Logical/Linear Operators for Image Curves.
AbstractWe propose a language for designing image measurement operators suitable for early vision. We refer to them as logical/linear (L/L) operators, since they unify aspects of linear operator theory and Boolean logic. A family of these operators appropriate for measuring the low-order differential structure of image curves is developed. These L/L operators are derived by decomposing a linear model into logical components to ensure that certain structural preconditions for the existence of an image curve are upheld. Tangential conditions guarantee continuity, while normal conditions select and categorize contrast profiles. The resulting operators allow for coarse measurement of curvilinear differential structure (orientation and curvature) while successfully segregating edge-and line-like features. By thus reducing the incidence of false-positive responses, these operators are a substantial improvement over (thresholded) linear operators which attempt to resolve the same class of features.
Introduction
There is no shortage of so-called "edge-detectors" and "line-detectors" in computer
vision. These are operators intended to respond to lines and edges in images. Many
different designs have been proposed, based on a range of optimality criteria (e.g
[1, 2]), and many of these designs exhibit properties in common with biological vision
systems [3]. While this agreement between mathematics and physiology is encourag-
ing, there is still dissatisfaction with these operators-despite their 'optimal' design,
they do not work sufficiently well to support subsequent analysis. Part of the problem
is undoubtedly the myopic perspective to which such operators are restricted,
suggesting the need for more global interactions [4]. But we believe that more can be
done locally, and that another significant part of the problem stems from the types
of models on which the operators are based and the related mathematical tools that
have been invoked to derive them. In this paper we introduce an approach to operator
design that differs significantly from the standard practice, and illustrate how it
can be used to design non-linear operators for locating lines and edges.
The usual model used in the design of edge operators involves two components:
an ideal step edge plus additive Gaussian noise. This model was proposed in one
of the first edge detector designs [5], and has continued through the most recent
[2, 6]. Thus it is no surprise that the solution resembles the product of two operators,
one to smooth the noise (e.g. a Gaussian) and the other to locate the edge (e.g. a
derivative).
While some of the limitations of the ideal step edge model have been addressed
elsewhere (e.g. [7, 8]), a perhaps more important limitation of the operator design
has not been considered. It is assumed that in viewing a small local region of the
image, only a single section of one edge is being examined. This may be a valid
simplification in some continuous limit, but it is definitely not valid in digital images.
Many of the systematic problems with edge and line detectors occur when structure
changes within the local support of the operator (e.g. several edges or lines coincide).
Since these singularities are not dealt with by the noise component of the model
either, the linear operator behaves poorly in their vicinity.
In particular, curve-detecting operators are usually designed to respond if a certain
intensity configuration exists locally (see Figure 1a). A signal estimation component
of the operator is then incorporated in the design to filter local noise (Figure 1b).
However, significant contrast changes are rarely noise-they are more likely to be the
result of a set of distinct objects whose images project to coincident image positions
1c). An operator which claims to 'detect' or `select' a certain class of image
features must continue to do so in the presence of such confounding information.
We propose that image operators should be designed to respond positively to
the expected image structure, and to not respond at all when such structure is not
(a)
(b)
(c)

Figure

1: A set of potential image curve configurations which must be
considered in the design of operators. An ideal image (a) of a black curve
on a white background; a noisy image (b) of a lower-contrast version of the
same curve; an obscured version (c) of the ideal image. The oval in each
image represents the spatial support of a local operator. A negative contrast
line operator should respond positively in all three cases.
present. Simple linear operators achieve the first of these goals, but in order to fulfill
the second we must incorporate a more direct verification of the existence conditions
for a given feature into the operator itself. We accomplish this by decomposing
the linear operator into components which correspond to assertions of the logical
preconditions for a given feature. When the expected image structure is present, a
boolean combination of these responses produces a linear response, whereas if any
of the conditions are violated the response will be suppressed non-linearly. Because
these operators unite elements from boolean logic and linear operator theory, we refer
to them as Logical/Linear (L/L) operators.
A. Image Curves
For consistency we shall adopt the following terminology. Edges are the curves which
separate lighter and darker areas of an image-the perceived discontinuities in the
intensity surface; lines are those curves which might have been drawn by a pen or
pencil (sometimes referred to as bars in other work [9]). Image curves are either
of these. Two independent properties describe such image curves: their structure
along the tangential and in the normal directions. Tangentially, both lines and edges
are projections of space curves; it is the cross-sectional structure in the image which
differentiates them.
Formally, let I: be an analytic intensity surface (an image) and ff:
curve parameterized by arc length (see Figure 2). The
orientation '(s) is the direction of the tangent - (s), a unit vector in the direction of
ff 0 (s), and the normal vector n(s) is a unit vector in the direction ff 00 (s).

Figure

2: An image curve ff: with the tangent - (s) and
normal n(s).
Formally, an image curve is defined by a set of local structural conditions
on the image in the directions tangential and normal to the curve. The normal
cross-section fi s at the point ff(s) is given by
image curve is a map ff: S ! I such that
(Tangent) ff is C 1 continuous on S, and (1a)
(Normal) a condition N(fi s ) holds for all s 2 S: (1b)
s ), the normal condition, determines the classification of the curve.
For the purposes of this paper, we concentrate on three kinds of image curve:
1. ff is an Edge 1 in I iff ff is an image curve with normal condition
2. ff is a Positive Contrast Line in I iff ff is an image curve with normal
1 Note that the definition of a line is independent of curve orientation, while a rising edge will only
be seen looking along ff in one direction. Thus lines need only be parameterized over - orientations
while edges require 2- orientations.
(a)
(b)
(c)

