Position, rotation, and scale invariant optical correlation David Casasent and Demetri Psaltis with the conventional A new optical transformation that combines geometrical coordinate transformations to both scale and rota- optical Fourier transform is described. The resultant transformations are invariant optical pattern recognition tional changes in the input object or function. Extensions of these operations to and initial experimental demonstrations are also presented. schemes do not have the Introduction with a digital computer. Such real-time parallel processing advantages of an optical Considerable attention has recently been focused storage requirements. possible processor and have extensive data on extending the flexibility of the operations only coherent optical implementations of has involved various Therefore, in an optical processor. This work to these problems will be considered both to preprocessing of imagery solutions geometrical transformations, enhance the practicality of these processors and because screens,2 nonlinear operations using optical by halftone 4 5 of their higher potential processing capacity. feedback,3 and Mellin transforms, among others. The Mellin transform has been shown to be quite useful Transformations since in the correlation of imagery that differs in scale, of overcoming these scale and ro- paper, a new Several methods this transform is scale invariant. In this have been considered. One approach to both a scale and a tational problems transformation that is invariant involves the storage of many multiplexed holographic rotational change in the input image is discussed. scale changes and imple- spatial filters of the object at various Experimental demonstrations of the optical theoretically feasible, Its use various rotational angles. While mentation of this transformation are presented. suffers from a severe loss in diffraction discussed, and the ex- this approach in optical pattern recognition is proportional to the square of the number of correlation that is scale efficiency perimental results of an optical stored filters. In addition, a precise synthesis system and rotationally invariant are included. the filter bank as well as a high and operations is is required to fabricate The need for such transformations recording media. The second approach 2. In Fig. storage density quite apparent when one considers Figs. 1 and suggested involves positioning the input correlation peak that has been 1, the SNR (signal-to-noise ratio) of the the transform lens. As the input is moved along change between the input behind is plotted vs the percent scale the optical axis, the transform is scaled. While useful and holographic matched filter function. The analo- is only appropriate for the in the laboratory, this method gous plot of SNR vs the rotation angle between Since this method involves 2. These plots smaller 20% scale changes. input and filter function is shown in Fig. movement of components, it is not com- a 35-mm transparency mechanical were experimentally obtained on patible with the real-time capability of an optical pro- of an aerial image with about 5-10-lines/mm resolution. of the input can also the space cessor. Similar mechanical rotation The rate of decrease of SNR increases with for orientation errors. However, The SNR is seen to be used to compensate bandwidth product of the image. limitations similar to those in the mechanical scaling decrease from 30 dB to 3 dB with a 2% scale change and These repre- system are still present. from 30 dB to 3 dB with a 3.50 rotation. The novel method to be discussed overcomes the sent severe losses and demonstrate the practical prob- present in the above implementa- recognition practical problems lems associated with an optical pattern tions by utilizing a transformation which is itself in- system. to scale and orientational changes in the input. can be addressed variant These scale and rotation problems first step in the synthesis of such a transformation digitizing the The by scanning the image point by point, is to form the magnitude of the Fourier transform to be described data, and performing the operations IF(w),wy)lof the input function f(x,y). This eliminates the effects of any shifts in the input and centers the light distribution on the optical axis. A University, Department of resultant The authors are with Carnegie-Mellon rotation of f(x,y) rotates IF(w, wy)lby the same angle, Engineering, Pittsburgh, Pennsylvania 15213. Electrical and a scale change in f(x,y) by a scales IF(coxwy)l by I/a. Received 8 January 1976. July 1976 / Vol. 15, No. 7 / APPLIEDOPTICS 1795 30 M'(w0 ,O) = a-J'M(w,0), (2) from which the magnitudes I- of the two transforms are seen to be identical. The optical implementation of the Mellin transform has been previously discussed,4 and 0 it has been shown that 0LU20 M(w,,O) = f F(expp,0) exp(-j&,p)dp, (3) 0 where p = nr. From Eq. (3), it can be seen that the 10 realization of the required optical Mellin transform z simply requires a logarithmic scaling of the r coordinate 0 followed by a one-dimensional Fourier transform in r. This follows from Eq. (3) since M(wp,0) is the Fourier transform 0 I I I of F(expp,O). 