Laurent Younes

Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms

AbstractThis paper examine the Euler-Lagrange equations for the solution of the large deformation diffeomorphic metric mapping problem studied in Dupuis et al. (1998) and Trouvé (1995) in which two images I0, I1 are given and connected via the diffeomorphic change of coordinates I0○ϕ−1=I1 where ϕ=Φ1 is the end point at t= 1 of curve Φt, t∈[0, 1] satisfying .Φt=vt (Φt), t∈ [0,1] with Φ0=id. The variational problem takes the form

Geodesic Shooting for Computational Anatomy

\mathop {\arg {\text{m}}in}\limits_{\upsilon :\dot \phi _t = \upsilon _t \left( {\dot \phi } \right)} \left( {\int_0^1 {\left\| {\upsilon _t } \right\|} ^2 {\text{d}}t + \left\| {I_0 \circ \phi _1^{ - 1} - I_1 } \right\|_{L^2 }^2 } \right),

Statistics on diffeomorphisms via tangent space representations

where ‖vt‖V is an appropriate Sobolev norm on the velocity field vt(·), and the second term enforces matching of the images with ‖·‖L2 representing the squared-error norm.In this paper we derive the Euler-Lagrange equations characterizing the minimizing vector fields vt, t∈[0, 1] assuming sufficient smoothness of the norm to guarantee existence of solutions in the space of diffeomorphisms. We describe the implementation of the Euler equations using semi-lagrangian method of computing particle flows and show the solutions for various examples. As well, we compute the metric distance on several anatomical configurations as measured by ∫01‖vt‖Vdt on the geodesic shortest paths.

Evidence of Structural Remodeling in the Dyssynchronous Failing Heart

This paper constructs metrics on the space of images I defined as orbits under group actions G. The groups studied include the finite dimensional matrix groups and their products, as well as the infinite dimensional diffeomorphisms examined in Trouvé (1999, Quaterly of Applied Math.) and Dupuis et al. (1998). Quaterly of Applied Math. Left-invariant metrics are defined on the product G × I thus allowing the generation of transformations of the background geometry as well as the image values. Examples of the application of such metrics are presented for rigid object matching with and without signature variation, curves and volume matching, and structural generation in which image values are changed supporting notions such as tissue creation in carrying one image to another.

A metric on shape space with explicit geodesics

Studying large deformations with a Riemannian approach has been an efficient point of view to generate metrics between deformable objects, and to provide accurate, non ambiguous and smooth matchings between images. In this paper, we study the geodesics of such large deformation diffeomorphisms, and more precisely, introduce a fundamental property that they satisfy, namely the conservation of momentum. This property allows us to generate and store complex deformations with the help of one initial “momentum” which serves as the initial state of a differential equation in the group of diffeomorphisms. Moreover, it is shown that this momentum can be also used for describing a deformation of given visual structures, like points, contours or images, and that, it has the same dimension as the described object, as a consequence of the normal momentum constraint we introduce.

Texture classification using windowed Fourier filters

A three‐dimensional (3D) diffusion‐weighted imaging (DWI) method for measuring cardiac fiber structure at high spatial resolution is presented. The method was applied to the ex vivo reconstruction of the fiber architecture of seven canine hearts. A novel hypothesis‐testing method was developed and used to show that distinct populations of secondary and tertiary eigenvalues may be distinguished at reasonable confidence levels (P ≤ 0.01) within the canine ventricle. Fiber inclination and sheet angles are reported as a function of transmural depth through the anterior, lateral, and posterior left ventricle (LV) free wall. Within anisotropic regions, two consistent and dominant orientations were identified, supporting published results from histological studies and providing strong evidence that the tertiary eigenvector of the diffusion tensor (DT) defines the sheet normal. Magn Reson Med, 2005. Published 2005 Wiley‐Liss, Inc.

Parametric Inference for imperfectly observed Gibbsian fields

In this paper, we present a linear setting for statistical analysis of shape and an optimization approach based on a recent derivation of a conservation of momentum law for the geodesics of diffeomorphic flow. Once a template is fixed, the space of initial momentum becomes an appropriate space for studying shape via geodesic flow since the flow at any point along the geodesic is completely determined by the momentum at the origin through geodesic shooting equations. The space of initial momentum provides a linear representation of the nonlinear diffeomorphic shape space in which linear statistical analysis can be applied. Specializing to the landmark matching problem of Computational Anatomy, we derive an algorithm for solving the variational problem with respect to the initial momentum and demonstrate principal component analysis (PCA) in this setting with three-dimensional face and hippocampus databases.

Metamorphoses Through Lie Group Action

Ventricular remodeling of both geometry and fiber structure is a prominent feature of several cardiac pathologies. Advances in MRI and analytical methods now make it possible to measure changes of cardiac geometry, fiber, and sheet orientation at high spatial resolution. In this report, we use diffusion tensor imaging to measure the geometry, fiber, and sheet architecture of eight normal and five dyssynchronous failing canine hearts, which were explanted and fixed in an unloaded state. We apply novel computational methods to identify statistically significant changes of cardiac anatomic structure in the failing and control heart populations. The results demonstrate significant regional differences in geometric remodeling in the dyssynchronous failing heart versus control. Ventricular chamber dilatation and reduction in wall thickness in septal and some posterior and anterior regions are observed. Primary fiber orientation showed no significant change. However, this result coupled with the local wall thinning in the septum implies an altered transmural fiber gradient. Further, we observe that orientation of laminar sheets become more vertical in the early-activated septum, with no significant change of sheet orientation in the late-activated lateral wall. Measured changes in both fiber gradient and sheet structure will affect both the heterogeneity of passive myocardial properties as well as electrical activation of the ventricles.