Carola-Bibiane Schönlieb

A Combined First and Second Order Variational Approach for Image Reconstruction

In this paper we study a variational problem in the space of functions of bounded Hessian. Our model constitutes a straightforward higher-order extension of the well known ROF functional (total variation minimisation) to which we add a non-smooth second order regulariser. It combines convex functions of the total variation and the total variation of the first derivatives. In what follows, we prove existence and uniqueness of minimisers of the combined model and present the numerical solution of the corresponding discretised problem by employing the split Bregman method. The paper is furnished with applications of our model to image denoising, deblurring as well as image inpainting. The obtained numerical results are compared with results obtained from total generalised variation (TGV), infimal convolution and Euler’s elastica, three other state of the art higher-order models. The numerical discussion confirms that the proposed higher-order model competes with models of its kind in avoiding the creation of undesirable artifacts and blocky-like structures in the reconstructed images—a known disadvantage of the ROF model—while being simple and efficiently numerically solvable.

Cahn-Hilliard Inpainting and a Generalization for Grayvalue Images

The Cahn-Hilliard equation is a nonlinear fourth order diffusion equation originating in material science for modeling phase separation and phase coarsening in binary alloys. The inpainting of binary images using the Cahn-Hilliard equation is a new approach in image processing. In this paper we discuss the stationary state of the proposed model and introduce a generalization for grayvalue images of bounded variation. This is realized by using subgradients of the total variation functional within the flow, which leads to structure inpainting with smooth curvature of level sets.

Subspace Correction Methods for Total Variation and

This paper is concerned with the numerical minimization of energy functionals in Hilbert spaces involving convex constraints coinciding with a seminorm for a subspace. The optimization is realized by alternating minimizations of the functional on a sequence of orthogonal subspaces. On each subspace an iterative proximity-map algorithm is implemented via oblique thresholding, which is the main new tool introduced in this work. We provide convergence conditions for the algorithm in order to compute minimizers of the target energy. Analogous results are derived for a parallel variant of the algorithm. Applications are presented in domain decomposition methods for degenerate elliptic PDEs arising in total variation minimization and in accelerated sparse recovery algorithms based on

\ell_1

\ell_1

-Minimization

-minimization. We include numerical examples which show efficient solutions to classical problems in signal and image processing.

Phase reconstruction from velocity-encoded MRI measurements--a survey of sparsity-promoting variational approaches.

In recent years there has been significant developments in the reconstruction of magnetic resonance velocity images from sub-sampled k-space data. While showing a strong improvement in reconstruction quality compared to classical approaches, the vast number of different methods, and the challenges in setting them up, often leaves the user with the difficult task of choosing the correct approach, or more importantly, not selecting a poor approach. In this paper, we survey variational approaches for the reconstruction of phase-encoded magnetic resonance velocity images from sub-sampled k-space data. We are particularly interested in regularisers that correctly treat both smooth and geometric features of the image. These features are common to velocity imaging, where the flow field will be smooth but interfaces between the fluid and surrounding material will be sharp, but are challenging to represent sparsely. As an example we demonstrate the variational approaches on velocity imaging of water flowing through a packed bed of solid particles. We evaluate Wavelet regularisation against Total Variation and the relatively recent second order Total Generalised Variation regularisation. We combine these regularisation schemes with a contrast enhancement approach called Bregman iteration. We verify for a variety of sampling patterns that Morozovs discrepancy principle provides a good criterion for stopping the iterations. Therefore, given only the noise level, we present a robust guideline for setting up a variational reconstruction scheme for MR velocity imaging.

A convergent overlapping domain decomposition method for total variation minimization

In this paper we are concerned with the analysis of convergent sequential and parallel overlapping domain decomposition methods for the minimization of functionals formed by a discrepancy term with respect to the data and a total variation constraint. To our knowledge, this is the first successful attempt of addressing such a strategy for the nonlinear, nonadditive, and nonsmooth problem of total variation minimization. We provide several numerical experiments, showing the successful application of the algorithm for the restoration of 1D signals and 2D images in interpolation/inpainting problems, respectively, and in a compressed sensing problem, for recovering piecewise constant medical-type images from partial Fourier ensembles.

Imaging with Kantorovich--Rubinstein Discrepancy

We propose the use of the Kantorovich-Rubinstein norm from optimal transport in imaging problems. In particular, we discuss a variational regularisation model endowed with a Kantorovich-Rubinstein discrepancy term and total variation regularization in the context of image denoising and cartoon-texture decomposition. We point out connections of this approach to several other recently proposed methods such as total generalized variation and norms capturing oscillating patterns. We also show that the respective optimization problem can be turned into a convex-concave saddle point problem with simple constraints and hence, can be solved by standard tools. Numerical examples exhibit interesting features and favourable performance for denoising and cartoon-texture decomposition.

Learning to Diversify Deep Belief Networks for Hyperspectral Image Classification

In the literature of remote sensing, deep models with multiple layers have demonstrated their potentials in learning the abstract and invariant features for better representation and classification of hyperspectral images. The usual supervised deep models, such as convolutional neural networks, need a large number of labeled training samples to learn their model parameters. However, the real-world hyperspectral image classification task provides only a limited number of training samples. This paper adopts another popular deep model, i.e., deep belief networks (DBNs), to deal with this problem. The DBNs allow unsupervised pretraining over unlabeled samples at first and then a supervised fine-tuning over labeled samples. But the usual pretraining and fine-tuning method would make many hidden units in the learned DBNs tend to behave very similarly or perform as “dead” (never responding) or “potential over-tolerant” (always responding) latent factors. These results could negatively affect description ability and thus classification performance of DBNs. To further improve DBN’s performance, this paper develops a new diversified DBN through regularizing pretraining and fine-tuning procedures by a diversity promoting prior over latent factors. Moreover, the regularized pretraining and fine-tuning can be efficiently implemented through usual recursive greedy and back-propagation learning framework. The experiments over real-world hyperspectral images demonstrated that the diversity promoting prior in both pretraining and fine-tuning procedure lead to the learned DBNs with more diverse latent factors, which directly make the diversified DBNs obtain much better results than original DBNs and comparable or even better performances compared with other recent hyperspectral image classification methods.

Total variation regularization in measurement and image space for PET reconstruction

The aim of this paper is to test and analyze a novel technique for image reconstruction in positron emission tomography, which is based on (total variation) regularization on both the image space and the projection space. We formulate our variational problem considering both total variation penalty terms on the image and on an idealized sinogram to be reconstructed from a given Poisson distributed noisy sinogram. We prove existence, uniqueness and stability results for the proposed model and provide some analytical insight into the structures favoured by joint regularization. For the numerical solution of the corresponding discretized problem we employ the split Bregman algorithm and extensively test the approach in comparison to standard total variation regularization on the image. The numerical results show that an additional penalty on the sinogram performs better on reconstructing images with thin structures.