BIOMEDICAL IMAGING GROUP (BIG)
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Recursive Regularization Filters: Design, Properties, and Applications

M. Unser, A. Aldroubi, M. Eden

IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, no. 3, pp. 272–277, March 1991.

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. 

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