CIRCUITSSYSTEMS SIGNAL PROCESS VOL 3, NO. 3, t983 GENERALIZED SCHUR-COLESKI FACTORIZATION WITH APPLICATIONS TO IMAGE PROCESSING* Romano M. DeSantis 1 and William A. Porter 2 Abstract. A special type of factorization for operators defined on partially ordered Hilbert resolution spaces is considered. The main result includes, as a particular case, the classical Schur-Coleski triangular factorization. Connections with stochastic op- timization and image-processing problems are established. 1. Introduction In recent studies [1], [5] the authors have generalized certain factorization results, associated with problems of optimal control and optimal filtering, to multidimensional form. In particular, the "special factorization" of Gohberg and Krein [3] has been extended to the multidimensional setting. The present study is motivated by problems in optimal signal extraction of multidimensional images. Here we take up the generalization of the classical Schur-Coleski matrix factorization [2]. The generalized S-C fac- torization results are demonstrated in the context of both matrix factoriza- tion and digital image processing. As in references [1] and [5], we shall use the partially ordered Hilbert resolution space, PHRS, structure to advantage. Since digitial image pro- cessing is the principal motivation for our study, our attention focuses primarily on large but finite sample sets. In Section 5 we sketch extensions to infinite samples sets and to the multidimensional S-C factorization of operators on arbitrary spaces. SOME DEFINITIONS. It is convenient to summarize the notation and defini- tions associated with the PHRS structure. For this, H denotes a Hilbert space * Received December20, 1982; revised December 7, 1983. This research was sponsored in part under NSF grant 78/88/71, AFOSR grant 78-3500and Canadian Research Council grant CNRC-A-8244. ' Departmentof Electrical Engineering, EcolePolytechniquede Montreal, Montreal, Quebec, Canada H3C 3A7. Department of Electrical Engineering, LouisianaState University, Baton Rouge, Louisiana 70803, USA.