Study On Anamorphic Fractional Correlation And Its Optical And Digital Implementation

Fractional Fourier optics is regarded as the development and extension of Fourier optics and it has become a new branch of modern optics, thus a new mathematical and optical tool is achieved. With the tool, we can study the light wave propagation, diffraction, imagery and information processing. Many applications may be exploited while Fourier transform is replaced by fraction Fourier transform.Correlation based on Fourier transform (FT) is one of the most important mathematical operations that can be optically implemented because it can be used for pattern recognition and image feature extraction, and so on. However, since it is a shift-invariant operation, conditional correlation is limited sometimes. Fractional correlation (FC) is the general notion of conditional correlation, because it is based on fractional Fourier transform (FRT). FC can be used in shift-variant object recognition because it is a shift-variant operation.Anamorphic fractional correlation is proposed in this paper when we combine FC and anamorphic optical system and simultaneously its mathematical expression is presented. Through theoretical analysis, the condition of maximum correlation peak appearing is obtained, which provides the basis that how to search the fractional orders that concerning to the correlation peak.The discrete expression of FRT is studied and presented, and program is composed according to the fast algorithm, so that the digital implementation of anamorphic FC becomes possible. The connection of anamorphic FC peak and fractional orders is a problem of higher dimensional optimization. Based on theoretical analysis and simulated annealing algorithm and programs we compose, the optimized orders and correlation peaks are gained while the patterns detected have dilation. Shift pattern correlation is also analyzed.An optical setup implementation of anamorphic FC is suggested. In order to simplify it, we propose to use computer-generated holography (CGH) to implement anamorphic FC.Since more fractional orders are involved in it, anamorphic FC can be used in more complex pattern recognition.