Figure

3: A set of image curve configurations which may generate false
positive operator responses. An image of an contrast edge (a) should not
stimulate a line operator; a improperly oriented operator (b) should not be
stimulated; an operator which does not lie on the curve (c) should not be
stimulated. The oval in each image represents the spatial support of the local
operator. A negative contrast line operator should not respond positively in
any of these cases.
condition
3. ff is a Negative Contrast Line in I iff ff is an image curve with normal
condition
Thus, edges are order 0 discontinuities (steps) in cross-section, while positive and
negative contrast lines are order 1 discontinuities (creases) which are also maxima or
respectively.
Note that in contrast to traditional definitions, the tangent and normal conditions
above are both point conditions, which must hold at every point in the trace of
the curve. We thus have a basis for designing purely local operators to locate and
categorize such curves in images.
Linear operators do respond when these conditions are met. However, they also
respond in situations in which the conditions are not met. These responses are referred
to as false-positives. The current analysis will focus on a mechanism for avoiding three
kinds of false-positives typical of linear operators:
1. Merging or interference between nearby curves: The local continuity of image
curves is important for resolving and separating nearby features. Linear operators
interfere with testing continuity by filling in gaps between nearby features
and responding significantly to curves which are far from their preferred orientations
(Fig. 3b).
2. Smoothing out discontinuities or failing to localize line-endings: The locations
of the discontinuities and end-points of a curve are fundamental to higher-level
descriptions [10, 11, 12]. Linear operators systematically interfere with their
localization by responding whenever the receptive field of the operator at all
overlaps the curve (Fig. 3c).
3. Confusion between lines and edges: Lines and edges are differentiated by their
cross-sections. For accurate identification the logical conditions on the cross-section
must be satisfied, and in each case we will show that a linear operator
tests them incompletely (Fig. 3a).
II. A Logical/Linear Framework for Image
Operators
The three qualitatively different kinds of image curves defined in xI.A. imply three
distinct sets of preconditions for the existence of an image curve. As noted previ-
ously, the curve description process must respect these distinctions. Focusing for the
moment on the line condition of (2b), we begin by adopting an oriented, linear line
operator similar to the one described in [2].
Canny adopted the assumption of linearity to facilitate noise sensitivity analy-
sis, and relied on post-processing to guarantee locality and selectivity of response.
He arrived at a line operator whose cross-section is similar to a Gaussian second-
derivative, and an edge operator similar to a Gaussian first derivative. Neurophysiologists
[13, 14, 3] and psychophysicists [15] have adopted such linear models to capture
many of the functional properties of the early visual system, and the mathematics for
analyzing them is widely known (e.g. Fourier analysis). These models are also attractive
from a computational point of view because they exhibit most of the properties
required of a measurement operator for image curves. However, they also exhibit the
false-positive responses described above (partially shown in Figures 7 and 8).
To limit these false-positives, we will relax the assumption of linearity and test
the necessary structural conditions explicitly. This is accomplished by developing an
algebra of Logical/Linear (L/L) Operators which allow these conditions to be
tested as the operator's response is being constructed. The resulting responses will
appear to be linear as long as all of these conditions are fulfilled.
A. Logical/Linear Combinators
As stated above, we wish to retain as much as possible of the desirable properties of
the linear operator approach while allowing for the kind of structural analysis which
can be used to categorize curves and verify continuity. We pursue these apparently
contradictory goals by starting with a nearly optimal linear operator, and then decomposing
it in a way that allows for it's reconstruction, provided that the structural
design conditions are verified.
In particular, we
1. Begin with a linear operator and decompose it into a set of linear component
operators whose sum is identical to the initial operator.
2. These linear components represent measurement operators for the logical pre-conditions
of the designed feature's existence.
3. The overall operator response is to be positive only if these structural preconditions
are satisfied.
4. For the range of inputs generating positive responses, the operator should act
identically to the original linear operator.
The combination of operator responses to fulfill the third and fourth conditions
above can be derived from a mapping of the real line to logical values. Assume
that positive operator responses represent confirmation of a logical condition (logical
True) and that negative responses represent rejection (logical False). To derive the
numeric)logical mapping, we adopt the principle that all confirming evidence should
be combined if the logical condition holds, and contradictory evidence combined if the
condition fails. This leads to the following set of logical/linear combinators:
Definition 2 The Logical/Linear Combinators
- are given by
Before we descend into technical detail, it should be noted that these operators
can be thought of as accumulating evidence for or against a particular hypothesis,
with positive values being evidence 'for' and negative values evidence `against'. Thus
if an hypothesis h requires that both of two prior hypotheses (x and y) be true
then consistent positive evidence from these inputs, represented by positive values,
is required to produce a positive output
y. Should this combined hypothesis
instead be rejected, all evidence for this rejection is combined. In all cases, the logical
truth or falsehood of an hypothesis is contained in the sign of the value, while the
strength of the evidence for or against the hypothesis is represented by the magnitude.
It should be apparent that reasoning about the signs of derivatives (see xI.A.) will be
natural with this formalism.
B. A Logical/Linear Algebra
We now proceed to develop general properties of these L/L combinators and define a
class of operators which embody these properties. With this background established,
we can then move on to the development of the specialized operators we will use for
early vision.
Using the combinators
-, we define a generative syntax for L/L expressions.
Logical/Linear Operator on the vector space X
any function L in the language L defined by the following grammar L:
where each a i is a real constant and each / i (x) is a bounded, real-valued, linear
function.
Example 1: The expression
defines a L/L operator which is positive only if both x and y are positive,
in which case it evaluates as F (x; y. An equivalent description of F as an
operator is given by
is the projection operator which selects the i th dimension of X.
There are two fundamental properties which justify the use of the term "Logi-
cal/Linear expressions" to describe these forms: they comprise a Boolean Algebra,
and they are linear on certain subspaces of their entire domain.
To show the first of these, consider the universe of vectors U in IR n excluding the
and the subspaces f L(x)
2 For real-valued variables, the exclusion of the axes needed to demonstrate logical equivalence is
not problematic because it is a subset of measure 0.
Theorem 1 (Logical) For the language of L/L operators L 2 L, the set of all
sets f L(x) g + and their complements f L(x)
Algebra with
- and complement \Gamma.
Proof The following equivalences can be derived directly from Definition 2, for all
It is easy to verify that these sets form a field with the help of the equivalences above
(e.g. the equivalence of
ensures that if X and Y are members then X
-Y is
also). Furthermore, these meet, join and complement operators are clearly isomorphic
to the standard set-theoretic ", [ and complement. The further observation that ;
and U are the bounds of this field ensure that this system is a Boolean algebra. ([16],
p.
The following equivalences can also be derived directly:
f a L(x)
if a - 0
These demonstrate that the constant weights a i in the language L act as either
identity or complement and thus do not disturb the Boolean algebra.
Corollary 2 Each L/L operator has an associated Boolean function created by substituting
- and - for
- respectively, and by replacing each a i constant with
either the identity function if positive or : (negation) if negative. The truth value of
each expression / i (x) is True if /
Thus, continuing Ex. 1, the equivalent logical function -
F for F is
The second fundamental property of these operators, their conditional linearity,
is revealed by considering the minimal polynomials
in the binary representation of j is zero, q i
if bit i is one. Then,
Theorem 3 (Linear) Any L/L operator L is linear on the subspace f P j
of
any minimal polynomial P j (x).
Proof Any Boolean polynomial can be equivalently expressed as the join of minimal
polynomials or the lower bound ; ([17], p. 370). Thus f L(x)
can be expressed
as the
of a group of such minimal polynomials of the / i (x)'s (the disjunctive
canonical form (DCF) of L(x)). Without loss of generality, consider a particular such
polynomial Noting that every element / i (x) for x
has a fixed
value and thus fixed sign, Definition 2 guarantees that
- is linear on the subspace
defined
(for fixed sign arguments, the branch chosen in the
- is fixed).
Thus, any minimal polynomial P j (x) is linear on f P j
Consider now the DCF of L(x). We know that each P j (x) in this DCF is both
linear and of constant sign on f P j
. By the same reasoning as for
above, we
can state that +
- is linear if its arguments have constant sign, and thus the DCF of
L(x) is a linear combination of expressions which are guaranteed linear on f P j
Therefore, L(x) is also linear on every f P j
C. Logical/Linear Image Operators
By extension from the arithmetic operators, the L/L operators are applied pointwise
to sequences of vectors or images. Thus, reconsidering Ex. 1 above, the operator F
becomes
We are now ready to develop the class of L/L operators that we shall require to reason
about images, and begin with an example.
Example 2: Suppose that the linear operators / 1 and / 2 provide a pointwise
measure of two image properties (e.g.
x and
y , the second directional
derivatives) which are components of a more complex image property (e.g. locating
local convexity, the points where D 2
0). If this aggregate
property can be expressed as a logical combination of the signs of the linear properties,
then we can build a L/L operator \Psi on the image such that
positive, if x is a locally convex point;
negative, otherwise.
In this case, we would define
y   I)(x):
This example reveals a class of L/L operators appropriate for reasoning about
images.
Definition 4 A Logical/Linear Convolution Operator \Psi is a L/L operator
on an image I such that all / i (I) are linear convolutions of the form
Z
The operation of such an operator on an image is termed the Logical/Linear
Convolution of I by \Psi, and is written
Note that the linear convolution /   I is a special case.
Returning to Ex. 2, we can assert that
y   I)
thus justifying the notation we will use for describing L/L convolution operators:
y
This operator will produce an image whose elements are positive only for convex
points of the input image.
An important relationship we will use to design image operators is that between
a L/L operator and its linear reduction.
Definition 5 The Linear Reduction / of a L/L operator \Psi is that linear operator
which is produced by substituting + for each L/L combinator in the L/L operator
description.
Corollary 4 Given the linear reduction /(x) of a L/L operator \Psi(x), a L/L convolution
of \Psi   I is exactly equal to the linear convolution of /   I if the logical expression
corresponding to the L/L expression is True.
Thus in fulfilling our goal of developing L/L image operators which retain some of
the optimal behaviour of a particular linear operator, we will seek to design L/L
operators which reduce to 'optimal' linear operators.
Before we move on to actual design, it will be important to examine a second,
equivalent definition of the L/L combinators which has useful computational consequences

