Combinatorial Continuous Maximal Flows

Maximum flow (and minimum cut) algorithms have had a strong impact on computer vision. In particular, graph cuts algorithms provide a mechanism for the discrete optimization of an energy functional which has been used in a variety of applications such as image segmentation, stereo, image stitching and texture synthesis. Algorithms based on the classical formulation of max-flow defined on a graph are known to exhibit metrication artefacts in the solution. Therefore, a recent trend has been to instead employ a spatially continuous maximum flow (or the dual min-cut problem) in these same applications to produce solutions with no metrication errors. However, known fast continuous max-flow algorithms have no stopping criteria or have not been proved to converge. In this work, we revisit the continuous max-flow problem and show that the analogous discrete formulation is different from the classical max-flow problem. We then apply an appropriate combinatorial optimization technique to this combinatorial continuous max-flow CCMF problem to find a null-divergence solution that exhibits no metrication artefacts and may be solved exactly by a fast, efficient algorithm with provable convergence. Finally, by exhibiting the dual problem of our CCMF formulation, we clarify the fact, already proved by Nozawa in the continuous setting, that the max-flow and the total variation problems are not always equivalent.Comment: 26 page

TV-Stokes And Its Variants For Image Processing

The total variational minimization with a Stokes constraint, also known as the TV-Stokes model, has been considered as one of the most successful models in image processing, especially in image restoration and sparse-data-based 3D surface reconstruction. This thesis studies the TV-Stokes model and its existing variants, proposes new and more effective variants of the model and their algorithms applied to some of the most interesting image processing problems. We first review some of the variational models that already exist, in particular the TV-Stokes model and its variants. Common techniques like the augmented Lagrangian and the dual formulation, are also introduced. We then present our models as new variants of the TV-Stokes. The main focus of the work has been on the sparse surface reconstruction of 3D surfaces. A model (WTR) with a vector fidelity, that is the gradient vector fidelity, has been proposed, applying it to both 3D cartoon design and height map reconstruction. The model employs the second-order total variation minimization, where the curl-free condition is satisfied automatically. Because the model couples both the height and the gradient vector representing the surface in the same minimization, it constructs the surface correctly. A variant of this model is then introduced, which includes a vector matching term. This matching term gives the model capability to accurately represent the shape of a geometry in the reconstruction. Experiments show a significant improvement over the state-of-the-art models, such as the TV model, higher order TV models, and the anisotropic third-order regularization model, when applied to some general applications. In another work, the thesis generalizes the TV-Stokes model from two dimensions to an arbitrary number of dimensions, introducing a convenient form for the constraint in order it to be extended to higher dimensions. The thesis explores also the idea of feature accumulation through iterative regularization in another work, introducing a Richardson-like iteration for the TV-Stokes. Thisis then followed by a more general model, a combined model, based on the modified variant of the TV-stokes. The resulting model is found to be equivalent to the well-known TGV model. The thesis introduces some interesting numerical strategies for the solution of the TV-Stokes model and its variants. Higher order PDEs are turned into inhomogeneous modified Helmholtz equations through transformations. These equations are then solved using the preconditioned conjugate gradients method or the fast Fourier transformation. The thesis proposes a simple but quite general approach to finding closed form solutions to a general L1 minimization problem, and applies it to design algorithms for our models.Doktorgradsavhandlin

ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing

We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the H−1-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation

A function space framework for structural total variation regularization with applications in inverse problems

In this work, we introduce a function space setting for a wide class of structural/weighted total variation (TV) regularization methods motivated by their applications in inverse problems. In particular, we consider a regularizer that is the appropriate lower semi-continuous envelope (relaxation) of a suitable total variation type functional initially defined for sufficiently smooth functions. We study examples where this relaxation can be expressed explicitly, and we also provide refinements for weighted total variation for a wide range of weights. Since an integral characterization of the relaxation in function space is, in general, not always available, we show that, for a rather general linear inverse problems setting, instead of the classical Tikhonov regularization problem, one can equivalently solve a saddle-point problem where no a priori knowledge of an explicit formulation of the structural TV functional is needed. In particular, motivated by concrete applications, we deduce corresponding results for linear inverse problems with norm and Poisson log-likelihood data discrepancy terms. Finally, we provide proof-of-concept numerical examples where we solve the saddle-point problem for weighted TV denoising as well as for MR guided PET image reconstruction

A primal-dual active-set method for non-negativity constrained total variation deblurring problems

Master'sMASTER OF SCIENC

A Two-Level Method for Image Denoising and Image Deblurring Models Using Mean Curvature Regularization

The mean curvature (MC)-based image denoising and image deblurring models are used to enhance the quality of the denoised images and deblurred images respectively. These models are very efficient in removing staircase effect, preserving edges and other nice properties. However, high order derivatives appear in the Euler–Lagrange equations of the MC-based models which create problems in developing an efficient numerical algorithm. To overcome this difficulty, we present a robust and efficient Two-Level method for MC-based image denoising and image deblurring models. The Two-Level method consists of solving one small problem and one large problem. The small problem is a nonlinear system, having high order derivative, on Level I (image having small number of pixels). The large problem is one less expensive system, having low order derivative, on Level II (image having large number of pixels). The derivation of the optimal regularization parameter of Level II is studied and formula is presented. Numerical experiments on digital images are presented to exhibit the performance of the Two-Level method