Several performance criteria are described to enable a fair comparison among the various correlation filter designs: signal-to-noise ratio, peak sharpness, peak location, light efficiency, discriminability, and distortion invariance. The trade-offs resulting between some of these criteria are illustrated with the help of a new family of filters called fractional power filters (FPFs). The classical matched filter, phase-only filter (POF), and inverse filter are special cases of FPFs. Using examples, we show that the POF appears to provide a good compromise between noise tolerance and peak sharpness.

Performance measures for correlation filters.

Filters that have optimal trade-offs among the criteria of noise robustness, sharpness of the correlation peak, and Horner efficiency are presented, and an explicit mathematical expression is provided. Owing to their optimality, these filters provide a figure of merit and then permit a rigorous characterization of filter performances for optical pattern recognition.

Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency.

Progress in Fourier optical processing techniques is reviewed. Particular emphasis is placed on real-time pattern recognition, which recently has received increased interest as the result of new filter formulations that can be implemented with existing spatial light modulators. Architectures for coherent optical correlation are reviewed, as are the recently reported phase-only filters. Smart filters that attack the inherent distortion sensitivity of correlation, including those suitable for implementation with phase-only modulation, are reviewed. Correlation experiments implementing real-time variation of both input and reference patterns are also reviewed. The potential for successful near-term application of these techniques is examined. >

Fourier optical signal processors

An efficient algorithm for designing a ternary valued filter yielding the highest signal to noise ratio (SNR) is utlined. Numerical evaluations using the image of a tank indicate that using such a filter can provide an improvement in SNR of ~5 dB over the conventional binary phase-only filter (BPOF). This is superior to the 1-dB improvement obtained for that image by varying the threshold line angle (TLA) in filter binarization. Simulation results are presented. They agree with the numerically computed SNRs.

Efficient algorithm for designing a ternary valued filter yielding maximum signal to noise ratio.

A new measure, termed the peak location error (PLE), is introduced to characterize the ability of a correlation filter to locate a target in the input. It is shown that the filter minimizing the PLE is the classical matched filter. Comparison is made between the minimal PLE and the error achievable with phase-only filters.

Correlation filters minimizing peak location errors

A binary phase-only filter (BPOF) bandwidth and, correspondingly, the performance with respect to stochastic noise are introduced as filter design parameters. A BPOF figure of merit is defined which references the matched filter. Analytical bounds on the BPOF signal-to-noise ratio are derived. The noise performance is illustrated with simulation results. It is demonstrated through analysis and simulation that BPOFs can be designed to perform well with respect to stochastic noise.

Bandwidth considerations for binary phase-only filters

Binary phase-only filters (BPOF) are a viable candidate for the replacement of matched filters in real-time image processing and pattern recognition applications. The original BPOFs were binarized versions of the real or imaginary parts of the Fourier transform. Recently, a filter has been proposed which is the binarized Hartley transform. In this paper, the noise performance of the binarized Hartley phase-only filter is analyzed and compared to other BPOFs as well as the phase-only and matched filter. It is well known that the matched filter optimizes signal-to-noise ratio. Further, it can be shown that the matched filter phase is the optimum phase-only filter. A bound on the BPOF signal-to-noise ratio relative to that of the phase-only filter is derived. Finally, the analysis is applied to the generalized form of the binarized Hartley transform.

K. T. Stalker

Papers

Bandwidth considerations for binary phase-only filters

Analysis Of Binarized Hartley Phase-Only Filter Performance With Respect To Stochastic Noise