Méthodes Variationnelles pour le Recalage Multimodal
Gerardo Hermosillo Valadez, FAUGERAS, Olivier · OpenGrey (Institut de l'Information Scientifique et Technique) · 2002
During the past few years, the use of the theory of partial differential equations has provided a solid formal approach to image processing and analysis research, and has yielded provably well-posed algorithms within a set of clearly defined hypotheses. These algorithms are the state-of-the-art in a large number of application fields such as image de-noising, segmentation and matching. At the same time, the combination of stochastic and variational approaches has led to powerful algorithms which may also be described in terms of partial differential equations. This is the approach followed in the present work, which studies the problem of dense matching between two images using statistical dissimilarity criteria. Two classes of algorithms are considered, corresponding to these criteria being calculated globally for the entire image, or locally within corresponding regions. In each case, three dissimilarity criteria are studied, defined as the opposite of the following similarity measures: mutual information (well adapted to a general statistical dependence between the grey-level intensities), correlation ratio (adapted to a functional dependence), and cross correlation (adapted to an affine dependence). The minimization of the sum of the dissimilarity term and a regularization term defines, through the associated Euler-Lagrange equations, a set of coupled functional evolution equations. Particular emphasis is put in establishing the conditions under which these evolution equations are well posed, i.e. they have a unique solution. It is shown that the proposed algorithms satisfy these conditions for two classes of linear regularization terms, including one which encourages discontinuities of the solution at the contours of the reference image. The discretization and the numerical implementation of the matching algorithms is discussed in detail and their performance is illustrated through several real and synthetic examples, both with 2D and 3D images. As these examples show, the described algorithms are of interest in applications which do not necessarily involve sensors of multiple modalities. They are also of special interest to the medical imaging community, where data fusion between different imaging sensors often requires correcting for nonlinear distortions.