Topology preservation for image-registration-related deformation fields

Abstract

In this paper, we address the issue of designing a theoretically well-motivated and computationally efficient method ensuring topology preservation on image-registration-related deformation fields. The model is motivated by a mathematical characterization of topology preservation for a deformation field mapping two subsets of Z 2 , namely, positivity of the four approximations to the Jacobian determinant of the deformation on a square patch. The first step of the proposed algorithm thus consists in correcting the gradient vector field of the deformation (that does not comply with the topology preservation criteria) at the discrete level in order to fulfill this positivity condition. Once this step is achieved, it thus remains to reconstruct the deformation field, given its full set of discrete gradient vectors. We propose to decompose the reconstruction problem into independent problems of smaller dimensions, yielding a natural parallelization of the computations and enabling us to reduce drastically the computational time (up to 80 in some applications). For each subdomain, a functional minimization problem under Lagrange interpolation constraints is introduced and its well-posedness is studied: existence/uniqueness of the solution, characterization of the solution, convergence of the method when the number of data increases to infinity, discretization with the Finite Element Method and discussion on the properties of the matrix involved in the linear system. Numerical simulations based on OpenMP parallelization and MKL multi-threading demonstrating the ability of the model to handle large deformations (contrary to classical methods) and the interest of having decomposed the problem into smaller ones are provided.

Image comparison and scaling via nonlinear elasticity

“…Our work is closest in spirit to that of Droske & Rumpf [14], Rumpf [27] and Rumpf & Wirth [26], who like us make use of the existence theory for polyconvex energies in [4], as also do Burger, Modersitski & Ruthotto [9], Debroux et al [12], Iglesias, Rumpf & Scherzer [16] and Iglesias [15]. Other nonlinear elasticity approaches are due to Lin, Dinov, Toga & Vese [19], Ozeré, Gout & Le Guyader [22], Ozeré & Le Guyader [23], Simon, Sheorey, Jacobs & Basri [28] and Debroux & Le Guyader [13]. Key differences with these works are: (i) that we minimize among homeomorphisms of the image domains rather than applying Dirichlet or other boundary conditions, (ii) technical improvements as regards the regularity of the intensity maps, and (iii) a novel analysis of linearly related images.…”

Joint Segmentation/Registration Model by Shape Alignment via Weighted Total Variation Minimization and Nonlinear Elasticity

“…An alternative method would consist of applying a correction step when the Jacobian is not positive, as done by Ozeré and Le Guyader in [41].…”

Robust cardiac motion estimation using ultrafast ultrasound data: a low-rank topology-preserving approach

“…The Jacobian determinant has been used in problems involving deformable structures in order to achieve more realistic transformations (e.g. Christensen et al (1996), Ashburner et al (1999), Ozere and Le Guyader (2015)). However, in this section we cover only those that have a relation with the modeling of deformable objects.…”