SPATIAL RANK AND APPROXIMATE SYMMETRIES IN

SEQUENTIAL RECONSTRUCTION OF DENSE PACKINGS

Alexander Vinogradov

Dorodnitsyn Computing Center of Russian Academy of Sciences, Moscow, Russia

Keywords: Hough transform, spatial shape, Parzen window, symmetry group, sequential trials, ergodicity, k-point

configuration, group manifold, spatial isomer.

Abstract: General rotation group manifold is used as a base structure for representation of k-point configuration

clusters in Hough-type parametric space. This yields to introduce efficiently spatial ranks inside k-point trial

set and arrange in multiple dimensions Parzen-like windows with properties analogous to the linear ones’.

As a result, asymptotically optimal dense packings of clusters are automatically produced for arbitrary

spatial shapes via independent sequential trials.

1 INTRODUCTION

Hough transform is widely used as a powerful

technique in various IP applications. (Hough P.V.C.,

1962). Recently it appears in many forms,

generalized versions, and specialized hard and soft

implements (Chen L.1 et al, 2004, Daries C.J. and

Nixon M.S., 1998, Monari J. et al, 2006). Essential

reduction of dimensionality as main advantage is

achieved on this way when more or less complicated

spatial shapes are transformed to point-wise clusters

in appropriate parametric space (Leavers V.F., 1992,

Torii A. et al, 2005, Zhang S.-C., Liu Z.-Q., 2005).

Single point is really good representation in

many respects, but it implies a sort of indifference to

all movements inside the shape and usually ignores

initial feature distribution on it, in particular. In this

paper we develop an approach to preserve this

information and propose a method for its efficient

usage in the case when the set of all inside-shape

movements forms a group structure. We describe an

application of the approach to one of the typical

clustering tasks.

2 GROUP STRUCTURES OF

HOUGH TRANSFORM

In its simplest authentic form the Hough transform

already uses some group properties of spatial data

(Figure 1). Really, population of any Hough cluster

representing a line L is indifferent with respect to

operations of one-parametric translation group of

this line T

(L). Similarly, clusters of double Hough

transform, representing concentric rings, are

indifferent with respect to operations of one-

parametric central rotation group SO

. In the case of

generalized 4-parametric Hough transform an

indifference of this type takes place on spheres with

respect to SO

-movements produced by three

independent generators, and so on.

−

Image

Hough(Image)

Hough(Hough(Image))

−

Image

Hough(Image)

Hough(Hough(Image))

−

Figure 1: Authentic Hough and double Hough transforms.

An important object appears after first step of

double Hough transform in the second example, i.e.,

a line clusters resulting from rings of initial Image

plane. Any line cluster of this type preserves empiric

density distribution of detected line details of the

ring, but the distribution is represented now on the

211

Vinogradov A. (2007).

SPATIAL RANK AND APPROXIMATE SYMMETRIES IN SEQUENTIAL RECONSTRUCTION OF DENSE PACKINGS.

In Proceedings of the Second International Conference on Computer Vision Theory and Applications, pages 211-214

DOI: 10.5220/0002071102110214

 SciTePress

angle

scale. Moreover, since zero rotation angle is

defined and thus fixes the unit of Lie group SO

, this

scale turns out as corresponding one-dimensional

group manifold itself. The subject matter of the

paper is an implementation of the idea that initial

density distribution of characteristic features

revealed on symmetric spatial shape could be

preserved on group manifold in a form of

frequencies, with which elements of the group are

used in the transform. Below, we describe details of

the formal construction along with investigation of

some special task of spatial clustering, where

representation of such type is essentially needful.

3 DENSE PACKINGS

We consider a class of dense packings with

coefficient 1, in which the shapes of elements may

change while all the elements have the same volume.

Examples in dimensions 2 and 3 are given by the

decomposition of asymmetric region into compact

domains of equal volume and the arrangement of

elastic reservoirs with identical filling in a bounded

volume of a space. For different ways of defining

admissible shapes of elements (constraints on the

linear dimensions and surface area, elasticity,

internal potentials; etc.), the analysis of the variants

of dense packings turns out to be related to the

solution of complex optimization problems. The

goal is to find ways to reduce the computational

complexity of problems of this type by using the

extremal properties of the process of sequential

random choice in which a sample

RX ⊂

is tested

by small sub-samples. Below, we describe the

application of a special sequential trial scheme in

which the problems of enumeration of close-to-

optimal variants of the arrangement of clusters,

finding approximate symmetries, as own discrete

symmetries of packings as hidden internal

symmetries of the domain X, and sequential filtering

of optimal solutions and exact symmetries among

them are considered from a unified point of view.

