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Optimal transport :
Theory and Applications
to cosmological Reconstruction and Image
processing
Reconstruction of peculiar velocities of galaxies from large-scale redshift catalogues provides an exemplary application of optimal transport. Here the cosmological Zeldovich approximation leads to either Monge–Ampère or Hamilton–Jacobi equation (also called the Bernouilli equation in cosmologists' parlance), which is solved by optimization of transport cost for data sets that include up to millions of objects, using continuous as well as combinatorial strategies; the resulting reconstructed velocities may further be used to constrain various quantities of interest for cosmologists.
In image analysis and processing, distributions of various quantities appear naturally: light and shadow on the physical surface of a picture, intensity in colour spaces, etc. Here transport optimization allows to measure the similarity (the Monge–Kantorovich distance, alternatively known as the Kantorovich–Wasserstein or earth-mover distance). Transport in physical space corresponds to problems of morphing, automatic recognition, classification, and retrieval of images and of image registration (reducing a set of likenesses of the same object to one coordinate system). Transport in colour space is a highly nonlinear transformation useful for automatic equalization of the colouring between images (say different reproductions of works of the same painter) and other adjustment of colour, such as the effacement of unwanted details or the “day for night” effect in filmmaking.