PATTERN RECOGNITION FEATURE AND IMAGE

PROCESSING THEORY ON THE BASIS OF STOCHASTIC

GEOMETRY

Nikolay G. Fedotov, Lyudmila A. Shulga, Alexander V. Moiseev, Andrey S. Kol’chugin

Department of Economical Cybernetics, Penza State University, 40, Krasnaya St., Penza, 440026, Russia

Keywords: Pattern recognition, Image understanding, Stochastic geometry, Trace-transform, Triple features.

Abstract: Application of stochastic geometry methods to pattern recognition is analysed. The paper is based on Trace-

transformations of original images into images on the Möbius band. Based on the new geometric

transformation, a new approach towards the construction of features, independent of images’ motions or

their linear transformations, is put forward. A prominent characteristics of the group of features under

consideration is that we can represent each of them as a consecutive composition of three functionals. The

paper considers the application of three-functional structure of recognition feature to image pre-processing.

Feature can be invariant or sensitive to the group of all motions transformation and linear deformation of

objects depending of functionals selection. Thus sensitive features are suitable to determine the parameters

of translation. It is an important task for robotics.

1 INTRODUCTION

In the field of pattern recognition we traditionally

distinguish feature construction and decision

procedure. In literature on cybernetics a vast

majority of works on the pattern recognition have

been historically devoted to decision rules, there

actually being no works on feature construction.

There has been general agreement that it could be

explained by the fact that the process of constructing

features is empirical and dependent on the intuition

of the recognition system designer.

The approach of stochastic geometry, developed

in (Fedotov, 1990), allows us to bridge the gap and

create processes to generate great many new features

for image recognition, along with a constitutive

theory of features. Such a prominent shift of stress

from decision procedures to new recognition

features gives the approach a strong resemblance to

neuro-computing.

In (Fedotov, 1990), the author suggests using

probabilities of geometrical events understood as the

result of geometrical objects interaction

(intersections, overlapping and so on), as image

recognition features. Geometrical objects here are,

on the one hand, complex scanning trajectories with

random parameters (segments, lines, curves, figures,

etc.), and on the other hand, fragments of an image

being recognized. The structure of similar

recognition systems and examples of particular

technical implementations, are considered. Possible

extensions of the fundamental recognition process

on stochastic geometry are considered as well. One

of the extensions deals with a complication of

observing a random event (an intersection of a scan

trace and an image), i.e. with the application of more

complicated recognition features. The article

presents the basics of a new theory to construct

recognition features based on stochastic geometry

and functional analysis joint application. The key

element of this theory is new geometric Trace-

transform concerned with scanning of image by

complicated trajectories (Fedotov, 1990) (Fedotov et

al., 1995).

Trace-transform can be both the base for the new

class of recognition features – the triple ones – and

the effective tool of image preprocessing.

This work is supported by Human Capital Foundation

This work is supported by the Russian Foundation for Basic Research (RFBR), project N 05-01-00991

187

G. Fedotov N., A. Shulga L., V. Moiseev A. and S. Kol’chugin A. (2005).

PATTERN RECOGNITION FEATURE AND IMAGE PROCESSING THEORY ON THE BASIS OF STOCHASTIC GEOMETRY.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Robotics and Automation, pages 187-192

DOI: 10.5220/0001160601870192

 SciTePress

2 TRACE-TRANSFORM

Let

denote a finite image. Given scanning

straight line

characterizing the location of

and the image as to each other, is to be computed

according to a certain rule

; map

is called a functional. The property required for us

here is the computational independence of an object

motion. Therefore the only requirement imposed on

is stated as follows. Let the image have been

shifted and rotated, and

T ),( Flg T=

′

be the new one. For the

same shift and rotation,

will become a straight line

, thus remaining, "frozen" into the image. It is

required that

′

),(),( FlFl

′

=T T

. The equality is to

hold for all straight lines and all acceptable images.

We may call the property a complete invariance of

functional

. It should be noted that the concept of

complete invariance extends pattern recognition

capabilities substantially, for it is not necessarily be

the number of intersections, intersection length, etc.

Functionals can be selected to describe finer

properties of neighborhood, such as neighborhood

morphology, or topological characteristics. For a

full-color image of a variable brightness a great

number of such functionals could be selected.

Hence, the range of functionals and images to be

processed widens considerably.

Just like in stochastic geometry, random value

is defined, its distribution being

independent of image shifts and rotations. Therefore,

numeric characteristics of the random value may

again serve as image features, which are to be

established with the help of special engineering

devices and systems. The limitation of the new

family of features is that they originally lack an

explicit geometrical meaning, and their

differentiating capability is a priori unknown.

