Abstract This survey is intended to inform non-expert readers about the field of recommender systems, particularly collaborative filtering, through the lens of the impactful Netflix Prize competition. Readers will quickly be brought up to speed on pivotal recommender systems advances through the Netflix Prize, informing their prospective state-of-the-art research with meaningful historic artifacts. We begin with the pivotal FunkSVD approach early in the competition. We then discuss Probabilistic Matrix Factorization and the importance and extensibility of the model. We examine the strategies of the Netflix Prize winner, providing comparisons to the Probabilistic Matrix Factorization framework as well as commentary as to why one approach became extensively used in research while another did not. Collectively, these models help to understand the progression of collaborative filtering through the Netflix Prize era. In each topic, we include ample discussion of results and background information. Finally, we highlight major veins of research following the competition.

In this paper, a graph-based nonlocal total variation method is proposed for unsupervised classification of hyperspectral images (HSI). The variational problem is solved by the primal-dual hybrid gradient algorithm. By squaring the labeling function and using a stable simplex clustering routine, an unsupervised clustering method with random initialization can be implemented. The effectiveness of this proposed algorithm is illustrated on both synthetic and real-world HSI, and numerical results show that the proposed algorithm outperforms other standard unsupervised clustering methods, such as spherical K -means, nonnegative matrix factorization, and the graph-based Merriman-Bence-Osher scheme.

In this paper, we provide a short review of Retinex and then present a unifying framework. The fundamental assumption of all Retinex models is that the observed image is a multiplication between the illumination and the true underlying reflectance of the object. Starting from Morel's 2010 PDE model, where illumination is supposed to vary smoothly and where the reflectance is thus recovered from a hard-thresholded Laplacian of the observed image in a Poisson equation, we define our unifying Retinex model in two similar, but more general, steps. We reinterpret the gradient thresholding model as variational models with sparsity constraints. First, we look for a filtered gradient that is the solution of an optimization problem consisting of two terms: a sparsity prior of the reflectance and a fidelity prior of the reflectance gradient to the observed image gradient. Second, since this filtered gradient almost certainly is not a consistent image gradient, we then fit an actual reflectance gradient to it, subject to further sparsity and fidelity priors. This generalized formulation allows making connections with other variational or kernel-based Retinex implementations. We provide simple algorithms for the optimization problems resulting from our framework. In particular, in the quadratic case, we can link our model to a plausible neural mechanism through Wilson--Cowan equations. Beyond unifying existing models, we derive entirely novel Retinex flavors by using more interesting non-local versions for the sparsity and fidelity priors. Eventually, we define within a single framework new Retinex applications to shadow detection and removal, nonuniformity correction, cartoon-texture decomposition, as well as color and hyperspectral image enhancement.

We propose a segmentation method based on the geometric representation of images as 2-D manifolds embedded in a higher dimensional space. The segmentation is formulated as a minimization problem, where the contours are described by a level set function and the objective functional corresponds to the surface of the image manifold. In this geometric framework, both data-fidelity and regularity terms of the segmentation are represented by a single functional that intrinsically aligns the gradients of the level set function with the gradients of the image and results in a segmentation criterion that exploits the directional information of image gradients to overcome image inhomogeneities and fragmented contours. The proposed formulation combines this robust alignment of gradients with attractive properties of previous methods developed in the same geometric framework: 1) the natural coupling of image channels proposed for anisotropic diffusion and 2) the ability of subjective surfaces to detect weak edges and close fragmented boundaries. The potential of such a geometric approach lies in the general definition of Riemannian manifolds, which naturally generalizes existing segmentation methods (the geodesic active contours, the active contours without edges, and the robust edge integrator) to higher dimensional spaces, non-flat images, and feature spaces. Our experiments show that the proposed technique improves the segmentation of multi-channel images, images subject to inhomogeneities, and images characterized by geometric structures like ridges or valleys.

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.

