Automatyczne rozpoznawanie obiektów z obrazów dyfrakcyjnych Automatic Target Recognition from Diffraction Patterns

Tadeusz Niedziela

Warsaw University of Technology, Faculty of Transport

AUTOMATIC TARGET RECOGNITION FROM

DIFFRACTION PATTERNS

The manuscript delivered, April 2013

Abstract: The paper presents the method of automatic diffraction pattern recognition. The proposed method is based on Fourier transform properties. It creates a possibility of bringing the problem to Fourier transform, extraction of characteristic feature vectors and classification. The method has been illustrated by a simple difference algorithm of the of ring-based ordering of diffraction patterns. Keywords: diffraction patterns, classification, artificial neural networks

1. INTRODUCTION

Automatic Target Recognition (ATR) from diffraction patterns constitutes a significant current research problem aiming at creation of simple and efficient systems that would give measurable practical advantages [1-14]. Classic algorithms of automatic recognition encompass two stages: extraction of feature vector from a real environment, in which a high level of noise can exist, as well as classification or identification. An important role among such systems is played by optoelectronic systems.

Current methods of pattern recognition can be generally divided into electronic, optical and hybrid [3].

Characteristic features of the target can be extracted directly from the image or from its transform. Whether the features are obtained from the image or from its transform depends on the type of the image. Images of a complex internal structure (texture) are more easily recognized within the plane of their transforms, whereas images of a poorer structure are easier to recognize within the image plane (directly from the image). A proper construction of a feature vector allows for a significant reduction of information amount, which is subject to further processing, and accelerates recognition process. Identicalness or similarity of recognized patterns can be concluded on the basis of the difference of feature vectors, both within image plane and within the transform one.

Target recognition in the present paper is understood as recognition of their diffraction patterns. The method is based on Fourier transform properties, sampling of diffraction patterns by feature extractor and on classification on the basis of statistical uncertainty

model. Characteristic features obtained from diffraction pattern constitute input data for the classifier.

2. SYSTEM OF AUTOMATIC TARGET RECOGNITION

FROM DIFFRACTION PATTERNS

Fig. 1 illustrates the idea of optoelectronic (hybrid) pattern recognition system.

Fig.1. Schematic diagram of pattern recognition within diffraction plane

The system is based on two main blocks: a feature extractor in the form of a ring-wedge detector (RWD) and a classifier in the form of artificial neural network (ANN). Input image transform to the Fourier plane is realized by a simple optical system (lens) and thus it is performed immediately (without any time delay). Feature extraction from a diffraction pattern is realized by proper sampling of Fourier spectrum (RWD). The advantage of this solution is a significant compression of feature vector dimensionality, and thus the data which is subject to further processing. Since the noise in the patterns is a sum of elementary noises caused by independent physical phenomena and has Gaussian distribution, it seems that classification should be realized on the basis of statistical uncertainty model [4]. In view of the above, neural networks can be efficiently used as classifiers, because after proper training, at the output they generate conditional probability

P(Ck/x) that the observed target is of class Ck if at the network input, feature vector x was

given. Therefore, classification on the basis of the above conditional probability is easy to

P1 P2

fL

Coherent light

Input plane L Fourier plane F (u, v)

f (x, y)

Optical part Electronic part Feature vector Digital classifier k1 k2 kn Recognized abstraction class

realize in practice on the basis of a decision criterion of recognition minimizing average losses caused by wrong classification.

Data included in diffraction pattern, as well as in the real image are identical, however presented in different forms. When speaking of the real image, we are using such notions as pixel, its coordinates (x, y) and intensity g(x, y). Whereas, Fourier pattern G(f_x ,f_y) is

described by such notions as image structures, their spatial frequencies (f_{x ,fy} expressed in lines/mm) and directions of these structures. Data included in Fourier pattern is synthetic information, because it describes not a single pixel, but their structures called structures. When using Fourier transform, we should remember that it is invariant because of translations, but it depends on the change of scale and rotation.

Fourier transform can be realized in an optical, digital or electronic way. Optical methods of realization of this transform are simple and currently find wider uses.

2.1. FEATURE EXTRACTOR IN THE FORM OF A

RING-WEDGE DETECTOR MATRIX (RWD)

The proposed feature extractor of diffraction patterns is a circular matrix of photodiodes, in one part of the half-plane in the form of half-rings, whereas in the other part of the half-plane in the form of wedges (fig.2).

