CT Image Reconstruction by Spatial-Radon Domain Data-Driven Tight Frame Regularization

This paper proposes a spatial-Radon domain CT image reconstruction model based on data-driven tight frames (SRD-DDTF). The proposed SRD-DDTF model combines the idea of joint image and Radon domain inpainting model of \cite{Dong2013X} and that of the data-driven tight frames for image denoising \cite{cai2014data}. It is different from existing models in that both CT image and its corresponding high quality projection image are reconstructed simultaneously using sparsity priors by tight frames that are adaptively learned from the data to provide optimal sparse approximations. An alternative minimization algorithm is designed to solve the proposed model which is nonsmooth and nonconvex. Convergence analysis of the algorithm is provided. Numerical experiments showed that the SRD-DDTF model is superior to the model by \cite{Dong2013X} especially in recovering some subtle structures in the images

Spectrum-Adapted Tight Graph Wavelet and Vertex-Frequency Frames

We consider the problem of designing spectral graph filters for the construction of dictionaries of atoms that can be used to efficiently represent signals residing on weighted graphs. While the filters used in previous spectral graph wavelet constructions are only adapted to the length of the spectrum, the filters proposed in this paper are adapted to the distribution of graph Laplacian eigenvalues, and therefore lead to atoms with better discriminatory power. Our approach is to first characterize a family of systems of uniformly translated kernels in the graph spectral domain that give rise to tight frames of atoms generated via generalized translation on the graph. We then warp the uniform translates with a function that approximates the cumulative spectral density function of the graph Laplacian eigenvalues. We use this approach to construct computationally efficient, spectrum-adapted, tight vertex-frequency and graph wavelet frames. We give numerous examples of the resulting spectrum-adapted graph filters, and also present an illustrative example of vertex-frequency analysis using the proposed construction

Sparse Representation on Graphs by Tight Wavelet Frames and Applications

In this paper, we introduce a new (constructive) characterization of tight wavelet frames on non-flat domains in both continuum setting, i.e. on manifolds, and discrete setting, i.e. on graphs; discuss how fast tight wavelet frame transforms can be computed and how they can be effectively used to process graph data. We start with defining the quasi-affine systems on a given manifold \cM that is formed by generalized dilations and shifts of a finite collection of wavelet functions $\Psi:=\{\psi_j: 1\le j\le r\}\subset L_2(\R)$. We further require that $\psi_j$ is generated by some refinable function $\phi$ with mask $a_j$. We present the condition needed for the masks $\{a_j: 0\le j\le r\}$ so that the associated quasi-affine system generated by $\Psi$ is a tight frame for L_2(\cM). Then, we discuss how the transition from the continuum (manifolds) to the discrete setting (graphs) can be naturally done. In order for the proposed discrete tight wavelet frame transforms to be useful in applications, we show how the transforms can be computed efficiently and accurately by proposing the fast tight wavelet frame transforms for graph data (WFTG). Finally, we consider two specific applications of the proposed WFTG: graph data denoising and semi-supervised clustering. Utilizing the sparse representation provided by the WFTG, we propose $\ell_1$-norm based optimization models on graphs for denoising and semi-supervised clustering. On one hand, our numerical results show significant advantage of the WFTG over the spectral graph wavelet transform (SGWT) by [1] for both applications. On the other hand, numerical experiments on two real data sets show that the proposed semi-supervised clustering model using the WFTG is overall competitive with the state-of-the-art methods developed in the literature of high-dimensional data classification, and is superior to some of these methods

Dynamic brain networks explored by structure-revealing methods

The human brain is a complex system able to continuously adapt. How and where brain activity is modulated by behavior can be studied with functional magnetic resonance imaging (fMRI), a non-invasive neuroimaging technique with excellent spatial resolution and whole-brain coverage. FMRI scans of healthy adults completing a variety of behavioral tasks have greatly contributed to our understanding of the functional role of individual brain regions. However, by statistically analyzing each region independently, these studies ignore that brain regions act in concert rather than in unison. Thus, many studies since have instead examined how brain regions interact. Surprisingly, structured interactions between distinct brain regions not only occur during behavioral tasks but also while a subject rests quietly in the MRI scanner. Multiple groups of regions interact very strongly with each other and not only do these groups bear a striking resemblance to the sets of regions co-activated in tasks, but many of these interactions are also progressively disrupted in neurological diseases. This suggests that spontaneous fluctuations in activity can provide novel insights into fundamental organizing principles of the human brain in health and disease. Many techniques to date have segregated regions into spatially distinct networks, which ignores that any brain region can take part in multiple networks across time. A more natural view is to estimate dynamic brain networks that allow flexible functional interactions (or connectivity) over time. The estimation and analysis of such dynamic functional interactions is the subject of this dissertation. We take the perspective that dynamic brain networks evolve in a low-dimensional space and can be described by a small number of characteristic spatiotemporal patterns. Our proposed approaches are based on well-established statistical methods, such as principal component analysis (PCA), sparse matrix decompositions, temporal clustering, as well as a multiscale analysis by novel graph wavelet designs. We adapt and extend these methods to the analysis of dynamic brain networks. We show that PCA and its higher-order equivalent can identify co-varying functional interactions, which reveal disturbed dynamic properties in multiple sclerosis and which are related to the timing of stimuli for task studies, respectively. Further we show that sparse matrix decompositions provide a valid alternative approach to PCA and improve interpretability of the identified patterns. Finally, assuming an even simpler low-dimensional space and the exclusive temporal expression of individual patterns, we show that specific transient interactions of the medial prefrontal cortex are disturbed in aging and relate to impaired memory

Graph Signal Processing: Overview, Challenges and Applications

Research in Graph Signal Processing (GSP) aims to develop tools for processing data defined on irregular graph domains. In this paper we first provide an overview of core ideas in GSP and their connection to conventional digital signal processing. We then summarize recent developments in developing basic GSP tools, including methods for sampling, filtering or graph learning. Next, we review progress in several application areas using GSP, including processing and analysis of sensor network data, biological data, and applications to image processing and machine learning. We finish by providing a brief historical perspective to highlight how concepts recently developed in GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE