Page 1
Completion Energies and Scale
Eitan Sharon, Achi Brandt
Dept. of Applied Math
The Weizmann Inst. of Science
Rehovot, 76100, Israel
?, Ronen Basri
y
Abstract
The detection of smooth curves in images and their com-
pletion over gaps are two important problems in perceptual
grouping. In this paper we examine the notion of com-
pletion energy and introduce a fast method to compute the
most likely completions in images. Specifically, we develop
two novel analytic approximations to the curve of least en-
ergy. In addition, we introduce a fast numerical method to
compute the curve of least energy, and show that our ap-
proximations are obtained at early stages of this numerical
computation. We then use our newly developed energies to
find the most likely completions in images through a gener-
alized summation of induction fields. Since in practice edge
elements are obtained by applying filters of certain widths
and lengths to the image, we adjust our computationto take
these parameters into account. Finally, we show that, due
to the smoothness of the kernel of summation, the process
of summing induction fields can be run in time that is linear
in the number of different edge elements in the image, or in
O
using multigrid methods.
?N log
N? where
N is the number of pixels in the image,
1. Introduction
The smooth completion of fragmented curve segments
is a skill of the human visual system that has been demon-
stratedthroughmanycompellingexamples. Duetothisskill
people often are able to perceive the boundaries of objects
eveninthelackofsufficientcontrastorinthepresenceofoc-
clusions. Anumberofcomputationalstudieshaveaddressed
the problem of curve completion in an attempt to both pro-
vide a computational theory of the problem and as part of a
processofextractingthesmoothcurvesfromimages. These
studies commonly obtain two or more edge elements (also
referredto as edgels)andfindeitherthe most likelycomple-
tionsthatconnecttheelementsorthesmoothestcurvestrav-
?Research supported in part by Israel Ministry of Science Grant 4135-
1-93 and by the Gauss Minerva Center for Scientific Computation.
yResearch supported in part by the Unites States-Israel Binational Sci-
ence Foundation, Grant No. 94-00100.
elingthroughthem. The methodsproposedforthisproblem
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 exist-
ing approaches. Below we present such a method that di-
rectlyrelatestoanumberofrecentstudiesofcompletionand
curvesalience [9, 18, 5, 19, 12, 7] (see also [2, 6, 8, 14, 15]).
Along with simplifying the computationsproposed 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 de-
termining the smoothest completion between pairs of edge
elements [17, 14, 2, 9, 18, 5]. 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
introducedin [18, 5]. While the definitionof scale-invariant
elastica is intuitive, there exists no simple analytic expres-
sionto calculateits shapeoritsenergy,andexistingnumeri-
cal computationsare orders-of-magnitudetoo 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
Page 2
anglesformedby the edgelsandthe 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 com-
pletion and salience as follows. Given
thespaceofallcurvesconnectingpairsofelementsisexam-
ined 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 val-
ues induced by the fixed element on the rest of the image
is obtained. The system finds the most likely completions
for the
summation of the induction fields for all
In the second part of this paper we use our newly devel-
oped completion energies to define an affinity measure that
encourages smoothness and penalizes for gap length. We
thenuse the inductionfieldsdefinedby thisaffinity measure
to solve the problem of finding the most likely completions
for
dimensionless, because they are usually obtained by apply-
ing 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
ouraffinitymeasureis asymptoticallysmooth,andso can be
implementedusing multigrid methods and run efficiently in
time complexity
and
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.
M edge elements,
M elements by applying a process that includes a
M elements.
M elements. Since in practice edge elements are never
O?nm? (where
n is the number of pixels
m is the numberof discrete orientationsat every pixel).
2. Elastica
Considertwo edgeelements
e1
?e2positionedat
P1
?P2
?
R2with directed orientations Ψ1and Ψ2respectively mea-
sured from the right-hand side of the line passing through
P1and
that Ψ1
may conveniently assume that
This is illustrated in Fig. 1(a).
P2. Below we shall confine ourselves to the case
?Ψ2
???
?
2
?
?
2
?. Denote by
r?kP2
?P1
k, we
P1
?
Let
?0 ?0 ? and
P2
??r?0 ?.
C12denote the set of
C2curves through
e1and
e2. Denote such a curve by its
orientation representation Ψ ?s?, where 0
arclength along the curve. That is,
and
the curve at
?s?L is the
x?s??
R
s
0cos
?Ψ ?ˆ
s ??dˆ
s
y?s??