The known Fraunhofer approximation of Fresnel – Kirchhoff integral realized by the lens can be expressed as (6):



_{, ,}



iE

0

i

z

2

2



_,



i

2

z

x

y

F

z

e

f x y e

dxdy

z

S

_{[ K}

S

_{[ K}

O

O

[ K

_O

§ · _{§ ·} ¨ ¸ ¨ ¸ ¨ ¸ _{© ¹} © ¹





f f



³ ³

ff

(2.1.1)

which as a result gives Fourier transform of the input function f (x, y) together with multiplication of the result by spherical phase parameter. Since only intensity can be directly observed and registered in practical applications, therefore the parameter of spherical phase can be neglected. Intensity is Fourier spectral power F2(u, v) of input function transmittance f (x, y), which is referred to as Fraunhofer diffraction pattern and is observed from infinity to the back-focal-plane of a spherical lens.

Photodiodes integrate Fourier spectral power (F2) and convert it to the form of discrete electric signals, which are a reflection of the value of single elements of a feature vector of a diffraction pattern.

If Fraunhofer diffraction pattern (F) is expressed in polar coordinates (Z, I), then characteristic features corresponding with rings

R

_i and the features corresponding with wedges

W

_i are given by the expression:

(2.1.2)





1 2

_,

0

r

i

R

_i

F

d

d

r

_i

S

Z M Z M

§ · ¨ ¸ ¨ ¸ ¨ ¸ © ¹



³ ³

1 2



,



0

R i

W

F

d

d

i

i

M

Z M M Z

M

§ · ¨ ¸ ¨ ¸ ¨ ¸ © ¹



³ ³

where: r iand r i+1 indicate respectively internal and external radius of a half-ring RING (i), Ii and I i+1 indicate respectively initial and final angle of a wedge WEDGE (i),

(

1,...,

)

j

j

N

W

M

, R is the radius of a ring-wedge detector (RWD),

N

_R and

N

_W are

respectively a total number of rings and wedges.

The values obtained of characteristic features Ri are invariant with respect to shift and

rotation of an input image, but dependent on scale (increase of an input image by a times causes a times decrease of a spectral image). Whereas, characteristic feature values Wi are

invariant with respect to shift and scale change of an input image but dependent on rotation. Which invariance and which dependence within a feature vector is important in practice, constitutes a problem in itself.

As a result of integration, a characteristic feature Ri is the information about shift and

rotation, whereas, Wi is the information about shift and scale of an input image. Therefore,

the total feature vector obtained is invariant with respect to shift, rotation and change of scale of an input image.

Fig. 2. Schematic diagram of a ring-wedge detector (RWD)

If RWD (fig. 2) is placed in the back-focal-plane of the lens (referred to as Fourier plane), then the possibility arises of sampling of Fraunhofer diffraction patterns – sampling of spectral power, which is focalized on rings and wedges. Thanks to this, RWD becomes a generator of a feature vector of a diffraction pattern.

2.2. CLASSIFIER BASED ON ARTIFICIAL NEURAL

NETWORK

Noise in images is most often a sum of elementary noises caused by numerous independent phenomena and has Gaussian distribution, hence the classification should be realized on the basis of statistical uncertainty model. In this model, the expected conditional probability (a’posteriori) is used as a criterion for decision of minimizing average losses caused by wrong classification. If G indicates classification operations, then classification criterion is the following:

 

x

 

x

 

x

,

j

C

P

k

C

P

k

j

k

C

!

z



œ

G

(2.2.1)

where: x is a feature vector, Ck and Cj are respectively recognized classes k and j.

Multi-layer perceptors are very frequently used as classifiers, because after proper training at the input they realize yk approximation of conditional probability (a’posteriori) P (Ck | x) of appearing a given class Ck if at their input, a feature vector x is given. This is

the most desirable feature in recognition with the model of random uncertainty, because these probabilities are used in the decision rule (2.2.1). Here, as it can be easily seen, from a statistical classifier, a binary classifier can be obtained.

In the present paper, the statistical classifier is proposed as probabilistic neural network (PNN). Such networks, constituting nuclear estimators of probability density function, are built as a special kind of radial neural networks, dedicated to estimation of probability density function.

From the point of view of network architecture, probabilistic neural network is a three-layer network (fig. 3) without feedback. It consists of an input three-layer, a pattern three-layer and a summation layer. ... ...

... ...

...

...

Summation layer Pattern layer Input layer

Characteristic features of patterns generated by HRWD

N

M

Class of recognized pattern

Fig. 3. Probabilistic neural network, classifying characteristic features generated by optimized RWD

The input layer consists of N elements so as to process N-dimensional feature vectors generated by RWD (N = NR + NW) .

