Siegel Descriptors for Image Processing

We introduce the Siegel upper half-space with its symplectic geometry as a framework for low-level image processing. We characterize properties of images with the help of six parameters: two spatial coordinates, the pixel value, and the three parameters of a symmetric positive-definite (SPD) matrix such as the metric tensor. We construct a mapping of these parameters into the Siegel upper half-space. From the general theory, it is known that there is a distance on this space that is preserved by the symplectic transformations. The construction provides a mapping that has relatively simply transformation properties under spatial rotations, and the distance values can be computed with the help of closed-form expressions which allow an efficient implementation. We illustrate the properties of this geometry by considering a special case where we compute for every pixel its symplectic distance to its four spatial neighbors and we show how spatial distances, pixel value changes, and texture properties are described in this unifying symplectic framework.