Trigonometric Transforms for Image Reconstruction

摘要

This dissertation demonstrates how the symmetric convolution- multiplication property of discrete trigonometric transforms can be applied to traditional problems in image reconstruction with slightly better performance than Fourier techniques and increased savings in computational complexity for symmetric point spread functions. The fact that the discrete Fourier transform a circulant matrix provides an alternate way to derive the symmetric convolution- multiplication property for 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-multiplication property allows for linear filtering of degraded images via point-by-point multiplication in the transform domain of trigonometric transforms. Specifically in the transform domain of a type-II discrete cosine transform, there is an asymptotically optimum energy compaction about the low-frequency indices of highly correlated images which has advantages in reconstructing images with high-frequency noise. The symmetric convolution- multiplication property allows for well-approximated scalar representations in the trigonometric transform domain for linear reconstruction filters such as the Wiener filter. An analysis of the scalar Wiener filter's improved mean-squared error performance in the trigonometric transform domain is given.