detection
of boundaries of features with a dimension of 2–4 pixels poses
some problems.
PELLEGRINO et al.: EDGE DETECTION REVISITED
1503
Fig. 2. Detection of step edges with odd and even filters in 2-D images. (a), (d), (g): details from interior images. (b), (e), (h) Step edges detected as local maxima
using odd and even filters. (c), (f), (j) Step edges detected using only odd filters. Eight orientations and nine scales, varying from three pixel/cycle to 18 pixel/cycle
were used. The ratio of to was 1.2 for odd filters and 2 for even filters.
A. Combination of Odd and Even Filters
Let us consider an ideal 1-D intensity profile with a 128 gray
and a value of 1 elsewhere.
values between pixel and
and in
The energy (2) has three local maxima in
, with the first two maxima corresponding to step edges
and the maximum at
corresponding to a roof edge.
When becomes close to three or four, the three local maxima
coalesce and cannot be reliably identified. Therefore, the detection of bars with a width between two and four pixels must be
carefully investigated. Let us consider the 1-D intensity profile
shown in Fig. 1(a), composed by thin bars of increasing width,
with one sharp and one smooth boundary. When the energy (1)
is computed by summing the output of odd and even gaussian
filters the profile shown in Fig. 1(b) is obtained, where only one
peak is present for bars with a width of less than four pixels.
Two peaks become visible when the bar width is larger than five
pixels. On the contrary, if only odd filters are used, two peaks
corresponding to the two boundaries of the bar are obtained [as
shown in Fig. 1(c)] even when the bar width is just two pixels.
Performances of the different filters are compared in Fig. 1(e)
and (f) for the intensity profile shown in Fig. 1(d), corresponding
to the second bar in Fig. 1(a). Fig. 1(e) shows the energy profile
obtained from (1) using even and odd Gabor (dotted line), log-
Gabor (dash dotted line) and gaussian filters (continuous line).
The two edges delimiting the bar are not well detected and one
large maximum is present, irrespectively on how small and large
filters are combined (see later section for further details). On the
contrary, if only odd filters are used [see Fig. 1(f)], maxima are
clearlydetected atthebarboundaries.WhenGaborandlog-Gabor
are used, additional maxima are seen [indicated by the arrows in
Fig. 1(f)]. These spurious additional maxima are produced by the
ripples of these filters, which become evident when the filters are
not used in quadrature pair. For this reason gaussian filters are
preferred to Gabor and log-Gabor filters.
When 2-D images are considered, edges are detected as local
maxima of the energy (2) summed over eight equally spaced
directions and nine different scales (see next section). Local
maxima were detected by a modification of the well-known
[46] nonmaximal suppression algorithm (performed, in each
point, along the direction perpendicular to the filter providing
the highest response). Such modification consists of an energy-driven thinning which removes possible double edges,
i.e., boundaries with a thickness larger than one pixel, which
may occur using the standard nonmaximal suppression.
Fig. 2(a), (d), and (g) illustrates three details of interior images, with thin bars of a width of two to four pixels. Edges detected as local maxima of (2) using odd and even gaussian filters
1504
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 34, NO. 3, JUNE 2004
Fig. 3. Detection of step edges with filters at different scales. (a) and (b) Fourier transform of filters at different scales normalized to the same energy but with
. (c) Fourier transform of filters normalized to the same maximal response. In (a)–(c) the dot line is the sum of all filters energy.
different distribution of !
(d) One-dimensional intensity profile with representative roof and step edges with different strength and smoothness. (e) The corresponding total energy for the
intensity profile of panel (d) for the three cases shown in (a) (round dot line), (b) (continuous thin line) and in (c) (continuous thick line).
are shown in Fig. 2(b), (e), and (h). These edges are not as complete as when only odd filters are considered, as clearly shown in
panels (b), (c), (e), (f), (h), and (j). When gaussian odd filters are
used, step edges are properly localized at the feature boundaries
also for the thin bars, such as those shown in Fig. 2(a) and (d). It
is not surprising that the detection of step edges is best obtained
by using only odd filters, as they are the natural matched filters
for these features [5]. As a consequence only odd gaussian filters will be used to detect step edges.
B. Combination of Different Scales
In order to detect step edges at fine and coarse scales it is necessary to use small and larger filters, with different frequency
properties, but it is not obvious how to weight their relative
contribution. In what follows, gaussian filters with nine scales
(see Section III) ranging from three
were considered with
pixel/cycle to 18 pixel/cycle. The choice of these nine scales
was suggested by a large experimentation with natural images,
as those analyzed in the present paper. The scales of odd filters
in (2) can be combined by using filters with the same energy (see
Fig. 3(a) and (b), where different distributions of center frequencies are shown) or with the same maximal response [Fig. 3(c)].
These filters sample the frequency content of the image in a
slightly different way, as shown by the summed total energy
drawn the corresponding panels (dotted lines). Fig. 3(d) illustrates an ideal 1-D intensity profile containing a variety of step
and roof edges: a smooth step edge, a strong roof edge flanking
two smoothed roof edges, a strong bar followed by a sharp step
edge. The corresponding total energy for the three sets of filters
as in a, b, and c is shown in panel e in round-dot, square-dot
and continuous line, respectively. With filters having the same
maximal response [as those shown in Fig. 3(c)] the total energy
has ten local maxima [continuous thin line in Fig. 3(e)] corresponding to the ten features to be detected. When the filters are
normalized in order to have the same energy [as in Fig. 3(a) and
(b) several local maxima are lost, as shown by the dot line and
the continuous thin line in Fig. 3(e)].
When edges with different contrast and opposite sign are
present in the same region of the image and the output of fine
and coarse filters superimpose spatially, their simultaneous
detection may be difficult. Extensive experimentation has
shown that some details [as those shown in Fig. 4(a), (d), and
(g)] are best detected by combining scales with same maximal
response [see Fig. 4(c), (f), and (j)], instead of when filters
having the same energy are used [see Fig. 4(b), (e), and (g)].
