Karhunen-Loeve transform-based fast algorithms for image restoration

In this work, we study restoration of digital images degraded by white and colored noise using the Wiener filtering technique. It has been established that the Karhunen-Loeve transform (KLT) of the covariance matrix of a first order Markov process can be closely approximated by the discrete cosine transform (DCT) [1]. This property is used to establish that the Wiener filtering equations for an (n × n) image (a linear system of order n²) degraded by white noise can be transformed using the DCT to obtain sets of linear equations, each of order . The solution for each of the systems can be computed in computations. Thus, the complete algorithm requires computations, which is approximately half of the computations required for the previously described algorithms [2]. Also, it is shown that the approximation of the KLT by DCT may be interpreted as a modification in one of the boundary conditions (horizontal or vertical) of the noisy image. The algorithms obtained are generalized to include the case of images degraded by colored noise. The rise in computational complexity of the algorithm for such a case is marginal.