Completion Energies and Scale.
AbstractThe detection of smooth curves in images and their completion over gaps are two important problems in perceptual grouping. In this study, we examine the notion of completion energy of curve elements, showing, and exploiting its intrinsic dependence on length and width scales. We introduce a fast method for computing the most likely completion between two elements, by developing novel analytic approximations and a fast numerical procedure for computing the curve of least energy. We then use our newly developed energies to find the most likely completions in images through a generalized summation of induction fields. This is done through multiscale procedures, i.e., separate processing at different scales with some interscale interactions. Such procedures allow the summation of all induction fields to be done in a total of only $O(N \log N)$ operations, where $N$ is the number of pixels in the image. More important, such procedures yield a more realistic dependence of the induction field on the length and width scales: The field of a long element is very different from the sum of the fields of its composing short segments.
Introduction
The smooth completion of fragmented curve segments
is a skill of the human visual system that has been demonstrated
through many compelling examples. Due to this skill
people often are able to perceive the boundaries of objects
even in the lack of sufficient contrast or in the presence of oc-
clusions. A number of computational studies have addressed
the problem of curve completion in an attempt to both provide
a computational theory of the problem and as part of a
process of extracting the smooth curves from images. These
studies commonly obtain two or more edge elements (also
referred to as edgels) and find either the most likely completions
that connect the elements or the smoothest curves trav-
Research supported in part by Israel Ministry of Science Grant 4135-
1-93 and by the Gauss Minerva Center for Scientific Computation.
y Research supported in part by the Unites States-Israel Binational Science
Foundation, Grant No. 94-00100.
eling through them. The methods proposed for this problem
generally require massive computations, and their results
strongly depend on the energy function used to evaluate the
curves in the image. It is therefore important to develop
methods which simplify the computation involved in these
methods while providing results competitive with the existing
approaches. Below we present such a method that directly
relates to a number of recent studies of completion and
curve salience [9, 18, 5, 19, 12, 7] (see also [2, 6, 8, 14, 15]).
Along with simplifying the computations proposed in these
studies our method also takes into account the size of edge
elements, allowing for a proper computation of completion
and saliency at different scales.
A number of studies have addressed the problem of determining
the smoothest completion between pairs of edge
elements These studies seek to define
a functional that, given two edge elements defined by their
location and orientation in the image, selects the smoothest
curve that connects the two as its minimizing curve. The
most common functional is based on the notion of elas-
tica, that is, minimizing the total squared curvature of the
curve [9]. Scale invariant variations of this functional were
introduced in [18, 5]. While the definition of scale-invariant
elastica is intuitive, there exists no simple analytic expression
to calculate its shape or its energy, and existing numerical
computations are orders-of-magnitude too expensive, as
will be shown below.
In the first part of this paper we revisit the problem
of determining the smoothest completion between pairs
of edges and introduce two new analytic approximations
to the curve of least energy. The first approximation is
obtained by assuming that the deviation of the two input
edgels from the straight line connecting them is relatively
small. This assumption is valid in most of the examples
used to demonstrate perceptual completions in humans and
monkeys [10, 11]. We show that under this simplifying
assumption the Hermite spline (see, e.g., [13]) provides a
good approximation to the curve of least energy and a very
good approximation to the least energy itself. We further
develop a second expression, which directly involves the
angles formed by the edgels and the straight line connecting
them. The second expression is shown to give extremely
accurate approximations to the curve of least energy even
when the input edgels deviate significantly from the line
connecting them. We then introduce a new, fast numerical
method to compute the curve of least energy and show that
our analytic approximations are obtained at early stages of
this numerical computation.
Several recent studies view the problems of curve completion
and salience as follows. Given M edge elements,
the space of all curves connecting pairs of elements is examined
in an attempt to determine which of these completions
is most likely using smoothness and length considerations.
For this purpose [7, 19] define an affinity measure between
two edge elements that grows with the likelihood of these
elements being connected by a curve. By fixing one of the
elements and allowing the other element to vary over the
entire image an induction field representing the affinity values
induced by the fixed element on the rest of the image
is obtained. The system finds the most likely completions
for the M elements by applying a process that includes a
summation of the induction fields for all M elements.
In the second part of this paper we use our newly developed
completion energies to define an affinity measure that
encourages smoothness and penalizes for gap length. We
then use the induction fields defined by this affinity measure
to solve the problem of finding the most likely completions
for M elements. Since in practice edge elements are never
dimensionless, because they are usually obtained by applying
filters of a certain width and length to the image, we
adjust our affinity measure to take these parameters into ac-
count. We do so by relating the scale of these filters to the
range of curvatures which can be detected by them and to
the orientational resolution needed. Finally, we show that
our affinity measure is asymptotically smooth, and so can be
implemented using multigrid methods and run efficiently in
time complexity O(nm) (where n is the number of pixels
and m is the number of discrete orientations at every pixel).
The paper is divided as follows. In Section 2 we review
the notion of elastica and its scale invariant variation. In
Section 3 we introduce the two analytic approximations to
the curve of least energy. Then, in Section 4 we develop a
fast numerical method to compute the curve of least energy
and compare it to our analytic approximations. Finally,
in Section 5 we construct an affinity measure taking into
account the length and width of the edge filters applied to
the image. We then discuss a multiscale (multigrid) method
for fast summation of induction fields.
2. Elastica
Consider two edge elements positioned at
R 2 with directed orientations Y 1 and Y 2 respectively measured
from the right-hand side of the line passing through
Below we shall confine ourselves to the case
that
may conveniently assume that
This is illustrated in Fig. 1(a). Let C 12 denote the set of
curves through e 1 and e 2 . Denote such a curve by its
orientation representation Y(s), where 0 - s - L is the
arclength along the curve. That is,
R s
s
and
R s
s. Also denote the curvature of
the curve at s by dY(s)=ds.
r
(a)
Y
y
F
F
F 1-
r
(b)