V. DETECTION OF THIN ROOF EDGES
In this section the detection of thin bars will be investigated. It
is well known [18], [24] that the detection of thin details in real
images, as the thin bars shown in Fig. 5(d) is best obtained by
using both even and odd filters [see Fig. 5(f)], when the output
of even filters dominates at the center of the bar and the output
of odd filters reduces and eliminates spurious maxima near the
bar boundaries. If only even filters are used, multiple maxima
are detected at the center of the bar and at its boundaries. In this
section, it will be shown that the detection of thin bars, however,
is best obtained considering only the energy along the direction
of maximal response and not the energy summed over all directions, like it was done instead for the detection of step edges (see
Section IV).
PELLEGRINO et al.: EDGE DETECTION REVISITED
1505
Fig. 4. Detection of step edge with filters at different scales in 2-D images. (a), (d), and (g) are details of an interior scene; (b), (e), and (h) show the detected
step edges summing the energy over filters having the same energy and in (c), (f), and (j) with same maximal response. Eight orientations and nine scales, varying
from 3 pixel/cycle to 18 pixel/cycle were used. The ratio of to was 1.2 for odd filters.
A. Different Directions Are not Summed
B. Detection at Different Scales
Therefore, combining even and odd filters seems the best solution for the detection of thin bars, but it is not obvious whether
it is convenient to sum the energy over a battery of even and odd
filters with different orientations.
The profile of the summed energy of the output of even and
odd filters perpendicularly oriented as shown in Fig. 5(a) to a
thin bar [with the intensity profile indicated by the solid line
in Fig. 5(c)] has one large central peak, correctly located at the
center of the thin bar [see line in Fig. 5(c)]. On the contrary,
when even and odd filters are not perpendicular to the thin bar
[see Fig. 5(b)] the profile of the summed energy has two smaller
peaks [see line Fig. 5(c)]. As a consequence, summing over all
directions is not advantageous for the detection of thin bars with
a low contrast as the one shown in Fig. 5(d) and of multiple thin
bars in different directions as those shown in Fig. 5(g). The thin
bar of Fig. 5(d) has an average contrast of only five gray levels
out of 256 and it is better detected and localized when considering the energy only along the direction providing maximal
response [see Fig. 5(f)]. Similarly, the intricate pattern of thin
bars of Fig. 5(g) is reasonably well detected when the energy
considered is only in the maximal direction. When the energy is
summed over all directions [see Fig. 5(e) and (h)] thin bars are
not well detected.
Roof edges with a different width were detected independently. Very thin roof edges, with a width of one or two pixels
were detected by using gaussian filters with a width of four
pixels cycle and are shown in green in Fig. 6(b). Medium size
roof edges, with a width of three or four pixels were detected
by using gaussian filters with a width of eight pixels/cycle and
are shown in blue in Fig. 6(b). Larger roof edges with a width of
five pixels or so were detected with a larger filter with a width
of 18 pixels/cycle and are shown in Fig. 6(b) in red. Directional
filters were longer than those used for detecting step edges and
to
was two for both even and odd filters. The
the ratio of
energy was computed at the three scales according to (1). Local
maxima were extracted and a threshold was applied as described
in the previous section.
C. Classification of Roof Edges
Once maxima of the energy have been located and identified,
it is still necessary to classify them in roof and step edges. The
usual way to classify edges [24] is to compare the absolute value
of the response of the odd and even filters: a maximum of the
energy is a step edge when the contribution of the odd filters to
the energy is greater than the one given by the even filters. This
1506
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 34, NO. 3, JUNE 2004
Fig. 5. Detection of thin roof edges. (a) and (b): even and odd filters perpendicular (a) and oblique (b) to a thin bar. (c) One-dimensional energy profile (thick
line) and the energy profile only along the direction providing maximal response (line) and energy profile summed over all directions. (d) A detail of the same
image as in Fig. 6(a). (e) Thin roof edges detected as local maxima of the energy summed over all directions. (f) Same as in (e) but with the energy only along the
direction providing maximal response. (g), (h), (j) as in (d), (e), (f) for a particular of an outdoor scene.
Fig. 6. Detection of roof edges of different width. (a) Original image. (b) The map of roof edges extracted as described in the text. Roof edges have an increasing
width from green, blue, and red.
classification procedure is adequate for classifying roof edges
with a width larger than one pixel. The detection and classification of roof edges with a width of one pixel is very sensitive to
noise and therefore unstable.
Once a pixel has been classified as a roof edge, a scale must be
assigned to it. If the edge occurs only at a given scale, that scale is
assigned to it. Otherwise, the smallest scale is assigned. Fig. 6(b)
illustrates roof edges extracted from the original image shown in
PELLEGRINO et al.: EDGE DETECTION REVISITED
1507
Fig. 7. Recovery of trihedral junctions. (a) Detail of a cube. (b) Detected step edges. (c) Normalized curvature. (d) Regions of interest of high curvature. (e) End
points of the edge map within the regions of interest. (f) Restored sides of the trihedral junctions.
Fig. 6(a). The great majority of roof edges are detected at the correct scale. However, the resulting edge map is more noisy than that
obtained forstep edges because edges are obtained from an energy
map which is not summed over different scales and directions and
therefore, they are more sensitive to noise.
times
. In order to achieve invariance from the local contrast, should be equal to zero. However, with this choice there
are many false detections. Extensive experimentation has suggested to use the quantity
(10)
VI. DETECTION OF TRIHEDRAL JUNCTIONS
It is well known that edge detection based on the extraction
of local maxima fails in the presence of trihedral junctions, such
as those shown in Fig. 7(a) and edges are usually interrupted at
junctions [see Fig. 7(b)]. By combining the output of V1-like
filters in an appropriate way, it is possible to restore these important image features, which are inevitably lost using (2). Regions where trihedral junctions are likely to be present can be
detected in several ways [1], [3], [8], [9], [17], [35], primarily
by evaluating the local curvature and detecting its maxima. By
using a modification of the method proposed in [17], high curvature regions are localized [Fig. 7(c)] where trihedral junctions
are likely to be present [Fig. 7(d)]. End points of edge lines—in
these regions—are detected [see Fig. 7(e)] and sides of trihedral junctions [see Fig. 7(f)] can be restored by using suitably
the maps of the directional local maxima, i.e., the binary map of
local maxima of the energy of individual directional filters. This
procedure is able to appropriately restore missing junctions (indicated by arrows in Fig. 8) in complex and cluttered images.