0 I 2 3 Let us now consider the effects of a rotation of the SCALE CHANGE "a" (%) input function f(x,y) by an angle 0. The rotation will Fig. not 1. SNR of the correlation peak vs the percent scale change be- affect the r coordinate in the (r,O)plane. The ef- tween the input and holographic spatial filter functions. fects on the 0 coordinate in the (r,O)plane are best seen with reference to Fig. 3 in which the original input F(, coy) is partitioned into two sections F(wx,wy) and F2(,coWy)for convenience, The effects of rotation and scale changes in the re- where F2 is the segment of sultant light distribution can be separated F that subtends an angle 00as shown in Fig. 3(a). The by per- polar forming a polar transformation on F(w,,wy)I from transformation of this function is shown in Fig. 3(b). The Fl(r,0) 2 (W.,Wy) coordinates to (r,0) coordinates. Since 0 section of F(rO) extends from Oto r 2 2 - 0 in the 0 direction, tan-(coy/) and r = (C, + y )1/2, a scale change in while F 2 (r,0) occupies the 27r- 00 to 2 7rportion of the ITl by a does not effect the 0 coordinate and scales the 0 axis. r coordinate directly to r' = ar. A two-dimensional A version of Fig. 3(a) rated clockwise by 0 is shown in scaling of the input function is thus reduced to a Fig. 3(c). For convenience, the sections of this ro- scaling tated in only one dimension (the r coordinate) in this trans- version are denoted by F'(wxwy) and F2(xy). formed F(r,O) function. The polar coordinate transformation of this F' function If a one-dimensional Mellin transform4 5 in r is shown in Fig. 3(d). The effect of a rotation by Oois is now seen to performed on F(r,0), a completely scale invariant be an upward shift in F(r,0) by 00and a down- transformation results. This is due to the scale in- variant property of the Mellin transform.4'5 The one-dimensional Mellin transform of F(r,0) in r is given by M(co,0) = ' F(r,0)r-i-ldr, (1) where p = lnr. The Mellin transform of the scaled function F(ar,0) is then (a) (b) a I.- C 10 1 I 2 0 20 (c) (d) Fig. D RK1S OF ROTATION 3. Effects of a rotation by 0 in the input function F(x y)on the transformed function F(r,0): (a) input function; (b) polar coor- Fig. 2. SNR of the correlation peak vs the rotational angle Oobetween dinate transform of (a); (c) rotated input function; (d) polar coordinate the input and holographic spatial filter functions. transform of (c). 1796 APPLIED OPTICS / Vol. 15, No. 7 / July 1976 the transformation of the function f'(x,y), which is scaled by a and rotated by 00, is given by M'(c,,oo) = M(w,wo) exp[-j(p lna + woho)] - - 00)]. (9) + M 2(w,wo) expt-jump lna coo(2ir Correlations rotational, and scale in- Fig. 4. Block diagram of the positional, rotational, and scale invariant (PRSI) variant PRSI transformation system. FT denotes Fourier The positional, transform. correlation is based on the form of Eqs. (8) and (9). If the product MM' is formed, we obtain M'1M = M*Mj exp[-j(cp lna + cooo)] + MM 2 expt-j[wp lna - wo(27r- 0)]. (10) The Fourier transform of Eq. (10) then consists of two terms: (a) the cross-correlation F1 (expp, 0) * F(expp, 0) lo- cated at p' = lna and 0' = 00; (b) the cross-correlation F 2(expp, 0) * F(expp, 0) lo- cated at p' = lna and 0' = -2ir + 0o,where the coordi- nates of this output Fourier transform plane are (p', 0'). I SYSTEM If the intensities of these two cross-correlation peaks are of F(expp, 0). Fig. 5. Block diagram of the real-time implementation of a PRSI summed, the result is the autocorrelation transformation using a real-time optically or electron beam addressed Therefore, the cross correlation of two functions that spatial light modulator. are scaled and rotated versions of one another can be obtained. Most important, the amplitude of this cross correlation will be equal to the amplitude of the auto- 2 this ward shift in F2 (r,0) by x - 00.