Several recent revolutions in imaging rely on numerical computations. One can think of single molecule localization microscopy (Nobel Prize 2014) or cryo-electron microscopy (Nobel Prize 2017). What they have in common is the need to perform prior mathematical modeling and calibration of the system. Although they have made it possible to observe phenomena that were previously out of reach, their expansion is currently limited by an important problem: it is difficult to precisely control the imaging conditions (e.g. temperatures, wavelengths, refractive indices). This results in modeling errors that can have disastrous repercussions on the quality of the images produced. Thus, these technologies are currently reserved for a handful of research centers possessing state-of-the-art equipment and considerable interdisciplinary experience. The objective of this project is to bring new theoretical and numerical solutions to overcome these difficulties, and then to apply them to different optical microscopes. This should allow to democratize their use, to reduce their cost and the preparation time of the experiments. The central idea is to characterize a measurement device, not by a single operator (e.g. a convolution), but by a small dimensional family allowing to model all possible states of the system. To our knowledge, this idea has been very little explored so far and opens many difficult questions: how to evaluate this family experimentally and numerically? How to identify the state of the system from indirect noisy observations? How to exploit this information to reconstruct images in short computing times? We have begun to explore these questions in recent works and wish to continue this effort using tools from optimization, harmonic analysis, probability and statistics, algebraic geometry, machine learning and massively parallel computing. We hope to make significant advances in the field of blind inverse problems. We will validate them on photonic microscopy problems in collaboration with opticians, responsible for two microscopy platforms. This will allow us to obtain direct feedback for real problems in biology. We will particularly study the problems of super-resolution by single molecule, multi-focal localization and blind structured illumination. Moreover, several companies in the Toulouse area (INNOPSYS, IMACTIV-3D, AGENIUM), will provide us with data from their microscopes (line scanning microscope, light sheet fluorescence microscope), which will ensure direct transfers to industry. The impact of this project goes far beyond optical microscopy, since similar problems exist in astrophysics, earth observation, non-destructive control and seismology. The funding of this project would promote the creation of an interdisciplinary team in computational imaging at the Center for Integrative Biology in Toulouse (CBI) and would encourage interactions and knowledge exchanges between the MORPHEME team in Sophia-Antipolis who is already a reference in the Côte d’Azur region and the new team in Toulouse.