The paper presents a time optimal parallel architecture for the inversion of a special class of Range-Hermitian matrices. In particular, the paper derives recursive equations for the computation of the covariance matrices, which are a sub-class of 1i;inge-Hermitian matrices. The derived recursive equations uptlatcs the covariance matrix and its inverse taking into account ;\I1 thc previous parameters. These equations apply for the singul;ir and nonsingular cases. A unique feature of the architecture is the capability of on-line updating of the covariance matrices.The proposed architecture is capable of updating an NxN covariance matrix in N+1 cycles. It features full use of symmetry properties to speed up computations and to reduce storage requirements.
I . IntroductionMany image processing and pictorial pattern analysis opplications require the computation of the covariance matrix and its inverse. This process is compute intensive if it is camed on a gcneral purpose sequential machine. The complexity of the computations is 0 (N ' 1, where N is the dimension of the covariance matrix. This compfexity hinders the on-line training capability of a real time system. For demanding real time applications, the need arise for special structures for updating the covariance matrix and its inverse to meet the application dcadl ine.In real-time image processing, there is a need for spccially designed dedicated processors that can have on-line response for a given application [l-SI. Recently, several algorithm specific parallel processing structures have been developed [ 1 -51. Typically, structures based on one-dimensional and two-dimensional arrays are the most common parallel nrchitectures used in image processing. The connectivity of these arrays is simple and regular [6]. The tree architecture is popular with applications that have a growth pattern that increases exponentially. Tree arrays are better suited for applications that employ search algorithms.In addition to algorithm specific structures, several general purpose machines were proposed in the literature for increasing the computational speed for real time image processing applications [5]. The majority of these machines use an array of general purpose processors for multi-purpose image processing. Examples of SIMD machines include DAP, CLIP 4/5, MPP, and NTT. Although these machines offer an improved computational speed, but they are better suited at the development stage.0094-2898/91/0000/0123$01 .OO 0 1991 IEEE In this paper, a specialized parallel structure, with online training capability, is proposed for updating the covariance matrix and its inverse. Here, it is assumed that the covariance matrix will be computed using a set of sample vectors, each consisting of N-measurements. With the assumption of an initial off-line computation of the covariance matrix, the paper derives recursive expressions for updating the matrix and its inverse, taking into consideration all previous parameters. The equations are derived for the singular and non-singular cases. The ...