Notes on Discrete Gaussian Scale Space

Gaussian scale space is a well-known linear multi-scale representation for continuous signals. The exploration of its so-called deep structure by tracing critical points over scale has various theoretical applications and allows for the construction of a scale space hierarchy tree. However, implementation issues arise, caused by discretization and quantization errors. In order to develop more robust scale space based algorithms, the discrete nature of computer processed signals has to be taken into account. We propose suitable neighborhoods, boundary conditions, and sampling methods. In analogy to prevalent approaches and inspired by Lindeberg's scale space primal sketch, a discretized diffusion equation is derived, including requirements imposed by the chosen neighborhood and boundary condition. The resulting discrete scale space respects important topological invariants such as the Euler number, a key criterion for the successful implementation of algorithms operating on critical points in its deep structure. Relevant properties of the discrete diffusion equation and the Eigenvalue decomposition of its Laplacian kernel are discussed and a fast and robust sampling method is proposed. We finally discuss properties of topological graphs under the influence of smoothing, setting the stage for more robust deep structure extraction algorithms.