However for pattern recognition, it proves not very

important, experimental testing being decisive.

),( Flg T=

To understand that the generalization proposed

in a certain aspect exhaust its own possibilities, we

are going to state the theory of Trace-transform.

Polar coordinates introduced to the plane,

characterized by distance

from the origin to l ,

and by angle

(up to 2

) of its directional vector:

}sincos:),{(

pyxyxl

),(

pll

, where

, are Cartesian coordinates on the plane. If we

allow

to take negative values, too, then

),(),(

plpl

−

Thus, a set of all directed straight lines

intersecting a circle of radius

with the center in

the origin (the "retina"), is unambiguously

parameterized by set

},0:),{( RpRp ≤

≤

−

≤

, which

provided parameters

and

),0( p ),( p−

define the

same straight line. The set of straight lines on the

retina are clearly seen to be topologically nothing

but a Möbius band. Thus, the set of numbers

)),,(( Fpl

T , depending on a point on Möbius band

, is a certain image transform, which we may call

a Trace-transform. If, for instance, a matrix

represents a Trace-transform in numerical analysis,

we may call it a Trace-matrix. If axis

is directed

horizontally, and axis

vertically, matrix element,

indicated

, i.e. value

),(

)),,(( Fpl

, is in point

, .

and are here certain values of

uniform discrete grids on the axes mentioned. Along

the horizontal axis, matrix is 2

-periodic, its

columns rotating within each interval of length

In addition, let us consider, that if

does not

intersect the image,

is a given number (say,

),( FlT

Figure 1: Example of calculation a binary function ),,( tpf

for given image a scanning line l and corresponding

Trace-Transform

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0), or another fixed element, if is nonnumeric. In

this case, a new image

corresponds to the

original image

)(

)),,(((

Fpl

may be treated as

an image which characteristics at

),(

are its

Trace-image.

Figure 1,a explains the computation of a Trace -

transform. It shows how to obtain a binary

function

),,( tpf

of a real variable for a scanning

line

. Function

),,(

•

equals 1 within the

interval

and

Within other precise it

equals 0. Let

stand for a functional applied to the

function, its independent variable being designated

by . Thus we get

),(

),,(),( tpfpg

T= . We call

function

result of Trace–transform.

For a two–level image, such a Trace–transform

could be obtained in case

),,( •pf

is the total of

all the intervals from the domain of the function to

be defined. For Figure 1,a it is the value of

−

and

segments’ total. If we determine a

similar

tt −

),,( •pf

for an aggregate of scanning

lines intersecting the image of a image at various

angles

and various distances

, we can get its

Trace–image shown in Figure 1,b.

Let

),,(

tpf

be maximum interval within

function

),,(

•

domain. In Figure 1,a it is the

value of

−

(max G). This functional get us

another Trace-image.

Figure 2: Nonlinear filtration by means of Trace-transform

Note that the famous Radon transform (Helgason

, 1980) can be viewed as an example of a Trace–

transform. Brightness integral value summed over

all scan lines for all directions, is Radon transform.

In case

— brightness integral

value

∫

∞

∞−

=Τ dttff

)(

θθ

),(

along a scan line with parameters

),(

, the collection

]2,0[,},{

∈∈Τ Rpf

bears all information on the image.

In the context of two-level images, Radon

transform will lie in summing up blackened

elements along each scan line. Hence, choosing an

appropriate functional, we can make Trace-

transform realize Radon transform. This is a

particular case, though. In other particular cases one

can both make Trace–transform match other well-

known geometric transforms — Fourier, Hough, etc.

— or outstep (Fedotov et al., 2000) (Fedotov et al.,

2005).

Property of Trace-transform. Trace-transform

proves convenient to study objects’ movements and

similarity transform within the retina.

Let us briefly consider how image

)(FT

Figure 3: The experiment on a doubly-connected and single-connected images and corresponding Trace-matrices

PATTERN RECOGNITION FEATURE AND IMAGE PROCESSING THEORY ON THE BASIS OF STOCHASTIC

GEOMETRY

189

changes after initial image

has been shifted and

rotated. If the original image rotates, its Trace-image

shifts along the horizontal axis

. If the original

image shifts to a certain vector, its Trace-image

undergoes changes as follows. For convenience they

are to be stated in terms of Trace-matrices. Columns

remain unchanged and stay in their places, though

may shift up or down. Shift vector specifies numbers

and

such that a column with coordinate a

shifts vertically to

)cos(

−

. It should be noted

that the description is to be totally rigorous if only

we consider Trace-matrix continuous, i.e.,

and

are continuous parameters.

