A snake is an energy-minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges. Snakes are active contour models: they lock onto nearby edges, localizing them accurately. Scale-space continuation can be used to enlarge the cap-ture region surrounding a feature. Snakes provide a unified account of a number of visual problems, in-cluding detection of edges, lines, and subjective contours; motion tracking; and stereo matching. We have used snakes successfully for interactive interpretation, in which user-imposed constraint forces guide the snake near features of interest.

We propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution. We focus on a semi-supervised framework that incorporates labeled and unlabeled data in a general-purpose learner. Some transductive graph learning algorithms and standard methods including Support Vector Machines and Regularized Least Squares can be obtained as special cases. We utilize properties of Reproducing Kernel Hilbert spaces to prove new Representer theorems that provide theoretical basis for the algorithms. As a result (in contrast to purely graph-based approaches) we obtain a natural out-of-sample extension to novel examples and so are able to handle both transductive and truly semi-supervised settings. We present experimental evidence suggesting that our semi-supervised algorithms are able to use unlabeled data effectively. Finally we have a brief discussion of unsupervised and fully supervised learning within our general framework. 1.

We are interested in regularizing fields of orthonormal vector sets, using constraint-preserving anisotropic diffusion PDE's. Each point of such a field is defined by multiple orthogonal and unitary vectors and can indeed represent a lot of interesting orientation features such as direction vectors or orthogonal matrices (among other examples). We first develop a general variational framework that solves this regularization problem, thanks to a constrained minimization of -functionals. This leads to a set of coupled vector-valued PDE's preserving the orthonormal constraints. Then, we focus on particular applications of this general framework, including the restoration of noisy direction fields, noisy chromaticity color images, estimated camera motions and DT-MRI (Di usion Tensor MRI) datasets.

This paper deals with the problem of regularizing noisy fields of diffusion tensors, considered as symmetric and semi-positive definite n n matrices (as for instance 2D structure tensors or DT-MRI medical images). We first propose a simple anisotropic PDE-based scheme that acts directly on the matrix coefficients and preserve the semipositive constraint thanks to a specific reprojection step. The limitations of this algorithm lead us to introduce a more effective approach based on constrained spectral regularizations acting on the tensor orientations (eigenvectors) and diffusivities (eigenvalues), while explicitely taking the tensor constraints into account. The regularization of the orientation part uses orthogonal matrices diffusion PDE's and local vector alignment procedures and will be particularly developed. For the interesting 3D case, a special implementation scheme designed to numerically fit the tensor constraints is also proposed. Experimental results on synthetic and real DT-MRI data sets finally illustrates the proposed tensor regularization framework.

Linear Discriminant Analysis (LDA) has been a popular method for extracting features which preserve class separability. The projection vectors are commonly obtained by maximizing the between class covariance and simultaneously minimizing the within class covariance. In practice, when there is no sufficient training samples, the covariance matrix of each class may not be accurately estimated. In this paper, we propose a novel method, called Semisupervised Discriminant Analysis (SDA), which makes use of both labeled and unlabeled samples. The labeled data points are used to maximize the separability between different classes and the unlabeled data points are used to estimate the intrinsic geometric structure of the data. Specifically, we aim to learn a discriminant function which is as smooth as possible on the data manifold. Experimental results on single training image face recognition and relevance feedback image retrieval demonstrate the effectiveness of our algorithm. 1.

We address the problem of restoring, while preserving possible discontinuities, fields of noisy orthonormal vector sets, taking the orthonormal constraints explicitly into account. We develop a variational solution for the general case where each image feature may correspond to multiple n-D orthogonal vectors of unit norms. We first formulate the problem in a new variational framework, where discontinuities and orthonormal constraints are preserved by means of constrained minimization and -functions regularization, leading to a set of coupled anisotropic diffusion PDE's. A geometric interpretation of the resulting equations, coming from the field of solid mechanics, is proposed for the 3D case. Two interesting restrictions of our framework are also tackled : the regularization of 3D rotation matrices and the Direction diffusion (the parallel with previous works is made). Finally, we present a number of denoising results and applications.

In this paper we propose a new way to iteratively solve the image reconstruction problem from noisy images or noisy data linearly related to the pixel intensities. This is done by exploiting the relation between Tikhonov regularization and multiobjective optimization to obtain iteratively approximations to the Tikhonov L-curve and its corner. Monitoring the change of the approximate L-curves allows us to adjust the regularization parameter adaptively during a preconditioned conjugate gradient iteration, so that the desired image can be reconstructed with a low number of iterations. Nonnegativity constraints are taken into account automatically. We present test results on image reconstruction in positron emission tomography (PET). Keywords: Tikhonov regularization, multiobjective optimization, ill-posed, L-curve, envelope, preconditioned conjugate gradients, image reconstruction, positron emission tomography (PET) 1991 MSC Classification: primary 65F10, secondary 65R30, 68U10, 90C29, 9...

The deconvolution of blurred and noisy satellite images is an ill-posed inverse problem. Donoho has proposed to deconvolve the image without regularization and to denoise the result in a wavelet basis by thresholding the transformed coefficients. We have developed a new filtering method, consisting of using a complex wavelet packet basis. Herein, the thresholding functions associated to the proposed method are automatically estimated. The estimation is performed within a Bayesian framework, by modeling the subbands using Generalized Gaussian distributions, and by applying the Maximum A Posteriori (MAP) estimator on each coefficient. Compared to real wavelet-packet-based algorithms, the proposed method is shift invariant, provides good directionality properties and remains of complexity O(N). Y=HX+NwhereHX=h?X 1.

Recently, molecular imaging has been rapidly developed to study physiological and pathological processes in vivo at the cellular and molecular levels. Among molecular imaging modalities, optical imaging has attracted a major attention for its unique advantages. In this paper, we establish a mathematical framework for multispectral bioluminescence tomography that allows simultaneous studies of multiple optical reporters. We show solution existence, uniqueness and continuous dependence on data, as well as the limiting behaviors when the regularization parameter approaches zero or when the penalty parameter approaches infinity. Then, we propose two numerical schemes for multispectral bioluminescence tomography and derive error estimates for the corresponding solutions.

What we believe images are determines how we take actions in image and lowlevel vision analysis. In the Bayesian framework, it is known as the importance of a good image prior model. This paper intends to give a concise overview on the vision foundation, mathematical theory, computational algorithms, and various classical as well as unexpected new applications of the BV (bounded variation) image model, first introduced into image processing by Rudin, Osher, and Fatemi in 1992 [Physica D, 60:259-268].