Compressed sensing for longitudinal MRI: An adaptive-weighted approach

Purpose: Repeated brain MRI scans are performed in many clinical scenarios, such as follow up of patients with tumors and therapy response assessment. In this paper, the authors show an approach to utilize former scans of the patient for the acceleration of repeated MRI scans. Methods: The proposed approach utilizes the possible similarity of the repeated scans in longitudinal MRI studies. Since similarity is not guaranteed, sampling and reconstruction are adjusted during acquisition to match the actual similarity between the scans. The baseline MR scan is utilized both in the sampling stage, via adaptive sampling, and in the reconstruction stage, with weighted reconstruction. In adaptive sampling, k-space sampling locations are optimized during acquisition. Weighted reconstruction uses the locations of the nonzero coefficients in the sparse domains as a prior in the recovery process. The approach was tested on 2D and 3D MRI scans of patients with brain tumors. Results: The longitudinal adaptive CS MRI (LACS-MRI) scheme provides reconstruction quality which outperforms other CS-based approaches for rapid MRI. Examples are shown on patients with brain tumors and demonstrate improved spatial resolution. Compared with data sampled at Nyquist rate, LACS-MRI exhibits Signal-to-Error Ratio (SER) of 24.8dB with undersampling factor of 16.6 in 3D MRI. Conclusions: The authors have presented a novel method for image reconstruction utilizing similarity of scans in longitudinal MRI studies, where possible. The proposed approach can play a major part and significantly reduce scanning time in many applications that consist of disease follow-up and monitoring of longitudinal changes in brain MRI

Video Compressive Sensing for Dynamic MRI

We present a video compressive sensing framework, termed kt-CSLDS, to accelerate the image acquisition process of dynamic magnetic resonance imaging (MRI). We are inspired by a state-of-the-art model for video compressive sensing that utilizes a linear dynamical system (LDS) to model the motion manifold. Given compressive measurements, the state sequence of an LDS can be first estimated using system identification techniques. We then reconstruct the observation matrix using a joint structured sparsity assumption. In particular, we minimize an objective function with a mixture of wavelet sparsity and joint sparsity within the observation matrix. We derive an efficient convex optimization algorithm through alternating direction method of multipliers (ADMM), and provide a theoretical guarantee for global convergence. We demonstrate the performance of our approach for video compressive sensing, in terms of reconstruction accuracy. We also investigate the impact of various sampling strategies. We apply this framework to accelerate the acquisition process of dynamic MRI and show it achieves the best reconstruction accuracy with the least computational time compared with existing algorithms in the literature.Comment: 30 pages, 9 figure

A Novel Fourier Theory on Non-linear Phases and Applications

Positive time varying frequency representation for transient signals has been a hearty desire of signal analysts due to its theoretical and practical importance. During approximately the last two decades there has formulated a signal decomposition and reconstruction method rooted in harmonic and complex analysis giving rise to the desired signal representation. The method decomposes any signal into a few basic signals that possess positive instantaneous frequencies. The theory has profound relations with classical mathematics and can be generalized to signals defined in higher dimensional manifolds with vector and matrix values, and in particular, promotes rational approximation in higher dimensions. This article mainly serves as a survey. It also gives a new proof for a general convergence result, as well as a proof for the necessity of multiple selection of the parameters. Mono-components are crucial to understand the concept instantaneous frequency. We will present several most important mono-component function classes. Decompositions of signals into mono-components are called adaptive Fourier decompositions (AFDs). We note that some scopes of the studies on the 1D mono-components and AFDs can be extended to vector-valued or even matrix-valued signals defined on higher dimensional manifolds. We finally provide an account of related studies in pure and applied mathematics, and in signal analysis, as well as applications of the theory found in the literature.Comment: 33 page

Identification of Smart Jammers: Learning based Approaches Using Wavelet Representation

Smart jammer nodes can disrupt communication between a transmitter and a receiver in a wireless network, and they leave traces that are undetectable to classical jammer identification techniques, hidden in the time-frequency plane. These traces cannot be effectively identified through the use of the classical Fourier transform based time-frequency transformation (TFT) techniques with a fixed resolution. Inspired by the adaptive resolution property provided by the wavelet transforms, in this paper, we propose a jammer identification methodology that includes a pre-processing step to obtain a multi-resolution image, followed by the use of a classifier. Support vector machine (SVM) and deep convolutional neural network (DCNN) architectures are investigated as classifiers to automatically extract the features of the transformed signals and to classify them. Three different jamming attacks are considered, the barrage jamming that targets the complete transmission bandwidth, the synchronization signal jamming attack that targets synchronization signals and the reference signal jamming attack that targets the reference signals in an LTE downlink transmission scenario. The performance of the proposed approach is compared with the classical Fourier transform based TFT techniques, demonstrating the efficacy of the proposed approach in the presence of smart jammers

A multilevel based reweighting algorithm with joint regularizers for sparse recovery

