Variational problems on flows of diffeomorphisms for image matching

Abstract

Abstract.This paper studies a variational formulation of the image matching problem. We consider a scenario in which a canonical representative image T is to be carried via a smooth change of variable into an image that is intended to provide a good fit to the observed data. where ||v|| is an appropriate norm on the velocity field v(-, •), and the second term attempts to enforce fidelity to the data. In this paper we derive conditions under which the variational problem described above is well posed. The key issue is the choice of the norm. Conditions are formulated under which the regularity of v(-, ■) imposed by finiteness of the norm guarantees that the associated flow is supported on a space of diffeomorphisms. The problem (0.2) can

Mapping cortical change in Alzheimer's disease, brain development, and schizophrenia

“…This distortion can be expressed using an operator, here denoted by L, which is typically the CauchyNavier differential operator L = mj 2 + (l + m)j(j T ! ), or powers of the Laplacian on space or space time (Christensen et al, 1996;Dupuis et al, 1998;Grenander and Miller, 1998;Toga, 1998;Toga and Thompson, 2003a,b). These operators tend to penalize high values of the Laplacian j 2 u(r) of the flow, as well as high values of the gradient of the divergence of the flow j(j T !…”

“…If not, the specific triangulations of the surfaces will affect how the surfaces are matched. In the covariant PDE approach (Christensen et al, 1996;Dupuis et al, 1998;Grenander and Miller, 1998;Thompson et al, 2000a,b;Toga, 1998;Toga and Thompson, 2003a,b), correction terms (Christoffel symbols, C i jk ) make the necessary adjustments for fluctuations in the metric tensor of the mapping procedure. In the partial differential equations (Eq.…”

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“…(We call this the p-admissibility condition.) Then the following can be shown ( [14], [55]): If u t is a time-dependent family of elements of g such that…”

“…These methods have enabled the systematic measurement and comparison of anatomical shapes and structures in biomedical imagery leading to better understanding of neurodevelopmental, neuropsychiatric and neurological disorders in recent years [5], [6], [11], [13], [17], [20], [42]- [48], [50], [52], [59]. The mathematical theory of Grenander's deformable template models, when applied to these problems, involves smooth invertible maps (diffeomorphisms), as presented in this context in [55], [56], [14], [41], [36] and [33]. In particular, the template matching approach involves Riemannian metrics on the diffeomorphism group and employs their projections onto specific landmark shapes, or image spaces.…”

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