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<p>Least squares approximation problems that are regularized with specified highpass stabilizing kernels are discussed. For each problem, there is a family of discrete regularization filters (R-filters) which allow an efficient determination of the solutions. These operators are stable symmetric lowpass filters with an adjustable scale factor. Two decomposition theorems for the z-transform of such systems are presented. One facilitates the determination of their impulse response, while the other allows an efficient implementation through successive causal and anticausal recursive filtering. A case of special interest is the design of R-filters for the first- and second-order difference operators. These results are extended for two-dimensional signals and, for illustration purposes, are applied to the problem of edge detection. This leads to a very efficient implementation (8 multiplies+10 adds per pixel) of the optimal Canny edge detector based on the use of a separable second-order R-filter.</p>
causal recursive filters; first-order difference operators; least squares approximation; highpass stabilizing kernels; discrete regularization filters; R-filters; stable symmetric lowpass filters; adjustable scale factor; decomposition theorems; z-transform; impulse response; anticausal recursive filtering; second-order difference operators; two-dimensional signals; edge detection; optimal Canny edge detector; separable second-order R-filter; filtering and prediction theory; least squares approximations; low-pass filters; pattern recognition; Z transforms
A. Aldroubi, M. Unser, M. Eden, "Recursive Regularization Filters: Design, Properties, and Applications", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 13, no. , pp. 272-277, March 1991, doi:10.1109/34.75514
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