Figure

1. (a) The planar relation between two edge ele-
ments, This relation is governed by
are
measured from the line
(b) The more general relation between F i and Y i .
The most common functional used to determine the
smoothest curve traveling through P 1 and P 2 with respective
orientations Y 1 and Y 2 is the elastica functional. Namely,
the smoothest curve through e 1 and e 2 is the curve Y(s)
which minimizes the functional G el (Y) def
R L
Elastica was already introduced by Euler. It was first applied
to completion by Ullman [17], and its properties were
further investigated by Horn [9].
One of the problems with the classical elastica model is
that it changes its behavior with a uniform scaling of the
image. In fact, according to this model if we increase r,
the distance between the two input elements, the energy
of the curve connecting them proportionately decreases,
as can be easily seen by rescaling s (cf. [1]). This is
somewhat counter-intuitive since psychophysical and neurobiological
evidence suggests that the affinity between a
pair of straight elements drops rapidly with the distance
between them [11]. Also, the classical elastica does not
yield circular arcs to complete cocircular elements. To solve
these problems Weiss [18, 5] proposed to modify the elastica
model to make it scale invariant. His functional is defined
as G inv (Y)
R L
We believe that a proper
adjustment of the completion energy to scale must take into
account not only the length of the curve (or equivalently the
distance between the input elements), but also the dimensions
of the input edge elements. Both the elastica functional
and its scale invariant version assume that the input elements
have no dimensions. In practice, however, edge elements
are frequently obtained by convolving the image with filters
of some specified width and length. A proper adjustment
of the completion energy as a result of scaling the distance
between the elements should also consider whether a corresponding
scaling in the width and length of the elements has
taken place. Below we first develop useful approximations
to the scale invariant functional. (These approximations can
also readily be used with slight modifications to the classical
elastica measure.) Later, in Section 5, we develop an affinity
measure between elements that also takes into account both
the distance between the elements and their dimensions.
3. Analytic simplification of G inv
Although the definition of both the classical and the scale
invariant elastica functionals is fairly intuitive, there is no
simple closed-form expression that specifies the energy or
the curve shape obtained with these functionals. In this
section we introduce two simple, closed-form approximations
to these functionals. Our first approximation is valid
when the sum of angles jF relatively small. This
assumption represents the intuition that in most psychophysical
demonstrations gap completion is perceived when the
orientations of the curve portions to be completed are nearly
collinear. With this assumption we may also restrict for now
the range of applicable orientations to Y
The second approximation will only assume that jF
is small, i.e., that the curve portions to be completed are
nearly cocircular.
Since the curve of least energy is supposed to be very
smooth, it is reasonable to assume that within the chosen
range of Y the smoothest curve will not wind much.
Consequently, it can be described as a function
in Fig. 1(a). Expressing the curvature in terms of x and y we
obtain that G inv
R L
For we get that L ! r, and that the variation
of unimportant for the comparison of
G inv (Y) over different curves Y 2 C 12 , so that G inv (Y) '
r
R r
Hence
G inv (Y) ' r min
The minimizing curve is the appropriate cubic Hermite
spline (see [1])
r
so that
Evidently, this simple approximation to E inv is scale-
independent. This leads us to define the scale-invariant
spline completion energy as: E spln
Although the spline energy provides a good
approximation to the scale invariant elastica measure for
small values of jF the measure diverges for large
values. An alternative approximation to E inv can be constructed
by noticing that for such small values tan F 1 ' F 1
and tan F 2 ' F 2 . Thus, we may define:
We refer to this functional as the scale-invariant angular
completion energy. This measure does not diverge for large
values of jF In fact, when F
In Section 4 below we show
that this angular energy is obtained in an early stage of the
numeric computation of E inv , and that it provides extremely
accurate approximations to the scale invariant least energy
functional even for relatively large values of jF 1 j+jF 2 j, especially
for small jF for the range of nearly
cocircular elements. Using the numeric computation we
can also derive the smoothest curve according to
The angular completion energy can be generalized as
follows:
where Eq. (4) is identical to Eq. (6) with 2. That
is, the angular completion energy is made of an equal sum
of two penalties. One is for the squared difference between
F 1 and F 2 , and the other is for the growth in each of them.
This suggests a possible generalization of E ang to other
weights a - 0 and b - 0. One can also think of using
energies such as E circ
more elaborate study of these types of energies and their
properties is presented in [1].
Finally, we note that the new approximations at small
angles can also be used to approximate the classical elastica
energy, since
el
G el (Y) ' 1
4. Computation of E inv
We use the scale-invariance property of G inv in order to
