1 June 2005 Novel signal-to-noise ratio analysis for binary phase-only filters
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In this paper I present a novel analysis of the signal-to-noise ratio (SNR) for the complex output of a correlator obtained by Fourier filtering an input image with a binary phase-only filter (BPOF). Rather than defining the output variance as being the variance of the real and imaginary components added together, it is defined as being the variance of the complex magnitude. An expression for the complex magnitude variance is obtained using a Taylor series expansion in which higher order terms are discarded. The variance is found to be related to the ratio of eveness to oddness of the BPOF, which in turn is related to the choice of threshold line angle. For a purely even BPOF, the correlator output is real, hence the variance is the same as for the conventional case. As the degree of oddness increases the higher frequency complex amplitudes of the noise are rotated out of phase with the dc component. Consequently, the complex magnitude variance decreases. For a BPOF that is odd the variance is a minimum as the higher frequency noise variations are orthogonal to the dc component. Furthermore, the SNR for the BPOF in this case is an order of magnitude greater than that obtained with a phase-only filter (POF). This last result contradicts the perceived wisdom that the SNR ratio for a BPOF and POF is in the range 4/π2→1. The analysis results are confirmed through simulation.
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Paul C. Miller, Paul C. Miller, "Novel signal-to-noise ratio analysis for binary phase-only filters", Proc. SPIE 5823, Opto-Ireland 2005: Imaging and Vision, (1 June 2005); doi: 10.1117/12.605136; https://doi.org/10.1117/12.605136

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