Generalized Karhunen-Loeve Transformation I (Theoretical Consideration)

A generalized approach to obtain a family of the reversible linear transform functional sets for the transform compression is proposed in this work. The present method is formulated so as to include the conventional Karhunen-Loeve (KL) transform method in itself, and shows how one might take account of a spatially inhomogeneous and correlated error measure. These new functional sets can be derived minimizing a generalized measure defined in a skew linear error space instead of the conventional orthogonal one. The generalized error measure is minimized to obtain an optimal functional set which is capable of making the errors spatially inhomogeneous and reducing the correlation between them. For a strongly correlated input data sequence, an algebraic equation for the sinusoidal inhomogeneity of errors is analytically solved and found to yield the discrete Mathieu functions (DMF´s) expanded m terms of the discrete cosine functions (DCF´s). As an example, the simulation results for the DMF´s are shown in comparison with the performances of the KL and the DC transform compressions. Since, as is well known, the spatially, highly correlated, and localized errors are very visible to the human eye, one may expect that the present method is effectively applicable to the block transform compression of image data to reduce the visible distortion.