The discrete scale space as a base for robust scale space algorithms

Linear or gaussian scale space is a well known 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, implementational 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 believe that the most suitable way to do this is via porting concepts and algorithms from the continuous to the spatially discrete scale space with continuous scale parameter. Aiming at a computationally practicable implementation of the discrete scale space framework, suitable neighborhoods, boundary conditions and sampling methods are proposed. In analogy to prevalent approaches, a discretized diffusion equation is derived from the continuous scale space axioms adapted to discrete two-dimensional images or signals, 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 successfull implementation of algorithms operating on its deep structure. Relevant and promising 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 promising properties of topological graphs under the influence of smoothing, setting the stage for more robust deep structure extraction algorithms.