1 December 1977 Optical Cross-Correlation of Complex Exponentiated Input Data
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Optical cross-correlation to determine relative signal displacements and degree of similarity between two signals is commonly implemented by matched filter techniques using absorption transparencies as inputs. The problems associated with absorption inputs include low correlation-signal intensities due to the light absorbing nature of the input and low signal-to-noise ratios. Without considerable preprocessing, positive correlation peak detection is not always readily achievable. These limitations are largely overcome by complex exponentiation of the inputs. For the optical analog this means phasing the input transparencies by a bleaching process to yield phase transparencies. The cross-correlation function of these complex exponentiated inputs has two striking properties. One, the correlation signal approaches a delta function. Two, the correlation signal is not affected by a difference in bias levels (average densities) of the two inputs in-asmuch as only differential phase differences are used for detecting correlation. This means constant phase shifts will not contaminate the correlation signal. Hence, extensive data preprocessing is not required. One- and two-dimensional digital simulation experiments were carried out to demonstrate these properties. Simulated density functions were defined by computer generated random numbers. Random noise distortions were added to study their impact on the correlation-signal shape and intensity. In order to have a standard for comparison, the commonly used (statistical) correlation coefficient was computed along with the correlation of the complex exponentiated inputs. The results indicate that complex exponentiation provides a means to obtain extremely reliable correlation peak positions having very high peak intensity and very high signal-to-noise ratio (SNR). Since the correlation function is a narrow well defined spike, a threshold detector can be employed for signal detection.
W. M. Gopfert, W. M. Gopfert, A. A. Read, A. A. Read, } "Optical Cross-Correlation of Complex Exponentiated Input Data," Optical Engineering 16(6), 166605 (1 December 1977). https://doi.org/10.1117/12.7972169 . Submission:


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