Definition 6 The ae-approximate L/L combinators are given by:
where
0; otherwise.
The oe ae (x) function is used as a partition function since it is everywhere infinitely
differentiable but is only non-zero for values ? \Gamma1=ae: The 'logistic' sigmoid function
of [18] is another option for oe ae (x), but the fact that the function chosen is non-singular
(i.e. means that the "hard" logic of
xII.A. still applies for values outside of this region. All that is required for predictable
behaviour is that oe ae (x) has range [0; 1], is monotonic, and that lim ae/1 oe ae
the step function below. In fact, a linear ramp in the interpolation region if eq.
seems to be adequate for practical purposes.
The notion of ae-approximate is clarified by the following.
Theorem 5
lim
lim
Proof Note that
lim
0; otherwise.
This function is a choice operator pivoting around zero, and as such it can be used
to directly define the L/L combinators above. If this limit is substituted in eqs. (8),
then they can be rephrased as
fy

Figure

4: Graphs of the ae-approximate L/L combinators varying ae: (a),
(b), and (c) show x
y, (d), (e), and (f) show x
y. Note that as ae varies
between 0 and 1, the combinators vary from purely linear to purely logical,
with smooth interpolation in between.
It can be verified that these are equivalent to Def. 2.
The approximations represented by
expose another relationship between
the linear sum and the L/L combinators. Since oe 0 substitution of
this value into eq. (8) simplifies both L/L combinators to a linear combination
Thus, the ae-approximates
ae form a continuous deformation from a linear
combination to the absolute L/L operations as ae goes from 0 to 1 (see Fig. 4).
III. Designing L/L Operators for Image Curves
Using the framework defined above, we now proceed to derive a family of L/L image
operators to locate and describe image curves as defined in Definition 1. We begin
by observing that the conditions expressed in eqs. (2b,2c,2a) segregate into independent
one-dimensional conditions in orthogonal directions-along the tangent and
the normal. The normal condition selects the proper contrast cross-section to define
a (positive or negative contrast) line or edge, and the tangential condition ensures
local C 1 continuity of the inferred curve. Thus, our solution is a separable family
of two-dimensional operators expressed as the Cartesian product of orthogonal, one-dimensional
L/L operators, one normal N(y) and the other tangential T(x) to some
preferred direction. With (x; y) defining a local, orthonormal coordinate system, we
seek
Moreover, the tangential condition (C 1 continuity), and thus the tangential operator
T, is identical for all three image curve types.
Thus, we divide the design into three stages:
ffl First, derive a set of one-dimensional L/L operators N fP;N;Eg which verify the
cross-sectional (normal) conditions of Definition 1, while avoiding the pitfalls
discussed in xI.
derive a one-dimensional L/L operator T which is capable of discriminating
between locally continuous and discontinuous curves along their tangent
direction.
ffl Finally, form a family of direction-specific two-dimensional L/L image curve
operators by forming the Cartesian product of the two one-dimensional operators

