Fast Geodesic Active Fields for Image Registration Based on Splitting and Augmented Lagrangian Approaches

In this paper, we present an efficient numerical scheme for the recently introduced geodesic active fields (GAF) framework for geometric image registration. This framework considers the registration task as a weighted minimal surface problem. Hence, the data-term and the regularization-term are combined through multiplication in a single, parametrization invariant and geometric cost functional. The multiplicative coupling provides an intrinsic, spatially varying and data-dependent tuning of the regularization strength, and the parametrization invariance allows working with images of nonflat geometry, generally defined on any smoothly parametrizable manifold. The resulting energy-minimizing flow, however, has poor numerical properties. Here, we provide an efficient numerical scheme that uses a splitting approach; data and regularity terms are optimized over two distinct deformation fields that are constrained to be equal via an augmented Lagrangian approach. Our approach is more flexible than standard Gaussian regularization, since one can interpolate freely between isotropic Gaussian and anisotropic TV-like smoothing. In this paper, we compare the geodesic active fields method with the popular Demons method and three more recent state-of-the-art algorithms: NL-optical flow, MRF image registration, and landmark-enhanced large displacement optical flow. Thus, we can show the advantages of the proposed FastGAF method. It compares favorably against Demons, both in terms of registration speed and quality. Over the range of example applications, it also consistently produces results not far from more dedicated state-of-the-art methods, illustrating the flexibility of the proposed framework.

Fast Geodesic Active Fields for Image Registration based on Splitting and Augmented Lagrangian Approaches

Xavier Bresson, Jean-Philippe Thiran, Dominique Zosso

In this paper we present an efficient numerical scheme for the recently introduced Geodesic Active Fields (GAF) framework for geometric image registration. This framework considers the registration task as a weighted minimal surface problem. Hence, data-term and regularization term are combined trough multiplication in a single, parametrization invariant and geometric cost functional. The multiplicative coupling provides an intrinsic spatially varying and data-dependent tuning of the regularization strength, while the parametrization invariance allows working with images of non-flat geometry, generally defined on any smoothly parametrizable manifold. The directly resulting minimizing energy flow, however, has poor numerical properties. Here, we provide an efficient numerical scheme that uses a splitting approach: data and regularity term are optimized over two distinct deformation fields that are constrained to be equal via an augmented Lagrangian approach. Our approach is more flexible than standard Gaussian regularization, since one can interpolate freely between isotropic Gaussian and anisotropic TV-like smoothing. In this work, we compare the Geodesic Active Fields method against the popular and fast state-of-the-art Demons method. Overall, we can show the advantages of the proposed FastGAF method. It compares favorably against Demons, both in terms of registration speed and quality.

2011

Geodesic Active Fields - A Geometric Framework for Image Registration

Xavier Bresson, Jean-Philippe Thiran, Dominique Zosso

In this paper we present a novel geometric framework called geodesic active fields for general image registration. In image registration, one looks for the underlying deformation field that best maps one image onto another. This is a classic ill-posed inverse problem, which is usually solved by adding a regularization term. Here, we propose a multiplicative coupling between the registration term and the regularization term, which turns out to be equivalent to embed the deformation field in a weighted minimal surface problem. Then, the deformation field is driven by a minimization flow toward a harmonic map corresponding to the solution of the registration problem. This proposed approach for registration shares close similarities with the well-known geodesic active contours model in image segmentation, where the segmentation term (the edge detector function) is coupled with the regularization term (the length functional) via multiplication as well. As a matter of fact, our proposed geometric model is actually the exact mathematical generalization to vector fields of the weighted length problem for curves and surfaces introduced by Caselles-Kimmel-Sapiro. The energy of the deformation field is measured with the Polyakov energy weighted by a suitable image distance, borrowed from standard registration models. We investigate three different weighting functions, the squared error and the approximated absolute error for monomodal images, and the local joint entropy for multimodal images. As compared to specialized state-of-the-art methods tailored for specific applications, our geometric framework involves important contributions. Firstly, our general formulation for registration works on any parametrizable, smooth and differentiable surface, including non-flat and multiscale images. In the latter case, multiscale images are registered at all scales simultaneously, and the relations between space and scale are intrinsically being accounted for. Secondly, this method is, to the best of our knowledge, the first re-parametrization invariant registration method introduced in the literature. Thirdly, the multiplicative coupling between the registration term, i.e. local image discrepancy, and the regularization term naturally results in a data-dependent tuning of the regularization strength. Finally, by choosing the metric on the deformation field one can freely interpolate between classic Gaussian and more interesting anisotropic, TV-like regularization.

Institute of Electrical and Electronics Engineers2011

Nonideal sampling and regularized interpolation of noisy data

Sathish Ramani

Conventional sampling (Shannon's sampling formulation and its approximation-theoretic counterparts) and interpolation theories provide effective solutions to the problem of reconstructing a signal from its samples, but they are primarily restricted to the noise-free scenario. The purpose of this thesis is to extend the standard techniques so as to be able to handle noisy data. First, we consider a realistic setting where a multidimensional signal is prefiltered prior to sampling, and the samples are corrupted by additive noise. In order to counterbalance the effect of noise, the reconstruction problem is formulated in a variational framework where the solution is obtained by minimizing a continuous-domain Tikhonov-like L2-regularization subject to a ℓp-based data fidelity constraint. We present theoretical justification for the minimization of this cost functional and show that the global-minimum solution belongs to a shift-invariant space generated by a function that is generally not bandlimited. The optimal reconstruction space is characterized by a condition that links the generating function to the regularization operator and implies the existence of a B-spline-like basis. We also consider stochastic formulations – min-max and minimum mean-squared error (MMSE/Wiener) formulations – of the nonideal sampling problem and show that they yield the same type of estimators and point towards the existence of optimal shift-invariant spaces for certain classes of stochastic processes. In the stochastic context, we also derive an exact formula for the error of approximating a stationary stochastic signal in the presence of discrete additive noise and justify the noise-reducing effect of regularization through illustrations. Next, we focus on the use of a much wider class of non-quadratic regularization functionals for the problem of interpolation in the presence of noise. Starting from the afine-invariance of the solution, we show that the Lp-norm (p ≠ 2) is the most suitable type of non-quadratic regularization for our purpose. We give monotonically convergent numerical algorithms to carry out the minimization of the non-quadratic cost criterion. We also demonstrate experimentally that the proposed regularized interpolation scheme provides superior interpolation performance compared to standard methods in the presence of noise. Finally, we address the problem of selecting an appropriate value for the regularization parameter which is most crucial for the working of variational methods in general including those discussed in this thesis. We propose a practical scheme that is based on the concept of risk estimation to achieve minimum MSE performance. In this context, we first review a well known result due to Stein (Stein's unbiased risk estimate — SURE) that is applicable for data corrupted by additive Gaussian noise and also derive a new risk estimate for a Poisson-Gaussian mixture model that is appropriate for certain biomedical imaging applications. Next, we introduce a novel and efficient Monte-Carlo technique to compute SURE for arbitrary nonlinear algorithms. We demonstrate experimentally that the proposed Monte-Carlo SURE yields regularization parameter values that are close to the oracle-optimum (minimum MSE) for all methods considered in this work. We also present results that illustrate the applicability of our technique to a wide variety of algorithms in denoising and deconvolution.

EPFL2009

Geodesic Active Fields - A Geometric Framework for Image Registration

Xavier Bresson, Jean-Philippe Thiran, Dominique Zosso

Institute of Electrical and Electronics Engineers2011

Nonideal sampling and regularized interpolation of noisy data

Sathish Ramani

EPFL2009