Suppose that X

⊂ R

is a bounded domain, the

space (X,σ,μ) is an a priori distribution in X, and F is

a functional that defines the type of a K-cluster

packing O={O(x

), k=1,…,K} in X,

F(O) → max (1)

For nondegenerate distributions (X,σ,μ) with

density p

, a pair (X,F) defines a certain set of

variants of optimal packing, i.e., a certain set of

solutions to problem (1) of the form

O* = argmax F(O) = {O* (x*

)}, (2)

∪

O(x

)=X.

If it is uniquely specified how the centers are

ranked, then each set (2) defines a point in the space

that describes the arrangement of the centers of

optimal clusters in R

∗

=(x*

1,…,

)

∈

. (3)

We will refer to the sets χ

∈R

as configurations

and the sequential acts of choosing configurations as

trials.

We will seek a solution to the problem of

enumerating various kinds of packing in the class of

algorithms requiring constant resources for

computation in all trials. The example is the

computation of centers as cluster averages, where

the means can be permanently refined by the same

recurrence formula

+x =(M+1)

1+M

. (4)

Each portion of M

elements extracted from X

considered as the general population reflects the

form of the distribution (X,σ,μ). In one-dimensional

case M

+1 linear blocks known as “Parzen

windows” (Parzen E., 1962) are widely used in

nonparametric density estimations. Applying certain

specialization of F for finite sets, one can translate

this approximation into a K-point representation

∈

)) whatever the volume of the portion M

If we follow the model chosen, then, in order to

combine particular solutions of the form χ

into the

summarized result, we should apply the same

procedure in all trials carried out in a fixed space of

memory, which includes an array for storing this

summarized result. Here, we apply the following

standard procedure:

(a) choose a sufficiently large number of

initial trials M

;

(b) construct an appropriate set Y

) of

particular configurations of the form χ

∈Y

);

) and single out from R

a base set of clusters and the corresponding set

) of their central elements χ*

∈

) by

using certain functional F

in R

;

(d) fill the base clusters with the results of

further trials so that the next configuration χ is

related to the nearest center of the base cluster as a

template by a certain measure of proximity or the

metric ρ(χ, χ’). To avoid pathological situations, we

assume that the proximity is consistent with the

usual Euclidean metric: ρ(χ, χ’)→0, as ||χ- χ’||→0 in

;

(e) the solution being refined will

correspond to the filling levels of clusters as

neighborhoods of the central elements χ*.

There is extremely efficient implementation of

outlined scheme in one dimension that is based on

good asymptotic behavior of rank statistics.

Temporal means for ranks are normalized sums of

IID values and thus are consistent estimates for

VISAPP 2007 - International Conference on Computer Vision Theory and Applications

212

bounding points of Parzen windows. This estimates

can be calculated by the formula (4); their a priory

distributions are rapidly normalized in sequential

trials; and corresponding variance decreases with the

rate D=1/O(M) as M→∞. So, the sample X

composes “by itself” the optimal packing via always

the same process and regardless to the forms of the

sample X and initial distribution (X,σ,μ).

In multiple dimensions the scheme looses many

of these attractive properties. The main reason is the

luck of immediate analogues of the ranks in view of

the absence of a certain linear ordering in the sample

space (Vinogradov A.P., 2001). If elements of the

problem posing are insensitive or invariant with

respect to the action of some symmetry group of the

sample, regardless exact or approximate, it’s

impossible to establish a doubtless correspondence

between ranks in different trials, and thus it’s a

problem to arrange the addition (4) correctly. In the

same time, if symmetry G is exact, i.e. probability

space (X,σ,μ) is G-invariant, any action of g

∈

results in a metric isomorphism between measurable

spaces (X,σ,μ) and g(X,σ,μ), and we can use fiducial

equivalents of ranks. So, we can postulate as basic

some G-invariant method of ranking configurations

in accordance with (1) and thus factorize the entire

sequential scheme by the action of G. In the case of

approximate symmetry the same is valid for some

equivariant conditional distribution on X.

4 GENERALIZED HOUGH

PARAMETERIZATION FOR

CENTRAL SYMMETRIES

Here we take into consideration only the simplest

linear groups of central symmetry. To efficiently

look for exact and approximate symmetries we use a

special parametric space S of Hough type built of

|Y*

)| copies of SO

manifold, where centers

∈

) of configuration clusters (c) are used as

templates of ranking (Figure 2). This type of

parameterization is applicable in more general case

too, when N exceeds 3. If spatial isomers are treated

as separate templates in each case when N>2, then

the whole sequential scheme is automatically

factorized also with respect to the Weyl group of R

It was shown that for any particular symmetry

group G

⊂ SO

the orbit of each base configuration

fills a manifold of an adjacent G

-class (Vinogradov

A.P., Voracek J., Zhuravlev Yu.I., 2006). Absolutely

asymmetric solutions appear as salient domains in S.

