#### Technical Report USC-IPI-950

“Design of SVD/SGK Convolution Filters for Image Processing”

by Sang Uk Lee

January 1980

This dissertation describes a special-purpose signal processor for performing two-dimensional convolution with a minimum amount of hardware using the concepts of singular value decomposition (SVD) and small generating kernel (SGK) convolution. The SVD of the impulse response of a two-dimensional finite impulse response (FIR) filter is employed to decompose a filter into a sum of two-dimensional separable linear operators. These linear operators are themselves decomposed into a sequence of small kernel convolution operators. The SVD expansion can be truncated to a relatively few terms without significantly affecting the filter input.

A statistical analysis of finite word-length effects in SVD/SGK convolution is presented. Two important issues, related to the implementation of the filters in cascade form, scaling and section ordering, are also considered. Computer simulation of image convolution indicates that 12 bits are required for the SGK/SVD accumulator memory and 16 bits are required for quantization of filter coefficients to obtain results visually indistinguishable from full precision computation. A normalized mean square error between the SVD/SGK processed output and the direct processed output is chosen as an objective criterion function. It is shown that a subjective visual improvement is obtained by resetting the output mean to be equal to the input mean.

The transformation technique developed for the one-dimensional case is used to parametrically modify the cutoff frequency of a baseline SVD/SGK convolution filter. A detailed discussion of the one-dimensional case is presented, and its applicability to SVD/SGK convolution filters is described.

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