R
s
0sin ?Ψ ?ˆ
s ??dˆ
s. Also denote the curvature of
s by
??s??dΨ ?s??ds.
r
(a)
Ψ
p
ΦΦ
y
L
x
21
1
f(x)
1
Ψ
2
2
p
Ψ
1
Φ
Φ
θ
Ψ
1
2
2
1θ
=Φ2
Φ1
2
−Ψ1+ =Ψ2
P
L
r
(b)
P
Figure 1. (a) The planar relation between two edge ele-
ments,
Φ1
2222
measured from the line
(b) The more general relation between Φ
?P1
?Ψ1
? and
?P2
?Ψ2
?. This relation is governed by
???
?
?
?
?, Φ2
???
?
?
?
?, and
r, where Φ1and Φ2are
P1
P2, hence Φ1
? Ψ1and Φ2
??Ψ2.
iand Ψ
i.
The most common functional used to determine the
smoothestcurvetravelingthrough
orientations Ψ1and Ψ2is the elastica functional. Namely,
the smoothest curve through
which minimizes the functional Γ
Elastica was already introduced by Euler. It was first ap-
plied 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
the distance between the two input elements, the energy
of the curve connecting them proportionately decreases,
as can be easily seen by rescaling
somewhat counter-intuitive since psychophysical and neu-
robiological 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
yieldcirculararcstocompletecocircularelements. Tosolve
theseproblemsWeiss[18,5]proposedtomodifytheelastica
model to make it scale invariant. His functional is defined
as Γ
0
adjustment of the completion energy to scale must take into
account not only the length of the curve (or equivalentlythe
distance between the input elements), but also the dimen-
sionsoftheinputedgeelements. Boththeelasticafunctional
anditsscaleinvariantversionassumethattheinputelements
P1and
P2withrespective
e1and
e2is the curve Ψ ?s?
el
?Ψ ?
def
?
R
L
0
?2
?s?ds?
r,
s (cf. [1]).This is
inv
?Ψ ?
def
?L
R
L
?2
?s?ds? We believe that a proper
Page 3
have no dimensions. In practice, however, edge elements
are frequentlyobtainedby convolvingthe 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 corre-
spondingscalingin thewidthandlengthofthe elementshas
taken place. Below we first develop useful approximations
tothe scale invariantfunctional. (Theseapproximationscan
alsoreadilybeusedwithslightmodificationstotheclassical
elasticameasure.) Later,inSection5,wedevelopanaffinity
measure between elements that also takes into account both
the distance between the elements and their dimensions.
3. Analytic simplification of Γ
inv
Althoughthedefinitionofboththeclassical andthescale
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 approxima-
tions to these functionals. Our first approximation is valid
when the sum of angles
assumptionrepresentstheintuitionthatinmostpsychophys-
ical demonstrations gap completion is perceived when the
orientationsofthecurveportionstobe completedare nearly
collinear. Withthisassumptionwemayalsorestrictfornow
the range of applicable orientations to Ψ1
The second approximationwill only assume that
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 Ψ1
Consequently,it canbedescribedasafunction
inFig.1(a). Expressingthecurvatureintermsof
obtainthat Γ
0
For
of
Γ
jΦ1
j?jΦ2
j is relatively small. This
?Ψ2
???
?
2
?
?
2
?.
jΦ1
? Φ2
j
?Ψ2the smoothest curve will not wind much.
Ψ ?C12Γ
y?f?x?, as
xand
y we
inv
?Ψ ??L
R
L
?2
?s?ds?L
R
r
0
?f
??
?2
?1??f
?
?2
?
5
2
dx?
?Φ1
?Φ2
?? 0 we get that
L?r, and that the variation
?1
??f
?
?2
? becomes unimportant for the comparison of
inv
?Ψ ? over different curves Ψ
?C12, so that Γ
inv
?Ψ ??
r
R
r
0
?f
??
?2
dx? Hence
E
inv
def
? min
inv
?Ψ ??r min
Ψ ?C12
Z
r
0
?f
??
?2
dx?
(1)
The minimizing curve is the appropriate cubic Hermite
spline (see [1])
f?x??x?x?r?
?1
r2
?t1
?t2
?x?
t1
r
?
?
(2)
where
t1
? tanΦ1and
t2
? tanΦ2, so that
E
inv
? 4 ?t2
1
?t2
2
?t1
t2
??
