Space-Varying Coefficient Models for Diffusion Tensor Imaging using 3d Wavelets

In this paper, the space-varying coefficients model on the basis of B-splines (Heim et al., (2006)) is adapted to wavelet basis functions and re-examined using artificial and real data. For an introduction to diffusion tensor imaging refer to Heim et al. (2005, Chap. 2). First, wavelet theory is introduced and explained by means of 1d and 2d examples (Sections 1.1 { 1.3). Section 1.4 is dedicated to the most common thresholding techniques that serve as regularization concepts for wavelet based models. Prior to application of the 3d wavelet decomposition to the space-varying coe cient elds, the SVCM needs to be rewritten. The necessary steps are outlined in Section 2 together with the incorporation of the positive de niteness constraint using log-Cholesky parametrization. Section 3 provides a simulation study as well as a comparison with the results obtained through B-splines and standard kernel application. Finally, a real data example is presented and discussed. The theoretical parts are based on books of Gen cay et al. (2002, Chap. 1, 4-6), Härdle et al. (1998), Ogden (1997) and Jansen (2001) if not stated otherwise

Color graph based wavelet transform with perceptual information

International audienceIn this paper, we propose a numerical strategy to define a multiscale analysis for color and multicomponent images based on the representation of data on a graph. Our approach consists in computing the graph of an image using the psychovisual information and analysing it by using the spectral graph wavelet transform. We suggest introducing color dimension into the computation of the weights of the graph and using the geodesic distance as a means of distance measurement. We thus have defined a wavelet transform based on a graph with perceptual information by using the CIELab color distance. This new representation is illustrated with denoising and inpainting applications. Overall, by introducing psychovisual information in the graph computation for the graph wavelet transform we obtain very promising results. Therefore results in image restoration highlight the interest of the appropriate use of color information

Signal processing algorithms for enhanced image fusion performance and assessment

The dissertation presents several signal processing algorithms for image fusion in noisy multimodal conditions. It introduces a novel image fusion method which performs well for image sets heavily corrupted by noise. As opposed to current image fusion schemes, the method has no requirements for a priori knowledge of the noise component. The image is decomposed with Chebyshev polynomials (CP) being used as basis functions to perform fusion at feature level. The properties of CP, namely fast convergence and smooth approximation, renders it ideal for heuristic and indiscriminate denoising fusion tasks. Quantitative evaluation using objective fusion assessment methods show favourable performance of the proposed scheme compared to previous efforts on image fusion, notably in heavily corrupted images. The approach is further improved by incorporating the advantages of CP with a state-of-the-art fusion technique named independent component analysis (ICA), for joint-fusion processing based on region saliency. Whilst CP fusion is robust under severe noise conditions, it is prone to eliminating high frequency information of the images involved, thereby limiting image sharpness. Fusion using ICA, on the other hand, performs well in transferring edges and other salient features of the input images into the composite output. The combination of both methods, coupled with several mathematical morphological operations in an algorithm fusion framework, is considered a viable solution. Again, according to the quantitative metrics the results of our proposed approach are very encouraging as far as joint fusion and denoising are concerned. Another focus of this dissertation is on a novel metric for image fusion evaluation that is based on texture. The conservation of background textural details is considered important in many fusion applications as they help define the image depth and structure, which may prove crucial in many surveillance and remote sensing applications. Our work aims to evaluate the performance of image fusion algorithms based on their ability to retain textural details from the fusion process. This is done by utilising the gray-level co-occurrence matrix (GLCM) model to extract second-order statistical features for the derivation of an image textural measure, which is then used to replace the edge-based calculations in an objective-based fusion metric. Performance evaluation on established fusion methods verifies that the proposed metric is viable, especially for multimodal scenarios