The level set method is a popular technique for tracking moving interfaces in several disciplines, including computer vision and fluid dynamics. However, despite its high flexibility, the original level set method is limited by two important numerical issues. First, the level set method does not implicitly preserve the level set function as a distance function, which is necessary to estimate accurately geometric features, s.a. the curvature or the contour normal. Second, the level set algorithm is slow because the time step is limited by the standard Courant-Friedrichs-Lewy (CFL) condition, which is also essential to the numerical stability of the iterative scheme. Recent advances with graph cut methods and continuous convex relaxation methods provide powerful alternatives to the level set method for image processing problems because they are fast, accurate, and guaranteed to find the global minimizer independently to the initialization. These recent techniques use binary functions to represent the contour rather than distance functions, which are usually considered for the level set method. However, the binary function cannot provide the distance information, which can be essential for some applications, s.a. the surface reconstruction problem from scattered points and the cortex segmentation problem in medical imaging. In this paper, we propose a fast algorithm to preserve distance functions in level set methods. Our algorithm is inspired by recent efficient l(1) optimization techniques, which will provide an efficient and easy to implement algorithm. It is interesting to note that our algorithm is not limited by the CFL condition and it naturally preserves the level set function as a distance function during the evolution, which avoids the classical re-distancing problem in level set methods. We apply the proposed algorithm to carry out image segmentation, where our methods prove to be 5-6 times faster than standard distance preserving level set techniques. We also present two applications where preserving a distance function is essential. Nonetheless, our method stays generic and can be applied to any level set methods that require the distance information.

We propose a segmentation method based on the geometric representation of images as surfaces embedded in a higher dimensional space, handling naturally multichannel images. The segmentation is based on an active contour embedded in the image manifold, along with a set of image features. Hence, both data-fidelity and regularity terms of the active contour are jointly optimized minimizing a single Polaykov energy representing the hyper-surface of this manifold. Compared to previous methods, our approach is purely geometrical and does not require additional weighting of the energy functional to drive the segmentation to the image contours. The potential of such a geometric approach lies in the general definition of Riemannian manifolds, validating the proposed technique for scale-space methods, volumetric data or catadioptric images. We present here the segmentation technique called Harmonic Active Contours, give an implementation for multichannel images including gradient and region-based segmentation criteria and apply it to color images.

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 nonflat 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. Second, this method is, to the best of our knowledge, the first reparametrization 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.

In this paper, we propose a novel method to register images defined on spherical meshes. Instances of such spherical images include inflated cortical feature maps in brain medical imaging or images from omni directional cameras. We apply the Geodesic Active Fields (GAF) framework locally at each vertex of the mesh. Therefore we define a dense deformation field, which is embedded in a higher dimensional manifold, and minimize the weighted Polyakov energy. While the Polyakov energy itself measures the hyper area of the embedded deformation field, its weighting allows to account for the quality of the current image alignment. Iteratively minimizing the energy drives the deformation field towards a smooth solution of the registration problem. Although the proposed approach does not necessarily outperform state-of-the-art methods that are tightly tailored to specific applications, it is of methodological interest due to its high degree of flexibility and versatility.

In this paper we define a scale-space for cortical mean curvature maps on the sphere, that offers a hierarchical representation of the brain cortical structures, useful in multiscale registration and analysis algorithms. A spherical feature map was obtained through inflation of the cortical surface of one hemisphere, extracted from structural MR images. Using the Beltrami framework, we embedded this spherical mesh in a higher dimensional space and the feature assigned to a mesh vertex became an additional component of its coordinates. This enhanced mesh then evolved under Beltrami flow. Imposing an appropriate aspect ratio for the feature components, we thus minimized an interpolation between theL2and TV-norm of the map. The collection of all maps produced by this PDE formed a scale-space. Our results suggest that this scale-space provides a generalization of the brain map suitable for use e.g., within a multiscale registration framework.

Tandem mass spectrometry is a powerful technique for the identification of proteins contained in a biological sample. We present a grid-based application service for the analysis of tandem mass spectra, based on the open-source X/Tandem algorithm. The implemented parallelization strategy allows the distribution of analysis jobs to a grid of desktop PCs, utilizing their idle capacity. For this application service, a user-friendly project-oriented frontend was developed, implemented as a Web-based workspace. After analysis, results are visualized and presented to the user along with a set of powerful and novel tools to allow further interpretation and evaluation of search results.