reformulate the minimization problem into minimizing over
all C 12 curves of length Applying
Euler-Lagrange equations (see, e.g., [13]) we get that a necessary
condition for -
s) to be an extremal curve is that it
should satisfy for some -:
Y s:t: (8)
Considering the very nature of the original minimization
problem, and also by repeatedly differentiating both sides
of the ODE equation, it can be shown that its solution must be
very smooth. Hence, we can well approximate the solution
by a polynomial of the form
where n is small. (By comparison, the discretization of the
same problem presented in [5] is far less efficient, since it
does not exploit the infinite smoothness of the solution on
the full interval (0,1). As a result the accuracy in [5] is
only second order, while here it is "1-order", i.e., the error
decreases exponentially in the number of discrete variables
:::; an from Eq. (8) and Fixing
n, as well as two other integers -
n and p, we will build the
following system of n+2 equations for the n+2 unknowns
a an ; and -
collocating the ODE, and
n) are the weights of a p-order numerical
integration. Generally, we increase n gradually
and increase -
n and p as functions of n in such a way that
the discretization error will not be governed by the discretization
error of the integration. The nonlinear system of
equations is solved by Newton iterations (also called
Newton-Raphson; see, e.g. [13].) We start the Newton iterations
from a solution previously obtained for a system with
a lower n. Actually, only one Newton iteration is needed
for each value of n if n is not incremented too fast. In
this way convergence is extremely fast. At each step, in
just several dozen computer operations, the error in solving
the differential equation can be squared. In fact, due to the
smoothness of the solution for the ODE, already for the simple
2)-system and the Simpson integration rule
very good approximation to the accurate solution
and also to E inv is obtained, as can be seen in
[1]. The good approximations obtained already for small
values of (n, -
n) suggest that E inv can be well approximated
by simple analytic expressions, as indeed we show in [1] by
comparingbetween several simple approximations to E inv .
Fig. 2 illustrates some of the completions obtained using
inv and the two analytic approximations E ang and E spln .
It can be seen that the differences between the three curves
is barely noticeable, except in large angles where E spln
diverges. Notice especially the close agreement between
the curve obtained with the angular energy (Eq. (5)) and that
obtained with the scale-invariant elastica measure even in
large angles and when the angles deviate significantly from
cocircularity.
Note that although the spline curve does not approximate
the scale invariant elastica curve for large angles jF 1 j
and jF 2 j it still produces a reasonable completion for the
elements. In fact, when the two elements deviate from co-
circularity the elastica accumulates high curvature at one of
its ends, whereas in the spline curve continues to roughly
follow the tangent to the two elements at both ends (see, e.g,
Fig. 2(b)) . This behavior is desirable especially when the
elements segments (see Section 5.2).
5. Completion field summation
Until now we have considered the problem of finding the
smoothest completion between pairs of edge elements. A
natural generalization of this problem is, given an image
from which M edge elements are extracted, find the most
likely completions connecting pairs of elements in the image
and rank them according to their likelihoods. This problem
has recently been investigated in [7, 19]. In these studies
affinity measures relating pairs of elements were defined.
The measures encourage proximity and smoothness of com-
pletion. Using the affinity measures the affinities induced
by an element over all other elements in the image (referred
to as the induction field of the element) are derived. The
likelihoods of all possible completions are then computed
simultaneously by a process which includes summation of
the induction fields for all M elements.
An important issue that was overlooked in previous ap-
proaches, however, is the issue of size of the edge elements.
Most studies of curve completion assume that the edge elements
are dimensionless. In practice, however, edge elements
are usually obtained by convolving the image with
filters of certain width and length. A proper handling of
scale must take these parameters into account. Thus, for
example, one may expect that scaling the distance between
two elements would not result in a change in the affinity of
the two elements if the elements themselves are scaled by the
same proportion. Below we first present the general type of
non-scaled induction underlying previous works. We then
modify that induction to properly account for the width and
length of the edge elements.
Finally, the process of summing the induction fields may
be computationally intensive. Nevertheless, in the third part
of this section we show that the summation kernel obtained
with our method is very smooth. Thus, the summation of
our induction fields can be speeded up considerably using a
multigrid algorithm. This result also applies to the summation
kernels in [19, 16, 7], and so an efficient implementation
of these methods can be obtained with a similar multigrid
algorithm.
5.1. Non-scaled induction
In [12, 19] a model for computing the likelihoods of curve
completions, referred to as Stochastic Completion Fields,

