The course introduces to digital images, their mathematical structure, and to the mathematical theories that explain how images are perceived and how they can be automatically analysed and modified. The course will present and involve mathematical tools such as basic Fourier analysis, a few classic partial differential equations, and continuous and discrete probability. Their presentation, being specific of images, will be as self-contained as possible, given the somewhat nonstandard use given to these tools in image processing. Each chapter of the course ends up in a description of a practical powerful algorithm to process images.

We start with the question of linking by Fourier analysis the discrete object (the digital image) to its continuous representation, that enables the use of mathematical operators.

We continue with the theory and algorithms for image resampling, color and contrast manipulation, image retouching.

Then the theory for invariant image representation, shape recognition, and automatic image comparison.

The students will practice these algorithms by themselves on their own images.