Computational microscopy. (Microscopie computationnelle)

Valentin Debarnot

Abstract: The contributions of this thesis are numerical and theoretical tools for the resolution of blind inverse problems in imaging. We first focus in the case where the observation operator is unknown (e.g. microscopy, astronomy, photography). A very popular approach consists in estimating this operator from an image containing point sources (microbeads or fluorescent proteins in microscopy, stars in astronomy). Such an observation provides a measure of the impulse response of the degradation operator at several points in the field of view. Processing this observation requires robust tools that can rapidly use the data. We propose a toolbox that estimates a degradation operator from an image containing point sources. The estimated operator has the property that at any location in the field of view, its impulse response is expressed as a linear combination of elementary estimated functions. This makes it possible to estimate spatially invariant (convolution) and variant (product-convolution expansion) operators. An important specificity of this toolbox is its high level of automation: only a small number of easily accessible parameters allows to cover a large majority of practical cases. The size of the point source (e.g. bead), the background and the noise are also taken in consideration in the estimation. This tool, coined PSF-estimator, comes in the form of a module for the Fiji software, and is based on a parallelized implementation in C++. The operators generated by an optical system are usually changing for each experiment, which ideally requires a calibration of the system before each acquisition. To overcome this, we propose to represent an optical system not by a single operator (e.g. convolution blur with a fixed kernel for different experiments), but by subspace of operators. This set allows to represent all the possible states of a microscope. We introduce a method for estimating such a subspace from a collection of low rank operators (such as those estimated by the toolbox PSF-Estimator). We show that under reasonable assumptions, this subspace is low-dimensional and consists of low rank elements. In a second step, we apply this process in microscopy on large fields of view and with spatially varying operators. This implementation is possible thanks to the use of additional methods to process real images (e.g. background, noise, discretization of the observation).The construction of an operator subspace is only one step in the resolution of blind inverse problems. It is then necessary to identify the degradation operator in this set from a single observed image. In this thesis, we provide a mathematical framework to this operator identification problem in the case where the original image is constituted of point sources. Theoretical conditions arise from this work, allowing a better understanding of the conditions under which this problem can be solved. We illustrate how this formal study allows the resolution of a blind deblurring problem on a microscopy example.[...]