Laplacian eigenimages in discrete scale space

Linear or Gaussian scale space is a well known multi-scale representation for continuous signals. 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. Aiming at a computationally practicable implementation of the discrete scale space framework we used suitable neighborhoods, boundary conditions and sampling methods. 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 successful implementation of algorithms operating on its deep structure. In this paper, 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. One of the properties leads to Laplacian eigenimages in scale space: Taking a reduced set of images can be considered as a way of applying a discrete Gaussian scale space.