Figure

2. Completion curves: elastica in solid line, -
(Eq. (5)) in dotted line, and the cubic Hermite spline (Eq. (2)) in dashed line. (a)
was proposed. According to this model, the edge elements
in the image emit particles which follow the trajectories
of a Brownian motion. It was shown that the most likely
path that a particle may take between a source element and
a sink element is the curve of least energy according to
the Elastica energy function 1 . To compute the stochastic
completion fields a process of summing the affinity measures
representing the source and sink fields was used. In [1]
we show, by further analyzing the results in [16], that the
affinity measure used for the induction in [19, 16] is of
the general type: A(e 1
Fig. 1(b) ), where r 0 and oe 0 are strictly positive a-priori set
parameters. These parameters need to be adjusted properly
according to the scale involved (see Sec. 5.2). Note that
for small values of (jF 1 j,jF 2 el . Hence,
el =oe 0 .
Another method which uses summation of induction
fields to compute the salience of curves was presented in [7].
In their method the affinity between two edge elements
which are cocircular has the form: e \Gammafl r e \Gammaffi- , where fl and ffi
are strictly positive constants, - is the curvature of the circle
connecting e 1 and e 2 , and r is the distance between e 1 and
e 2 . A reasonable and straightforward definition in that spirit
is -
serves as an approximation
according to Eq. (3). Fig. 3 shows
an example of computing the "stochastic completion field,"
suggested by Williams and Jacobs in [19], while replacing
their affinity measure with the simple expression -
It can be verified by comparing the fields obtained with our
affinity measure with the fields presented in [19] that the
results are very similar although a much simpler affinity
measure was employed.
5.2. Induction and scale
Given an image, an edge element is producedby selecting
a filter of a certain length l and width w (e.g., rectangular
filters) and convolving the filter with the image at a certain
position and orientation. The result of this convolution is
a scalar value, referred to as the response of the filter. An
edge filter may, for example, measure the contrast along
its primary axis, in which case its response represents the
"edgeness level", or the likelihood of the relevant subarea
of the image to contain an edge of (l; w) scale. Similarly,
1 Actually, the path minimizes the energy functional
R L
for some predetermined constant -.
(a) (b)