(3)
Evidently, this simple approximation to
independent. This leads us to define the scale-invariant
E
inv is scale-
spline completion energy as:
Although the spline energy provides a good
approximation to the scale invariant elastica measure for
small values of
values. An alternative approximation to
structed by noticing that for such small values tanΦ1
and tanΦ2
E
spln
?Φ1
?Φ2
?
def
? 4 ?t2
1
?
t2
2
?t1
t2
??
jΦ1
j?jΦ2
j, the measure diverges for large
E
invcan be con-
? Φ1
? Φ2. Thus, we may define:
E
ang
?Φ1
?Φ2
?
def
? 4 ?Φ2
1
? Φ2
2
? Φ1Φ2
??
(4)
We refer to this functional as the scale-invariant angular
completion energy. This measure does not diverge for large
values of
obtain
that this angular energy is obtained in an early stage of the
numericcomputationof
accurate approximations to the scale invariant least energy
functional even for relatively large values of
pecially for small
cocircular elements. Using the numeric computation we
can also derive the smoothest curve according to
jΦ1
j?jΦ2
j. In fact, when Φ1
? Φ2
???2 we
E
ang
?E
inv
??2. In Section 4 below we show
E
inv,andthatitprovidesextremely
jΦ1
?
j+jΦ2
? 2. That
j, es-
jΦ1
? Φ2
j, i.e., for the range of nearly
E
ang:
¯Ψ ?s?? 3 ?Ψ1
? Ψ2
?s2
??4Ψ1
? 2Ψ2
?s? Ψ1
?
(5)
The angular completion energy can be generalized as
follows:
E
gang
?Φ1
?Φ2
??a?Φ2
1
? Φ2
2
??b?Φ1
? Φ2
?2
?
(6)
where Eq. (4) is identical to Eq. (6) with
is, the angular completion energy is made of an equal sum
of two penalties. One is for the squared difference between
Φ1and Φ2, and the other is for the growth in each of them.
This suggests a possible generalization of
weights
energies such as
more elaborate study of these types of energies and their
properties is presented in [1].
Finally, we note that the new approximations at small
anglescan also be used to approximatethe classical elastica
energy, since
essary condition for˜Ψ ?˜
should satisfy for some
2˜Ψ
ab
E
ang to other
a? 0 and
b? 0. One can also think of using
E
circ
?Φ1
?Φ2
?
def
?aΦ2
1
?b?Φ1
? Φ2
?2. A
E
el
def
? min
Ψ ?C12Γ
el
?Ψ ??1
r
E
ang
?1
r
E
spln
?
(7)
4. Computation of
We use the scale-invariance property of Γ
reformulatethe minimizationproblemintominimizingover
all
Euler-Lagrangeequations(see, e.g., [13]) we get that a nec-
E
inv
invin order to
C12curves of length
L? 1. (See also [5]). Applying
s? to be an extremal curve is that it
?:
??
??cos˜Ψ
s?t?
(8)
Z1
0
sin ?˜Ψ ?˜
s??d˜
s? 0 ?
˜Ψ ?0 ?? Ψ1
?
˜Ψ ?1 ?? Ψ2
?
Page 4
Considering the very nature of the original minimization
problem, and also by repeatedly differentiating both sides
oftheODEequation,itcanbeshownthatitssolutionmustbe
very smooth. Hence, we can well approximate the solution
by a polynomial of the form
˜Ψ
n
?s???1
?s?Ψ1
?sΨ2
?s?1
?s?
n
X
k?0
a
k
s
k
?
(9)
where
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 “
decreases exponentially in the number of discrete variables
n is small. (By comparison, the discretization of the
?-order”, i.e., the error
n+2 ( i.e.,
??a0
? ????a
n from Eq. (8) and (9) ).) Fixing
n, as well as two other integers ¯
followingsystem of
n and
p, we will build the
n?2equationsforthe
n?2unknowns
a0
?a1
? ????a
n
? and?
