Extraction of trajectories of moving objects from video is an important step towards activity classification in a surveillance system. It has been recently shown that multiple motion fields (MMF) estimated from trajectories can efficiently describe the movement of objects, and allow an automatic classification of activities in the scene. However, the object segmentation and tracking may introduce gaps in trajectories due to the sensor failures (i.e. miss detections) or pedestrian occlusions. This may hamper the performance of activity classification from the observed trajectories. In this paper, we propose a predictive motion model based on MMF to describe the trajectory of the object. Precomputed motion fields are used to predict the possible position of an object based on its location and trajectory up to that point and therefore, the method can deal with the case of a missed detection in one or more frames, thus able to perform trajectory completion. Experiments on real data show that the proposed method is remarkable in dealing with up to around 70% of the positions missing in the pedestrian trajectories.

In this paper, we address the problem of image reconstruction with missing pixels or corrupted with impulse noise, when the locations of the corrupted pixels are not known. A logarithmic transformation is applied to convert the multiplication between the image and binary mask into an additive problem. The image and mask terms are then estimated iteratively with total variation regularization applied on the image, and l0 regularization on the mask term which imposes sparseness on the support set of the missing pixels. The resulting alternating minimization scheme simultaneously estimates the image and mask, in the same iterative process. The logarithmic transformation also allows the method to be extended to the Rayleigh multiplicative and Poisson observation models. The method can also be extended to impulse noise removal by relaxing the regularizer from the l0 norm to the l1 norm. Experimental results show that the proposed method can deal with a larger fraction of missing pixels than two phase methods, which first estimate the mask and then reconstruct the image.

Estimation of velocity fields from a video sequence is an important step towards activity classification in a surveillance system. It has been recently shown that multiple motion fields estimated from trajectories are an efficient tool to describe the movement of objects, allowing an automatic classification of activities in the scene. However, the trajectory detection in noisy environments is difficult, usually requiring some sort manual editing to complete or correct them. This paper proposes two novel contributions. First, an automatic method for building pedestrian trajectories in far-field surveillance scenarios is presented not requiring user intervention. This basically comprises the detection of multiple moving objects in a video sequence through the detection of the active regions, followed by the estimation of the velocity fields that is accomplished by performing region matching of the above regions at consecutive time instants. This leads to a sequence of centroids and corresponding velocity vectors, describing the local motions presented in the image. A motion correspondence algorithm is then applied to group the centroids in a contiguous sequence of frames into trajectories corresponding to each moving object. The second contribution is a method for automatically finding the trajectories from a library of previously computed ones. Experiments on extensive video sequences from university campuses show that motion fields can be reliably estimated from these automatically detected trajectories, leading to a fully automatic procedure for the estimation of multiple motion fields.

We propose a new fast algorithm for solving one of the standard approaches to ill-posed linear inverse problems (IPLIP), where a (possibly nonsmooth) regularizer is minimized under the constraint that the solution explains the observations sufficiently well. Although the regularizer and constraint are usually convex, several particular features of these problems (huge dimensionality, nonsmoothness) preclude the use of off-the-shelf optimization tools and have stimulated a considerable amount of research. In this paper, we propose a new efficient algorithm to handle one class of constrained problems (often known as basis pursuit denoising) tailored to image recovery applications. The proposed algorithm, which belongs to the family of augmented Lagrangian methods, can be used to deal with a variety of imaging IPLIP, including deconvolution and reconstruction from compressive observations (such as MRI), using either total-variation or wavelet-based (or, more generally, frame-based) regularization. The proposed algorithm is an instance of the so-called alternating direction method of multipliers, for which convergence sufficient conditions are known; we show that these conditions are satisfied by the proposed algorithm. Experiments on a set of image restoration and reconstruction benchmark problems show that the proposed algorithm is a strong contender for the state-of-the-art.

Ill-posed linear inverse problems (ILIP), such as restoration and reconstruction, are a core topic of signal/image processing. A standard formulation for dealing with ILIP consists in a constrained optimization problem, where a regularization function is minimized under the constraint that the solution explains the observations sufficiently well. The regularizer and constraint are usually convex; however, several particular features of these problems (huge dimensionality, non-smoothness) preclude the use of off-the-shelf optimization tools and have stimulated much research. In this paper, we propose a new efficient algorithm to handle one class of constrained problems (known as basis pursuit denoising) tailored to image recovery applications. The proposed algorithm, which belongs to the category of augmented Lagrangian methods, can be used to deal with a variety of imaging ILIP, including deconvolution and reconstruction from compressive observations (such as MRI). Experiments testify for the effectiveness of the proposed method.

In some imaging inverse problems, it may be desired that the solution simultaneously exhibits a set of properties not enforceable by a single regularizer. To attain this goal, one may use a linear combinations of regularizers, thus encouraging the solution to simultaneously exhibit the characteristics enforced by each of them. This paper addresses the optimization problem associated with this type of compound regularization, using an alternating direction optimization algorithm. We illustrate the approach in two image deblurring problems - one in which the images are simultaneously sparse and piece-wise smooth, using a linear combination of the ℓ1and total variation regularizers, and the other for a natural image with a combination of frame-based synthesis and analysis ℓ1norm regularizers.

We propose a new fast algorithm for solving one of the standard formulations of image restoration and reconstruction which consists of an unconstrained optimization problem where the objective includes an l2 data-fidelity term and a nonsmooth regularizer. This formulation allows both wavelet-based (with orthogonal or frame-based representations) regularization or total-variation regularization. Our approach is based on a variable splitting to obtain an equivalent constrained optimization formulation, which is then addressed with an augmented Lagrangian method. The proposed algorithm is an instance of the so-called alternating direction method of multipliers, for which convergence has been proved. Experiments on a set of image restoration and reconstruction benchmark problems show that the proposed algorithm is faster than the current state of the art methods.