Astigmatic holographic processor for local pattern recognition

Volume 65, number 5

ASTIGMATIC HOLOGRAPHIC

OPTICS COMMUNICATIONS

1 March 1988

PROCESSOR FOR LOCAL PATTERN RECOGNITION

H.J. RABAL, W.D. F U R L A N and M. G A R A V A G L I A Centro de Investigaciones Opticas, (CIOp) Casilla de Correo 124, 1900 La Plata (B.A.), Argentina

Received 20 June 1987; revised manuscript received 26 October 1987

A coherent processor for local correlation is described. This device can be used to unmask local resemblances between object transparencies and a stored matched filter, which would be hidden in global correlation, as that of a classical Vander Lugt filter. This is achieved by means of a holographic filter that is an image hologram in one direction and a lensless Fourier transform hologram in the other. The hologram is registered with a line reference beam in the conjugate plane of a cylindrical lens and is reconstructed by means of a spherocylindrical optical system.

1. Introduction Since 1963, an i m p o r t a n t tool for pattern recognition d e v e l o p e d by Vander Lugt [1], as is the m a t c h e d filter, has deserved considerable attention to m a n y researches. Though this m a t c h e d filter has desirable properties, such as the fact that it can be recorded optically and as a pure a m p l i t u d e filter, and nevertheless control both a m p l i t u d e a n d phase o f the processed signal, it has some shortcomings as a pattern recognition filter which render it unpractical for most applications. M a i n shortcomings are due to high sensitivity to rotation or scale size o f the i n p u t functions [2]. M a n y efforts have been d e v o t e d to desensitize the system a n d / o r multiplex the filter to account for different orientations or scales in the input [ 3-5 ]. Besides, correlation is p e r f o r m e d between the input function a n d the point spread function synthesized in the filter in a what could be called a whole field operation. We think that a m o r e localized o p e r a t i o n could be interesting to u n h i d d e n regions o f the input which resemble other ones o f the p o i n t spread function which would be m a s k e d in a global whole field operation. In this p a p e r we propose an astigmatic processor which processes i n d e p e n d e n t l y a n d in parallel, narrow strips o f the input function, in one direction, with corresponding strips o f the matched filter in the same direction. In this way, if in some regions the input resembles the image stored in the m a t c h e d filter, but

Fig. 1. Astigmatic holographic processor. P~, P2 and P3 are the input, the hologram and the output planes respectively. a p p r e c i a b l y differs in the other, the observed correlation will show it as a narrow line in the regions o f resemblance and will depict a spread in those o f difference. This correlation line corresponds to a 1D extension o f the correlation peak o b t a i n e d with the classical Vander Lught correlator.

2. Description The astigmatic holographic processor we propose is sketched in fig. 1. It consists m a i n l y o f two steps. In the first step, a hologram is recorded between the input plane P~ a n d P2 plane. This hologram is an image hologram in one direction ( x ) and a lensless

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OPTICS COMMUNICATIONS

Fourier transform hologram in the other (y). In the second step, a spherocylindrical system of lenses performs the reconstruction. This processor performs in a similar fashion to the Vander Lught adapted filter correlator in the y direction while in the x direction it samples both the object and the adapted filter so that in the output P3 the light field distribution corresponds to the convolution and cross-correlation between narrow bands of almost 1-D regions of object and filter. The amplitude distribution in P~ plane due to the propagation of both the object and reference beams is

1 March 1988

8t

b

b~(x, y) = U(x, y) + R(x, y) =A(x, y) • 6(x, Y-Yo) +exp(ix2/4gf~2) (5(x, Y+ Yo) ,

(1)

where A(x, y) is the object field amplitude, centered in Yo by the first Dirac delta function, and the second term represents a line reference source produced by the cylindrical lens L2.2 is the wavelength of the coherent light illuminating the object. The cylindrical lens L2 conjugates planes P Land P2 in one direction (y) and spreads the light in the other, almost as free propagation but for a phase factor. By taking into account the phase delays introduced by L2 and the paraxial Fresnel diffraction formula, the intensity distribution in P2 is found to be

I(x', y') = (1/1623f 3)

x I f [V(-X, yo) U*(-x, yo) +R( - x , yo) R*( - x , y,)

+R(-x,y{,) U'*( - x , yl ) + U( - x , Y0) R*( - x , y, )] × exp[( - i~z/42~)(yo - y ~ ) ]

×exp[(-izry'/42f2)(yo-yl)] dyo dy~ ,

(2)

f2 being the focal length of L2. It is apparent that if a high resolution plate is exposed to such intensity distribution under the usual conditions of linearity of the t-E curve, a hologram is obtained and the last two terms ofeq. (2) are those which contribute to the holographic reconstruction 344

Fig. 2. (a) Object; (b) Anamorphic reconstructed image obtained in plane P~ of fig. 1 with a line reference. when the hologram is illuminated in the same setup by only the reference line source.