A. The Normal Operators: Categorization
For the purpose of illustration, we will begin with the analysis of a positive contrast
line (2b). The methodology developed will apply naturally to the two other image
curve types.
Since a necessary condition for the existence of such a line is a local extremum in
intensity (fig. 5 is a display of typical 1D cross-sections of lines and edges), we will
first consider the operator structure normal to its preferred orientation. This is just
the problem of locating extrema in the cross-section fi s .
G' * I
G" * I
G"' * I
(a)
G' * I
G" * I
G"' * I
(b)

Figure

5: Cross sections of image lines and edges. A line in an intensity
image (a) is located at the peak of its cross-section. Note that this coincides
with a zero in the derivative fi 0 and a negative second derivative fi 00 . An
intensity edge (b) occurs at peaks in the derivative fi 0 of the cross-section.
The derivatives shown are derived from convolution by G 0 and G 00 operators
with
A local extremum in a one-dimensional differentiable signal fi(x) exists only at
those points where
Estimating the location of such zeroes in the presence of noise is normally achieved
by locating zero-crossings, thus in practice these conditions become
for some ffl ? 0. An operator which can reliably restrict its responses to only those
points where these conditions hold will only respond to local maxima in a one-dimensional
signal.
A set of noise-insensitive linear derivative operators (or 'fuzzy derivatives' [19])
are the various derivatives of the Gaussian,
G oe
which will be expressed as G 0
oe (x), G 00
oe (x), etc. These estimators are optimal for
additive, Gaussian, i.i.d. noise.
When convolved over a one-dimensional signal these give noise-insensitive esti-
Sum
G'l
G'r
(a)
Sum
G"l
G"r
(b)

Figure

Central differences suggest that an approximation to the n th
derivative can be obtained from a difference between two displaced (n \Gamma 1) th
derivatives. Thus in (a) the sum of two G 0 operators approximate \GammaG 00 , and
in (b) the sum of two G 00 operators approximate G (3) .
mates of the derivatives of the signal, for example
oe
Theorem 6 For the one-dimensional signal fi(x), the following conditions on the
smoothed signal
are sufficient
to indicate a local maximum in the signal fi(x).
The identity in (12) shows that these conditions are necessary and sufficient for
the existence of a local maximum in fi oe (x). The maximum principle
for the heat equation ([20], p. 161) implies that convolution by a Gaussian cannot
introduce new maxima. Thus the above conditions imply the existence of a maximum
in fi(x).
This suggests a practical method for locating maxima in a noisy one-dimensional
signal. Comparing the results of convolutions by derivatives of Gaussians will allow
us to determine the points where Theorem 6 holds. The loci of such points will
form distinct intervals with widths - 2ffl. The parameter oe determines the amount of
smoothing used to reduce noise-sensitivity.
Observe by central limits that:
Thus for the derivative estimates fi oe , one would expect that
with the accuracy of the approximation a function of ffl. Thus the conditions in
Theorem 6 can be verified from examination of the derivative fi 0
oe (x)-a linear combination
of two points will give \Gammafi 00
oe (x). More specifically, we adopt the approximation
\GammaG 00
=2ffl, where ffl=oe - 1. Figure 6a shows that for
this is an acceptable approximation.
Thus, convolution by G 0 allows testing of all three conditions in Theorem 6 si-
multaneously. Using the L/L combinators of xII.A. we are now able to define a
one-dimensional operator which has a positive response only within a small interval
around a local maxima.
Operator 1 The one-dimensional normal operator for Local Maxima NM is
r
where
Clearly then, the key advantage of this NM operator is that:
Observation 7 The response NM (fi)(x) will be positive only if there is local maximum
in fi oe within the region [x \Gamma ffl; x
By reference to Definition 2 we can see that NM (fi)(x) ? 0 implies that both
Equation 12 then implies that
l (x)
r (x)
Thus a positive response ensures that fi 0
oe (x+ ffl) ! 0, which in turn
imply the presence of a local maximum on fi oe between x \Gamma ffl and x
3 Observe that although the local maximum in fi oe is guaranteed to fall within this region, the
corresponding maximum in fi is not necessarily so restricted. Qualitatively however, we can rely on
the observation that the maxima for a signal will converge on the centroid of that signal under heat
propagation (or as we convolve with larger and larger Gaussians). Considering the features of fi in
isolation then, we can state that the smoothing will cause the location of the local maximum in fi oe
-0.250.250.751.251.75I
NP * I
(a)
(b)
-0.250.250.751.251.75I
NP * I
(c)