˜Ψ
??
n
?
i? 1
n? 2
???
ncos˜Ψ
n
?
i? 1
n? 2
?? 0
??0
?i?n?
collocating the ODE, and
where
merical integration.
and increase ¯
the discretization error will not be governed by the dis-
cretization error of the integration. The nonlinearsystem of
P¯
n
j?0
w
jsin˜Ψ
n
?j?¯
n?? 0,
w
j
?0
?j? ¯
n? are the weights of a
Generally, we increase
p-order nu-
n gradually
n and p as functions of
n in such a way that
n
Newton-Raphson; see, e.g. [13].) We start the Newton iter-
ationsfroma solutionpreviouslyobtainedfora system with
a lower
for each value of
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
smoothnessofthesolutionfortheODE,alreadyforthesim-
ple
? 2 equations is solved by Newton iterations (also called
( p
n. Actually, only one Newton iteration is needed
values of ( n,¯
by simple analytic expressions, as indeed we show in [1] by
comparingbetweenseveralsimpleapproximationsto
Fig. 2 illustrates some of the completionsobtained using
n if
n is not incremented too fast. In
?n? 0 ? ¯
n? 2 ?-system andtheSimpsonintegrationrule
? 3), a very good approximationto the accurate solution
˜Ψ ?s??
[1].
? and also to
The good approximations obtained already for small
E
inv is obtained, as can be seen in
n) suggest that
E
invcan be well approximated
E
inv.
E
It can be seen that the differences between the three curves
is barely noticeable, except in large angles where
diverges. Notice especially the close agreement between
thecurveobtainedwiththeangularenergy(Eq.(5))andthat
obtained with the scale-invariant elastica measure even in
large angles and when the angles deviate significantly from
cocircularity.
invand the two analytic approximations
E
angand
E
spln.
E
spln
Note that although the spline curve does not approxi-
mate the scale invariant elastica curve for large angles
and
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
followthetangentto thetwo elementsat bothends(see, e.g,
Fig. 2(b)) . This behavior is desirable especially when the
elements represent long curve segments (see Section 5.2).
jΦ1
j
jΦ2
j it still produces a reasonable completion for the
5. Completion field summation
Until now we have consideredthe problemof findingthe
smoothest completion between pairs of edge elements. A
natural generalization of this problem is, given an image
from which
likelycompletionsconnectingpairsofelementsintheimage
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.
Themeasuresencourageproximityandsmoothnessofcom-
pletion. Using the affinity measures the affinities induced
by an element overall 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
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 el-
ements are dimensionless. In practice, however, edge ele-
ments 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
thetwoelementsiftheelementsthemselvesarescaledbythe
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 computationallyintensive. 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 summa-
tionkernelsin[19,16,7],andsoanefficientimplementation
of these methods can be obtained with a similar multigrid
algorithm.
M edge elements are extracted, find the most
M elements.
5.1. Non-scaled induction
In[12,19]amodelforcomputingthelikelihoodsofcurve
completions, referred to as Stochastic Completion Fields,
Page 5
0 0.2 0.4 0.6 0.81
0
0.05
0.1
(a)
0 0.2 0.40.6 0.81
0
0.2
0.4
0.6
0.8
1
(b)
Figure 2. Completion curves: elastica in solid line,¯Ψ?s? (Eq. (5)) in dotted line, and the cubic Hermite spline (Eq. (2)) in dashed line. (a)
Φ1
? 30
?
?Φ2
? 15
?, (b) Φ1
? 80
?
?Φ2
? 20
?.
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 function1. To compute the stochastic
completionfieldsaprocessofsummingtheaffinitymeasures
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:
Fig. 1(b) ), where
parameters. These parameters need to be adjusted properly
according to the scale involved (see Sec. 5.2). Note that
for small values of (
A?e1
?e2
?
def
?e
?r ?r0
e
?E
ang
??r?0
?(see
r0and
?0are strictly positive a-priori set
jΦ1
j,jΦ2
j):
E
ang
?r?E
el. Hence,
A?e1
?e2
??e
?r ?r0
e
?E
el
??0.
Another method which uses summation of induction
fieldstocomputethesalienceofcurveswaspresentedin[7].
In their method the affinity between two edge elements
whicharecocircularhastheform:
are strictly positiveconstants,
connecting
e
??r
e
???, where
? and
?
? is the curvatureof the circle
e1and
e2, and
r is the distance between
e1and
e2. A reasonableand straightforwarddefinitionin that spirit
is˜
proximation for
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.
A?e1
?e2
?
def
?e
??r
e
??E
splnwhere
E
splnserves as an ap-
E
inv according to Eq. (3). Fig. 3 shows
A?e1
?e2
?.
5.2. Induction and scale
Givenanimage,anedgeelementisproducedbyselecting
a filter of a certain length
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 and width
w (e.g., rectangular
?l?w? scale. Similarly,
1Actually, the path minimizes the energy functional
for some predetermined constant
R
L
0
?2
?s?ds? ?L
?.