Let us now see the different steps of the proposed procedure.
A. Computation of Local Curvature
Given the intensity profile
vature of it is
of an image, the local cur-
(9)
Kitchen and Rosenfeld [17] have suggested to identify possible junctions and corners as local maxima of the curvature
and where has been introduced in
corresponding to
order to reduce the distortion appearing when the gradient is
zero or very small. Equation (10) is equivalent to
(11)
is the second derivative in the direction orthogonal
where
is the maximal directional derivative.
to the gradient and
All these quantities are obtained by the output of odd and even
filters here used.
B. Computation of Regions of Interest and Endpoints
Points on the image with a high value of are obtained by
setting a fixed threshold that allows the detection of a trihedral junction having one side with a contrast of six gray levels
perpendicular to a strong sharp edge with a contrast of about
255 gray levels. This set of points is dilated by seven pixels,
so to have regions wide enough to contain possible endpoints
of edge lines. In these regions of interest, possible end points
are identified by means of a look-up table containing all possible endpoints in a 3 3 neighborhood. To each endpoint, corresponding to a “presumed junction,” a direction is associated.
Once end points and their “presumed junction” direction have
been identified, it is necessary to validate or not the presumed
junction, by considering the energy of directional filters. The
overall adopted procedure is
Given an end-point in a region of interest, the direction along
which the energy has the highest value is taken as the presumed
1508
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 34, NO. 3, JUNE 2004
Fig. 8. Example of trihedral junction recovery. (a) Detail of an interior image. (b) Detected step edges. (c) Step edges plus restored junctions. (d) Detail of an MRI
image of a brain slice. (e) Detected step edges. (f) Step edges plus restored junctions. (g) Synthetic image. (h) Detected roof edges. (j) Roof edges plus restored
junctions.
junction direction (with a versus pointing toward the region with
less edge pixels). The energy is obtained by summing the output
of directional filters, as those used in Section IV, summated over
nine different scales (see legend of Fig. 2 for further details).
The end-point is connected to another edge pixel (and hence
the junction is restored) if there exists a chain of edges which are
local maxima of directional filters and connect the two points.
The junction is finally validated if its local average contrast is
larger than eight gray levels and its total length is not more than
five pixels.
This procedure is able to recover correctly missing junctions
in complex real images, such as the interior detail of Fig. 8(a)
and the MRI image of a brain slice of Fig. 8(d). In both cases
the procedure is able to recover missing junctions [see Fig. 8(c)
and (f)], which are lost during the usual edge detection [see
Fig. 8(b) and (e)]. Although it is not clear how to evaluate the
performance of a junction detection/recovery algorithm, a possible (simple) way is to check if the procedure is able to extract the “correct” junctions in an artificial noisy image, without
introducing artifacts. We performed such an evaluation using
artificial images containing junctions of different contrast and
blurring, adding more and more noise in order to check the robustness. Our detector behaved very well, identifying and reconstructing all the correct junctions in images containing ad-
ditive normal noise up to a variance of nine gray levels. It is
useful to observe that missing junctions are restored correctly,
even when the direction where the end points seem to converge
is not correct: this robustness is primarily due to the validation
procedure using the output of directional filters. The proposed
technique for the recovery of trihedral junctions can be easily
extended also to restore junctions among thin bars or roof edges
[see Fig. 8(g), (h), and (j)].
VII. EXPERIMENTAL RESULTS
A major and well known problem for evaluating performances of edge detection algorithms is the absence of clear
ground truth data (see for example [21]). Nonetheless, there
are some quantitative criteria to evaluate edge detection: for
instance task dependent criteria have been suggested based on
the recovery of 3-D data on a structure from motion task [36],
the recognition of objects [37] and the recovery of the original
image from the edge map [28]. This last method is based on the
reconstruction algorithm proposed in [6] and it will be used for
a quantitative assessment of the proposed algorithm for edge
detection. This section is divided in three parts: in the first part
(Section VII-A) a qualitative comparison between previous
approaches and the present scheme for edge detection is made;
PELLEGRINO et al.: EDGE DETECTION REVISITED
in the second part (Section VII-B), the recovery procedure [6]
already adopted in [28] is used for a quantitative evaluation and
comparison of the proposed scheme; in the third part (Section
VII-C), performances of the proposed edge detector for the
task structure from motion [36] are discussed; finally, in the
fourth part (Section VII-D), a selection of experimental results
on cluttered real images is presented.
A. Qualitative Comparison With Previous Approaches
The proposed edge detector has been designed and constructed in such a way to 1) detect step edges at all scales; 2)
detect thin roof edges; 3) detect trihedral junctions; and 4) be
robust against noise and be almost parameter free. The detection of thin roof edges and of trihedral junctions is obtained by
separate modules, described in Sections V and VI. The detection of step edges at all scales and the robustness against noise
is primarily obtained by summing the energy over different
directions and scales. Step edges are then detected by looking
at local maxima. This procedure has two major advantages:
first, it allows the detection of step edges at multiple scales;
secondly, by averaging over multiple directions and scales,
edges originating from noise are eliminated and the resulting
detector has very good performances even with a low and fixed
threshold. This is the key feature of the proposed edge detector.
It is useful, now, to make a methodological comparison
with previous approaches. The classical edge detector proposed
almost 20 years ago by Canny [5] performs remarkably well
with its simplicity and elegance. Indeed, in recent comparisons
of edge detector performances [14]–[36], Canny detector was
the best or one of the best. Canny’s edge detector attempts to
maximize simultaneously localization and signal-to-noise ratio.
As a consequence, the best way to detect step edges is to filter
the image with odd directional filters of a shape similar to the
derivative of a gaussian function. This conclusion is very similar
to what it is argued here in Section IV. Canny’s edge detector
has two drawbacks: first, it cannot detect simultaneously step
edges at two very different scales; second, trihedral junctions
cannot be properly detected and are often broken in distinct
segments.
