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Discrete deep structure

: Tschirsich, Martin; Kuijper, Arjan


Kuijper, Arjan; Bredies, Kristin; Pock, Thomas; Bischof, Horst:
Scale space and variational methods in computer vision. 4th international conference, SSVM 2013 : Schloss Seggau, Leibnitz, Austria, June 2 - 6, 2013, Proceedings
Berlin: Springer, 2013 (Lecture Notes in Computer Science 7893)
ISBN: 3-642-38266-5
ISBN: 978-3-642-38266-6 (Print)
ISBN: 978-3-642-38267-3 (Online)
DOI: 10.1007/978-3-642-38267-3
International Conference on Scale Space and Variational Methods in Computer Vision (SSVM) <4, 2013, Leibnitz>
Conference Paper
Fraunhofer IGD ()
discrete images; partial differential equations; digital image processing; mathematics

The discrete scale space representation L of f is continuous in scale t. A computational investigation of L however must rely on a finite number of sampled scales. There are multiple approaches to sampling L differing in accuracy, runtime complexity and memory usage. One apparent approach is given by the definition of L via discrete convolution with a scale space kernel. The scale space kernel is of infinite domain and must be truncated in order to compute an individual scale, thus introducing truncation errors. A periodic boundary condition for f further complicates the computation. In this case, circular convolution with a Laplacian kernel provides for an elegant but still computationally complex solution. Applied in its eigenspace however, the circular convolution operator reduces to a simple and much less complex scaling transformation. This paper details how to efficiently decompose a scale of L and its derivative t L into a sum of eigenimages of the Laplacian circ ular convolution operator and provides a simple solution of the discretized diffusion equation, enabling for fast and accurate sampling of L.