Realization of 3-D Separable-Denominator Digital Filters With Low $l_2$ -Sensitivity

Three-dimensional (3-D) digital filters find applications in a variety of image and video signal processing problems. This paper presents a coefficient-sensitivity analysis for a wide class of 3-D digital filters with separable denominators in local state space that leads to an analytic formulation for sensitivity minimization, and to present two solution techniques for the sensitivity minimization problem at hand. To this end, a vector-matrix-vector decomposition of a given 3-D transfer function that separates the three variables and leads to a state-space realization in a form convenient for subsequent analysis. An
l
₂
-sensitivity analysis is then performed. The result is a computationally tractable formula of the overall
l
₂
-sensitivity for 3-D digital filters. The
l
₂
-sensitivity is minimized subject to
l
₂
-scaling constraints by using one of the two solution methods proposed-one relaxes the constraints into a single trace constraint and solves the relaxed problem with an effective matrix iteration scheme; while the other converts the contained optimization problem at hand into an unconstrained problem and solves it using a quasi-Newton algorithm. A case study is presented to illustrate the validity and effectiveness of the proposed techniques.