Figure

3. Stochastic completion fields (128 \Theta 128 pixels, 36
orientations) with the induction e \Gamma2r e \Gamma20E spln . (a) F
. The results closely
resemble those obtained in [19].
a filter may indicate the existence of fiber-like shapes in the
image, in which case its response represents the "fiberness
level" of the relevant subarea of the image. Below we use the
term "straight responses" to refer to the responses obtained
by convolving the image with an edge or a fiber filter.
Consider now the edge elements obtained by convolving
the image with a filter of some fixed length l and width w.
Every edge element now is positioned at a certain pixel P
and is oriented in two opposite directed orientations Y and
-. The number of edge elements required to faithfully
represent the image at this scale depends on l and
Thus, long and thin elements require finer resolution
in orientation than square elements. In fact, the orientational
resolution required to sample significantly different
orientations increases linearly with l=w (see [4]). Similarly,
elements of larger size require less spatial resolution than
elements of smaller size. Brandt and Dym ([4]) use these
observations in order to introduce a very efficient computation
(O(N log N), where N is the number of pixels in the
image) of all significantly different edge elements.
Given a particular scale determined by the length l and
width w of edge elements, we would like to compute a completion
field for this scale. Note that only curves within
a relevant range of curvature radii can arouse significant
responses for our l \Theta w elements. Denote the smallest curvature
radius that will arouse a still significant response by
curvature radii will arouse significant
responses also in larger l=w scales, implying there for
a farther-reaching and more orientation-specific continua-
tion.)
By Fig. 4(a) we see that
Consequently, we have (l=2ae) 2 -
implying that ae - l 2 =8w.
Next, consider a pair of straight responses. Assuming
these elements are roughly cocircular, then, using the relations
defined in Fig. 4(b) , the differential relation Y 0
1=ae(s) can be approximated by (Y
so that W - r=ae.
Hence, for completion at a particular scale (l; w), it is reasonable
to define for every pair of points P 1 and P 2 a scale for
the turning angle W given by r=ae(l; w). That is, in the scale
(l,w) we define the completion energy between the pair of
straight responses so as to depend on the scaled turning angle
Wae=r. Since is straightforward to show
that 0:5W -
reasonable definition
for the scaled angular energy, therefore, is a monotonically
decreasing function of (ae=r)
a
(a)
rcosa
r
Y
F 1Y
F
r
(b)
Y
r

Figure

4. (a) The relation between l,w, and the curvature radius
ae. (b) The turn W that a moving particle takes in its way between
two straight responses, characterized each by a planar location and
an orientation.
Obviously, in any given scale of straight responses,(l; w),
for every F 1 and F 2 , the induction of P 1 upon P 2 should
decrease with an increase of r=ae. Hence, we define the field
induced by an element e 1 of length l and width w at location
directed orientation Y 1 on a similar element e 2 at
where u 1 denotes the strength of response at e 1 ,
some appropriate function of this response, and
ae
r
F d and F t (the distance and turning attenuation functions,
are smoothly decreasing dimensionless functions
that should be determined by further considerations
and experience. Thus, our summation kernel is a product
of the orientational and the spatial components involved in
completing a curve between e 1 and e 2 . As we shall see
below, this definition has many computational advantages.
Let fu i g denote the set of straight responses for a given
scale (l; w), where each u i is associated with two directed
edges
-). The total
field induced at any element e
is expressed by

\Theta -
The total field induced at -
is given by
G (l;w)
\Theta

\Theta
Since in general the responses obtained by convolving the
image with edge filters are bi-directional we may want to
combine these two fields into one. This can be done in
various ways. The simplest way is to take the sum fv
as the completion field. Another possibility, in the spirit of
[19], is to take the product fv j -
as the completion field.
Note that the field of a long straight response should
be very different (farther-reaching and more orientation-
specific) than the sum of the fields of shorter elements
composing it, and should strongly depend on its width (see
Fig. 5). This suggests that for a comprehensive completion
process one must practice a multiscale process, performing
a separate completion within each scale. The scaled induction
field (10)-(11), avoids a fundamental difficulty of
non-scaled fields like [7, 19, 16]. The latter exhibit so weak
a completion for far elements, that it would be completely
masked out by local noise and foreign local features.103050(a)103050(b)

