ABSTRACT
This chapter discusses the theory of fast 2-D discrete Fourier transformation, which is based on the concept of partitions that reveals the two-dimensional and multi-dimensional transformations. The discrete Fourier transform has become a powerful technique in signal processing, and in particular in image processing. Effective methods, or fast algorithms of the Discrete Fourier transforms (DFT) are used for solving many problems in image processing in the frequency domain, such as image filtration, restoration, enhancement, compression, and image reconstruction by projections. The Hadamard transform has found useful applications in signal and image processing, communication systems, image coding, image enhancement, pattern recognition, and general two-dimensional filtering. The chapter describes the tensor approach and its improvement, for dividing the calculation of the 2-D DFT into the minimal number of short 1-D transforms. The approach is universal because it can be implemented to calculate other discrete unitary transforms, such as the Hadamard, cosine, and Hartley transforms, and transforms of high dimensions.
