Phase multiplexed ternary phase amplitude filters Gregory Gheen and Aed El-Saba

When this work was done both authors were with Univer­sity of Alabama in Huntsville, Electrical Engineering De­partment, Huntsville, Alabama 35899; G. Gheen is now with Lockheed Missiles & Space Company, P.O. Box 3504, Sunnyvale, California 94089. Received 3 July 1989. 0003-6935/90/294335-03$02.00/0. © 1990 Optical Society of America.

Phase encoding is used to obtain good target discrimina­tion for multiplexed ternary phase amplitude filters. The proposed technique reduces the number of filters that need to be stored in a filter bank. Keywords; Correlators, opti­cal processing.

This Letter proposes an approach for generating phase multiplexed ternary phase amplitude filters (TPAFs)1,2 or binary phase-only filters (BPOFs).3-5 However, the discus­sion is restricted to TPAFs for brevity. The term multi­plexed filter refers to a filter that is generated by adding together a small number of matched filters. Phase multi­plexing associates a unique phase with each image multi­plexed on a filter. Using phase causes contributions from individual filters to add out of phase and thus reduces the probability of false classification. The advantage of using phase in filter synthesis was first pointed out by Kallman who used it to formulate a low noise synthetic discriminant function (SDF) filter.6 Our concern is different. We are interested in combining a few images to obtain a high dis­crimination filter while significantly reducing the memory required to store a bank of filters.

The phase values used in the multiplexed filter should satisfy two constraints. First, the phase values should be uniformly distributed between 0 and 2π. Second, the phase values should be assigned so that the resulting filter impulse

Fig. 1. Impulse response used to construct phase multiplexed TPAF.

response has Hermitian symmetry. This satisfies the Fouri­er transform relationship of the real valued TPAF.4 To satisfy the second constraint, each image is paired with its reflection, and a conjugate phase term is assigned to this image. If the reflected image is not present in the training set, it is artificially added. Thus this multiplexing scheme is twice as efficient for a training set consisting of in-plane rotations of an image.

The proposed phase multiplexed image is best illustrated with an example. Consider the problem of detecting a trian­gle, independent of in-plane rotations. To accomplish this, we could store seventy-two filters, one for each 5° interval of rotation. However, this requires storing and circulating through seventy-two filters to determine if the triangle is present. To reduce the memory requirements and access time, we store eighteen multiplexed filters with four images on each filter. However, we could also phase multiplex 6, 8 , . . . , images.

The first step in generating a phase multiplexed TPAF is

10 October 1990 / Vol. 29, No. 29 / APPLIED OPTICS 4335

Fig. 2. (a) Target image; (b) clutter image.

to construct an impulse response that satisfies the two con­straints listed above. One realization of such an impulse response is shown in Fig. 1 where the phase term associated with each triangle is indicated. The TPAF is generated by taking the cosine transform of Fig. 1 and mapping each value into +1, 0, or - 1 . This mapping was determined to minimize the angle (in an Euclidean vector space) between the cosine transform and the TPAF.

Simulations were performed to compare the performance of a phase multiplexed TPAF and a nonphase multiplexed TPAF. In this simulation, a 128 × 128 sampling array was used to represent the data. The longest side of the triangle was around sixty pixels. The target and clutter objects are shown in Figs. 2(a) and (b), respectively. The correlation of the triangle with the phase multiplex and nonphase multi­plexed TPAF is shown in Figs. 3(a) and (b), respectively. The cross-correlation of the clutter object with the phase multiplexed and nonphase multiplex TPAF is shown in Figs. 3 (c) and (d), respectively. Comparing the response generat­ed by the clutter image for the two TPAFs illustrates the enormous advantage that phase multiplexing provides.

The improved multiplexing capabilities provided by phase encoding are best understood using the notion of a decision surface. It has been shown that a filter with a complex

Fig. 3. Output correlation produced between (a) target and phase multiplexed TPAF, (b) target and nonphase multiplexed TPAF, (c) clutter and phase multiplexed TPAF, and (d) clutter and nonphase multiplexed TPAF.

4336 APPLIED OPTICS / Vol. 29, No. 29 / 10 October 1990

impulse response. Note that two decision surfaces are im­plemented, one for positive correlations and the other for negative correlations. A filter with a complex valued im­pulse response provides an extra degree of freedom. This allows four different images to be multiplexed without a significant reduction in the performance of the filter. If we let fi for i = 1,2,3, and 4 be the four orthogonal images to be multiplexed, the real and imaginary part of the filter can be represented as hr = f1 - f2 and hi = f3 - f4 respectively. These two components of the filter define the axes of the hypercylinder decision surface shown in Fig. 4(b). The hy-percylinder can be thought of as approximating four hyper-planes which straddle the sides of the hypercylinder. If more than four images are phase multiplexed onto a filter, the performance deteriorates. Nevertheless, phase multi­plexing still helps the overall performance of the resulting filter.

Summarizing: We have presented a new technique for multiplexing a number of images onto a TPAF. By assign­ing a different and uniformly distributed phase to each im­age that is multiplexed, a high discrimination filter is ob­tained". Theory suggests that four images can be multiplexed onto the same filter without a significant reduc­tion in performance. However, the Hermitian symmetry constraint of the TPAF will limit this number to two unless the training images correspond to in-plane rotations. A larger number of images can be multiplexed with a corre­sponding reduction in performance. Since multiplexing is achieved without increasing the space-bandwidth product of the filter, the proposed technique can offer a savings in memory space and filter search time.

References 1. D. L. Flannery, J. S. Loomis, and M. E. Milkovich, "Transform-

Ratio Ternary Phase-Amplitude Filter Formation for Improved Correlation Discrimination," Appl. Opt. 27, 4079-0000 (1988).

2. B. V. K. Kumar and Z. Bahri, "Efficient Algorithm for Designing a Ternary Valued Filter Yielding Maximum Signal to Noise Ra­tio," Appl. Opt. 28, 1919-0000 (1989).

3. D. Psaltis, E. Paek, and S. Venkatesh, "Optical Image Correlation with a Binary Spatial Light Modulator," Opt. Eng. 23, 698-000 (1984).

4. J. L. Horner and J. R. Leger, "Pattern Recognition with Binary Phase-Only Filters," Appl. Opt. 24, 609-000 (1985).

5. D. L. Flannery, J. S. Loomis, M. E. Milkovich, and P. Keller, "Application of Binary Phase-Only Correlation to Machine Vi­sion," Opt. Eng. 27, 309-000 (1988).

6. R. R. Kallman, "Construction of Low Noise Optical Correlation Filters," Appl. Opt. 25, 1032-0000 (1986).

7. G. Gheen, "Distortion Invariant Pattern Recognition in Multi­channel Optical Correlators," Proc. Soc. Photo-Opt. Instrum. Eng. 1053, 000-000 (1989).

Fig. 4. Decision surface for (a) a filter with a real bipolar impulse response and (b) a filter with complex valued impulse response.

valued impulse response implements a hypercylinder deci­sion surface, while a filter with a real valued impulse re­sponse implements a hyperplane.7 The hypercylinder deci­sion surface provides an extra degree of freedom, which helps reduce false classifications. Typically, a filter with a real impulse response can multiplex two orthogonal images (i.e., images with little or no common area) without significantly reducing discrimination. This is accomplished by encoding one of the images with a positive value and the other image with a negative value. This is demonstrated in Fig. 4(a), where f1 and f2 are the two images and h = f1 - f2 is the filter

10 October 1990 / Vol. 29, No. 29 / APPLIED OPTICS 4337