Nelly Pustelnik

Parallel Proximal Algorithm for Image Restoration Using Hybrid Regularization

Regularization approaches have demonstrated their effectiveness for solving ill-posed problems. However, in the context of variational restoration methods, a challenging question remains, namely how to find a good regularizer. While total variation introduces staircase effects, wavelet-domain regularization brings other artefacts, e.g., ringing. However, a tradeoff can be made by introducing a hybrid regularization including several terms not necessarily acting in the same domain (e.g., spatial and wavelet transform domains). While this approach was shown to provide good results for solving deconvolution problems in the presence of additive Gaussian noise, an important issue is to efficiently deal with this hybrid regularization for more general noise models. To solve this problem, we adopt a convex optimization framework where the criterion to be minimized is split in the sum of more than two terms. For spatial domain regularization, isotropic or anisotropic total variation definitions using various gradient filters are considered. An accelerated version of the Parallel Proximal Algorithm is proposed to perform the minimization. Some difficulties in the computation of the proximity operators involved in this algorithm are also addressed in this paper. Numerical experiments performed in the context of Poisson data recovery, show the good behavior of the algorithm as well as promising results concerning the use of hybrid regularization techniques.

Proximal Algorithms for Multicomponent Image Recovery Problems

In recent years, proximal splitting algorithms have been applied to various monocomponent signal and image recovery problems. In this paper, we address the case of multicomponent problems. We first provide closed form expressions for several important multicomponent proximity operators and then derive extensions of existing proximal algorithms to the multicomponent setting. These results are applied to stereoscopic image recovery, multispectral image denoising, and image decomposition into texture and geometry components.

Parallel algorithm and hybrid regularization for dynamic PET reconstruction

To improve the estimation at the voxel level in dynamic Positron Emission Tomography (PET) imaging, we propose to develop a convex optimization approach based on a recently proposed parallel proximal method (PPXA). This class of algorithms was successfully employed for 2D deconvolution in the presence of Poisson noise and it is extended here to (dynamic) space + time PET image reconstruction. Hybrid regularization defined as a sum of a total variation and a sparsity measure is considered in this paper. The total variation is applied to each temporal-frame and a wavelet regularization is considered for the space+time data. Total variation allows us to smooth the wavelet artifacts introduced when the wavelet regularization is used alone. The proposed algorithm was evaluated on simulated dynamic fluorodeoxyglucose (FDG) brain data and compared with a regularized Expectation Maximization (EM) reconstruction. From the reconstructed dynamic images, parametric maps of the cerebral metabolic rate of glucose (CMRglu) were computed. Our approach shows a better reconstruction at the voxel level.

Temporal wavelet denoising of PET sinograms and images

The level of noise in PET dynamic studies makes it difficult to provide accurate and robust kinetic parameters from time activity curves, particularly at the voxel level. Several approaches have been followed to lower noise: denoising reconstructed images with spatial wavelets, adding a priori information during reconstruction about the signal without noise, including the temporal dimension during reconstruction. In this work, we propose to use a temporal wavelet denoising approach, based on the characteristics of PET time activity curves in sinograms (or reconstructed images). This approach has recently been proposed in image processing and relies on discriminating signal from noise by including relevant “a priori” information (on the statistical distribution of the wavelet coefficient for a whole sinogram or reconstructed image), as well as appropriate noise formation model in the time domain. This approach is tested in a 2D spatial + 1D time Monte Carlo simulation mimicking brain, and compared with a standard denoising approach : SUREShrink. Preliminary results indicate that better performances are obtained for sinogram denoising with the proposed approach compared with SUREShrink, and that the resulting sinograms can be reconstructed with a weighted least-squares (WLS) algorithm for all techniques. Denoising in the reconstructed images with this approach was also investigated.

Proximal method for geometry and texture image decomposition

We propose a variational method for decomposing an image into a geometry and a texture component. Our model involves the sum of two functions promoting separately properties of each component, and of a coupling function modeling the interaction between the components. None of these functions is required to be differentiable, which significantly broadens the range of decompositions achievable through variational approaches. The convergence of the proposed proximal algorithm is guaranteed under suitable assumptions. Numerical examples are provided that show an application of the algorithm to image decomposition and restoration in the presence of Poisson noise.

A wavelet-based quadratic extension method for image deconvolution in the presence of poisson noise

Iterative optimization algorithms such as the forward-backward and Douglas-Rachford algorithms have recently gained much popularity since they provide efficient solutions to a wide class of non-smooth convex minimization problems arising in signal/image recovery. However, when images are degraded by a convolution operator and a Poisson noise, a particular attention must be paid to the associated minimization problem. To solve it, we propose a new optimization method which consists of two nested iterative steps. The effectiveness of the proposed method is demonstrated via numerical comparisons.