Fast Transforms in Image Processing: Compression, Restoration, and Resampling

Table 1

Register of relevant fast transforms, their main characteristic features, and application areas.

	Relevance to imaging optics	Main characteristic features	Main application areas
Discrete Fourier Transforms	Represents optical Fourier Transform	Cyclic shift invariance vulnerable to boundary effects	(i) Analysis of periodicities.; (ii) Fast convolution and correlation.; (iii) Fast and accurate image resampling.; (iv) Image compression.; (v) Image denoising and deblurring.; (vi) Numerical reconstruction of holograms.
Discrete Cosine Transform	Represents optical Fourier Transform	Cyclic shift invariance (with double cycle); virtually not sensitive to boundary effects
Discrete Fresnel Transforms	Represent optical Fresnel Transform	Computable through DFT/DCT	Numerical reconstruction of holograms
Walsh-Hadamard Transform	No direct relevance	Binary basis functions.; Provides piece-wise constant separable image band-limited approximations	(i) Image compression (marginal).; (ii) Coded aperture imaging.
Haar Transform and other Discrete Wavelet Transforms	Subband decomposition	Binary basis functions.; The fastest algorithm.; Multiresolution.; Shift invariance in each particular scale.	(i) Signal/image wideband noise denoising.; (ii) Image compression.