Axioms and fundamental equations of image processing

Image-processing transforms must satisfy a list of formal requirements. We discuss these requirements and classify them into three categories: “architectural requirements” like locality, recursivity and causality in the scale space, “stability requirements” like the comparison principle and “morphological requirements”, which correspond to shape-preserving properties (rotation invariance, scale invariance, etc.). A complete classification is given of all image multiscale transforms satisfying these requirements. This classification yields a characterization of all classical models and includes new ones, which all are partial differential equations. The new models we introduce have more invariance properties than all the previously known models and in particular have a projection invariance essential for shape recognition. Numerical experiments are presented and compared. The same method is applied to the multiscale analysis of movies. By introducing a property of Galilean invariance, we find a single multiscale morphological model for movie analysis.