Figure

5. Induction fields (200 \Theta 200 pixels) in different scales
using F d
(a) The induction field of one
long directed orientations. (b) The
sum of induction fields of the three shorter elements composing
this long element, each consist of: directed
orientations.
5.3. Fast multigrid summation of induction-fields
be the number of sites (P ), and
the number of orientations (Y) at each site,
that are required in order to describe all the l \Theta w straight
responses that are significantly different from each other.
It can be shown (see [4]) that if l and w are measured in
pixel units then, for any N-pixel picture,
so the total number of l \Theta w elements
is O(N=w). Hence, for any geometric sequence of scales
(e.g., l=1,2,4,., and w=1,3,9,.) the total number of straight
elements is O(N log N). It has been shown (in [4]) that all
the responses at all these elements can be calculated in only
O(N log N) operations, using a multiscale algorithm that
constructs longer-element responses from shorter ones.
At any given scale l \Theta w, it seems that the summations
(Eqs. (12) and (13)), summing over
for each value of would require a total
of operations (even though some of them
can be performed in parallel to each other, as in [20]).
However, using the smoothness properties of the particular
kernel (11), the summation can be reorganized in a
multiscale algorithm that totals only O(nm) operations
(and the number of unparallelizable steps grows only logarithmically
in nm). Indeed, the functions in (11) would
usually take on the typical form F d
. For such choices of the functions,
and practically for any other reasonable choice, the kernel
has the property
of "asymptotic smoothness." By this we mean that any q-
order derivative of G with respect to any of its six arguments
decays fast with r
\Theta
and the
higher q is the faster the decay is. Also, for any fixed r ij
(even the smallest, i.e., r O(l)), G is a very smooth
function of Y i and of Y j .
Due to the asymptotic smoothness, the total contribution
to v j (and -
elements far from P j is a smooth function
of need not be computed separately for
each j, but can be interpolated (q-order interpolation, with as
small an error as desired by using sufficiently high q) from
its values at a few representative points. For this and similar
reasons, multiscale algorithms, which split the summations
into various scales of farness (see details in [3]) can perform
all the summations in merely O(nm) operations.
6. Conclusion
Important problems in perceptual grouping are the detection
of smooth curves in images and their completion
over gaps. In this paper we have simplified the computation
involved in the process of completion, exploiting the
smoothness of the solution to the problem, and have defined
affinity measures for completion that take into a proper account
the scale of edge elements. In particular, we have
introduced new, closed-form approximations for the elas-
tica energy functional and presented a fast numeric method
to compute the curve of least energy. In this method the
error decreases exponentially with the number of discrete
elements. We then have used our approximations to define
an affinity measure which takes into account the width
and length of the edge elements by considering the range
of curvatures that can be detected with corresponding filters
of the same scale. Finally, we have shown that solutions
to the problem of finding the most likely completions in an
image can be implemented using a multigrid algorithm in
time that is linear in the number of discrete edge elements
in the image. This last observation applies also to recent
methods for completion and salience [7, 19]. In the future
we intend to use the multigrid algorithm to simultaneously
detect completions at different scales in order to combine
these completions into a single saliency map.



--R

"Completion energies and scale,"
"Shape Encoding and Subjective Contours,"
"Multilevel computations of integral transforms and particle interactions with oscillatory kernels,"
"Fast computation of multiple line integrals,"
"On Minimal Energy Trajec- tories,"
"The Role of Illusory Contours in Visual Segmentation,"
"Inferring Global Perceptual Contours from Local Features,"
"A Computational Model of Neural Contour Processing: Figure-Ground Segregation and Illusory Contours,"
"The Curve of Least Energy,"
"Organization in Vision,"
"Improve- ment in visual sensitivity by changes in local context: Parallel studies in human observers and in V1 of alert monkeys,"
"Elastica and Computer Vision,"
"Handbook of applied mathematics - Second Edition,"
"Shape Completion,"
"Structural Saliency: The Detection of Globally Salient Structures Using a Locally Connected Network,"
"Analytic Solution of Stochastic Completion Fields,"
"Filling-In the Gaps: The Shape of Subjective Contours and a Model for Their Generation,"
"3D Shape Representation by Contours,"
"Stochastic Completion Fields: A Neural Model of Illusory Contour Shape and Salience,"
"Local Parallel Computation of Stochastic Completion Fields,"
--TR

--CTR
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Song Wang , Joachim S. Stahl , Adam Bailey , Michael Dropps, Global Detection of Salient Convex Boundaries, International Journal of Computer Vision, v.71 n.3, p.337-359, March     2007
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
Xiaofeng Ren , Charless C. Fowlkes , Jitendra Malik, Learning Probabilistic Models for Contour Completion in Natural Images, International Journal of Computer Vision, v.77 n.1-3, p.47-63, May       2008
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
elastica curve;multiscale;induction field;completion field;perceptual grouping;fast summation;scale;least-energy curve;curve completion;curve saliency