Alain Trouvé

Statistics on diffeomorphisms via tangent space representations

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.

POP: Patchwork of Parts Models for Object Recognition

We formulate a deformable template model for objects with an efficient mechanism for computation and parameter estimation. The data consists of binary oriented edge features, robust to photometric variation and small local deformations. The template is defined in terms of probability arrays for each edge type. A primary contribution of this paper is the definition of the instantiation of an object in terms of shifts of a moderate number local submodels—parts—which are subsequently recombined using a patchwork operation, to define a coherent statistical model of the data. Object classes are modeled as mixtures of patchwork of parts POP models that are discovered sequentially as more class data is observed. We define the notion of the support associated to an instantiation, and use this to formulate statistical models for multi-object configurations including possible occlusions. All decisions on the labeling of the objects in the image are based on comparing likelihoods. The combination of a deformable model with an efficient estimation procedure yields competitive results in a variety of applications with very small training sets, without need to train decision boundaries—only data from the class being trained is used. Experiments are presented on the MNIST database, reading zipcodes, and face detection.

Geodesic shooting and diffeomorphic matching via textured meshes

We propose a new approach in the context of diffeomorphic image matching with free boundaries. A region of interest is triangulated over a template, which is considered as a grey level textured mesh. A diffeomorphic transformation is then approximated by the piecewise affine deformation driven by the displacements of the vertices of the triangles. This provides a finite dimensional, landmark-type, reduction for this dense image comparison problem. Based on an optimal control model, we analyze and compare two optimization methods formulated in terms of the initial momentum: direct optimization by gradient descent, or root-finding for the transversality equation, enhanced by a preconditioning of the Jacobian. We finally provide a series of numerical experiments on digit and face matching.

The metric spaces, Euler equations, and normal geodesic image motions of computational anatomy

Over the past several years our group has been studying biological shape in the emerging new discipline of computational anatomy (CA). CA consists of several components: (i) the construction of coordinatized anatomical manifolds, (ii) comparison of anatomical manifolds, and (iii) inference of morphometric change on anatomical manifolds. In this paper we focus on (ii) the comparison of anatomical shapes and structures in imagery via metric mapping. The purpose of this paper is to examine the generation of the geodesics associated with the metric from several points of view, the first the Euler equation describing the geodesic diffeomorphic flow, and the second the variational formulation of the geodesic in terms of the minimizing flow of vector fields which generate them.

A Fast 3D Volume Reconstruction for Confocal Micro-rotation Cell Imaging

We propose a fast reconstruction algorithm for nonadherent living cell images from micro-rotational confocal microscopy. This procedure is called bi-protocol. Its key point consists in coupling micro-rotation data and conventional z-stack. Doing so, the estimation of the micro-rotation slice positions becomes a slice-to-volume registration allowing to reduce drastically the computing time. The validation of our procedure is performed on real data which gives significant improvement compared with the conventional z-stacking imaging.

Atlas-based shape analysis and classification of retinal optical coherence tomography images using the functional shape (fshape) framework

&NA; We propose a novel approach for quantitative shape variability analysis in retinal optical coherence tomography images using the functional shape (fshape) framework. The fshape framework uses surface geometry together with functional measures, such as retinal layer thickness defined on the layer surface, for registration across anatomical shapes. This is used to generate a population mean template of the geometry‐function measures from each individual. Shape variability across multiple retinas can be measured by the geometrical deformation and functional residual between the template and each of the observations. To demonstrate the clinical relevance and application of the framework, we generated atlases of the inner layer surface and layer thickness of the Retinal Nerve Fiber Layer (RNFL) of glaucomatous and normal subjects, visualizing detailed spatial pattern of RNFL loss in glaucoma. Additionally, a regularized linear discriminant analysis classifier was used to automatically classify glaucoma, glaucoma‐suspect, and control cases based on RNFL fshape metrics.

Detecting Curved Edges in Noisy Images in Sublinear Time

Detecting edges in noisy images is a fundamental task in image processing. Motivated, in part, by various real-time applications that involve large and noisy images, in this paper we consider the problem of detecting long curved edges under extreme computational constraints, that allow processing of only a fraction of all image pixels. We present a sublinear algorithm for this task, which runs in two stages: (1) a multiscale scheme to detect curved edges inside a few image strips; and (2) a tracking procedure to estimate their extent beyond these strips. We theoretically analyze the runtime and detection performance of our algorithm and empirically illustrate its competitive results on both simulated and real images.

Learning the kernel through examples: an application to shape classification

One important problem in any retrieval system is the design of good features and of good similarity measures between features. Usually these similarity functions are defined through ad-hoc distances between features. We propose a new way to design such distances, based on non-rigid deformation of nonlinear principal components, in the framework of semi-parametric statistical regression. The proposed approach is applied to the construction of new rotation invariant distance between planar curves.

Distortion Minimizing Geodesic Subspaces in Shape Spaces and Computational Anatomy

The estimation of finite dimensional nonlinear submanifold representing shape samples is of paramount importance in many applications. The Distortion Minimizing Geodesic Submanifold (DMGS) approach allows to select the most accurate submanifolds in term of distortion under a dimensionality constraint for shape spaces. We show that the computation of DMGS is widely compatible with the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework and the varifold distortion for application to computational anatomy. It allows the estimation of finite dimensional geodesic submanifolds in the difficult situation where we do not assume any one to one correspondance between shapes (parametrisation invariance). Unlike regular Tangent PCA, the computation of DMGS does not need to deal with the classical balance between the deformation cost from the template to target and the resulting distortion. On the contrary, the greedy minimization of the distortion under dimensionality constraints, hiding the deformation metric in the exponential map, suggests a new way to select between alternative metrics and shape spaces under the unifying point of view of the dimension/distortion curves in the spirit of the rate/distortion curves in information theory. Proof of concept on 2D and 3D experiments are discussed.