Different edge detectors have been discussed and compared
in details in [36] and here only a briefer qualitative comparison is presented. Edge detectors proposed by Deriche [7] and
by Shen and Castan [38] have the same structure of Canny’s
detector, but use different directional filters. In the case of Deand for Shen
riche, the directional filter is
. The filter used by
and Castan is
Shen and Castan is discontinuous at zero and therefore, is very
sensitive to noise. The detectors proposed in [34]–[38] use a
single scale and a gradient based approach. SUSAN detector
[40] has a rather different approach and is based on a nonlinear,
single-scale filtering.
Multiple scales were used by several authors [2], [10], [19].
Our approach differs from these by the way in which the different scales are processed and combined.
Kovesi [18] proposes to detect edges by looking at maxima of
the phase congruency and not of the energy. The phase congruency is the energy properly normalized. The energy is summed
over the output of filters with different scales and directions,
1509
provided that it is above a threshold related to the energy of
the output at the smallest filter. Contrary to us, however, Kovesi
sums over odd and even filters and therefore his approach suffers from the problems described in Figs. 2 and 3. The approach
of [33] is similar to that of [18], but at a single scale.
Recently, a multifeature technique has been proposed [21]
in which a classifier is trained using human labeled images as
ground truth. Human labeled images represent a segmentation
of the images in few regions (20–30) and the algorithm is more
properly a boundary detector. Moreover, it produces a probability map instead of a binary edge map. So, a comparison between the proposed detector and that one of Martin and coauthors [21] is inappropriate.
B. Quantitative Comparison With Previous Approaches:
Image Recovery From Edges [34]
Carlsson [6] has proposed the following procedure to recover
from the edge map . First, this edge
the original image
map is dilated by 1 pixel, so to obtain the set . Secondly, the
original gray values
with
are used as interpolating values for the minimization of the (discretized version of
the) functional
(12)
The function
straints
, which minimizes (12) under the con(13)
is the recovered image. The quality of the recovery can be evaluated by computing two errors, the mean absolute error (MAE)
and root mean square error (RMSE) defined as:
MAE
(14)
RMSE
(15)
These two errors will be small when the cardinality, i.e., the
total number, of the interpolating edges is large and when the
interpolating points occur at the most relevant features of the
original image
. This last criterion was used in [28] for
making a quantitative comparison between edge detectors. We
compared the procedure here proposed for edge detection with
that of Canny on a large class of images including those used in
[28] and kindly provided to us by the authors. The technique
proposed by Carlsson deals only with step edges and, therefore, only step edges and step junctions where used in (12)–(13).
Differences between our results and those reported in [28] are
likely to originate from a refinement of the minimization of the
functional (12) based on the advanced gradient descent (bold
driver; see [44]). Fig. 9 illustrates a comparison of the recovery
of the original image from Canny’s edge map (a), (c) and from
the proposed technique (b), (d). The parameters for the Canny’s
detector were optimized by an exhaustive search in the space
of parameters: the sigma of the gaussian filtering was sampled
from 0.2 to 3 by taking 29 equally spaced values and the high
1510
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 34, NO. 3, JUNE 2004
Fig. 9. Comparison between the Canny’s detector and the proposed detector with the recovery procedure [6]. (a) Constraints for the reconstruction algorithm
obtained from Canny’s edges. (b) Constraints from the proposed algorithm’s edges (c), (d) Reconstructed images, respectively from Canny and the proposed
algorithm.
threshold in the hysteresis step was sampled from two to 32 gray
levels by taking 16 equally spaced values. The third parameter,
i.e., the low threshold, was set in such a way that the cardinality of interpolating edges was the same for the two detecdifferent settings
tors. Therefore, for each image
of parameters were considered for Canny’s detector (not all of
them leading to feasible choices of the third parameter). The
lowest RMSE and MAE (possibly originating from different settings) were chosen. The spatial location of interpolating edges
is slightly different in the case of the proposed algorithm with
respect to that one of Canny, which produces maps that are more
dense near sharp edges and almost empty near smooth edges. As
a consequence, the error of the recovered image from our edge
map is lower than that one from Canny’s edges. A comparison
of performances between Canny’s and our edge detector, according to the recovery procedure of [6], is shown in Table I for
a variety of images. In six out of nine images, our edge detector
provides a lower recovery error with respect to Canny’s. For the
nine analyzed images, our detector has an average error of 11.62
(MAE) and 19.53 (RMSE), whereas Canny’s detector has an av-
erage error of 12.02 (MAE) and 20.17 (RMSE). The data shown
in Table I deserve some comments. Images recovered from very
sparse edge maps, such as in the case of the “nuts” image, have
large values of MAE and RMSE. The poor performance of the
recovery is primarily due to the very low density of constraints,
i.e., around 5% of the total pixels, and not because extracted
edges are wrong. The recovery of images with many fine details, such as image “Sherbrooke,” is obtained with a significant
error, even in the presence of a high density of constraints, i.e.,
almost 50%. This large error is due to the complexity of the images. In images primarily with sharp edges, Canny’s detector
performs rather well: in fact, in the well-studied image of Lena,
Canny’s edge detector performs slightly better than our detector.
Our edge detector is superior with images with many edges at
different scales.
C. Quantitative Comparison With Previous Approaches:
Structure From Motion [36]
Shin and co-authors [36] have proposed the recovery of structure from motion (SFM) as a way to obtain a quantitative com-
PELLEGRINO et al.: EDGE DETECTION REVISITED
1511
TABLE I.
