Number Theoretic Transforms and their Applications in Image Processing

Abstract

Publisher Summary This chapter introduces the concepts of number theoretic transforms (NTT) and their applications to 2-D convolutions. The use of NTT's for 2-D convolutions is discussed and a method for calculating them without matrix transpose and without overlap is explained with examples. Although the transform domain for 2-D NTTs is not as simple to interpret as for some other transforms, the 2-D Fermat number transform (FNT) domain does have special patterns of zeros and nonzero values that could find applications. These are discussed for a number of simple images and for 2-D periodic structures. The effects of small errors in the periodic data on the 2-D FNT are discussed and applications are considered. Another family of transforms based upon the Mersenne numbers is introduced together with its separable form. The transform is extended to the multidimensional case. The chapter provides an overview of how this transform can be combined with the 2-D FNT, using the 2-D mixed radix conversion, to provide extended dynamic range and great parallelism using simple and convenient moduli leading to an efficient method for parallel image-processing applications. Because of the somewhat inconvenient word lengths required by some NTTs, the use of very-large-scale integration (VLSI), where the word length can be chosen to suit the design, has had a dramatic effect on circuit design for NTTs. A number of designs for pipeline FNT processors, including a vector radix structure, are discussed in the chapter.