IEEE TENCON ‘931 B d j W
EDGE DETECTION USING SCALE SPACE KNOWLEDGE
Anoop Kulkarni, R K Shevgaonkar, S C Sahasrabudhe
Dept. of Electrical Engineering
Indian Institute of Technology, Powai,
Bombay, INDIA.
email: [email protected]
2-d image can only begin by first defining these objects
loosely in terms of a crude outline of its boundary. This
provides the motivation for the boundary or edge based
representations. These representations are compact and
the elegant description of the image therein allows more
efficient analysis of the contents of the image. Edges
are loosely defined as points where the image intensity
Abstract
Edge detection has acquired enormous importance in computer vision research. Gaussian filter has been widely used in the past to accomplish this task. Although the gaussian filter has
nice scaling behaviour and are computationally
elegant, it suffers from the disadvantage of delocalisation of edges when operated at higher
scales. Multi-scale processing of an image is
thus required.
In this paper, the process of edge detection has
been looked upon as a reasoning problem. In
the past, knowledge handling and reasoning have
been attributed to high level vision routines, but
from the results presented here, it can be argued
that reasoning does play an important role in
the process of edge detection. The knowledge
base required for this is formed out of the theory of scale spare. Particular emphasis is laid on
the behaviour of delocalised, missing and false
or spurious edges in the scale space. The scale
space is formed by operating the LOG filter of
different sizes on the input image. The dissimilarity among the zero crossings is measured
across the scales. It is useful in the removal
of false edges. A compatibility measure relates
the zero-crossing contour with the current scale
parameter, cr.
The rules are coded in NEXPERT OBJECT
2.0. The results are compared with Canny’s
multiple edge detection algorithm.
1
function changes significantly. As edges detected at this
stage are used by high-level processes, a correct edge
profile becomes a necessity.
Multi-resolution edge based image representations are
based on the principle of filtering an image with low
pass filters of various bandwidths, followed by an edge
detection process which localises the position of changes
in the intensity of the filtered image [l]. Multi-resolution
image representations have been suggested in the past
[2,3,4]. Marc and Hildreth [2] proposed a method for
edge detection by smoothing the image by using gaussian low pass filters and then using the zero crossings of
the second derivative t o localise the edges. The gaussian
filter has following impulse response in 2-dimensions.
Introduction
The pr,ocess of edge detection is of prime importance in
computer vision. This is because the most commonly
encountered image representation scheme is based on
the description of image edges. Also, the goal of recovering physical properties of objects in a scene from
..
In most cases, different physical processes are associated
with a typical behaviour across the scales. In images, a
spatial conincidence of zerecrossings in the laplacian of
the gaussian, indicates the existence of a distinct physical edge. Moreover, useful information can be obtained
by combining the descriptions effectively across scales.
The effective integration of information available at different scales thus poses a major problem. There have
been a few suggestions in the literature [5,6,7,8]. But,
at large, the problem has remained open.
Witkin[S] proposed a new way of describing the zerccrossings across the scales by defining a scale space.
Witkin proposed a one dimensional signal smoothing by
convolving the signal with the gaussian filter of all sizes
and by localising the zeroes of the second derivative to
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produce a plot of zerecrossings in the x- U plane, where
U is the size of the gaussian filter. He could thus effectively classify the zerecrossings using the scale space.
In gaussian scale space, the geometry of the zero crossing
contours is very simple. Babaud[lO] proved that there
are only two kinds of zero crossing contours in gaussian
scale space - either the ones which extend from zero scale
to infinite scale or those which after originating at zero
scale proceed to some finite scale and then gracefully
turn back to zero scale. Thus, the tracking of the zero
crossing from coarser to finer scale is possible.
All multi-resolution techniques look into two fundamental aspects, first of appropriate selection of the scales of
operators for an image and second of integrating information at different scale:r. The second has strong links
with the first. The selection of proper scale of operator is an open problem [I I]. Integration of information,
whereas, is largely a domain specific task.
Berzins[l2] reported three major problems with the gaussian scale space. They ale:
Fig. 1
The following knowledge base was developed out of the
theory of scale space [1,11,14,16].
Assertions:
1. Isolated zero crossing contours cannot disappear
at finite non-zero scale.
2. For linear edges, as U changes monotonously, the
diaplcement directions of the edges on the same
line are same.
edge delocalisation: edge location in filtered image.
