Inverse problems in signal and image processing

In many fields of applied physics, such as optics, radar, thermics, spectroscopy, geophysics, acoustics, radio astronomy, non-destructive testing, biomedical engineering, instrumentation and imaging in general, the problem arises of determining the spatial distribution of a scalar or vector quantity, often called the object, from measurements. Depending on the case, these measurements of the object may be direct – we call them images – or indirect – we call them projections in the case of tomography, or visibility in astronomy, for example. Solving such an imaging problem can usually be broken down into three stages:

• a direct problem where, knowing the object and mechanism of observation, we establish a mathematical description of the observed data. This model must be precise enough to provide a correct description of the observed physical phenomenon, yet simple enough to lend itself to subsequent numerical processing;

• an instrumentation problem, where we need to collect the most informative data possible in order to solve the imaging problem in the best possible conditions;

• an inverse problem, where we need to calculate an acceptable image of the object from the previous model and data.

Good object estimation obviously requires these three sub-problems to be studied in a coordinated way. However, the common characteristic of these image reconstruction or restoration problems is that they are often ill-posed or ill-conditioned. The higher-level problems encountered in computer vision, such as image segmentation, optical flow processing and shape reconstruction from shading, are also inverse problems, and suffer from the same difficulties.

Schematically, there are two main scientific communities interested in these inverse problems, from a methodological point of view:

• mathematical physics, which can be traced back to the seminal work of Phillips, Twomey and Tikhonov in the 1960s, pioneered in France by P.C. Sabatier (with his programmed thematic action of the same name), and whose representative journal is "Inverse problems";

• statistical data processing, which can be traced back to Franklin's work in the late 1960s, and whose gas pedals in image processing were the Geman brothers, whose representative journal is "IEEE Transactions on Image Processing".

Roughly speaking, we can say that some approach the problem in infinite dimension, with questions of existence, uniqueness and stability that become very complicated with direct nonlinear problems, and solve it numerically in finite dimension; whereas others start with a problem...