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The analysis, processing, evolution, optimization and/or regulation and control of shapes and images appear naturally in engineering (shape optimization, image processing, visual control), in numerical analysis (interval analysis), in physics (front propagation] problems), in biological morphogenesis, in population dynamics (migrations) and dynamic economic theory.

These problems are currently studied with tools forged out of differential geometry and functional analysis, thus requiring shapes and images to be smooth.

However, shapes and images are basically sets, most often not smooth. We propose another vision, where shapes and images are just any compact sets. Hence their evolution --- which requires a kind of differential calculus --- must be studied in the metric space of compact subsets. Fortunately, despite the loss of linearity, one can transfer most of the basic results of differential calculus and differential equations in vector spaces to mutational calculus and mutational equations in any mutational space, including naturally the space of nonempty compact subsets.