Diagonal forms of symmetric convolution matrices for asymmetric multidimensional sequences

Thomas M. Foltz; Byron M. Welsh

doi:10.1117/12.323174

1 October 1998 Diagonal forms of symmetric convolution matrices for asymmetric multidimensional sequences

Thomas M. Foltz, Byron M. Welsh

Proceedings Volume 3460, Applications of Digital Image Processing XXI; (1998) https://doi.org/10.1117/12.323174
Event: SPIE's International Symposium on Optical Science, Engineering, and Instrumentation, 1998, San Diego, CA, United States

Abstract

This paper presents diagonal forms of matrices representing symmetric convolution which is the underlying form of convolution for discrete trigonometric transforms. Symmetric convolution is identically equivalent to linear convolution for appropriately zero-padded sequences. These diagonal forms provide an alternate derivation of the symmetric convolution-multiplication property of the discrete trigonometric transforms. Derived in this manner, the symmetric convolution-multiplication property extends easily to multiple dimensions, and generalizes to multidimensional asymmetric sequences. The symmetric convolution of multidimensional asymmetric sequences can then be accomplished by taking the product of the trigonometric transforms of the sequences and then applying an inverse transform to the result. An example is given of how this theory can be used for applying a 2D FIR filter with nonlinear phase which models atmospheric turbulence.

Citation Download Citation

Thomas M. Foltz and Byron M. Welsh "Diagonal forms of symmetric convolution matrices for asymmetric multidimensional sequences", Proc. SPIE 3460, Applications of Digital Image Processing XXI, (1 October 1998); https://doi.org/10.1117/12.323174

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