Figure

7: Responses of L/L positive contrast line operator and the linear
operator \GammaG 00 which it reduces to, near a step edge whose local profile varies
from the ideal. The graphs show the image profiles being operated on,
covering (a) a simple step edge, (b) a compound step with slope above ? 0,
and (c) a compound step with slope above ! 0. It can be seen that the L/L
operator blocks the unwanted response near a step which is not also a local
maximum (a,b), but that when the edge is also a local maximum (c) it does
respond. The linear \GammaG 00 operator, however, responds positively in each of
these cases, exhibiting consistent (and erroneous) displacement of the peak
response.
The performance improvement from introducing this non-linearity is considerable.
The linear operator exhibits consistent patterns of false positive responses. The
simplest example is the response near a step (see fig. 7). The linear operator displays
a characteristic (false) peak response when the step is centered over one of the zeroes
in the operator profile. The logical/linear
operation prevents this error since both
halves of the operator register derivatives in the same direction and so do not
fulfill the conditions of (11). The L/L operator will respond positively only in the
case that the slope above the step is negative (i.e. only when the transition point is
also a local maximum).
A more specific operator can be derived by examining the implications of (2b)
beyond the local maxima. A discontinuous peak, such as that shown Fig. 5a is not
only a negative local minimum in fi 00
oe , but a positive local maximum in fi (4)
oe . Thus
two additional conditions are required
to shift towards the centroid of the local intensity distribution, a phenomenon observed in studies
of biological visual systems (e.g. the vernier acuity studies of [21]).
This can again be captured by central differences, combining two offset third-derivative
estimates.
Operator 2 The one-dimensional normal operator for Positive Contrast Lines
N P is
r (16)
where
Clearly, the addition of these two conditions to the NM operator will select a proper
subset of the positive responses to NM . But since these new conditions were derived
from an analysis of differential structure around roof discontinuities, the N P operator
responses will include these points and be more specific than the NM responses.
The extension of this analysis to the other curve types in xI.A. is straightforward.
The above analysis can be repeated in its entirety with a simple change of sign so as
to be specific for an identical feature of the opposite contrast.
Operator 3 The one-dimensional operator for Negative Contrast Lines NN is
\Gamman 0
\Gamman (3)
\Gamman (3)
Slightly more complicated is the case for edges. In the simplest case, a rising
discontinuity is signalled by a local maxima of the first derivative, thus imposing the
following conditions
or
This condition is just the familiar zero-crossing of a second derivative, exactly the
condition used by Haralick [22] and Canny [2]. Note that this operator actually selects
any inflection points in the signal.
Mirroring the analysis above, the verification of these conditions can be realized
in an L/L operator selecting inflection points N I :
Operator 4 The one-dimensional operator N I for Inflection Points is
where
Now as with the line operators, selection of more truly edge-like features is possible
by examination of other derivatives. Note that a blurred step edge has vanishing even
derivatives and sign-alternating non-zero odd derivatives (see fig. 5). The description
of an edge adopted in (17) is clearly consistent with this observation, but incomplete.
Note also that an "edge" is the derivative of a "peak," which was used for analysing
line-like images. With this additional information, we can adopt a more selective
operator for image edges which requires that all of the following conditions must be
verified
or
These conditions 4 can be verified in an L/L edge operator
Operator 5 The one-dimensional operator NE for Edges is
r
where
oe
This operator is significantly more selective than the 'zero-crossing of a second-
derivative,' [23] which is only one of the logical preconditions on which this operator
depends. One can therefore expect less of a problem of non-edge signals generating
edge-like responses with this operator than with these other less specific operators.
It is important to realize that the operator family which forms the basis for this
analysis is the Derivatives of Gaussian family of operators. Koenderink [24, 25] de-
4 The condition fi 0 (x) ? 0 is actually also implemented by Haralick [22] and Canny [2], since their
lateral maxima selection is followed by a threshold fi 0 (x) ? ', where ' is positive.
rived this family as one orthonormal solution of the problem of deriving size-invariant
spatial samplings of images. Members of this operator family can be transformed into
each other via a set of simple, unitary transformations. This has definite computational
advantages, since the higher derivatives and spatial offsets from pixel centers
may be derived from a small canonical set of operators by linear combinations. In ad-
dition, Young [26] has persuasively argued that these are exactly the basis functions
which are used by primate visual systems.
B. The Tangential Operator: Continuity
far only the normal image structure (fi s ) has been discussed. In order to extend
this result to two-dimensions, we must examine the tangential (curvilinear) structure
of the curves (ff). By Definition 1 we must verify the local continuity of candidate
curves. In addition, the extraction of orientation-specific measures was deemed essential
for further processing. In this section, these problems will be addressed by
imposing a further tangential structure on the operator. We will follow the same
course as for the normal cross-section-first a linear structure will be proposed which
will then be decomposed to reveal linear measurement operators for the components
of the structural preconditions. The emphasis again is placed on these preconditions
and their L/L combination.
Consider the image cross-section that is tangent to the image curve ff at every
point. Assume that the intensity variation along this curve is everywhere smooth
and corrupted only by additive Gaussian noise. The local contrast along the curve
as compared against its background is an acceptable measure of the curve's salience.
This suggests filtering image noise with a linear Gaussian operator
along
the tangential direction.
Near a curve end-point, however, the tangential section will exhibit an abrupt
discontinuity (see fig. 8). The indiscriminate smoothing of the Gaussian will obscure
this contrast discontinuity by, in effect, assuming that no discontinuity is present
before it is applied, thus violating the third criterion of xI.A. The local continuity of
the curve must be verified prior to smoothing.
To resolve this, consider a definition of the local continuity of a function. The
function f(x) is said to be continuous at x 0 iff
lim
For our purposes, assume that the non-linearities associated with the normal components
of the L/L image curve operators are evaluated before 5 those in the tangential
5 With a pure linear operator expressed as the Cartesian product of normal and tangential one-0.51
Tangent
Linear
Tangent
Desired
(b)

Figure

8: The signal is the tangential section of an image line near the
discontinuous termination of the line (the endline). Note that the linear operator
(a) exhibits a smooth attenuation of response around the line ending.
We seek an operator (b) whose response attenuates abruptly at or near the
endline discontinuity.
(a) (b) (c)

Figure

9: Schematic of the half-field decomposition and line endings. The
elliptic regions in each figure represent the operator positions as the operator
is placed beyond the end of an image line. In (a) the operator is centered on
the image line and the line exists in both half-fields. In (b) the operator is
centered on the end-point and whereas the line only exists in one half-field,
the other half-field contains the end-point. In (c) the operator is centered
off the line and the line only exists in one half-field.
L/L operators. Then a curve termination point in the image must be signalled by
a contrast sign reversal in the image section seen by the tangential L/L operator-a
transition from a region which has been confirmed to be of the given category (posi-
tive response) to a region which has been rejected (negative response). We will call
the behaviour which the tangential operator must exhibit End-Line Stability. A
one-dimensional operator is end-line stable if and only if it responds positively only
when centered on a uniformly positive region of the image.
Representing the intensity variation along the curve ff as a function of the arc-length
I ff (s), the worst-case line-ending (or beginning) is a step in intensity at
End-line stability requires that the operator's response T(I ff )(s) be non-positive for all
positive for s ? ffl. 6 Given the requirement for symmetric approach outlined
in eq. (19), from fig. 9 we observe that this can be achieved by separately considering
the behaviour of the curve in each tangential direction around the operator centre.
We therefore adopt a partition which divides the operator kernel into two regions
along its length. Using the step function oe(x) of eq. (II.C.) a partition of G(x) around
0 is given by
Operator 6 The one-dimensional operator for Tangential Continuity T is
Note that required. The smooth partition
operator oe ae (x) of eq. (9) can be used for a smooth, stable partition.
Observation 8 The operator T is end-line stable.
Consider the component responses in the neighbourhood of the step edge I ff
oe(s). The response of t + to this step is given by
operators, order-of-evaluation is unimportant, but with Logical/Linear operators order-
of-evaluation can be essential.
6 This property must also operate symmetrically at the other end of the curve.
G
(a)
G
(b)