(a) (b)
Figure 3. Stochastic completion fields (128
orientations) with the induction
and Φ2
resemble those obtained in [19].
? 128 pixels, 36
e
?2r
e
?20E
spln. (a) Φ1
? 30
?
? 30
?, (b) Φ1
? 30
?and Φ2
??30
?. The results closely
a filter may indicate the existence of fiber-like shapes in the
image, in which case its response represents the “fiberness
level”oftherelevantsubareaoftheimage. Belowweusethe
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
Every edge element now is positioned at a certain pixel
and is oriented in two opposite directed orientations Ψ and
Ψ
fully represent the image at this scale depends on
l and width
w.
P
??. The number of edge elements required to faith-
tion ( O
image) of all significantly different edge elements.
Given a particular scale determined by the length
width
pletion field for this scale. Note that only curves within
a relevant range of curvature radii can arouse significant
responses for our
vature radius that will arouse a still significant response by
l and
w. Thus, long and thin elements require finer resolution
in orientation than square elements. In fact, the orienta-
tional resolution required to sample significantly different
orientationsincreases linearlywith
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 computa-
l ?w (see [4]). Similarly,
?N log
N?, where
N is the number of pixels in the
l and
w of edgeelements, we wouldlike to computea com-
l?w elements. Denote the smallest cur-
?
cant responses also in larger
a farther-reaching and more orientation-specific continua-
tion.)
By Fig. 4(a) we see that
???l?w?. (Larger curvature radii will arouse signifi-
l ?w scales, implying there for
l?2
??sin
?? ?? and
w?
???cos
????2
?2. Consequently, we have
?l?2 ??2
?
Page 6
?2
( l,w) we define the completion energy between the pair of
straight responses so as to depend on the scaled turning an-
? 2 w
Next, consider a pair of straight responses. Assuming
these elements are roughly cocircular, then, using the rela-
tions defined in Fig. 4(b) , the differential relation Ψ
pΦ2
??, implying that
??l2
?8 w.
?
?s??
1 ???s?canbeapproximatedby
so that Ω
Hence,forcompletionataparticularscale
sonabletodefineforeverypairofpoints
the turning angle Ω given by
?Ψ1
?Ψ2
??r? 2 ???1
??2
?,
?r ??.
?l?w?,itisrea-
P1and
P2ascalefor
r ???l?w?. That is, in the scale
gleΩ??r. SinceΩ
? Φ1
?Φ2, it isstraightforwardto show
that0 ?5Ω
for the scaled angular energy, therefore, is a monotonically
decreasing function of
?
p
E
ang
?Φ1
?Φ2
?? Ω. A reasonabledefinition
???r?
1
? Φ2
2
? Φ1Φ2.
l
w
α
(a)
ρcosα
ρ
+
2
Ω =Ψ1−
−Φ11
= Ψ
Ψ
Φ2
Φ1
2
Ψ
2
Φ
r
(b)
2= Φ1+ Φ2
Ψ
Ψ1
Ω
ρ
ρ1
2
Figure 4. (a) The relation between
l, w,and the curvature radius
?. (b) The turn Ω that a moving particle takes in its way between
two straight responses, characterized each by a planar location and
an orientation.
Obviously,inanygivenscaleofstraightresponses,
for every Φ1and Φ2, the induction of
decreasewith an increaseof
inducedbyan element
?l?w?,
P1upon
P2should
r ??. Hence, we define the field
e1of length
l andwidth
w at location
P1and directed orientation Ψ1on a similar element
e2at
?P2
?Ψ2
? by
G
?l?w?
?e1;
e2
?f?u1
??
(10)
where
some appropriate function of this response, and
u1denotes the strength of response at
e1,
f?u1
? is
G
?l?w?
?e1;
e2
??F
d
?
r
?
?
F
t
?
?
r
q
Φ2
1
? Φ2
2
? Φ1Φ2
?
?
(11)
F
respectively) are smoothly decreasing dimensionless func-
tions 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
below, this definition has many computational advantages.
Let
scale
edges
dand
F
t(the distance and turning attenuation functions,
e1 and
e2. As we shall see
fu
i
g denote the set of straight responses for a given
?l?w?, where each
u
iis associated with two directed
e
i
??P
i
?Ψ
i
? and ¯
e
i
??P
i
?Ψ
i
???. The total
field induced at any element
e
j
??P
j
?Ψ
j
? by all elements
fe
i
? ¯
e
i
g is expressed by
v
j
def
?