Toward sparse and geometry adapted video approximations

Video signals are sequences of natural images, where images are often modeled as piecewise-smooth signals. Hence, video can be seen as a 3D piecewise-smooth signal made of piecewise-smooth regions that move through time. Based on the piecewise-smooth model and on related theoretical work on rate-distortion performance of wavelet and oracle based coding schemes, one can better analyze the appropriate coding strategies that adaptive video codecs need to implement in order to be efficient. Efficient video representations for coding purposes require the use of adaptive signal decompositions able to capture appropriately the structure and redundancy appearing in video signals. Adaptivity needs to be such that it allows for proper modeling of signals in order to represent these with the lowest possible coding cost. Video is a very structured signal with high geometric content. This includes temporal geometry (normally represented by motion information) as well as spatial geometry. Clearly, most of past and present strategies used to represent video signals do not exploit properly its spatial geometry. Similarly to the case of images, a very interesting approach seems to be the decomposition of video using large over-complete libraries of basis functions able to represent salient geometric features of the signal. In the framework of video, these features should model 2D geometric video components as well as their temporal evolution, forming spatio-temporal 3D geometric primitives. Through this PhD dissertation, different aspects on the use of adaptivity in video representation are studied looking toward exploiting both aspects of video: its piecewise nature and the geometry. The first part of this work studies the use of localized temporal adaptivity in subband video coding. This is done considering two transformation schemes used for video coding: 3D wavelet representations and motion compensated temporal filtering. A theoretical R-D analysis as well as empirical results demonstrate how temporal adaptivity improves coding performance of moving edges in 3D transform (without motion compensation) based video coding. Adaptivity allows, at the same time, to equally exploit redundancy in non-moving video areas. The analogy between motion compensated video and 1D piecewise-smooth signals is studied as well. This motivates the introduction of local length adaptivity within frame-adaptive motion compensated lifted wavelet decompositions. This allows an optimal rate-distortion performance when video motion trajectories are shorter than the transformation "Group Of Pictures", or when efficient motion compensation can not be ensured. After studying temporal adaptivity, the second part of this thesis is dedicated to understand the fundamentals of how can temporal and spatial geometry be jointly exploited. This work builds on some previous results that considered the representation of spatial geometry in video (but not temporal, i.e, without motion). In order to obtain flexible and efficient (sparse) signal representations, using redundant dictionaries, the use of highly non-linear decomposition algorithms, like Matching Pursuit, is required. General signal representation using these techniques is still quite unexplored. For this reason, previous to the study of video representation, some aspects of non-linear decomposition algorithms and the efficient decomposition of images using Matching Pursuits and a geometric dictionary are investigated. A part of this investigation concerns the study on the influence of using a priori models within approximation non-linear algorithms. Dictionaries with a high internal coherence have some problems to obtain optimally sparse signal representations when used with Matching Pursuits. It is proved, theoretically and empirically, that inserting in this algorithm a priori models allows to improve the capacity to obtain sparse signal approximations, mainly when coherent dictionaries are used. Another point discussed in this preliminary study, on the use of Matching Pursuits, concerns the approach used in this work for the decompositions of video frames and images. The technique proposed in this thesis improves a previous work, where authors had to recur to sub-optimal Matching Pursuit strategies (using Genetic Algorithms), given the size of the functions library. In this work the use of full search strategies is made possible, at the same time that approximation efficiency is significantly improved and computational complexity is reduced. Finally, a priori based Matching Pursuit geometric decompositions are investigated for geometric video representations. Regularity constraints are taken into account to recover the temporal evolution of spatial geometric signal components. The results obtained for coding and multi-modal (audio-visual) signal analysis, clarify many unknowns and show to be promising, encouraging to prosecute research on the subject

Statistical Diffusion Tensor Imaging

Magnetic resonance diffusion tensor imaging (DTI) allows to infere the ultrastructure of living tissue. In brain mapping, neural fiber trajectories can be identified by exploiting the anisotropy of diffusion processes. Manifold statistical methods may be linked into the comprehensive processing chain that is spanned between DTI raw images and the reliable visualization of fibers. In this work, a space varying coefficients model (SVCM) using penalized B-splines was developed to integrate diffusion tensor estimation, regularization and interpolation into a unified framework. The implementation challenges originating in multiple 3d space varying coefficient surfaces and the large dimensions of realistic datasets were met by incorporating matrix sparsity and efficient model approximation. Superiority of B-spline based SVCM to the standard approach was demonstrable from simulation studies in terms of the precision and accuracy of the individual tensor elements. The integration with a probabilistic fiber tractography algorithm and application on real brain data revealed that the unified approach is at least equivalent to the serial application of voxelwise estimation, smoothing and interpolation. From the error analysis using boxplots and visual inspection the conclusion was drawn that both the standard approach and the B-spline based SVCM may suffer from low local adaptivity. Therefore, wavelet basis functions were employed for filtering diffusion tensor fields. While excellent local smoothing was indeed achieved by combining voxelwise tensor estimation with wavelet filtering, no immediate improvement was gained for fiber tracking. However, the thresholding strategy needs to be refined and the proposed model of an incorporation of wavelets into an SVCM needs to be implemented to finally assess their utility for DTI data processing. In summary, an SVCM with specific consideration of the demands of human brain DTI data was developed and implemented, eventually representing a unified postprocessing framework. This represents an experimental and statistical platform to further improve the reliability of tractography

A Primal-Dual Proximal Algorithm for Sparse Template-Based Adaptive Filtering: Application to Seismic Multiple Removal

Unveiling meaningful geophysical information from seismic data requires to deal with both random and structured "noises". As their amplitude may be greater than signals of interest (primaries), additional prior information is especially important in performing efficient signal separation. We address here the problem of multiple reflections, caused by wave-field bouncing between layers. Since only approximate models of these phenomena are available, we propose a flexible framework for time-varying adaptive filtering of seismic signals, using sparse representations, based on inaccurate templates. We recast the joint estimation of adaptive filters and primaries in a new convex variational formulation. This approach allows us to incorporate plausible knowledge about noise statistics, data sparsity and slow filter variation in parsimony-promoting wavelet frames. The designed primal-dual algorithm solves a constrained minimization problem that alleviates standard regularization issues in finding hyperparameters. The approach demonstrates significantly good performance in low signal-to-noise ratio conditions, both for simulated and real field seismic data

Multidimensional Wavelets and Computer Vision

This report deals with the construction and the mathematical analysis of multidimensional nonseparable wavelets and their efficient application in computer vision. In the first part, the fundamental principles and ideas of multidimensional wavelet filter design such as the question for the existence of good scaling matrices and sensible design criteria are presented and extended in various directions. Afterwards, the analytical properties of these wavelets are investigated in some detail. It will turn out that they are especially well-suited to represent (discretized) data as well as large classes of operators in a sparse form - a property that directly yields efficient numerical algorithms. The final part of this work is dedicated to the application of the developed methods to the typical computer vision problems of nonlinear image regularization and the computation of optical flow in image sequences. It is demonstrated how the wavelet framework leads to stable and reliable results for these problems of generally ill-posed nature. Furthermore, all the algorithms are of order O(n) leading to fast processing