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Wavelet-based representations of point processes for modelling and statistical learning

  • Brochard, Antoine
Publication Date
Mar 31, 2022
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This dissertation presents a class of representations of spatial point processes. Inspired from the success of wavelet methods in signal processing, these descriptors rely on the convolution of a point process with a family of wavelet filters. From these convolutions are built sets of statistical descriptors of stationary point process, by applying non-linear operators, followed by a spatial averaging. Much like classical summary characteristics for point processes, these statistics are designed to extract information about the process with a relatively small number of numerical values, by describing its geometry. Their goal is to describe whether the atoms of the process tend to repel each other, or cluster together, and by doing so, form possibly complex geometric shapes. By construction, these descriptors enjoy several properties that make them suitable for statistical analysis and learning tasks. To illustrate the quality of these representations as statistical descriptors, we study several problems involving statistical analysis of point processes. In a first experiment, we seek to estimate an unknown function that takes as input a point pattern, and returns a marked version of this pattern, where a numerical value is associated to each atom of the pattern. We use a wavelet-based representation of point patterns to estimate the relation between their non-marked and marked version. We then study, in a second experiment, the ability of such representations to model the distribution of a point process, by defining a maximum entropy model defined by a set of wavelet-based statistics, computed on a single observation. For these two problems, we observe that our representations lead to better performance than summary statistics commonly used in the literature on point processes. Finally, to study to what extent such representations can capture geometric structures of texture images, we define a maximum entropy model relying on similar wavelet statistics, yielding syntheses of similar visual quality to state-of-the-art models based on deep convolutional neural networks representations.

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