3. For parallel edge contours, as U changes, edges on
the same line have same amount of delocalisation.
missing edge: edge in original image maynot correspond to any edge in filtered image;
4. False edge lines are formed if the edge contours
form a single or multiple staircase model.
false edge: edge in filtered image which doesnot
have a corresponding edge in the original image.
5. Scales at which false edge lines occur can be determined by knowing the distance between the edge
lines.
It has been proved by Lu and jain [Ill that LOG
scale space has all these three inconsistencies. Yuille
and Poggio [13] have reported interesting results about
gaussian scale space. Clark [I41 suggested a new formalism based on the Catastr,ophe theory for the description
of the behaviour of phantom edges in the scale space.
So far, high-level vision processes were using the explicit
knowledge and low-level vision processes weredominated
by data driven operations. Due to this, not much attention was paid to the scale space, wherein lay hidden large
chunks of knowledge. The cognitive sciences have since
proved [15] that reasoning plays a pivotal role in every
aspect of human perception. Hence, although edge detection is primarily a low level vision task, reasoning can
refine the solution greatly. The problem of edge detection has thus been viewed in this paper as a problem of
reasoning.
2
Fig. 2
6. A closed zero crossing contour expands as
creases.
U
in-
7. Two zero-crossing contours can merge into one
8. The zero crossings of closed edge curve can become
open.
9. An edge contour can partially or wholly disappear
at a larger scale.
10. A false edge contour exists only between two true
edge contours.
11. Contrast of a phantom zero crossing increaes as
U increases.
12. Contrast of a true zero crossing contour decreases
as U increases.
Knowledge in Scale Space
Scale space of an image is defined as the Cartesian product of the image plane with the U > 0 ray and involves
continuum of scales. Fig. 1 illustrates the typical scaling
behaviour observed by W1tikin[9]in one dimension. Fig.
2 illustrates the other possible contour shapes which are
not seen in the gaussian scale space [12]. Yuille and
Poggio 1131 give a mathematical proof of this peculiar
behaviour of edges in gaussian scale space.
13. Contrast of a fale edge that lies between two true
edge contours is less than that of either of two
edges.
14. A false edge always merges with a neighbouring
true edge contour at some scale and the pair disappears.
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Rules:
3
1. If two edge contours form a staircase model, then
maximum delocalisation of the edge contour is less
than or equal to w / 2 where w is the distance
between tose t.wo edge contours.
The knowledge based system
The overall schematic of the knowledge based system is
as shown in Fig 3. A rule interpreter or an inference
engine operates on a bank of images with zero crossing
contours created by operating the image
- with the LOG
operator at various scales. The inference engine uses the
knowledge base formed out of the facts and rules pertinent to the behaviour of edges in gaussian scale space.
A correct edge profile of the image is the result of the
reasoning and is thus a d a b l e at the output of the inference engine. A global database is necessary t o keep
the relevant facts and recent history for ready reference
of the inference engine. The present work doesnot assume the availability of any a priori information about
the desired features.
2. If an edge contour is non-linear, then edge points
on that contour have various amounts of delocalisation and displacement directions.
I
3. If an edge point is detected at ul, its delocalisation
is always less than or equal t o 61.
z1 is detected a t 01 and
another edge point located at 12 is detected at UZ,
then in most cases, 112 - 211 5 162 - 611.
4. If an edge point located a t
-
-
5. If zero crossing contours disappear at certain scale,
they do so in pairs.
6. If a zero crossing contour vanishes at U ] , then it
can always be recovered at a scale U 5 U].
7. If a closed edge contour forms a staicase model
with its neighbouring contours, then it can become open as a result of the segments of that contour vanishing with neighbouring false zero crossing contours.
ansettions
8. If a closed edge contour forms a staircase model
with its neighbouring contours, then it can split
into several curves at larger scales.
9. If a zero crossing contour exists at a larger scale
and disappears at a smaller scale, then it must be
a false one.
E d ye
Profile
ir
10. If false edge contour occurs at ul, then it will disappear at a U 5 u1.
11. If an edge contour changes shape significantly with
U ,then it is formed by noise.
Fig. 3
The knowledge base is coded in NEXPEFX OBJECT
2.0, an expert system buliding shell. The object oreiented paradiam
- of the representation of the knowledge
a lot of versatility to the system. The pllilos;
PhY of NEXPERT allows the facts t o be reprented in
12. If an edge contour keeps on splitting with &reasing U ,then it may have been formed by noise.
13. If the contrast of the false or spurious edge lying
between two true edges increases and the
of the true edges decreases, then, a t certain scale,
the false edge merges with the true one and the
pair disappears.
