Analysis of Descent-Based Image Registration

We present a performance analysis for image registration with gradient descent. We consider a typical multiscale registration setting where the global 2-D translation between a pair of images is estimated by smoothing the images and minimizing the distance between them with gradient descent. Our study particularly concentrates on the effect of noise and low-pass filtering on the alignment accuracy. We analyze the well-behavedness of the image distance function by estimating the neighborhood of translations for which it is free of undesired local minima. This is the neighborhood of translations that are correctly computable with a simple gradient descent minimization. We show that the area of this neighborhood increases at least quadratically with the smoothing filter size. We then examine the effect of noise on the alignment accuracy and derive an upper bound for the alignment error in terms of the noise properties and filter size. Our main finding is that the error increases at a rate that is at least linear with respect to the filter size. Therefore, smoothing improves the well-behavedness of the distance function; however, this comes at the cost of amplifying the alignment error in noisy settings. Our results provide a mathematical insight about why hierarchical techniques are effective in image registration, suggesting that the multiscale alignment strategy of these techniques is very suitable from the perspective of the trade-off between the well-behavedness of the objective function and the registration accuracy. To the best of our knowledge, this is the first such study for descent-based image registration.

Pascal Frossard, Elif Vural

We present a performance analysis for image registration with gradient descent methods. We consider a typical multiscale registration setting where the global 2-D translation between a pair of images is estimated by smoothing the images and minimizing the distance between them with gradient descent. Our study particularly concentrates on the effect of noise and low-pass filtering on the alignment accuracy. We adopt an analytic representation for images and analyze the well-behavedness of the image distance function by estimating the neighborhood of translations for which it is free of undesired local minima. This corresponds to the neighborhood of translation vectors that are correctly computable with a simple gradient descent minimization. We show that the area of this neighborhood increases at least quadratically with the smoothing filter size, which justifies the use of a smoothing step in image registration with local optimizers such as gradient descent. We then examine the effect of noise on the alignment accuracy and derive an upper bound for the alignment error in terms of the noise properties and filter size. Our main finding is that the error increases at a rate that is at least linear with respect to the filter size. Therefore, smoothing improves the well-behavedness of the distance function; however, this comes at the cost of amplifying the alignment error in noisy settings. Our results provide a mathematical insight about why hierarchical techniques are effective in image registration, suggesting that the multiscale coarse-to-fine alignment strategy of these techniques is very suitable from the perspective of the trade-off between the well-behavedness of the objective function and the registration accuracy. To the best of our knowledge, this is the first such study for descent-based image registration.

2013

Transformation-Invariant Analysis of Visual Signals with Parametric Models

Elif Vural

The analysis of collections of visual data, e.g., their classification, modeling and clustering, has become a problem of high importance in a variety of applications. Meanwhile, image data captured in uncontrolled environments by arbitrary users is very likely to be exposed to geometric transformations. Therefore, efficient methods are needed for analyzing high-dimensional visual data sets that can cope with geometric transformations of the visual content of interest. In this thesis, we study parametric models for transformation-invariant analysis of geometrically transformed image data, which provide low-dimensional image representations that capture relevant information efficiently. We focus on transformation manifolds, which are image sets created by parametrizable geometric transformations of a reference image model. Transformation manifolds provide a geometric interpretation of several image analysis problems. In particular, image registration corresponds to the computation of the projection of the target image onto the transformation manifold of the reference image. Similarly, in classification, the class label of a query image can be estimated in a transformation-invariant way by comparing its distance to transformation manifolds that represent different image classes. In this thesis, we explore several problems related to the registration, modeling, and classification of images with transformation manifolds. First, we address the problem of sampling transformation manifolds of known parameterization, where we focus on the target applications of image registration and classification in the sampling. We first propose an iterative algorithm for sampling a manifold such that the selected set of samples gives an accurate estimate of the distance of a query image to the manifold. We then extend this method to a classification setting with several transformation manifolds representing different image classes. We develop an algorithm to jointly sample multiple transformation manifolds such that the class label of query images can be estimated accurately by comparing their distances to the class-representative manifold samples. The proposed methods outperform baseline sampling schemes in image registration and classification. Next, we study the problem of learning transformation manifolds that are good models of a given set of geometrically transformed image data. We first learn a representative pattern whose transformation manifold fits well the input images and then generalize the problem to a supervised classification setting, where we jointly learn multiple class-representative pattern transformation manifolds from training images with known class labels. The proposed manifold learning methods exploit the information of the type of the geometric transformation in the data to compute an accurate data model, which is ignored in previous manifold learning algorithms. Finally, we focus on the usage of transformation manifolds in multiscale image registration. We consider two different methods in image registration, namely, the tangent distance method and the minimization of the image intensity difference with gradient descent. We present a multiscale performance analysis of these methods. We derive upper bounds for the alignment errors yielded by the two methods and analyze the variations of these bounds with noise and low-pass filtering, which is useful for gaining an understanding of the performance of these methods in image registration. To the best of our knowledge, these are the first such studies in multiscale registration settings. Geometrically transformed image sets have a particular structure, and classical image analysis methods do not always suit well for the treatment of such data. This thesis is motivated by this observation and proposes new techniques and insights for handling geometric transformations in image analysis and processing.

