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

Leonid P Yaroslavsky

doi:10.1155/2014/276241

Abstract

Transform image processing methods are methods that work in domains of image transforms, such as Discrete Fourier, Discrete Cosine, Wavelet, and alike. They proved to be very efficient in image compression, in image restoration, in image resampling, and in geometrical transformations and can be traced back to early 1970s. The paper reviews these methods, with emphasis on their comparison and relationships, from the very first steps of transform image compression methods to adaptive and local adaptive filters for image restoration and up to “compressive sensing” methods that gained popularity in last few years. References are made to both first publications of the corresponding results and more recent and more easily available ones. The review has a tutorial character and purpose.

Highlights

It will not be an exaggeration to assert that digital image processing came into being with introduction, in 1965 by Cooley and Tukey, of the Fast Fourier Transform algorithm (FFT, [1]) for computing the Discrete Fourier Transform (DFT)
Image Resampling and Building ‘‘Continuous’’ Image Models. As it was indicated in the introductory section, Discrete Fourier and Discrete Cosine Transforms occupy the unique position among other orthogonal transforms
Signal sampling and reconstruction devices, discrete frequency responses of digital filters for perfect differentiation and integrations should be, correspondingly, proportional and inversely proportional to the frequency index [18, 26]. This result directly leads to fast differentiation and integration algorithms that work in DFT or Discrete Cosine Transform DFT (DCT) domains using corresponding fast transforms with computational complexity O(log N) operation per signal sample for signals of N samples

Summary

Introduction

It will not be an exaggeration to assert that digital image processing came into being with introduction, in 1965 by Cooley and Tukey, of the Fast Fourier Transform algorithm (FFT, [1]) for computing the Discrete Fourier Transform (DFT). Being optimal in terms of the energy compaction capability, Karhunen-Loeve and Hotelling transforms have, generally, high computational complexity: the per pixel number of operations required for their computation is of the order of image size This is why for practical needs only fast transforms that feature fast transform algorithms with computational complexity of the order of the logarithm of image size are considered. DCT of a discrete signal is essentially, to the accuracy of an unimportant exponential shift factor, DFT for the same signal extended outside its border to a double length by means of its mirrored, in the order of samples, copy [15, 16] This way of signal extension has a profound implication on the speed of decaying to zero of signal DCT spectra and is a paramount and fundamental reason of the good energy compaction capability of DCT.

Dilemma

Transform Domain Filters for Image Restoration and Enhancement

Image Resampling and Building ‘‘Continuous’’ Image Models

Image Recovery from Sparse and Irregularly Sampled Data

Findings

Conclusion