14. If the contrast of the zero crossing contour decreases to zero with increasing scale, then the true
edge turns into a false edge at that scale.
15. If the contrast of a true edge contour decreases
with increasing scale, then it will drop t o zero at
some scale, and the true edge will become a false
one at that scale.
the form of an object network. The rule network o p
erates on this object network during the reasoning process, NEXPERT allows bi-directional reasoning strategies, i.e. a mix of both the forward and the backward
chaining mechanisms. In the implementation, a class,
zero-mowingsg, is created for every scale of LOG o p
erator, U,. Thus, there are as many classes as there
are zerc-crossing image files for analysis. All the zerocrossing contours found at a particular scale are repre-
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sented as instances or objects under that class. Fig. 8
represents the typical p.roperties possessed by the objects. These objects are dynamically either linked to a
class, edges, or are removed for further analysis. These
decisions are taken by t'he inference engine. Also, new
objects may be created and attached to the class, edges.
The dis-similarity measurement among the zero-crossing
contours was carried out across the scales. This helps
remove the false edges. This measurement is carried
through from the highe:t scale in the scale space, Uh,gh
to the current scale, U. '[f A; is the zero crossing contou
detected at ui and Aj is the zero crossing contour detected at u j , then the dis-similarity between A; and Aj
is defined as under:
use for obtaining the correct edge profile. This knowledge was coded in the form of assertions/rules. The
object oriented paradigm offered a very elegant knowledge representation scheme. Geometric reasoning can
be incorporated to further enhance the authenticity of
edges detected. The theory of scale space is evolving and
hence the knowledge base can he further enrichened by
incorporating more knowledge pertinent to scale space.
This knowledge base can also be used for some high level
processes that involve multi-scale analysis, such as finding the disparity map in stereo vision. If the a priori
information about the desired features is known, then
the performance can be greatly enhanced.
References
where A,,, is the area of the zero crossing contour obtained at U,. For highly dis-similar curves, DSM will
be close to unity whereas for very similar ones, DSM
will be close to zero. This DSM is repeatedly computed
and recorded for further analysis during the reasoning
session.
The compatibility between the zero crossing contour and
the current scale operator, U , was established to ascertain whether the contour was formed due to noise. This
measure was close to unity when the scale at which
the zero crossing contour was obtained was significantly
greater than the current scale parameter, U. This was
because small changes in the scale parameter wouldnot
affect the zero crossing contour. In case of noise, this
will not be the case.
The order in which the zero crossings were tracked was
from umazto 1, where umoz is the maximum scale with
which LOG was operatod upon the image.
4
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Results
The knowledge base was tried on a variety of images,
and the results of one such run on the image of Fig.4 are
presented in Fig 5. Fig. 6 illustrates the results of LOG
operation on the image of Fig. 4 at various scales. The
result of Canny's multi-:xale edge detector are presented
for comparison in Fig. '7.
The NEXPERT OBJECT was run under the MS Windows 3.1 environment 'on a 386 machine operating at
25MHz. The time taken by the entire reasoning was
around 2.5 minutes. This time doesnot include the time
taken for the creation of various zerecrossing files. Those
were off-line generated <onthe same machine.
5
[l] J J Clark, "Singularity theory and phantom edges
in scale space", IEEE Trans on PAMI, vol 10, no
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Summary
The problem of edge detection was viewed as a reasoning problem. The gaussian scale space of an image contains useful knowledge which can be effectively put to
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[lo] J F Canny, " A variational approach to edge detection", Proc A A A 1 Conj, Washington DC, sept 1983,
pp 54-57.
[ l l ] Yi Lu, R C Jain, "Behaviour of edges in scale
space", IEEE Trans PAMI, vol 11, no 4 , apl 1989,
pp 337-56.
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[12] V Berzins, ”Accuracy of Laplacian Edge Detectors,” Computer Vision, Graphics, Image Processing, 1984, pp 195-210.
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pp 15-25.
[I41 J J Clark, ”Authenticating edges produced by zero
crossing algorithms”, IEEE Trans on PAMI, vol 11,
no 1, Jan 1989, pp 43-57.
,
Fig. 6 (U)
[15] J F Canny, ”A computational approach to edge detection”, IEEE‘ Trans PAMI, vol 8, no 6, nov 1986,
pp 679-98.
[16] Lu, R C Jain, ”Reasoning about edges in scale
space”, IEEE Trans PAMI, vol 14, no 4, april 1992,
pp 450-68.
+.
Fig 6 Cc)
Fig <7>
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