Engineering presentation of the stochastic interpolation framework and its applications

The paper is an engineering exposition of the Stochastic Interpolation Framework, a novel mathematical approach to data regularization, which recovers a function from input data that is a representation of this data. The framework is an area-based method that comprises a two-step procedure: de-convolution and convolution, involving row-stochastic matrices. Varying the extent of convolution with respect to de-convolution in the framework obtains a gamut of functional recovery ranging from interpolation to approximation, to peak sharpening. Construction of the row stochastic matrices is achieved by means of a mollifier, a positive function which serves as the generator of the row space of these matrices. The properties of the recovered function will depend on the choice of this mollifier. For example, only if the mollifier is differentiable so is the recovered function, and the framework can obtain derivatives anywhere in the domain. The mollifier can be a probability distribution function. Thus, the framework connects interpolation to statistical analysis. Two novel applications in image analysis illustrate the potential of the framework for security applications: as an alternative method of lossy image compression, and as an alternative method to zoom-up an image.