TV-Stokes And Its Variants For Image Processing
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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.
Has partsPaper A: Bin Wu, Talal Rahman, and Xue-Cheng Tai, Sparse-Data Based 3D Surface Reconstruction for Cartoon and Map, Imaging, Vision and Learning Based on Optimization and PDEs, Springer, Cham, (2016): 47. The accepted article is available at: https://hdl.handle.net/11250/2732627
Paper B: Bin Wu, Xue-Cheng Tai, and Talal Rahman, Sparse-Data Based 3D Surface Reconstruction with Vector Matching. Available in the main thesis
Paper C: Bin Wu, Xue-Cheng Tai, and Talal Rahman, Multidimensional TV-Stokes for Image Processing. Available in the main thesis
Paper D: Bin Wu, Leszek Marcinkowski, Xue-Cheng Tai, and Talal Rahman, Iterative Regularization Algorithms for Image Denoising with the TV-Stokes Model. Available in the main thesis
Paper E: Bin Wu, Xue-Cheng Tai, and Talal Rahman, Alternating Minimization for a Single Step TV-Stokes Model for Image Denoising. Available in the main thesis