EPFL2013

Trust-region methods based on radial basis functions with application to biomedical imaging

Rodrigue Oeuvray

We have developed a new derivative-free algorithm based on Radial Basis Functions (RBFs). Derivative-free optimization is an active field of research and several algorithms have been proposed recently. Problems of this nature in the industrial setting are quite frequent. The reason is that in a number of applications the optimization process contains simulation packages which are treated as black boxes. The development of our own algorithm was originally motivated by an application in biomedical imaging: the medical image registration problem. The particular characteristics of this problem have incited us to develop a new optimization algorithm based on trust-region methods. However it has been designed to be generic and to be applied to a wide range of problems. The main originality of our approach is the use of RBFs to build the models. In particular we have adapted the existing theory based on quadratic models to our own models and developed new procedures especially designed for models based on RBFs. We have tested our algorithm called BOOSTERS against state-of-the-art methods (UOBYQA, NEWUOA, DFO). On the medical image registration problem, BOOSTERS appears to be the method of choice. The tests on problems from the CUTEr collection show that BOOSTERS is comparable to, but not better than other methods on small problems (size 2-20). It is performing very well for medium size problems (20-80). Moreover, it is able to solve problems of dimension 200, which is considered very large in derivative-free optimization. We have also developed a new class of algorithms combining the robustness of derivative-free algorithms with the faster rate of convergence characterizing Newtonlike-methods. In fact, they define a new class of algorithms lying between derivative-free optimization and quasi-Newton methods. These algorithms are built on the skeleton of our derivative-free algorithm but they can incorporate the gradient when it is available. They can be interpreted as a way of doping derivative-free algorithms with derivatives. If the derivatives are available at each iteration, then our method can be seen as an alternative to quasi-Newton methods. At the opposite, if the derivatives are never evaluated, then the algorithm is totally similar to BOOSTERS. It is a very interesting alternative to existing methods for problems whose objective function is expensive to evaluate and when the derivatives are not available. In this situation, the gradient can be approximated by finite differences and its costs corresponds to n additional function evaluations assuming that Rn is the domain of definition of the objective function. We have compared our method with CFSQP and BTRA, two gradient-based algorithms, and the results show that our doped method performs best. We have also a theoretical analysis of the medical image registration problem based on maximization of mutual information. Most of the current research in this field is concentrated on registration based on nonlinear image transformation. However, little attention has been paid to the theoretical properties of the optimization problem. In our analysis, we focus on the continuity and the differentiability of the objective function. We show in particular that performing a registration without extension of the reference image may lead to discontinuities in the objective function. But we demonstrate that, under some mild assumptions, the function is differentiable almost everywhere. Our analysis is important from an optimization point of view and conditions the choice of a solver. The usual practice is to use generic optimization packages without worrying about the differentiability of the objective function. But the use of gradient-based methods when the objective function is not differentiable may result in poor performance or even in absence of convergence. One of our objectives with this analysis is also that practitioners become aware of these problems and to propose them new algorithms having a potential interest for their applications.