COMPARISON BETWEEN THE PROPOSED ALGORITHM AND THAT OF CANNY FOR THE RECOVERY [6] OF IMAGES FROM EDGES
For each image, edges were extracted as here proposed and the threshold for selecting edges in the Canny detector (first column)
was set so as to have the same number of constraints for the two detectors (second column). For each recovered image MAE (third
column) and RMSE (fourth column) were computed. images named Lena, Back, Nuts, and Sherbrooke were kindly provided for us
by the authors of [34] and were the same used in their comparison. images named "La Cournouvaille," "Stairs," "Fruits," "Street
Scene #1," and "Street Scene #2," are, respectively, reproduced in Fig. 10(a), Fig. 6(a), Fig. 9(c), Fig. 11(a), and Fig. 11(e).
parison between edge detectors. This comparison is made between recovered parameters from detected edges and ground
truth “structure” and “motion” data, measured directly, leading
to a “structure error” and a “motion error.” The “structure error”
is the absolute difference in degrees between computed and
measured angles and the “motion error” is the percentage error
in the estimation of the rotation of a given point around a rotation axis.
Eight different edge detectors on 18 different image sequences of wood blocks and Lego houses were compared in
[36]. Performances of different detectors were optimized by
searching for the best set of parameters, characterizing each
detector. For each image sequence, performances obtained by
parameters minimizing the “structure error” and the “motion
error” were considered. In this comparison, only long and
straight edge chains are used, as the SFM algorithm is based on
long straight lines. As a consequence, short edge chains or long
rounded edge chains are not considered. In addition, the used
image sequences have primarily sharp edges, better detected at
a fine scale, therefore detectors are compared for their ability
to detect primarily long and sharp edges. The correspondence
between long edge chains, allowing the recovery of structure
and motion, is obtained automatically, based on a set of manually selected reference lines. The matching algorithm, essential
for the task of SFM, favors thick lines, making correspondence
easier. This comparison indicated that Canny’s edge detector
provided the lowest errors in this SFM task, followed by
Heitger’s detector.
The proposed edge detector and the Canny’s detector were
compared for the 18 sequences, using the software and the
images kindly provided by Prof. Shin. Initially, performances
of the proposed edge detector were not optimized, i.e., edges
used for the SFM task were those obtained as described in
Sections IV and VII-A. Performances of Canny’s detector
were those obtained in [36] after parameter optimization. This
comparison is shown in Table II.
In eight of 18 sequences, our detector, without any parameter
optimization, provides a lower error either in the structure or
1512
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 34, NO. 3, JUNE 2004
TABLE II.
COMPARISON BETWEEN THE PROPOSED ALGORITHM AND THAT OF CANNY FOR THE STRUCTURE FROM MOTION TASK [42]
motion recovery. Performances of the proposed detector significantly improved when the traditional nonmaximal suppression
algorithm was used and slightly thicker edge chains were obtained. The modified nonmaximal suppression algorithm introduced to obtain thinner edge chains reduced the number of correspondences between lines, leading to degraded performances.
With this, the “structure error” and the “motion error” of the proposed detector were drastically reduced. For the image sequence
LEGO1A where our detector had the worst performance, the
two errors decreased from 15.23 to 1.20 and from 12.27% to
1.80% for the structure and motion task respectively and our detector performed significantly better than Canny’s detector. As a
consequence, our detector, without any optimization, performs
comparably as well as Canny’s detector.
D. Selection of Experimental Results
Let us now see how the proposed edge detector performs in a
variety of natural complex images. Step edges were detected in
all images with the same parameters and in particular with the
same threshold
of four gray levels (see Section IV).
Fig. 10(a) is an interesting image (“La Cornouvaille”), where
ropes and poles produce many thin roofs with a different width.
Detected step edges are shown in black and recovered junction sides are in red in Fig. 10(b). Edges are remarkably clean
and occur—almost always—where an edge is perceptually observed. There is almost no edge in large perceptually homogeneous regions, such as the sky and the boat bottom. Similarly, the great majority of perceived edges are correctly de-
tected. Roof edges are well detected [Fig. 10(d)] from the detail [Fig. 10(c)] representing hanging ropes and poles. Observe
that the thin rope [see the white arrow in Fig. 10(d)] is not detected. As the energy is summed over many scales, the strong
edge originating from the near black vertical pole, obscure the
detection of the thin rope.
In fact, when a low contrast roof edge is close to a high contrast edge [as in Fig. 10(c)] the first one is very often lost. This
problem originates also when very small filters extending over
three or five pixels have to be used. In this case, gaussian filters are not well sampled and it is not surprising that the digital
filtering does not provide good results. This problem is significantly reduced and even eliminated [see Fig. 10(e)] when the
same image, but at a higher resolution, is analyzed (or simply
the original image is oversampled). Edges are better detected
and localized [Fig. 10(f)] in the 512 512 pixel image.
Fig. 11(a) and (e) reproduce two cluttered street scenes. Detected step edges and recovered junction sides are shown in
black and red, respectively, in Fig. 11(b) and (f).
Details of these two images are shown in Fig. 11(c) and (g).
Detected edges (step edges and recovered side junctions) are
shown in yellow superimposed to the original gray levels in
Fig. 11(e) and (h). Also in this case the proposed scheme detects
almost all visible features and is robust against noise. Fig. 12(a)
illustrates a rather complex satellite image with a fine texture,
originating from the terrain geology. Detected step edges and recovered junction sides are shown in black and red respectively
in Fig. 12(b). All relevant local luminosity changes are correctly
detected, as shown by the edges in yellow in Fig. 12(d).
PELLEGRINO et al.: EDGE DETECTION REVISITED
1513
2
Fig. 10. Edge detection of the Cornouaille boat. (a) Original image at 512 512 pixels of resolution. (b) Detected step edges in black and recovered trihedral
junctions in red from the sampled image at 256 256 pixels. (c) Detail from the image sampled at 256 256 pixels with many thin roofs of different width and
a smoothed step edge produced by the pale cloud. (d) Step edges in black and roof edges in blue (small width), green (medium width), and red (large roof edge).
Observe the thin roof edge, indicated by the arrow, not detected. (e) The same detail as in (c) but at 512 512 pixels. (f) Step edges detected from the detail shown
in (e). Observe that the thin roof edge is almost completely detected.
2
2
2
Fig. 11. Edge detection in two street images. (a) and (e) Original images. (b) and (f) Detected step edges in black and recovered junction sides in red. (c) and (g)
enlarged details from (a) and (e). (d) and (h) Detected step edges and recovered junctions in yellow superimposed to the images in (c) and (g).