X
i
?
G
?l?w?
?
e
i;
e
j
?
?G
?l?w?
?¯
e
i;
e
j
??
f?u
i
?? (12)
The total field induced at ¯
e
jby
fe
i
? ¯
e
i
g is given by
¯
v
j
def
?
X
i
?
G
?l?w?
?
e
i; ¯
e
j
?
?G
?l?w?
?¯
e
i; ¯
e
j
??
f?u
i
?? (13)
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
variousways. Thesimplestwayisto takethesum
as the completion field. Another possibility, in the spirit of
[19], is to take the product
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 in-
duction 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.
fv
j
? ¯
v
j
g
fv
j¯
v
j
g as the completion field.
0
10
20
30
40
50
60
(a)
0
10
20
30
40
50
60
(b)
Figure 5. Induction fields (200
using
?200 pixels) in different scales
F
d
????e
??1
?
?1,and
F
t
????e
??2
?
?2,where
?1
? 0 ?5,
?1=0.5,
long element:
sum of induction fields of the three shorter elements composing
this long element, each consist of:
orientations.
?2
? 128, and
?2
? 1 ?5. (a) The induction field of one
l? 9,
w? 1 ?2, 25 directed orientations. (b) The
l? 3,
w? 1 ?2, 12 directed
5.3. Fast multigrid summation of induction-fields
Let
n?n?l?w? be the number of sites (
P), and
m
that are required in order to describe all the
responses that are significantly different from each other.
It can be shown (see [4]) that if
pixel units then, for any N-pixel picture,
and
?m?l?w? the number of orientations (Ψ) at each site,
l?w straight
l and
w are measured in
n?O?N ?lw?
m?O?lN ?w?, so the total number of
l?w elements
Page 7
is
(e.g.,
elements is
the responses at all these elements can be calculated in only
O?N ?w?. Hence, for any geometric sequence of scales
l=1,2,4,...,and
w=1,3,9,...) thetotalnumberofstraight
O?N log
N?. It has been shown (in [4]) that all
O
constructs longer-element responses from shorter ones.
At any given scale
tions (Eqs. (12) and (13)), summing over
for each value of
tal of
can be performed in parallel to each other, as in [20]).
However, using the smoothness properties of the partic-
ular kernel (11), the summation can be reorganized in a
multiscale algorithm that totals only
(and the number of unparallelizable steps grows only log-
arithmically in
usually take on the typical form
For such choices of the functions,
and practically for any other reasonable choice, the kernel
?N log
N? operations, using a multiscale algorithm that
l?w, it seems that the summa-
i? 1 ?2 ?????nm
j? 1 ?2 ?????nm, would require a to-
O?n2
m2
? operations (even though some of them
O?nm? operations
nm). Indeed, the functions in (11) would
F
d
????e
??1
?
?1, and
F
t
????e
??2
?
?2.
G
of “asymptotic smoothness.” By this we mean that any
orderderivativeof
decays fast with
higher
(even the smallest, i.e.,
function of Ψ
Due to the asymptotic smoothness, the total contribution
to
of
eachj,butcanbeinterpolated(
small an error as desired by using sufficiently high
its valuesat a few representativepoints. Forthis and similar
reasons, multiscale algorithms, which split the summations
intovariousscalesof farness(see detailsin [3])canperform
all the summations in merely
6. Conclusion
?G?e
i;
e
j
??G?x
i
?y
i
?Ψ
i;
x
j
?y
j
?Ψ
j
? has the property
q-
G withrespecttoanyofitssixarguments
r
ij
?
?
?x
i
?x
j
?2
??y
i
?y
j
?2
?1
2, and the
q is the faster the decay is. Also, for any fixed
r
ij
r
ij
?
O
O
?nm? operations.
?l?), G is a very smooth
iand of Ψ
j.
v
j(and ¯
v
j)ofallelementsfarfrom
P
jisasmoothfunction
?x
j
?y
j
?Ψ
j
?, henceitneednotbecomputedseparatelyfor
q-orderinterpolation,withas
q) from
Important problems in perceptual grouping are the de-
tection of smooth curves in images and their completion
over gaps. In this paper we have simplified the computa-
tion involved in the process of completion, exploiting the
smoothnessof the solution to the problem,and have defined
affinity measures for completion that take into a proper ac-
count 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 de-
fine an affinity measure which takes into account the width
and length of the edge elements by considering the range
of curvaturesthat can be detected with correspondingfilters
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.
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