Figure

10: A one-dimensional Gaussian (representing the tangential operator
partitioned into two regions (a) to obtain the two half-field operators
defined by eq. (20). The addition of 'stabilizers' is shown in (b).
The L/L AND of t + and t \Gamma to produce T requires that both component responses
be strictly positive for a positive response, thus whenever s - 0 around the step
described above, the T response is also zero. It is obvious that the same analysis
applies to the symmetric t \Gamma component and the 1 \Gamma oe(s) step edge, which describes
behaviour around the other end of the line. Thus the T operator is end-line stable
symmetrically around a step edge.
The above proof, however, depends critically on the use of ideal L/L combinators,
while in most cases we would prefer to use non-ideal combinators
ae where ae ! 1).
When the non-ideal combinators are used, the 'end-line stable' operator described
above does not properly attenuate responses beyond the line ending (see Fig. 11a).
In order to achieve this attenuation, it is necessary to force the component responses
in the region just beyond a line ending significantly below zero.
This is achieved with the addition of the 'stabilizers' (shown in Fig. 10b):
Thus, a smooth partition of G(x) by oe a (x) is augmented with an overshoot \GammabG 0 (x).
The overshoot guarantees that when the center of the operator is near the line ending
(see Fig. 11b) one component will give a negative response over the region where the
Tangent
t- * I
(a)
Tangent
t- * I
(b)

Figure

11: Responses (including component responses) of an unstabilized
endline operator (a) and the stabilized version (b). Note that the L/L combination
of the unstabilized components (a) does not, in fact, reduce to zero
beyond the end of line. This is due to the use of the L/L
ae approximation
with ae ! 1. In order to produce stable attenuation at a line ending, inhibitory
regions (stabilizers) are added to the t \Gamma and t + components, which
have the effect of pushing the component responses 'near' but `off ' the line-
ending below zero (b).
operator is not centered on the line. Since the stabilizers are symmetric, it does not
matter whether the operator is near a rising or falling line-ending-if the operator
is centered over the positive region it will respond. Furthermore, since the integral
of the stabilizers is zero, they will have no effect whatsoever on a locally constant
signal. The candidate tangential operator is then the L/L AND of these stabilized
components. The parameters a and b are chosen so that the cutoff is exactly aligned
with the line ending.
We have, in addition, extended these principles to multiple (more than two) tangential
regions (e.g. four), whose responses are combined with the L/L AND com-
binator. In essence, this looks for more tangential structure than simple continuity.
Requiring that regions which do not overlap the center of the operator show positive
responses as well, can be used to impose a minimum-length criterion on detected
curves. Davis [27] suggested that such an approach would have the effect of decreasing
noise sensitivity. This technique is described in detail in [28]. Note the similarity
between this approach and the ANDing of LGN afferents proposed by Marr and
Hildreth [23]).
C. The Two-Dimensional Image Operators
Finally then, we can construct the two-dimensional image curve operators by taking
the Cartesian product of the normal and tangential components. Our original definition
of curvilinear image structure (Def. 1) points directly to this by defining an image
curve as the locus of points satisfying some normal condition along a differentiable
curve in the image. In order to test both normal and tangential conditions then we
construct the component two-dimensional operators by taking the cartesian products
of all of the tangential and normal components (see Fig. 12 for an example). For each
tangential region, we combine the outputs of the normal combinations producing a
confirmation of the hypothesis that the normal condition is satisfied within that tangential
region. These hypotheses are then combined using the Tangential Continuity
combination of Op. 6 to verify the local continuity hypothesis.
Operator 7 The Logical/Linear Image Curve Operators \Psi i (where i selects
the operator category) are given by
where
r for Positive Contrast Lines,
\Gamman
\Gamman (3) l
\Gamman (3) r for Negative Contrast Lines,
r for Edges.
Thus, the constructed two-dimensional operator, a L/L combination of linear
two-dimensional operators, has a positive response only when the normal conditions
(which categorize line types) are consistent through the tangential regions, thus verifying
local curvilinear continuity.
IV. Results
As per the decomposition into curve types described above, we create three different
classes of curve operator, for positive and negative contrast lines and for edges. The
operators in the following examples all have a tangential oe = 2:0 and a lateral
pixels. The ffl of the the lateral operator separations is 1:0. This ensures that all curves
are localized to connected regions with width - 2 pixels.
For the comparison images with Canny's algorithm, an upper threshold of - 15%
contrast was used. This value was adequate for suppressing most noise, although
some of the examples show that the noise has not been entirely eliminated. The low
threshold was set to 1% so as to come close to matching the sensitivity of the L/L
operators to very faint structures.
A natural but informal evaluation criterion for edge operators is the degree to
which the 'edge map' produced corresponds to a reasonable line drawing of an im-
age. We therefore use a test image of a statue "Paolina" (Fig. 13) not unlike the
subjects in Michelangelo's drawings (Fig. 14). This drawing is particularly suitable
because, as Koenderink has pointed out, the representation of creases and folds is
especially important for conveying a sense of three-dimensional structure [10, 11].
An examination of the Canny and L/L edge images for the statue reveals a marked
difference in the ability to distinguish perceptually salient edges from other kinds of
intensity changes. Comparison with Michelangelo's treatment reveals clearly that the
L/L operators represent more of the significant structure than the Canny operator.
Formal criteria for an image curve were established in xI.A., and these provide
less subjective demonstrations of where the Canny operator fails. We stress that
our goal here is not to focus on the shortcomings of the Canny operator, but rather
to indicate the shortcomings of the long tradition of edge operators that consist of
linear convolutions followed by thresholding, from Sobel [29] through Marr-Hildreth
[23] and most recently in Canny.
The first criterion, the need for predictable behaviour in the neighbourhood of multiple
image curves is examined in each of the details from the statue image (Figs. 15,
16, and 17). In these circumstances, The Canny operator either leaves large gaps
(Figs. 15b and 17b), or simply infers a smooth, undisturbed local contour (Fig. 16b).
This failure disrupts the ability to reconstruct the kind of information which gives a
sense of three-dimensional structure, since creases and folds involve the intersection

Figure

12: Illustration of the construction of the two-dimensional positive
contrast line operator. Each of the bottom row of operators is a linear
operator which is formed by one of the linear component operators n l;r \Theta t l;r .
The middle row represents the linear reduction of the operators t l;r \Theta N P ,
in other words the sum of the two operators below. The operator shown at
the top of the pyramid is the linear reduction of \Psi P , the sum of the middle
operators. The cross-hairs represent the centre of each operator and are
provided solely for purposes of alignment.
(a) (b)

Figure

13: Image of statue (a), provided by Pietro Perona, and edge maps
computed by: (b) Canny's algorithm (both
algorithms are run at the same scale). Compare these representations with
the human expert's line drawing in Fig. 14, especially around the chin and
neck. The Canny operator consistently signals non-salient 'edges', misses
edges in complex neighbourhoods (e.g. near the T-junction of the chin and
neck) and shows discontinuous orientation changes as smooth. (The boxes
represent the approximate locations of the details shown in subsequent fig-
ures).

