Challenges in the decomposition of 2D NMR spectra of mixtures of small molecules.

Analytical methods for mixtures of small molecules require specificity (is a certain molecule present in the mix?) and speciation capabilities. NMR spectroscopy has been a tool of choice for both of these issues since its early days, due to its quantitative (linear) response, sufficiently high resolving power and capabilities of inferring molecular structures from spectral features (even in the absence of a reference database). However, the analytical performances of NMR spectroscopy are being stretched by the increased complexity of the samples, the dynamic range of the components, and the need for a reasonable turnover time. One approach that has been actively pursued for disentangling the composition complexity is the use of 2D NMR spectroscopy. While any of the many experiments from this family will increase the spectral resolution, some are more apt for mixtures, as they are capable of unveiling signals belonging to whole molecules or fragments of it. Among the most popular ones, one can enumerate HSQC-TOCSY, DOSY and Maximum-Quantum (MaxQ) NMR spectroscopy. For multicomponent samples, the development of robust mathematical methods of signal decomposition would provide a clear edge towards identification. We have been pursuing, along these lines, Blind Source Separation (BSS). Here, the un-mixing of the spectra is achieved relying on correlations detected on a series of datasets. The series could be associated with samples of different relative composition or in a classically acquired 2D experiment by the mathematical laws underlying the construction of the indirect dimension, the one not recorded by the spectrometer. Many algorithms have been proposed for BSS in NMR spectroscopy since the seminal work of Nuzillard. In this paper, we use rather standard algorithms in BSS in order to disentangle NMR spectra. We show on simulated data (both 1D and 2D HSQC) that these approaches enable us to accurately disentangle multiple components, and provide good estimates for the concentrations of compounds. Furthermore, we show that after proper realignment of the signals, the same algorithms are able to disentangle real 1D NMR spectra. We obtain similar results on 2D HSQC spectra, where the BSS algorithms are able to successfully disentangle components, and provide even better estimates for concentrations.

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.

Satellite image restoration in the context of a spatially varying point spread function.

In this paper, we consider a deconvolution problem where the point spread function (PSF) of the optical imaging system varies between different spatial locations, thus leading to a spatially varying blur. This problem arises, for example, in synthetic aperture instruments and in wide-field optical systems. Unlike the classical deconvolution context where the PSF is assumed to be spatially invariant, the problem cannot be easily solved in the Fourier domain. We propose here an iterative algorithm based on convex optimization techniques and a wavelet frame regularization. This approach allows restoration of the image, taking into account the properties of the blur operator, the latter being known.

Image analysis using a dual-tree M-band wavelet transform.

We propose a two-dimensional generalization to the M-band case of the dual-tree decomposition structure (initially proposed by Kingsbury and further investigated by Selesnick) based on a Hilbert pair of wavelets. We particularly address: 1) the construction of the dual basis and 2) the resulting directional analysis. We also revisit the necessary pre-processing stage in the M-band case. While several reconstructions are possible because of the redundancy of the representation, we propose a new optimal signal reconstruction technique, which minimizes potential estimation errors. The effectiveness of the proposed M-band decomposition is demonstrated via denoising comparisons on several image types (natural, texture, seismics), with various M-band wavelets and thresholding strategies. Significant improvements in terms of both overall noise reduction and direction preservation are observed.