A Mathematical Perspective Of Image Denoising

Digital images are matrices of regularly spaced samples, the pixels, each containing a photon count. Each pixel thus contains a random sample of a Poisson variable. Its mean would be the ideal image value at this pixel. It follows that all images are random discrete processes and therefore "noisy". Ever since digital images exist, numerical methods have been proposed to recover the ideal mean from its random observed value. This problem is obviously ill posed and makes sense only if there is an underlying image model. Inventing or learning from data a consistent mathematically image model is the core of the problem. Images being 2D projections of our complex surrounding visual world, this is a challenging problem, which is nevertheless beginning to find simple but mathematically innovative answers. We shall distinguish four classes of denoising principles, relying on functional or stochastic image models. We show that each of these principles can be summarized in a single formula. In addition these principles can be combined efficiently to cope with the full image complexity. This explains their immediate industrial impact. All current cameras and imaging devices rely directly on the simple formulas explained here. In the past ten years the image quality delivered to users has increased fast thanks to this exemplary mathematical modeling.As an illustration of the universality and simplicity reached by the theory, most image denoising algorithms discussed in this paper can be tested directly on any digital image at Image Processing On Line, http://www.ipol.im/. In this web journal, each paper contains a complete algorithmic description, the corresponding source code, and can be run online on arbitrary images.