3 IMAGE PREPROCESSING

BASED ON TRACE-

TRANSFORM

3.1 Nonlinear filtration

Through a Trace-transform, one can implement

nonlinear filtration of image to reduce its noise and

quantization.

Figure 2,a shows the original image, while

Figure 2,b shows the same image distorted by

additive noise. If we try to obtain their Trace-

transforms using the functionals considered above

we get different Trace-matrices, the latter containing

information both about original image and the noise.

For getting information only about the original

image we propose to use more complicated

T-functionals. For example, correspondent

T-functional (1) let us to get information about

original image excepting noise information:

⎩

⎨

⎧

0if ,lg10

0if , 0

ijij

(1)

where

are number of intersections scanning line

of given image.

The images and its Trace-transforms obtained by

functional (1) depict in Figure 2. One can see that

pure (Figure 2,a) and noisy (Figure 2, b) images

have identical Trace-matrices.

Figure 4: The result of applying Diametrical functional to a Trace–matrix and the calculation of triple feature by

means of Circus functional

This approach was successfully applied in

diagnostics of welded joints and pores. Through the

Trace-transform, we get a matrix of an original

dimension, which elements almost unexceptionally

differ from zero if they correspond to a pore.

Other types of T-functional have been found to

replace certain types of image pre-processing, such

as inversion, and filling closures.

3.2 Evaluation of objects’ relative

position and number

Let us consider a valuable feature of Trace-

transforms to tackle certain problems of image

analysis normally solved structurally: through

segmentation, or evaluation of objects’ relative

position and number. Figure 3,b shows an object to

be viewed as having a single-connected, and its

Trace-matrix, Figures 3,a showing segmented

images and their Trace-matrices.

The example, which proves structuralistic

features of Trace-transform, applies to the field of

technical diagnostics. A problem of recognizing

welded joints’ pores was being solved based on the

results of their radiographic inspection. Figure 3

shows one of the defections typical for welded joints

— spherical pores. Instructions on radiographic

inspection request that chains of pores be

differentiated from single pores and other defections,

such as slag inclusions, incomplete penetration, etc.

Moreover, this type of defection is suggested to

be differentiated according to the relative position of

pores within the chain, as well as to the number, and

the size (the diameter) of pores.

It should be noted that one could enhance non-

linear filtration applying the filtering capabilities of

both the Trace-functional and other functional of

triple structure described below.

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4 TRIPLE FEATURES

The most valuable feature of the Trace-transform is

that it proves a source of a new class of recognition

features which have a triple structure (triple features)

(Fedotov, 1990) (Fedotov et al., 1995).

Analysis of great many formulas of stochastic

geometry, as well as well-known geometric

transformations — those of Fourier, Chough, Radon

— demonstrate that they can be represented as a

composition of three functionals. From this

conclusion, the authors construct recognition

features as a composition of three functionals. Their

recognition power matters when solving

classification problems.

Let us consider formation of triple features

which are a consecutive composition of three

functionals:

)),,(()( tpLFF

ooo TPΘ=Π

. (2)

Each functional

, and effects the

function of one variable

Ρ Τ

and

correspondingly. For each of the three functionals it

is easy to think out dozens of various concretizations

which comply with the conditions required. Hence,

we may at once get thousands of new features

invariant to motions.

Functional

, corresponding to a Trace-

transformation, has been above considered in detail.

In a discrete variant of computation the result of the

transformation, or the Trace-transform,

)),,(( tplF

oΤ

is a matrix, which elements are, say,

values of brightness parameter for image

at the

intersections with the scanning line

),( pl

Parameters of the scanning line

specify the

position of the element within the matrix.

Computation of feature to follow involves a

consecutive processing of the matrix columns with

the help of functional

, which we call diametrical.

Functional “Norm”, a standard Euclidean norm

∫

=Ρ dpphh )(

has been used as functional

other instances of diametrical functionals applied

may be the functional called “Max”, which is the

maximum value of the function in a Trace–matrix

column; and the “Mid” functional. It is a standard

middle point computed through

)(

dpph

dppph

∫

=Ρ

(3)

The result of applying

(“Norm”) functional to

a Trace–matrix (Figure 1,b) is a 2

- periodic curve

shown in Figure 4.

Next stage is to perform transformations on the

curve with the help of

functional, which we call

a circus within

. The “Log” functional has been

used as a variant of

functional, being computed

through

∫

+=Θ .1)(ln

θθ

dhh

(4)

The triplex recognition features considered may

be computed through a highly parallel process. Like

features formed by neuron nets, the given features

have no pre-assigned meanings, their selection being

realized during a machine experiment, considering

their being useful for classification only.

