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Publication# Image alignment with rotation manifolds built on sparse geometric expansions

Résumé

In this paper we discuss the problem of alignment of patterns under arbitrary rotation. When a generic image pattern is geometrically transformed, it typically spans a (possibly nonlinear) manifold in a high dimensional space. When the pattern of interest is given by a sparse approximation over a structured dictionary of geometric atoms, we show that the rotation manifold can be expressed analytically as a function of the transformation parameters. At the same time, its high order derivatives are also given in a closed form when the pattern is represented as a sparse linear combination of a few differentiable basis functions. In this framework, the alignment problem is formulated as the minimization of the distance between the reference pattern and the manifold, which boils down to a nonlinear least squares optimization problem. We propose to solve this problem by a Newton-type method, whose solution is facilitated by the analytical expressions of the manifold derivatives. We further derive a global optimization heuristic algorithm based on Newton, and provide sufficient conditions for computing the global minimizer. Experimental results demonstrate the effectiveness of the proposed methodology for image alignment and rotation invariant pattern recognition.

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Manifold models provide low-dimensional representations that are useful for processing and analyzing data in a transformation-invariant way. In this paper, we study the problem of learning smooth pattern transformation manifolds from image sets that are observations of geometrically transformed signals. In order to construct a manifold, we build a representative pattern whose transformations accurately fit various input images. The pattern is formed by selecting a good common sparse approximation of the images with parametric and smooth atoms. We examine two aspects of the manifold building problem, where we first target an accurate transformation-invariant approximation of the input images, and then extend this solution for their classification. For the approximation problem, we propose a greedy method that constructs a representative pattern by selecting analytic atoms from a continuous dictionary manifold. We present a DC (Difference-of-Convex) optimization scheme which is applicable for a wide range of transformation and dictionary models, and demonstrate its application to transformation manifolds generated by the rotation, translation and anisotropic scaling of a reference pattern. Then, we generalize this approach to a setting with multiple transformation manifolds, where each manifold represents a different class of signals. We present an iterative multiple manifold building algorithm such that the classification accuracy is promoted in the joint selection of atoms. Experimental results suggest that the proposed methods yield high accuracy in the approximation and classification of data in comparison with some reference methods, while achieving invariance to geometric transformations due to the transformation manifold model.

We address the problem of building a manifold in order to represent a set of geometrically transformed images by selecting a good common sparse approximation of them with parametric atoms. We propose a greedy method to construct a representative pattern such that the total distance between the transformation manifold of the representative pattern and the input images is minimized. In the progressive construction of the pattern we select atoms from a continuous dictionary by optimizing the atom parameters. Experimental results suggest that the representative pattern built with the proposed method provides an accurate representation of data, where the invariance to geometric transformations is achieved due to the transformation manifold model.

2011Pascal Frossard, Philippe Jost, Pierre Vandergheynst

This paper proposes a tree-based pursuit algorithm that efficiently trades off complexity and approximation performance for overcomplete signal expansions. Finding the sparsest representation of a signal using a redundant dictionary is, in general, a NP-Hard problem. Even sub-optimal algorithms such as Matching Pursuit remain highly complex. We propose a structuring strategy that can be applied to any redundant set of functions, and which basically groups similar atoms together. A measure of similarity based on coherence allows for representing a highly redundant sub-dictionary of atoms by a unique element, called molecule. When the clustering is applied recursively on atoms and then on molecules, it naturally leads to the creation of a tree structure. We then present a new pursuit algorithm that uses the structure created by clustering as a decision tree. This tree-based algorithm offers important complexity reduction with respect to Matching Pursuit, as it prunes important parts of the dictionary when traversing the tree. Recent results on incoherent dictionaries are extended to molecules, while the true highly redundant nature of the dictionary stays hidden by the tree structure. We then derive recovery conditions on the structured dictionary, under which tree-based pursuit is guaranteed to converge. Experimental results finally show that the gain in complexity offered by tree-based pursuit does in general not have a high penalty on the approximation performance. They show that the dimensionality of the problem is reduced thanks to the tree construction, without significant loss of information at hand.

2005