3. Experimental results Fig. 2 shows an object and its reconstructed image obtained in plane P3 by means of the spherocylindrical lens systems shown in the right side of fig. 1. Such system cancels the astigmatism produced by the lens L~ ands conjugates planes P1 and P3 but for an eventual anamorphosis. The holograms recorded in this fashion can be used to perform convolution and correlation operations between images in only direction. In the other direction, the area on which these operations are performed is highly localized and determined mainly by the width of the point spread function associated with the pupil limiting L2. So, the comparison between corresponding almost 1-D regions (in the x-direction) of the object and the image stored in the hologram can be performed without this comparison being masked by the cross-correlation between noncorresponding areas. In such sense the processing can be thought as almost 1-D localized. Fig. 3 shows the result of a correlation experiment

Volume 65, number 5

OPTICS COMMUNICATIONS

1 March 1988

b

Fig. 3. (a) Correlation between two identical curves; (b) Correlation between two curves which are identical only in the rightmost part, between two pairs o f curves. In the first case the two c o m p a r e d curves are identical while in the second one, they are identical only in the rightmost half. The correlation function, in the first case resembles a reconstruction o f the line reference beam, while in the second this only can be seen in the region where two curves are identical. Fig. 4 shows the o u t p u t plane intensity distributions in a character recognition experiment. The m a t c h e d filter was registered with some ancient Hebrew characters. In fig. 4 ( a ) , those characters are c o m p a r e d with themselves. Three rows can be distinguished. The u p p e r m o s t corresponds to convolution, the m i d d l e one is an a n a m o r p h i c image o f the input signal and the lower row is the correlation between the input and the image stored in the m a t c h e d filter. It can be seen that the correlation is c o n t a i n e d in a bright line showing resemblance. The intensity distribution along this line is uneven due to the fact that the filter function is not normalized. This resuits in bright dots in those regions where the characters have lines paralell to the correlation direction.

In fig. 4 ( b ) the input differs from the filter in the third character, as can be seen c o m p a r i n g the m i d d l e rows o f fig. 4 ( a ) and 4 ( b ) . The correlation corresponding to the substituted character is lower, has m o r e spread a n d departs from the straight line. In both examples, conventional whole field operation should give a single point-like correlation peak with a certain spread where the i n f o r m a t i o n on local resemblances should be lost. The correlation peak should be smaller than that which should be obt a i n e d if the input function were identical to the function stored in the filter [2]. On the other h a n d as, in our processor, localization in the x-diffraction is preserved, only corresponding areas o f the matched filter a n d the input are correlated and therefore, careful x-direction positioning is required.

4. Conclusions We propose an astigmatic holographic processor which performs a kind o f localized convolution and 345

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l March 1988

Fig. 4. (a) Correlation between two identical sets of characters. (b) Correlation between two sets of characters which differ in the right side character. c r o s s - c o r r e l a t i o n . T h i s o p e r a t i o n keeps track o f local (in o n e d i r e c t i o n ) r e s e m b l a n c e s b e t w e e n corres p o n d i n g regions o f the i n p u t a n d the s p r e a d function. We h a v e s h o w n that the c o r r e l a t i o n p e a k that appears in a classical V a n d e r Lugt e x p e r i m e n t is ext e n d e d here to a c o r r e l a t i o n line. T h i s c o r r e l a t i o n d e g r a d e s in the regions w h e r e the object a n d the m a t c h e d filter differ, w h i l e m a i n t a i n i n g high v a l u e s w h e r e b o t h f u n c t i o n s are identical. Also t w o e x a m ples are s h o w n o f an e x p e r i m e n t in c h a r a c t e r recognition. We e x p e c t that this k i n d o f p r o c e s s o r can also be used w i t h c o m p u t e r g e n e r a t e d d e t o u r phase holog r a m s as m a t c h e d filters [6].

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Acknowledgements T h i s w o r k was s u p p o r t e d w i t h a C O N I C E T g r a n t P I D 0812.

References [ 1] A. Vander Lugt, IEEE Trans. Inform. Theory IT-10 (1964) 139. [2] J.W. Goodman, Introduction to Fourier optics (New York, Mc. Graw-Hill, 1968) chap. IV. [3] D. Casasent and D. Psaltis, Appl. Optics 15 (1976) 1795. [4] T. Szoplik, J. Opt. Soc. Am. A 2 (1985) 1419. [ 5 ] T. Szoplik and H. Arsenault, Appl. Optics 24 (1985) 3179. [ 61 A.W. Lohnmann and D.P. Paris, Appl. Optics 7 ( 1968 ) 651.