Sparsity is one of the key concepts that allows the recovery of signals that are subsampled at a rate significantly lower than required by the Nyquist-Shannon sampling theorem. Our proposed framework uses arbitrary multiscale transforms, such as those build upon wavelets or shearlets, as a sparsity promoting prior which allow to decompose the image into different scales such that image features can be optimally extracted. In order to further exploit the sparsity of the recovered signal we combine the method of reweighted $\ell^1$, introduced by Cand\`es et al., with iteratively updated weights accounting for the multilevel structure of the signal. This is done by directly incorporating this approach into a split Bregman based algorithmic framework. Furthermore, we add total generalized variation (TGV) as a second regularizer into the split Bregman algorithm. The resulting algorithm is then applied to a classical and widely considered task in signal- and image processing which is the reconstruction of images from their Fourier measurements. Our numerical experiments show a highly improved performance at relatively low computational costs compared to many other well established methods and strongly suggest that sparsity is better exploited by our method

A Local Spectral Exterior Calculus for the Sphere and Application to the Shallow Water Equations

We introduce $\Psi\mathrm{ec}$, a local spectral exterior calculus for the two-sphere $S^2$. $\Psi\mathrm{ec}$ provides a discretization of Cartan's exterior calculus on $S^2$ formed by spherical differential $r$-form wavelets. These are well localized in space and frequency and provide (Stevenson) frames for the homogeneous Sobolev spaces $\dot{H}^{-r+1}( \Omega_{\nu}^{r} , S^2 )$ of differential $r$-forms. At the same time, they satisfy important properties of the exterior calculus, such as the de Rahm complex and the Hodge-Helmholtz decomposition. Through this, $\Psi\mathrm{ec}$ is tailored towards structure preserving discretizations that can adapt to solutions with varying regularity. The construction of $\Psi\mathrm{ec}$ is based on a novel spherical wavelet frame for $L_2(S^2)$ that we obtain by introducing scalable reproducing kernel frames. These extend scalable frames to weighted sampling expansions and provide an alternative to quadrature rules for the discretization of needlet-like scale-discrete wavelets. We verify the practicality of $\Psi\mathrm{ec}$ for numerical computations using the rotating shallow water equations. Our numerical results demonstrate that a $\Psi\mathrm{ec}$-based discretization of the equations attains accuracy comparable to those of spectral methods while using a representation that is well localized in space and frequency

The curvelet transform for image denoising

We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourier-domain computation of an approximate digital Radon transform. We introduce a very simple interpolation in the Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudo-polar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of a` trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with "state of the art" techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including tree-based Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than wavelet-based reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement

Analyzing transient-evoked otoacoustic emissions by concentration of frequency and time

The linear part of transient evoked (TE) otoacoustic emission (OAE) is thought to be generated via coherent reflection near the characteristic place of constituent wave components. Because of the tonotopic organization of the cochlea, high frequency emissions return earlier than low frequencies; however, due to the random nature of coherent reflection, the instantaneous frequency (IF) and amplitude envelope of TEOAEs both fluctuate. Multiple reflection components and synchronized spontaneous emissions can further make it difficult to extract the IF by linear transforms. In this paper, we propose to model TEOAEs as a sum of {\em intrinsic mode-type functions} and analyze it by a {nonlinear-type time-frequency analysis} technique called concentration of frequency and time (ConceFT). When tested with synthetic OAE signals {with possibly multiple oscillatory components}, the present method is able to produce clearly visualized traces of individual components on the time-frequency plane. Further, when the signal is noisy, the proposed method is compared with existing linear and bilinear methods in its accuracy for estimating the fluctuating IF. Results suggest that ConceFT outperforms the best of these methods in terms of optimal transport distance, reducing the error by 10 to {21\%} when the signal to noise ratio is 10 dB or below

Shannon wavelet approximations of linear differential operators

Recent works emphasized the interest of numerical solution of PDE's with wavelets. In their works, A.Cohen, W.Dahmen and R.DeVore focussed on the non linear approximation aspect of the wavelet approximation of PDE's to prove the relevance of such methods. In order to extend these results, we focuss on the convergence of the iterative algorithm, and we consider different possibilities offered by the wavelet theory: the tensorial wavelets and the derivation/integration of wavelet bases. We also investigate the use of wavelet packets. We apply these extended results to prove in the case of the Shannon wavelets, the convergence of the Leray projector algorithm with divergence-free wavelets.Comment: preprint IMPAN (19 pages

Rigid-Motion Scattering for Texture Classification

A rigid-motion scattering computes adaptive invariants along translations and rotations, with a deep convolutional network. Convolutions are calculated on the rigid-motion group, with wavelets defined on the translation and rotation variables. It preserves joint rotation and translation information, while providing global invariants at any desired scale. Texture classification is studied, through the characterization of stationary processes from a single realization. State-of-the-art results are obtained on multiple texture data bases, with important rotation and scaling variabilities.Comment: 19 pages, submitted to International Journal of Computer Visio