Figure

14: Line drawings, such as this Michelangelo, demonstrate in a
clear and compelling manner the significance of image curves for the visual
system. A well-executed line drawing depends critically on curvature, line
terminations and junctions for its visual salience. Koenderink has stressed
how the "bifurcation structures" define the arm and shoulder musculature
and the manner in which the chin occludes the neck. Observe the similarity
between this and the L/L operator responses, and differences with the Canny
operator.
and joining of just such multiple image curves. In the worst case, nearby curves can
interfere with the Canny operator's ability to extract much meaningful structure at
all (Fig. 19b).
This leads us to the second criterion, the need to preserve line terminations and
discontinuities. In our approach to early vision, we take curve discontinuities to be
represented by multiple, spatially coincident edges [30, 31, 32]. This holds for both
"corners" and "T-junctions"-such discontinuities are inadequately captured by the
Canny operator. Where there are clear discontinuities and junctions in the image
curves, the Canny operator either leaves gaps or gives smooth output curves (see in
particular the detail in Fig. 16b). The L/L operators represent such curve crossings
and junctions by supporting multiple independent orientations in a local neighbour-
hood, just the representation we require. So not only do the L/L operators respond
stably in the neighbourhood of multiple coincident curves, but they are also able
to adequately represent this coincidence. Preceding attempts at edge operators have
relied on the a priori assumption (usually implicit) that only one edge need be considered
in each local neighbourhood, and thus that only one edge need be represented at
each point in the output image. By rejecting that assumption and ensuring that the
L/L operators perform stably in the neighbourhood of edge conjunctions, we provide
a stable, complete representation of these fundamental image structures.
Recently, there has been some attempt to define "steerable filters" for edge detection
[33, 34, 35], and to have them provide a representation for image curve discontinuities
analogous to ours (i.e., as multiple orientations at the same position).
However, the linear spatial support of these operators again causes problems, in this
case a "smearing" or blurring of the corner energy over a neighbourhood. An additional
search process is therefore introduced to find the locations and directions of
maximal response [35], analogous to what we called "lateral maxima selection" in
earlier implementations of our system [4]. While such search processes provide some
of the necessary non-linear behaviour, they introduce additional interpretative difficulties
that do not arise with the L/L decomposition. Search also further complicates
parallel implementations by introducing sequential bottlenecks. Finally, the standard
steerable filters still exhibit mislocalization of line endings (which led in [35] to the
introduction of end-line detectors). The steerable filters approach is useful, however,
for reasons of computational efficiency, and we suggest that they may be used as a
basis set for the linear components of our L/L operators.
Finally, the third criterion, the potential confusion between lines and edges, is
seen to be addressed by the L/L operator approach. This problem is acute with
the Canny operator, and is deliberately confounded by the "edge energy" methods
[36, 34], thus necessitating a second stage of analysis that refers back to absolute image
intensities to fully describe the local structure of the image curve. The fingerprint
(Fig. 19) and the composite image of the statue (Fig 18) show the utility and richness
of a representation which separates edge and line information. The fingerprint is
clearly more appropriately and parsimoniously represented by the line image, while
the highlights on the statue are only revealed by the line image. It has been argued
that most line-like structures can be revealed by looking for locally parallel edge
responses, but clearly not all (e.g. the many highlights on the statue's surface). We
submit that parsimonious representations will combine features from both edge and
line images and interpret them as appropriate.
It is also important to note that computing Canny's algorithm on a parallel architecture
requires a number of iterations of dilation in order to implement the 'hys-
teretic threshold'. Thus it's time complexity on a fully parallel implementation is
O(n), where n is the maximum length of a curve. Worst case, this is proportional to
the number of pixels in the image, thus representing a significant sequential bottle-neck
in an otherwise parallelizable algorithm. The L/L operator implementation is,
however, O(1) in both time and processors.
V. Conclusions
One of the major problems with linear operator approaches to detecting image curves
is their false-positive responses to uncharacteristic stimuli. After outlining the necessary
structural conditions for the existence of an image curve, we developed an
operator decomposition which allows for the efficient testing of these conditions, and
the elimination of the associated false-positive responses. To achieve this, it was essential
to consider both the cross-section of the intensity image and the low-order
differential structure of the curve itself.
The operator families which are used as a basis set for these computations consists
of spatial derivatives of Gaussians. It is a widely held assumption in the field that
measurements of higher order derivatives are unstable and therefore unusable. We
have deliberately chosen to highlight these higher order derivatives and have demonstrated
that in the context of L/L operators these measurements are not only stable
but extremely useful, even at high resolutions.
The output of the operators is unconventional since we chose at an early stage to
not impose a functional mapping between image points and local linear structure. Not
only may there be multiple line types at a single image position, there may even be
multiple lines of a single type. In [28] this representation is referred to as a Discrete
Image Trace. It it is independently justified for its ability to implicitly represent
continuity, intersection and some topological properties of differentiable structures
representable as fibre bundles on the image (e.g. image curves). Moreover, it is
shown how these discrete traces may be refined using relaxation labelling to verify
more global constraints and begin to construct higher-level representations.

Figure

15: Detail of statue (a) from lower left near jaw and neck, and edge
maps computed by: (b) Canny's algorithm, and (c) L/L operators (both
algorithms are run at the same scale). Note that Canny's algorithm does
not connect the two edges which join at the T-junction. The L/L operator
responses represent the discontinuity by supporting two independent
orientations in the same local neighbourhood.

Figure

16: Detail of statue (a) from upper right, and edge maps computed
by: (b) Canny's algorithm, and (c) L/L operators (both algorithms are run
at the same scale). The Canny operator misses much of the rich structure in
this small region as a result of interference between the nearby edges and the
choice of high threshold. A lower threshold would have the effect of exposing
more structure, but then the noisy responses seen in Fig. 13a would also be
expanded. The L/L operator exposes this structure and also represents the
discontinuities and bifurcations in the underlying edge structure.