The layer of patterns consists of M groups of neurons of patterns, associated with M classes. In this layer, radial neurons with Gaussian shift function are proposed. Each neuron of the pattern layer is connected with each neuron of the input layer, and weight vectors of the pattern layer are equal to feature vectors used in a teaching set. In contrast with the pattern layer, the summation layer consisting of M neurons is organized in such a way that only one output neuron is connected with neurons from any group of patterns.

3. EXAMPLE. DIFFERENCE ALGORITHM OF RING

ORDERING OF DIFFRACTION PATTERNS

This algorithm compares the values of feature vectors of real images obtained from wedges with a feature vector of a pattern image. A pattern target characterizes of unique values and proper proportions of data in a feature vector form the wedges.

The real target belonging to the same class as the pattern target has a similar value of a feature vector, differing, however in its position, the level of particular components and distractions connected with image acquisition.

Let us define a feature vector of a pattern image as:

1 1

1

r = r , r , …,rn

₁

§¨_¨

_{1 2}

·¸_¸

© ¹ (3.1)

and a feature vector of a real image (amplitudes of signals for a recognized diffraction pattern):

2

2

2

r = r

_{,r , …,rn}

2

1

2

§ · ¨ ¸ © ¹ (3.2)

Then let us change the amplitude ordering and assume the sequence of amplitude ordering (from the highest to the lowest amplitude) for a diffraction image of the pattern image.

1

_,

1

_,

_,

1

1

1

2

s

§¨_¨

r

r

_rn

·¸_¸

©



¹ (3.3)

Analogically, the sequence of amplitude ordering (from the highest to the lowest amplitude) for the recognized diffraction image.

2

_,

2

_,

_,

2

2

1

2

s

§¨_¨

r

r

_rn

·¸_¸

©



¹ (3.4)

Let us define the measurement of similarity of two diffraction images as:

1

2

,

1

2

1

n

S s s

r

_w

r

_w

w

§ · ¨ ¸ © ¹

¦



(3.5)

where: w – wedge index

Minimum distance between the pattern vector and the tested one is:



1 2





1 2



ˆ _{, min @}

In order to find the minimum distance between vectors, one of the vectors should be rotated by one wedge per every measurement. This can be done by changing indexes of a feature vector of a real image. Each index indicating a subsequent vector position should be decreased by one:



1 1 1



1 1, 2, , n x m m m (3.7) 1 1 1 k k n k k n m otherwise k k Ÿ   ­ ® _ ¯ (3.8)

where: k – value of the index describing element number of the feature vector. Algorithm of the wedge-difference sum experimentally proved its quality in practice and can be successfully applied in many domains. The algorithm is convenient and efficient because it compares feature vectors. These vectors include much less data than diffraction patterns, therefore this method is very fast from the point of view of calculation.

Table 1

Sample values of a pattern (I) and a real (II) feature vector

Sample wedge detector

Wedge

Pattern (I) Real image (II)

Amplitude Order according to value Amplitude Order according to value 1 400 3 2000 1 2 50 6 150 5 3 1000 1 400 4 4 75 5 1000 2 5 200 4 800 3 7 500 2 100 6 Table 2

Algorithm of similarity searching between feature vectors

A L G O R I T H M I II ' Rotation of a feature vector of a real images by two

wedges I II ' High similarity 3 1 2 3 3 0 6 5 1 6 6 0 1 4 3 1 1 0 5 2 3 5 5 0 4 3 1 4 4 0 2 6 4 2 2 0 6 14 6 0

CONCLUSION

Optoelectronic system of image recognition of complex internal texture has been proposed. The system can be easily realized in a programmable version. Initial experiments conducted have indicated high efficiency of the system. A simple algorithm has been proposed of ring ordering for diffraction pattern recognition.

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AUTOMATYCZNE ROZPOZNAWANIE OBIEKTÓW Z OBRAZÓW DYFRAKCYJNYCH

Streszczenie: Referat prezentuje metod automatycznego rozpoznawania obrazów dyfrakcyjnych. Proponowana metoda, oparta na jest waciwociach transformaty Fouriera. Stwarza ona moliwo sprowadzenia problemu do transformaty Fouriera, ekstrakcji wektora cech charakterystycznych i klasyfikacji. Metod zilustrowano prostym algorytmem rónicy uporzdkowania piercieniowego obrazów dyfrakcyjnych Sowa kluczowe: obrazy dyfrakcyjne, klasyfikacja, sztuczne sieci neuronowe