1514
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 34, NO. 3, JUNE 2004
Fig. 12. Edge detection of a satellite image. (a) Original image. (b) Detected step edges in black and recovered junction sides in red. (c) Enlarged details from
image (a). (d) Detected step edges and recovered junctions in yellow superimposed to the images in (c).
Localization, accuracy and sensitivity to local changes of image
intensity is rather remarkable. Finally, Fig. 13(a) and (c) illustrates two natural, highly textured, images with many details and
sharp and smooth edges. All edges are almost perfectly detected
as shown in Fig. 13(b) and (d). Notice the complete absence of
detected edges in the flat background in both images.
VIII. SUMMARY
In this paper four classical problems of edge detection were
addressed: 1) the detection of all step edges from a fine to a
coarse scale; 2) the detection of thin bars, i.e., of roof edges;
3) the detection of trihedral junctions and of their sides; and 4)
the development of an almost parameter free edge detector. The
proposed scheme for edge detection uses a large variety of filters
with different orientation, shape and size as already suggested
by several previous works [4], [11], [18], [19], [25], [29], [30],
[39]. The detection and restoration of trihedral junctions, here
proposed, is based on the technique already proposed for computing the local curvature [17] and makes use of an appropriate
validation procedure.
As discussed in Section VII, the proposed edge detector has
been inspired from previous approaches and is a refinement
of the original Canny’s detector. It is indeed able to detect
trihedral junctions and edges at different scales, features that
the original Canny’s detector cannot spot. A quantitative
PELLEGRINO et al.: EDGE DETECTION REVISITED
1515
Fig. 13. Edge detection in complex natural scenes. (a) and (b) Original images. (b) and (d) Detected step edges and recovered junctions in yellow superimposed
to the images in (a) and (c).
1516
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 34, NO. 3, JUNE 2004
comparison based on the SFM task (see Section VII-B) and
on the recovery of the original image from the edge map (see
Section VII-C) indicates that the proposed detector performs
better than Canny’s detector. The extensive experimentation
performed on images of different kind (see Section VII),
indicates that the proposed scheme detects all edges (with
some occasional exceptions) which are perceptually present in
the original image. On the other hand, all detected edges are
visually present in the original image. Therefore, we believe
that the proposed scheme represents a significant improvement
over existing edge detectors. Let us now summarize the main
issues and conclusions.
A. Detection of Step Edges
Step edges are detected by looking at local maxima of the
summed energy (2), where the summation extends only over
odd gaussian filters, along eight different directions and over
nine scales ranging from 3 to 18 pixel/cycle. Filters at different
scales are normalized so to have equal maximal response [see
Fig. 3(c)] and not equal energy [see Section IV]. It is assumed
that the noise of imaging devices has a standard deviation
around 1.3 gray levels [10]; as a consequence, all edges with
gray levels cannot originate from
a contrast larger than
random fluctuations and must be a meaningful perceptual feagray levels are
ture. Edges with a contrast lower than
rejected and those with a contrast between four and eight gray
levels are selected according to the hysteresis criterion discussed
in Section IV. This fixed threshold has been used for extracting
edges in all images presented in the paper and it is the only relevant parameter of the proposed edge detector. The possibility
of using such a low threshold is a consequence of the noise removal obtained by summing the output of directional filters over
many directions and scales. A drawback of this approach is the
possible interference between edges at different scales: indeed,
a high contrast smooth edge can make the detection of a nearby
low contrast sharp edge more difficult (see Fig. 10).
The reasons for detecting step edges with only odd filters and
not with both odd and even filters [24], [18] are substantially
the same used by Canny in 1986 for detecting edges with directional filters having the shape of the first derivative of a gaussian
function (see Section VII-A).
B. Detection of Roof Edges
The detection of thin bars or roof edges is more sensitive
to noise than the detection of step edges. In fact, the detection
of roof edges with a width of one pixel or so can be obtained
with very narrow filters with a small scale around three or four
pixels/cycle. These narrow filters cannot be well sampled and
are not able to remove noise, as larger filters can. On the other
side, the detection of larger bars cannot be obtained with filters
in all orientations (see Fig. 6) and therefore there is less noise
removal. In agreement with previous approaches [18], [33] roof
edges are best detected by using both odd and even filters (see
Section V).
C. Detection of Trihedral Junctions
The proposed detection of trihedral junctions and recovery of
missing sides is largely inspired by previous approaches [17].
The major difference is the validation step based on the analysis
of the output of directional filters. This procedure seems able to
restore correctly the great majority of missing sides.
D. Noise Robustness, Parameter Sensitivity, and
Computation Time
The proposed edge detector is not simple and has many parameters, such as the number of different directions (eight directions), the size of the smallest and largest filters (three and 19
and
for the construction
pixels/cycle), the ratio between
of directional filters (1.2). Edge maps obtained by increasing the
largest filter up to 40 pixels/cycle, increasing the number of directions of oriented filters up to 16 or changing the ratio and
to two or modifying by 50% the parameter values for junction
detection, were almost indistinguishable. In fact, at least 95% of
edges were common to all these edge maps. When step edges
at a very large scale have to be detected (or when dealing with
very high resolution images), it may be necessary to extend the
size of the largest filter.
As already discussed, there is a threshold on the contrast of
detected edges, corresponding to four gray levels. Changing this
value will significantly change the number of detected edges.
However, a rather extensive experimentation with a large variety
of images shows that this threshold can be consistently used and
taken as fixed.1 As a consequence, the proposed edge detector
can be considered to be almost parameter free.
The proposed edge detector requires about 120s for extracting
edges in 256 256 pixel images, using Matlab and a PC with an
Athlon 1.1 GHz processor under O.S. Windows 2000. By using
a low level language and optimizing the code (and considering
that most of the computations can be performed in a parallel
way), times can be drastically reduced.