Figure

17: Detail of statue (a) from lower right near shoulder, and edge
maps computed by: (b) Canny's algorithm, and (c) L/L operators (both
algorithms are run at the same scale). Again the Canny operator does not
represent the conjunction of edges in this neighbourhood, while the L/L
operators show the edge bifurcation clearly.

Figure

18: The statue as represented by the three categories of L/L opera-
tors. The black lines show the edge responses while the white and grey lines
show the bright and dark lines respectively. Note that some features, such
as the bottom of the palm of the hand, are only clearly represented by the
line images.
(a) (b)
(c) (d)

Figure

19: Fingerprint image (a), and edge maps computed by (b) Canny's
algorithm, and (c) L/L edge operators. The most appropriate representation
(d) is the L/L positive contrast line operator. The complexity of display
and the proximity between nearby image features are the most significant
contributors to the breakdown of Canny's algorithm in this case. These
problems are dealt with in the L/L operators by the explicit testing of local
consistency before combining component inputs. This serves to isolate features
even when other nearby structures exist within the spatial support of
the operator.
More generally, we have introduced a flexible language for describing a useful
class of non-linearities in operators. This language of Logical/Linear operators
serves to combine existing linear models with logical descriptions of structure to
produce operators which have guaranteed stable behaviour. This class of operators
represents a new approach to the problem of translating linear measures into logical
categories. Thus they may prove essential in the eventual solution of a wide variety
of classification problems, and in the principled and realistic modelling of neural
networks.



--R

"An operator which locates edges in digital pictures,"
"A computational approach to edge detection,"
"An evaluation of the two-dimensional gabor filter model of simple receptive fields in cat striate cortex,"
"The organization of curve detection: Coarse tangent fields and fine spline coverings,"
"On boundary detection,"
"Using Canny's criteria to derive a recursively implemented optimal edge detector,"
"Understanding image intensities,"
"The local structure of image discontinuities in one dimension,"

"The singularities of the visual mapping,"
"The shape of smooth objects and the way contours end,"
"The internal representation of solid shape and visual explo- ration,"

"Length summation in simple cells of cat striate cortex,"
"Spatial frequency analysis in the visual system,"
New York
A Survey of Modern Algebra (fourth edition).
"Learning internal representations by error propagation,"
"Representation of local geometry in the visual system,"
Maximum Principles in Differential Equa- tions
"Mechanisms responsible for the assessment of visual location: theory and evidence,"
"Digital step edges from zero-crossing of second directional derivatives,"
"Theory of edge detection,"
"Operational significance of receptive field assemblies,"
"Receptive field families,"
"The gaussian derivative theory of spatial vision: Analysis of cortical receptive field line-weighting profiles,"
"On models for line detection,"
"Toward Discrete Geometric Models for Early Vision,"
Pattern Classification and Scene Analysis.
"Corner detection in curvilinear dot grouping,"
"The computational connection in vision: Early orientation selec- tion,"
"Two stages of curve detection suggest two styles of visual computation,"
"The design and use of steerable filters for image analysis, enhancement and wavelet decomposition,"
"Detecting and localizing edges composed of steps, peaks and roofs,"
"Steerable-scalable kernels for edge detection and junction analysis,"
"Feature detection in human vision: a phase dependent energy model,"
--TR

--CTR
Song Wang , Feng Ge , Tiecheng Liu, Evaluating edge detection through boundary detection, EURASIP Journal on Applied Signal Processing, v.2006 n.1, p.213-213, 01 January
Doug DeCarlo , Adam Finkelstein , Szymon Rusinkiewicz , Anthony Santella, Suggestive contours for conveying shape, ACM Transactions on Graphics (TOG), v.22 n.3, July
P. Meer , B. Georgescu, Edge Detection with Embedded Confidence, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.23 n.12, p.1351-1365, December 2001
Jonas August , Steven W. Zucker, Sketches with Curvature: The Curve Indicator Random Field and Markov Processes, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.25 n.4, p.387-400, April
Stefano Casadei , Sanjoy Mitter, Hierarchical Image SegmentationPart I: Detection of Regular Curves in a Vector Graph, International Journal of Computer Vision, v.27 n.1, p.71-100, March, 1998
Gang Li , Steven W. Zucker, Contextual Inference in Contour-Based Stereo Correspondence, International Journal of Computer Vision, v.69 n.1, p.59-75, August    2006
Ohad Ben-Shahar , Steven W. Zucker, The Perceptual Organization of Texture Flow: A Contextual Inference Approach, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.25 n.4, p.401-417, April
Peter N. Belhumeur , James S. Duncan , Gregory D. Hager , Drew V. Mcdermott , A. Stephen Morse , Steven W. Zucker, Computational Vision at Yale, International Journal of Computer Vision, v.35 n.1, p.5-12, Nov. 1999
Benoit Dubuc , Steven W. Zucker, Complexity, Confusion, and Perceptual Grouping. Part I: The Curve-like Representation, International Journal of Computer Vision, v.42 n.1-2, p.55-82, April/May 2001
Benoit Dubuc , Steven W. Zucker, Complexity, Confusion, and Perceptual Grouping. Part I: The Curve-like Representation, Journal of Mathematical Imaging and Vision, v.15 n.1-2, p.55-82, July/October 2001
Jacob Feldman, Perceptual Grouping by Selection of a Logically Minimal Model, International Journal of Computer Vision, v.55 n.1, p.5-25, October
Song Wang , Toshiro Kubota , Jeffrey Mark Siskind , Jun Wang, Salient Closed Boundary Extraction with Ratio Contour, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.27 n.4, p.546-561, April 2005
Blaise Agera y Arcas , Adrienne L. Fairhall , William Bialek, Computation in a single Neuron: Hodgkin and Huxley revisited, Neural Computation, v.15 n.8, p.1715-1749, August
James H. Elder, Are Edges Incomplete?, International Journal of Computer Vision, v.34 n.2-3, p.97-122, Nov. 1999
Vicent Caselles , Bartomeu Coll , Jean-Michel Morel, Topographic Maps and Local Contrast Changes in Natural Images, International Journal of Computer Vision, v.33 n.1, p.5-27, Sept. 1999
Matthias S. Keil, Smooth Gradient Representations as a Unifying Account of Chevreul's Illusion, Mach Bands, and a Variant of the Ehrenstein Disk, Neural Computation, v.18 n.4, p.871-903, April 2006
nonlinear operators;edge detection;image processing;feature extraction;computer vision