First order algorithms in variational image processing

Variational methods in imaging are nowadays developing towards a quite universal and flexible tool, allowing for highly successful approaches on tasks like denoising, deblurring, inpainting, segmentation, super-resolution, disparity, and optical flow estimation. The overall structure of such approaches is of the form ${\cal D}(Ku) + \alpha {\cal R} (u) \rightarrow \min_u$ ; where the functional ${\cal D}$ is a data fidelity term also depending on some input data $f$ and measuring the deviation of $Ku$ from such and ${\cal R}$ is a regularization functional. Moreover $K$ is a (often linear) forward operator modeling the dependence of data on an underlying image, and $\alpha$ is a positive regularization parameter. While ${\cal D}$ is often smooth and (strictly) convex, the current practice almost exclusively uses nonsmooth regularization functionals. The majority of successful techniques is using nonsmooth and convex functionals like the total variation and generalizations thereof or $\ell_1$-norms of coefficients arising from scalar products with some frame system. The efficient solution of such variational problems in imaging demands for appropriate algorithms. Taking into account the specific structure as a sum of two very different terms to be minimized, splitting algorithms are a quite canonical choice. Consequently this field has revived the interest in techniques like operator splittings or augmented Lagrangians. Here we shall provide an overview of methods currently developed and recent results as well as some computational studies providing a comparison of different methods and also illustrating their success in applications.Comment: 60 pages, 33 figure

X-ray CT Image Reconstruction on Highly-Parallel Architectures.

Model-based image reconstruction (MBIR) methods for X-ray CT use accurate models of the CT acquisition process, the statistics of the noisy measurements, and noise-reducing regularization to produce potentially higher quality images than conventional methods even at reduced X-ray doses. They do this by minimizing a statistically motivated high-dimensional cost function; the high computational cost of numerically minimizing this function has prevented MBIR methods from reaching ubiquity in the clinic. Modern highly-parallel hardware like graphics processing units (GPUs) may offer the computational resources to solve these reconstruction problems quickly, but simply "translating" existing algorithms designed for conventional processors to the GPU may not fully exploit the hardware's capabilities. This thesis proposes GPU-specialized image denoising and image reconstruction algorithms. The proposed image denoising algorithm uses group coordinate descent with carefully structured groups. The algorithm converges very rapidly: in one experiment, it denoises a 65 megapixel image in about 1.5 seconds, while the popular Chambolle-Pock primal-dual algorithm running on the same hardware takes over a minute to reach the same level of accuracy. For X-ray CT reconstruction, this thesis uses duality and group coordinate ascent to propose an alternative to the popular ordered subsets (OS) method. Similar to OS, the proposed method can use a subset of the data to update the image. Unlike OS, the proposed method is convergent. In one helical CT reconstruction experiment, an implementation of the proposed algorithm using one GPU converges more quickly than a state-of-the-art algorithm converges using four GPUs. Using four GPUs, the proposed algorithm reaches near convergence of a wide-cone axial reconstruction problem with over 220 million voxels in only 11 minutes.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113551/1/mcgaffin_1.pd

Sparse approximate inverse preconditioners on high performance GPU platforms

Simulation with models based on partial differential equations often requires the solution of (sequences of) large and sparse algebraic linear systems. In multidimensional domains, preconditioned Krylov iterative solvers are often appropriate for these duties. Therefore, the search for efficient preconditioners for Krylov subspace methods is a crucial theme. Recent developments, especially in computing hardware, have renewed the interest in approximate inverse preconditioners in factorized form, because their application during the solution process can be more efficient. We present here some experiences focused on the approximate inverse preconditioners proposed by Benzi and Tůma from 1996 and the sparsification and inversion proposed by van Duin in 1999. Computational costs, reorderings and implementation issues are considered both on conventional and innovative computing architectures like Graphics Programming Units (GPUs)

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

Advanced Denoising for X-ray Ptychography

The success of ptychographic imaging experiments strongly depends on achieving high signal-to-noise ratio. This is particularly important in nanoscale imaging experiments when diffraction signals are very weak and the experiments are accompanied by significant parasitic scattering (background), outliers or correlated noise sources. It is also critical when rare events such as cosmic rays, or bad frames caused by electronic glitches or shutter timing malfunction take place. In this paper, we propose a novel iterative algorithm with rigorous analysis that exploits the direct forward model for parasitic noise and sample smoothness to achieve a thorough characterization and removal of structured and random noise. We present a formal description of the proposed algorithm and prove its convergence under mild conditions. Numerical experiments from simulations and real data (both soft and hard X-ray beamlines) demonstrate that the proposed algorithms produce better results when compared to state-of-the-art methods.Comment: 24 pages, 9 figure