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UNITARY TRANSFORMS Two-dimensional unitary transforms have found two major applications in image processing. Transforms have been utilized to extract features from images. For example, with the Fourier transform, the average value or dc term is proportional to the average image amplitude, and the high-frequency terms (ac term) give an indication of the amplitude and orientation of edges within an image. | Digital Image Processing PIKS Inside Third Edition. William K. Pratt Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-37407-5 Hardback 0-471-22132-5 Electronic 8 UNITARY TRANSFORMS Two-dimensional unitary transforms have found two major applications in image processing. Transforms have been utilized to extract features from images. For example with the Fourier transform the average value or dc term is proportional to the average image amplitude and the high-frequency terms ac term give an indication of the amplitude and orientation of edges within an image. Dimensionality reduction in computation is a second image processing application. Stated simply those transform coefficients that are small may be excluded from processing operations such as filtering without much loss in processing accuracy. Another application in the field of image coding is transform image coding in which a bandwidth reduction is achieved by discarding or grossly quantizing low-magnitude transform coefficients. In this chapter we consider the properties of unitary transforms commonly used in image processing. . GENERAL UNITARY TRANSFORMS A unitary transform is a specific type of linear transformation in which the basic linear operation of Eq. is exactly invertible and the operator kernel satisfies certain orthogonality conditions 1 2 . The forward unitary transform of the N1 x N2 image array F n1 n2 results in a N1 x N2 transformed image array as defined by N N2 F m1 m2 z z F np n2 A ftp n2 m-1 m2 1 1 n2 1 185 186 UNITARY TRANSFORMS where A n1 n2 m1 m2 represents the forward transform kernel. A reverse or inverse transformation provides a mapping from the transform domain to the image space as given by N 2 F n1 n2 z z F m m2 B np n2 m m2 mi 1 m2 1 where B n1 n2 m1 m2 denotes the inverse transform kernel. The transformation is unitary if the following orthonormality conditions are met Z ZA n1 n2 m1 m2 A j1 j2 mv m2 8 n1 -j1 n2 -j2 m1 m2 ZZB n1 n2 m1 m2 B j1 j2 m1 m2