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Discrete determinantal point processes and their application to image processing

Abstract : Determinantal point processes (DPPs in short) are probabilistic models that capture negative correlations or repulsion within a set of elements. They tend to generate diverse or distant subsets of elements. This notion of similarity or proximity between elements is defined and stored in the kernel associated with each DPP. This thesis studies these models in a discrete framework, defined on a discrete and finite set of elements. We are interested in their application to image processing, when the initial set of points corresponds to the pixels or the patches of an image. Chapter 1 and 2 introduce determinantal point processes in a general discrete framework, their main properties and the algorithms usually used to sample them, i.e. used to select a subset of points distributed according to the chosen DPP. In this framework, the kernel of a DPP is a matrix. The main algorithm is a spectral algorithm based on the computation of the eigenvalues and the eigenvectors of the DPP kernel. In Chapter 2, we present a sampling algorithm based on a thinning procedure and a Cholesky decomposition but which does not require the spectral decomposition of the kernel. This algorithm is exact and, under certain conditions, competitive with the spectral algorithm. Chapter 3 studies DPPs defined over all the pixels of an image, called Determinantal Pixel Processes (DPixPs). This new framework imposes periodicity and stationarity assumptions that have consequences on the kernel of the process and on properties of the repulsion generated by this kernel. We study this model applied to Gaussian textures synthesis, using shot noise models. In this chapter, we are also interested in the estimation of the DPixP kernel from one or several samples. Chapter 4 explores DPPs defined on the set of patches of an image, that is the family of small square images contained in the image. The aim is to select a proportion of these patches, diverse enough to be representative of the information contained in the image. Such a selection can speed up certain patch-based image processing algorithms, or even improve the quality of existing algorithms that require patch subsampling. We present an application of this question to a texture synthesis algorithm.
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Claire Launay. Discrete determinantal point processes and their application to image processing. Probability [math.PR]. Université Paris Cité, 2020. English. ⟨NNT : 2020UNIP7034⟩. ⟨tel-03189384⟩

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