E. When the Proposed Scheme Fails
Although it is claimed that the proposed scheme for edge detection is almost ideal, it fails in two circumstances. When a
low contrast edge is very close to a high contrast edge [as in
Fig. 10(c)] the low contrast edge is very often lost. Also roof
edges with a width of just one pixel are not reliably detected.
These problems originate when very small filters extending over
three or five pixels have to be used. In this case, gaussian filters
are not well sampled and it is not surprising that the digital filtering does not provide good results. These two problems are almost completely eliminated when the same image but at a higher
resolution is used or, more simply, when the original image is
oversampled.
1In the case of very noisy images, such as those obtained during an echocardiography, this threshold could be increased. More generally, when dealing with
a particular class of images, a fine tuning of the threshold could be made. One
possible solution would be, in these cases, to estimate the sensor noise and set
the threshold accordingly. A number of methods have been proposed to estimate the noise: for example, in [10] the sensor noise n(x; y ) is modeled as a
stationary additive zero-mean white noise process: then the standard deviation
of the distribution is computed by high-pass filtering an “almost flat” region of
the image.
PELLEGRINO et al.: EDGE DETECTION REVISITED
ACKNOWLEDGMENT
The authors would like to thank C. Morrone for many helpful
discussions and useful suggestions and to G. Musso and A.
Verri for having read and commented on the preliminary version of the manuscript. They also thank T. Nguyen, D. Ziou, D.
Goldgof, and M. Shin for providing the code and the images for
the comparison of edge detectors. Finally, they are grateful to L.
Hall who kindly solved a complicated file transfer problem.
REFERENCES
[1] P. R. Beaudet, “Rotational invariant image operators,” in Proc. Int. Conf.
Pattern Recognition, 1978, pp. 579–583.
[2] F. Bergholm, “Edge focusing,” IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-9, pp. 726–741, 1987.
[3] K. Brunnstrom, T. Lindeberg, and J. O. Eklundh, “Active detection and
classification of junctions,” in Proc. 2nd Eur. Conf. Computer Vision, St.
Margherita Ligure, Italy, 1992, pp. 701–709.
[4] D. Burr and M. C. Morrone, “A nonlinear model of feature detection,”
in Nonlinear Vision, Determination of Receptive Field, Function and
Networks. Boca Raton, FL: CRC, 1992.
[5] J. F. Canny, “A computational approach to edge detection,” IEEE Trans.
Pattern Anal. Machine Intell., vol. 8, pp. 679–698, Sept. 1986.
[6] S. Carlsson, “Sketch based coding of gray level images,” Signal
Process., no. 15, pp. 57–83, 1988.
[7] R. Deriche, “Using Canny’s criteria to derive a recursively implemented
optimal edge detector,” Int. J. Comput. Vis., vol. 1, no. 2, pp. 167–187,
1987.
[8] R. Deriche and G. Giraudon, “A computational approach for corner and
vertex detection,” Int. J. Comput. Vis., vol. 10, no. 2, pp. 102–124, 1993.
[9] L. Dreschler and H. H. Nagel, “On the selection of critical points and
local curvature extrema of region boundaries for interframe matching,”
in Proc. Int. Conf. Pattern Recognition, 1982, pp. 542–544.
[10] J. H. Elder and S. W. Zucker, “Local scale control for edge detection and
blur estimation,” IEEE Trans. Pattern Anal. Machine Intell., vol. 20, pp.
699–716, July 1998.
[11] W. T. Freeman and E. H. Adelson, “The design and use of steerable
filters,” IEEE Trans. Pattern Anal. Machine Intell., vol. 13, pp. 891–906,
Sept. 1991.
[12] W. T. Freeman, “Steereable filters and local analysis of image structures,” Ph.D. dissertation, Dept. Media Arts Sci., Mass. Inst. Technol.,
Cambridge, 1992.
[13] A. Grossman, “Wavelet transforms and edge detection,” in Stochastic
Processes in Physics and Engineering, S. Albeverio, P. Blanchard, M.
Hezewinkel, and L. Streit, Eds. Amsterdam, The Netherlands: Reidel,
1988, pp. 149–157.
[14] M. Heath, S. Sarkar, T. Sanocki, and K. W. Bowyer, “A robust visual
method for assessing the relative performance of edge-detection
algorithms,” IEEE Trans. Pattern Anal. Machine Intell., vol. 19, pp.
629–639, Dec. 1997.
[15] M. Heath, S. Sarkar, T. Sanocki, and K. W. Bowyer, “Comparison of
edge detectors: A methodology and initial study,” Comput. Vis. Image
Understand., vol. 69, no. 1, pp. 38–54, 1998.
[16] D. H. Hubel and T. N. Wiesel, “Receptive fields and functional architecture in two nonstriate visual areas (18 and 19) of the cat,” J. Neurophysiol., no. 28, pp. 229–289, 1965.
[17] L. Kitchen and A. Rosenfeld, “Gray-level corner detection,” Pattern
Recognit. Lett., vol. 1, no. 2, pp. 95–102, 1982.
[18] P. Kovesi, “Image features from phase congruency,” in
Videre. Cambridge, MA: MIT Press, 1999, vol. 1, pp. 1–26.
[19] T. Lindeberg, “Edge detection and ridge detection with automatic scale
selection,” Int. J. Comput. Vis., vol. 30, no. 2, 1998.
[20] D. Marr and E. Hildreth, “Theory of edge detection,” in Proc. Royal Soc.
London B, 1980, pp. 187–217.
[21] D. R. Martin, C. C. Fowlkes, and J. Malik, “Learning to detect natural
image boundaries using brightness and texture,” in Proc. NIPS, Vancouver, BC, Canada, 2002.
[22] Y. Meyer, Wavelets—Algorithms and Applications. Philadelphia, PA:
SIAM, 1993.
[23] J. N. Morel and S. Solimini, Variational Methods in Image Segmentation. Boston, MA: Birkhauser, 1995.
[24] M. C. Morrone and D. Burr, “Feature detection in human vision: A
phase-dependent energy model,” in Proc. Royal Soc. London B, 1988,
pp. 221–245.
1517
[25] M. C. Morrone and R. A. Owens, “Feature detection from local energy,”
Pattern Recognit., no. 6, pp. 303–313, 1987.
