Analysis of Image Registration with Tangent Distance

The computation of the geometric transformation between a reference and a target image, known as image registration or alignment, corresponds to the projection of the target image onto the transformation manifold of the reference image (the set of images generated by its geometric transformations). It often takes a nontrivial form such that exact computation of projections on the manifold is difficult. The tangent distance method is an effective alignment algorithm that exploits a linear approximation of the transformation manifold of the reference image. As theoretical studies about the tangent distance algorithm have been largely overlooked, we present in this work a detailed performance analysis of this useful algorithm, which can eventually help the selection of algorithm parameters. We consider a popular image registration setting using a multiscale pyramid of lowpass filtered versions of the (possibly noisy) reference and target images, which is particularly useful for recovering large transformations. We first show that the alignment error has a nonmonotonic variation with the filter size, due to the opposing effects of filtering on manifold nonlinearity and image noise. We then study the convergence of the multiscale tangent distance method to the optimal solution. We finally examine the performance of the tangent distance method in image classification applications. Our theoretical findings are confirmed by experiments on image transformation models involving translations, rotations and scalings. Our study is the first detailed study of the tangent distance algorithm that leads to a better understanding of its efficacy and to the proper selection of design parameters

Image registration with sparse approximations in parametric dictionaries

We examine in this paper the problem of image registration from the new perspective where images are given by sparse approximations in parametric dictionaries of geometric functions. We propose a registration algorithm that looks for an estimate of the global transformation between sparse images by examining the set of relative geometrical transformations between the respective features. We propose a theoretical analysis of our registration algorithm and we derive performance guarantees based on two novel important properties of redundant dictionaries, namely the robust linear independence and the transformation inconsistency. We propose several illustrations and insights about the importance of these dictionary properties and show that common properties such as coherence or restricted isometry property fail to provide sufficient information in registration problems. We finally show with illustrative experiments on simple visual objects and handwritten digits images that our algorithm outperforms baseline competitor methods in terms of transformation-invariant distance computation and classification

Towards multiple 3D bone surface identification and reconstruction using few 2D X-ray images for intraoperative applications

This article discusses a possible method to use a small number, e.g. 5, of conventional 2D X-ray images to reconstruct multiple 3D bone surfaces intraoperatively. Each bone’s edge contours in X-ray images are automatically identified. Sparse 3D landmark points of each bone are automatically reconstructed by pairing the 2D X-ray images. The reconstructed landmark point distribution on a surface is approximately optimal covering main characteristics of the surface. A statistical shape model, dense point distribution model (DPDM), is then used to fit the reconstructed optimal landmarks vertices to reconstruct a full surface of each bone separately. The reconstructed surfaces can then be visualised and manipulated by surgeons or used by surgical robotic systems

Part-to-whole Registration of Histology and MRI using Shape Elements

Image registration between histology and magnetic resonance imaging (MRI) is a challenging task due to differences in structural content and contrast. Too thick and wide specimens cannot be processed all at once and must be cut into smaller pieces. This dramatically increases the complexity of the problem, since each piece should be individually and manually pre-aligned. To the best of our knowledge, no automatic method can reliably locate such piece of tissue within its respective whole in the MRI slice, and align it without any prior information. We propose here a novel automatic approach to the joint problem of multimodal registration between histology and MRI, when only a fraction of tissue is available from histology. The approach relies on the representation of images using their level lines so as to reach contrast invariance. Shape elements obtained via the extraction of bitangents are encoded in a projective-invariant manner, which permits the identification of common pieces of curves between two images. We evaluated the approach on human brain histology and compared resulting alignments against manually annotated ground truths. Considering the complexity of the brain folding patterns, preliminary results are promising and suggest the use of characteristic and meaningful shape elements for improved robustness and efficiency.Comment: Paper accepted at ICCV Workshop (Bio-Image Computing

Curvature and Concentration of Hamiltonian Monte Carlo in High Dimensions

In this article, we analyze Hamiltonian Monte Carlo (HMC) by placing it in the setting of Riemannian geometry using the Jacobi metric, so that each step corresponds to a geodesic on a suitable Riemannian manifold. We then combine the notion of curvature of a Markov chain due to Joulin and Ollivier with the classical sectional curvature from Riemannian geometry to derive error bounds for HMC in important cases, where we have positive curvature. These cases include several classical distributions such as multivariate Gaussians, and also distributions arising in the study of Bayesian image registration. The theoretical development suggests the sectional curvature as a new diagnostic tool for convergence for certain Markov chains.Comment: Comments welcom