Thomas Pock

A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging

In this paper we study a first-order primal-dual algorithm for non-smooth convex optimization problems with known saddle-point structure. We prove convergence to a saddle-point with rate O(1/N) in finite dimensions for the complete class of problems. We further show accelerations of the proposed algorithm to yield improved rates on problems with some degree of smoothness. In particular we show that we can achieve O(1/N2) convergence on problems, where the primal or the dual objective is uniformly convex, and we can show linear convergence, i.e. O(ωN) for some ω∈(0,1), on smooth problems. The wide applicability of the proposed algorithm is demonstrated on several imaging problems such as image denoising, image deconvolution, image inpainting, motion estimation and multi-label image segmentation.

A duality based approach for realtime TV-L 1 optical flow

Variational methods are among the most successful approaches to calculate the optical flow between two image frames. A particularly appealing formulation is based on total variation (TV) regularization and the robust L1 norm in the data fidelity term. This formulation can preserve discontinuities in the flow field and offers an increased robustness against illumination changes, occlusions and noise. In this work we present a novel approach to solve the TV-L1 formulation. Our method results in a very efficient numerical scheme, which is based on a dual formulation of the TV energy and employs an efficient point-wise thresholding step. Additionally, our approach can be accelerated by modern graphics processing units. We demonstrate the real-time performance (30 fps) of our approach for video inputs at a resolution of 320 × 240 pixels.

Total Generalized Variation

The novel concept of total generalized variation of a function

PROST: Parallel robust online simple tracking

u

Anisotropic Huber-L1 Optical Flow.

is introduced, and some of its essential properties are proved. Differently from the bounded variation seminorm, the new concept involves higher-order derivatives of

An Improved Algorithm for TV-L1 Optical Flow

u

Diagonal preconditioning for first order primal-dual algorithms in convex optimization

. Numerical examples illustrate the high quality of this functional as a regularization term for mathematical imaging problems. In particular this functional selectively regularizes on different regularity levels and, as a side effect, does not lead to a staircasing effect.

A convex relaxation approach for computing minimal partitions

Tracking-by-detection is increasingly popular in order to tackle the visual tracking problem. Existing adaptive methods suffer from the drifting problem, since they rely on self-updates of an on-line learning method. In contrast to previous work that tackled this problem by employing semi-supervised or multiple-instance learning, we show that augmenting an on-line learning method with complementary tracking approaches can lead to more stable results. In particular, we use a simple template model as a non-adaptive and thus stable component, a novel optical-flow-based mean-shift tracker as highly adaptive element and an on-line random forest as moderately adaptive appearance-based learner. We combine these three trackers in a cascade. All of our components run on GPUs or similar multi-core systems, which allows for real-time performance. We show the superiority of our system over current state-of-the-art tracking methods in several experiments on publicly available data.

A Globally Optimal Algorithm for Robust TV-L 1 Range Image Integration

TV regularization is an L1 penalization of the flow gradient magnitudes, and due to the tendency of the L1 norm to favor sparse solutions (i.e. lots of ‘zeros’), the fill-in effect caused by the regularizer leads to piecewise constant solutions in weakly textured areas. This effect, known as ‘staircasing’ in a 1D setting, can be reduced significantly by using a quadratic penalization for small gradient magnitudes while sticking to linear penalization for larger magnitudes to maintain the discontinuity preserving properties known from TV. A comparison of isotropic TV and isotropic Huber regularity is shown in Fig. 1 by means of rendering the disparities u1 of the Dimetrodon dataset. The color coded flow (cf. Fig. 1(a)) is superimposed as texture. Based on the two observations that motion discontinuities often occur along object boundaries and that in turn object boundaries often coincide

Second order total generalized variation (TGV) for MRI.

A look at the Middlebury optical flow benchmark [5] reveals that nowadays variational methods yield the most accurate optical flow fields between two image frames. In this work we propose an improvement variant of the original duality based TV-L 1 optical flow algorithm in [31] and provide implementation details. This formulation can preserve discontinuities in the flow field by employing total variation (TV) regularization. Furthermore, it offers robustness against outliers by applying the robust L 1 norm in the data fidelity term. Our contributions are as follows. First, we propose to perform a structure-texture decomposition of the input images to get rid of violations in the optical flow constraint due to illumination changes. Second, we propose to integrate a median filter into the numerical scheme to further increase the robustness to sampling artefacts in the image data. We experimentally show that very precise and robust estimation of optical flow can be achieved with a variational approach in real-time. The numerical scheme and the implementation are described in a detailed way, which enables reimplementation of this high-end method.