[26] D. Mumford and J. Shah, “Optimal approximation of piecewise smooth
function and associated variational problems,” Commun. Pure Applicat.
Math., no. 42, pp. 577–685, 1989.
[27] D. Nair, A. Mitiche, and J. K. Aggarwal, “On comparing the performance of object recognition systems,” in Proc. Int. Conf. Image Processing, 1995, pp. 631–634.
[28] T. B. Nguyen and D. Ziou, “Contextual and noncontextual performance
evaluation of edge detectors,” Pattern Recognit. Lett., no. 21, pp.
805–816, 2000.
[29] P. Perona and J. Malik, “Scale-space and edge detection using
anisotropic diffusion,” IEEE Trans. Pattern Anal. Machine Intell., vol.
12, pp. 629–639, 1990.
[30] P. Perona and J. Malik, “Detecting and localizing edges composed of
steps, peaks and roofs,” in Proc. 3rd Int. Conf. Computer Vision, Osaka,
Japan, 1990, pp. 55–57.
[31] B. Robbins and R. Owens, “2D feature detection via local energy,”
Image Vis. Comput., no. 15, pp. 353–368, 1997.
[32] R. Rodriguez-Sanchez, J. A. Garcia, J. Fdez-Valdivia, and X. R.
Fdez-Vidal, “The RGFF representational model: A system for the
automatically learned partioning of visual patterns in digital images,”
IEEE Trans. Pattern Anal. Machine Intell., vol. 21, pp. 1044–1073,
Oct. 1999.
[33] L. Rosenthaler, F. Heitger, O. Kubler, and R. von der Heydt, “Detection
of general edges and keypoints,” in Proc. ECCV, 1992, pp. 78–86.
[34] C. A. Rothwell, J. L. Mundy, W. Hoffman, and V. D. Nguyen, “Driving
vision by topology,” in Proc. SCV, 1995, pp. 395–400.
[35] M. A. Ruzon and C. Tomasi, “Edge, junction, and corner detection using
color distributions,” IEEE Trans. Pattern Anal. Machine Intell., vol. 23,
pp. 1281–1295, Nov. 2002.
[36] M. C. Shin, D. B. Goldgof, K. W. Bowyer, and S. Nikiforou, “Comparison of edge detection algorithms using a structure from motion task,”
IEEE Trans. Syst., Man, Cybern. B, vol. 31, pp. 589–601, Aug. 2001.
[37] M. C. Shin, D. B. Goldgof, and K. W. Bowyer, “Comparison of edge detector performance through use in an object recognition task,” Comput.
Vis. Image Understand., vol. 84, no. 1, pp. 160–178, 2001.
[38] J. Shen and S. Castan, “An optimal linear operator for edge detection,”
Comput. Vis. Graph. Image Process., vol. 54, no. 2, pp. 122–133, 1992.
[39] E. P. Simoncelli and H. Farid, “Steerable wedge filters for local orientation analysis,” IEEE Trans. Image Processing, vol. 5, pp. 1377–1382,
Sept. 1996.
[40] S. M. Smith and J. M. Brady, “SUSAN—A new approach to low level
image processing,” Int. J. Comput. Vis., vol. 1, no. 23, pp. 45–78, 1997.
[41] V. Torre and T. A. Poggio, “On edge detection,” IEEE Trans. on Pattern
Anal. Machine Intell., vol. 8, pp. 148–163, Mar. 1986.
[42] W. Vanzella, F. A. Pellegrino, and V. Torre, Regularization and segmentation of natural images, in IEEE Trans. Pattern Anal. Machine Intell.,
June 2004, to be published.
[43] S. Venkatesh and R. A. Owens, “On the classification of image features,”
Pattern Recognit. Lett., no. 11, pp. 339–349, 1990.
[44] T. P. Vogl, J. W. Mangis, A. K. Rigler, W. T. Zink, and D. L. Alkon,
“Accelerating the convergence of the back-propagation method,” Biol.
Cybern., no. 59, pp. 257–263, 1988.
[45] A. Witkin, “Scale-space filtering,” in Proc. Int. Joint Conf. Artificial Intelligence, 1983, pp. 1019–1022.
[46] D. Ziou and S. Tabbone, “Edge detection techniques-An overview,”
Dept. Math Comput. Sci., Univ. Sherbrooke, Sherbrooke, QC, Canada,
Tech. Rep. 195, 1997.
Felice Andrea Pellegrino was born in Conegliano,
Italy, in 1974. He received the Laurea degree in engineering from the University of Udine, Udine, Italy, in
June 2000 and is currently working toward the Ph.D.
degree in systems theory at the same university.
He began his research activity at the Department
of Mathematics and Computer Science (DIMI),
University of Udine. From 2001 to 2003 he was
Research Fellow at the International School for
Advanced Studies, Trieste, Italy. His research
interests include control theory, computer vision,
and pattern recognition.
1518
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 34, NO. 3, JUNE 2004
Walter Vanzella received the Laurea degree in computer science from the University of Udine, Udine,
Italy in 1998.
Since 2001, he has been Research Fellow at the
International School of Advanced Studies, Trieste,
Italy. His main interests are related to variational
methods applied to computer vision, biological
inspired computer vision, and pattern recognition.
Vincent Torre received the degree in theoretical
physics from Genoa University, Genoa, Italy, in
1973.
In the early 1980s, along with T. Poggio and C.
Koch, he pioneered in the area of computational
neuroscience. For several years he was a Post Doc
and Visiting Scientist in Sir Alan Hodgkin Lab,
Cambridge, U.K., where he learned a variety of
electrophysiological methods and became expert in
modeling biological process. In the late 1980s and
early 1990s, he worked on computer vision and in
particular, edge detection and motion perception. Since 1997 he has been a
Full Professor at the International School of Advanced Studies, Trieste, Italy,
and is the Head of the Department of Neurobiology. He has a well-established
expertise in electrophysiology, computer vision, computational neuroscience,
and modeling biological process. He is the author of more than 100 full papers.