Optical microscopy, an invaluable tool in biology and medicine to observe and quantify cellular function, organ development, or disease mechanisms, requires constant trade-offs between spatial, temporal, and spectral resolution, invasiveness, acquisition time, and post-processing effort. Deep learning technologies have enabled multiple applications that are transforming our day-to-day routines, including the way we approach microscopy. Yet despite the ever-increasing computational power, it is often the lack of labeled training data that is the limiting factor for wide adoption in this domain. Annotating data is often a lengthy and expensive task, since it involves tedious work, generally by skilled experts.

In this thesis, I explored "weakly supervised" learning methods targeted at a variety of applications to enhance microscopy images and extract physical information from a single image. The specificity of these "weakly supervised" methods is the fact that they use very little prior information about the image in order to keep the effort to annotate training data as low as possible. Specifically, I reduced the dimensionality of the learning problem by targeting the experiment towards estimating the parameters of a spatially-variant point-spread function (PSF) model using a convolutional neural network (CNN), which does not require instrument- or object-specific calibration. Using such a model permitted to simulate realistically accurate training data that could be generalized, once the model was trained, to real microscopy images. I extensively benchmarked different network architectures, training datasets and simulation modalities towards the optimal PSF prediction performance and robustness to image degradation.

Starting from the estimated PSF model parameters, I developed a variety of applications, such as a semi-blind spatially-variant deconvolution method for image deblurring and enhancement, a robust and fast microscopy auto-focus, a method for the estimation of the object surface from a single 2D image, and a method for the estimation of the object velocity in a fluid, all of them with minimal need for a priori knowledge about the optical setup.