Short communication: the fast DCT-IV/DST-IV computation via the MDCT

The trend of low-powered discrete orthogonal sinusoidal transforms has become essential due to the explosion of wireless technologies, especially for applications which run on batteries. The discrete cosine transforms (DCTs) and discrete sine transforms (DSTs) are real-valued transforms that map integer-valued signals to floating-point coefficients. Although the fast algorithms for their computation reduce the number of arithmetic operations significantly, they still need floating-point operations. The DCTs and DSTs with integer coefficients are of great interest since they lead to low-cost systems that can be designed more simply and implemented more efficiently with low-power consumption. The DCT and DST implementations based on integer approximation are currently modern transform technologies that are adaptable to the available power resources and are suitable for transform-based lossless coding. The fast integer DCTs and DSTs in the form of generalized signal flow graphs are ready to be used in practical applications. The chapter concludes with a summary, problems and exercises, and comprehensive references related to integer DCT and DST.

CHAPTER 5 – Integer Discrete Cosine/Sine Transforms

Pipelined VLSI/WSI architectures supporting image coding transforms are defined and evaluated in the paper. The structures proposed in the paper have been derived by considering a common algorithmic kernel of the set of examined transforms. The possibility of reducing the computations to a common algorithmic version allows definition of flexible structures characterized by a “basic” pipeline - performing the common kernel of computation - and by transform-dependent input and output stages. Three different pipelines have been defined and figures of merit, such as silicon area occupation and throughput, have been evaluated for architectures comparisons.

Multiple-transform pipelines for image coding

This chapter discusses the fast 1-D and 2-D DCT/DST algorithms for all even types of the discrete cosine transform (DCT) and discrete sine transform (DST). Almost all are direct algorithms defined by the recursive sparse matrix factorization of the transform matrix. DCTs and DSTs are members of the class of sinusoidal unitary transforms. A sinusoidal unitary transform is an invertible linear transform whose kernel is defined by a set of complete, orthogonal/orthonormal discrete cosine and/or sine basis functions. The fast 1-D DCT/DST algorithms are numerically stable, have the regular computational structures and employ real arithmetic only. The generalized signal flow graphs corresponding to the sparse matrix factorization of the transform matrix for the fast DCT/DST computation have also been provided in the chapter and they are ready to be used in practical applications.

Chapter 4 - Fast DCT/DST Algorithms

Trigonometric transforms like the Fourier transform or the discrete cosine transform (DCT) are of immense importance in signal and image processing, physics, engineering, and data processing. The research of past decades has provided us with runtime optimal algorithms for these transforms. Significant runtime improvements are only possible if there is additional a priori information about the sparsity of the signal. In the first part of this thesis we develop sublinear algorithms for the fast Fourier transform for frequency sparse periodic functions. We investigate three classes of sparsity: short frequency support, polynomially structured sparsity and block sparsity. For all three classes we present new deterministic, sublinear algorithms that recover the Fourier coefficients of periodic functions from samples. We prove theoretical runtime and sampling bounds for all algorithms and also investigate their performance in numerical experiments. 
In the second part of this thesis we focus on the reconstruction of vectors with short support from their DCT of type II. We present two different new, deterministic and sublinear algorithms for this problem. The first method is based on inverse discrete Fourier transforms and uses complex arithmetic, whereas the second one utilizes properties of the DCT and only employs real arithmetic. We show theoretical runtime and sampling bounds for both algorithms and compare them numerically in experiments. Furthermore, we generalize the techniques for recovering vectors with short support from their DCT of type II using only real arithmetic to the two-dimensional setting of recovering matrices with block support, also providing theoretical runtime and sampling complexities for the obtained new two-dimensional algorithm.

Sparse Fast Trigonometric Transforms

A new matrix factorization is proposed for DCT-IV, which is the basis of fast algorithms for many sinusoidal transforms. A new fast algorithm for complex-data DFT based on the new factorization requires the same number of multiplications and far fewer additions than the Preuss algorithm. A new fast algorithm for real-data DFT based on a new algorithm for the discrete Hartley transform is also proposed.

M. Hatori

Papers

Fast algorithms for the discrete Fourier transform and for other transforms