5 FUNCTIONALS INVARIANT

AND SENSITIVE TO THE

AFFINE TRANSFORMATIONS

One can obtain dozens of features using three-

functional structure. But it is a fact that features

useful for practical application can be classified into

two types: invariant to affine transformation of an

image and sensitive to them.

Let us provide more strict definitions of the

invariant and sensitive functionals.

We call functional

invariant if

))(( ubxu

for all

R∈

We call functional

sensitive if

))(( bubxu

−

o for all .

Rb ∈

The following theorem is proven: if functional

,, are invariant, then feature does not

depend upon the group of all motions.

)(FΠ

Therefore, we can rapidly and successfully find

identical images in the sequence of images

regardless of their affine transformations with the

help of invariant functionals (for example, when we

investigate the structure of micro objects, we

determine the structure elements regardless of their

location or scaling up the microscope).

We use sensitive functionals to compute the

coordinates of object motions (it can be helpful in

machine vision system for exact positioning of robot

tools). Efficiency of designed algorithms and

methods was proved by means of experiments in the

field of nanotechnology and in technical and

medicine diagnostic systems designing.

Thus, triple feature theory allows us to create

new features of two types: those are invariant to

affine transformation for given images (such as

translation, rotation and scale transforms) and their

linear deformations, and those are sensitive to affine

transformations.

PATTERN RECOGNITION FEATURE AND IMAGE PROCESSING THEORY ON THE BASIS OF STOCHASTIC

GEOMETRY

191

One can easily propose tens of different specific

realizations of each of the three functionals which

satisfy the required conditions. Hence, without much

effort, we get thousands of new features which are

invariant or sensitive to image motion. This only

proves the value of the considered theory for the

problems of pattern recognition with multiple

structure of the classes, like the problem of the

recognition of hieroglyphs or textures. In the

following, some functionals are presented which we

use to solve different problems and which are

implemented on a PC.

a) Invariant functionals:

1. Maximum value of the function.

2. The number of extrema of the function.

3. Total variation of the function.

4. Standard Euclidean norm of the function.

5. Any functional determined by the distribution

of the values of the function.

b) Sensitive functionals:

1. Standard center of gravity of the masses

calculated by formula (3).

2. The phase of the second Fourier harmonics of

the function.

3. Absolute value of the Fourier coefficient of

the third harmonics.

4. Probability-theoretic median.

6 CONCLUSION

Trace-transform has been considered, the latter

applying all-direction scanning.

The most valuable property of the Trace-

transform is that it establishes the new class of

recognition features with a triple structure (triple

features).

It is possible to form great many features (in fact,

thousands of them), as it needs to recognize patterns

with a great number of classes. A vast number of

features helps to understand image better and to

increase recognition flexibility and fidelity.

Besides, it has been established that the theory

based on triple features yields a stable result when

analyzing biological micro and nano objects, and, in

particular, when recognizing images in the field of

nanotechnology. Results have been obtained in

successful recognition of leucocytes and

erythrocytes, which саn be found in (Fedotov et al.,

2000). Moreover, the theory of triple features stated

above has been tested on and proved suitability for

technical diagnostics and flaw detection problems.

Applicability of the given ideas has been

experimentally proved in (Fedotov et al., 2002).

The theory created allows us to obtain features

independent of object motions or linear

deformations. Additionally, we can use the theory to

obtain features which depend on the transformations

mentioned above in a simple way giving us a

possibility to compute the parameters of motions and

motions or linear deformations transformed objects

(it can be helpful in machine vision system for exact

positioning of robot tools).

A composition of three functionals is applicable

at the same time to construct recognition features

and to perform non-linear filtration, to reduce noise

within the image and to segment the objects, to

smooth the image etc. It should be noted that one

could enhance non-linear filtration applying the

filtering capabilities of both the Trace-functional and

another functional of triple structure. Having the

opportunity to construct features and to perform

image pre-processing simultaneously and by the

same technique, one can make recognition system

faster, pre-processing and feature construction being

realized at the same stage of the scanning.

REFERENCES

Fedotov N.G., 1990. Metody Stokhasticheskoi Geometrii

v Raspoznavanii Obrazov (Methods of Stochastic

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Fedotov N.G., Kadyrov A.A., 1995. Image scanning in

machine vision leads to new understanding of image.

In Proc. of 5th International Workshop on Digital In

Processing and Computer Graphics, Proc. International

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pp. 256-261.

Helgason S., 1980. The Radon Transform, Birkhauser,

Basel and Boston, Massachusetts.

Fedotov N.G., Shulga L.A., 2000. New Theory of Pattern

Recognition on the Basis of Stochastic Geometry. In

WSCG'2000 Conference Proceedings, Vol. 1, р. 373-

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