If an essential approximate symmetry takes place,

the filling of the adjacent class must be continuous;

for exact symmetry it must be either uniform. For a

ranking template j

∈

) and a number m

of its

realizations in trials performed the population of the

-orbit represents instant sample distribution that

approximates associated fiducial distribution on the

scale of parameter g

∈

. In particular, Figure 2

represents the scale space for G

-orbit sample being

refined in trials, where number of realizations m

(of

symmetry G

for ranking template j) serves as semi-

group parameter

U(Σm

,p(m

))=U(m

max

,…,U(m

,p(m

)…). (5)

g G

∈

p(m,K

)

g G

max

g G

∈

p(m,K

)

g G

max

Figure 2: Hough-type parameterization, left. A scale space

for samples refined on the G

-orbit manifold K

, right. K

depicts some dotted manifold of a discrete symmetry. 0

points the SO

unit.

One can watch the process of filling orbits in S

and test their continuity or uniformity. The process

of filling corresponding clusters chosen on the step

(e) in configuration space R

still obeys the IID

condition mentioned above. It was shown, that

configuration clusters in R

can be processed with

basic set-theoretical operations without the lost of

this condition.

In the same time, if the size of the portion M

small, then the deviations of particular solutions

from the optimal one may be significant; the clusters

may be hardly separable; and the set Y*

), which

represents the entire set of noticeable variants, may

be too large. It is impossible to change the situation

by increasing the parameter M

, although the

primary structure of similarity (b), which is strongly

determined by the size of the portion M

, will be

displayed with increasing accuracy as M→∞.

The situation can be changed by increasing the

parameter M

: in more representative samples,

foreign configurations will more rarely be chosen as

the centers χ*

∈

). In particular, approximate

symmetries and other particular solutions, which

represent the local extrema of F, will have less

pronounced representatives in the primary structure

of similarity in R

. Nevertheless, the ergodicity of

the scheme is preserved; namely, the following

assertion holds: whatever the parameter M

and the

approximate symmetry G with maximal support X*

in X, if μ(X*

)>0, then, for any finite M

and M→∞,

the configuration of the symmetry G fills its own

SPATIAL RANK AND APPROXIMATE SYMMETRIES IN SEQUENTIAL RECONSTRUCTION OF DENSE

PACKINGS

213

cluster in R

with nonzero probability. Moreover,

for any approximate (exact) symmetry G the filling

of the G-orbit manifold in S becomes strongly

continuous (respectively, strongly uniform) as

M→∞, whatever the corresponding configuration

template χ* was chosen on the step (c).

5 CONCLUSIONS

We have considered above only the most important

questions related to the application of a sequential

scheme to the construction of optimal packings and

to the determination of symmetries on the base of

some group-theoretical generalization of the Hough

transform. As usual, direct solution of such problems

requires complex optimization computations. The

key point of the approach proposed consists in the

application, instead of the laborious direct solution,

of simple algorithm, that uses this generalization

essentially, during a long enough period of time.

We have introduced a certain parameterization

of Hough type that is able to preserve information

about feature distribution on spatial shapes in the

case of central symmetries, as exact as approximate.

On this base an invariant definition was done for

spatial ranks, which are used in multi-dimensional

sequential scheme in a manner analogous to efficient

usage of boundary points of Parzen windows in

nonparametric density estimation in one dimension.

Some special structure S was constructed, in which

exact and approximate symmetries of arbitrary

spatial shape are revealed along with sequential

reconstruction of optimal K-cluster packings. All

symmetries are sought for via the generalized Hough

transform adjusted to certain types of adjacent

classes on the SO

group manifold equipped with

indices j of ranking templates. It was shown that

optimal solutions can be efficiently filtered out due

to good asymptotic behavior of the independent

sequential choice procedure.

The introduction of additional structures into the

sequential scheme that enhance the handling of

detected approximate and exact symmetries could

serve as a development of the approach. Certain

variants of this kind appear due to the presence of

natural scale-spaces associated with maximal filters

of approximate symmetries in the standard subgroup

lattice of SO

. The construction of a priori estimates

that describe the dynamics of filling these scale

spaces by the realizations of approximate and exact

symmetries seems to be very important. These

estimates would enable one to obtain an integral idea

about the subordination between symmetries, about

the difference between the approximate and exact

solutions found, and about the number of trials

necessary to reliably filter out exact solutions and

exact symmetries from among them.

ACKNOWLEDGEMENTS

This work was supported by the Russian Foundation

for Basic Research (projects 05-01-00332, 05-07-

90333, 06-01-00492, and 06-01-08-045).

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