Recovering missing slices of the discrete fourier transform using ghosts

The discrete Fourier transform (DFT) underpins the solution to many inverse problems commonly possessing missing or unmeasured frequency information. This incomplete coverage of the Fourier space always produces systematic artifacts called Ghosts. In this paper, a fast and exact method for deconvolving cyclic artifacts caused by missing slices of the DFT using redundant image regions is presented. The slices discussed here originate from the exact partitioning of the Discrete Fourier Transform (DFT) space, under the projective Discrete Radon Transform, called the discrete Fourier slice theorem. The method has a computational complexity of O(n\log-{2}n) (for an n=N\times N image) and is constructed from a new cyclic theory of Ghosts. This theory is also shown to unify several aspects of work done on Ghosts over the past three decades. This paper concludes with an application to fast, exact, non-iterative image reconstruction from a highly asymmetric set of rational angle projections that give rise to sets of sparse slices within the DFT

Spatial Implementation for Erasure Coding by Finite Radon Transform

International audienceFault-tolerance has been widely studied these years in order to fit new kinds of applications running on unreliable systems such as the Internet. Erasure coding aims at recovering information that has been lost during a transmission (e.g. congestion). Considered as the alternative to the Automatic Repeat-reQuest (ARQ) strategy, erasure coding differs by adding redundancy to recover lost information without the need to retransmit data. In this paper we propose a new approach using the Finite Radon Transform (FRT). The FRT is an exact and discrete transformation that relies on simple additions to obtain a set of projections. The proposed erasure code is Maximal Distance Separable (MDS). We detail in this paper the systematic and nonsystematic implementation. As an optimization, we use the same algorithm called "row-solving" for creating the redundancy and for recovering missing data

AliasNet: Alias Artefact Suppression Network for Accelerated Phase-Encode MRI

Sparse reconstruction is an important aspect of MRI, helping to reduce acquisition time and improve spatial-temporal resolution. Popular methods are based mostly on compressed sensing (CS), which relies on the random sampling of k-space to produce incoherent (noise-like) artefacts. Due to hardware constraints, 1D Cartesian phase-encode under-sampling schemes are popular for 2D CS-MRI. However, 1D under-sampling limits 2D incoherence between measurements, yielding structured aliasing artefacts (ghosts) that may be difficult to remove assuming a 2D sparsity model. Reconstruction algorithms typically deploy direction-insensitive 2D regularisation for these direction-associated artefacts. Recognising that phase-encode artefacts can be separated into contiguous 1D signals, we develop two decoupling techniques that enable explicit 1D regularisation and leverage the excellent 1D incoherence characteristics. We also derive a combined 1D + 2D reconstruction technique that takes advantage of spatial relationships within the image. Experiments conducted on retrospectively under-sampled brain and knee data demonstrate that combination of the proposed 1D AliasNet modules with existing 2D deep learned (DL) recovery techniques leads to an improvement in image quality. We also find AliasNet enables a superior scaling of performance compared to increasing the size of the original 2D network layers. AliasNet therefore improves the regularisation of aliasing artefacts arising from phase-encode under-sampling, by tailoring the network architecture to account for their expected appearance. The proposed 1D + 2D approach is compatible with any existing 2D DL recovery technique deployed for this application

Chaotic Sensing

We propose a sparse imaging methodology called Chaotic Sensing (ChaoS) that enables the use of limited yet deterministic linear measurements through fractal sampling. A novel fractal in the discrete Fourier transform is introduced that always results in the artefacts being turbulent in nature. These chaotic artefacts have characteristics that are image independent, facilitating their removal through dampening (via image denoising) and obtaining the maximum likelihood solution. In contrast with existing methods, such as compressed sensing, the fractal sampling is based on digital periodic lines that form the basis of discrete projected views of the image without requiring additional transform domains. This allows the creation of finite iterative reconstruction schemes in recovering an image from its fractal sampling that is also new to discrete tomography. As a result, ChaoS supports linear measurement and optimisation strategies, while remaining capable of recovering a theoretically exact representation of the image. We apply the method to simulated and experimental limited magnetic resonance (MR) imaging data, where restrictions imposed by MR physics typically favour linear measurements for reducing acquisition time

Curvelets and Ridgelets

International audienceDespite the fact that wavelets have had a wide impact in image processing, they fail to efficiently represent objects with highly anisotropic elements such as lines or curvilinear structures (e.g. edges). The reason is that wavelets are non-geometrical and do not exploit the regularity of the edge curve. The Ridgelet and the Curvelet [3, 4] transforms were developed as an answer to the weakness of the separable wavelet transform in sparsely representing what appears to be simple building atoms in an image, that is lines, curves and edges. Curvelets and ridgelets take the form of basis elements which exhibit high directional sensitivity and are highly anisotropic [5, 6, 7, 8]. These very recent geometric image representations are built upon ideas of multiscale analysis and geometry. They have had an important success in a wide range of image processing applications including denoising [8, 9, 10], deconvolution [11, 12], contrast enhancement [13], texture analysis [14, 15], detection [16], watermarking [17], component separation [18], inpainting [19, 20] or blind source separation[21, 22]. Curvelets have also proven useful in diverse fields beyond the traditional image processing application. Let’s cite for example seismic imaging [10, 23, 24], astronomical imaging [25, 26, 27], scientific computing and analysis of partial differential equations [28, 29]. Another reason for the success of ridgelets and curvelets is the availability of fast transform algorithms which are available in non-commercial software packages following the philosophy of reproducible research, see [30, 31]

Projections et distances discrètes

Le travail se situe dans le domaine de la géométrie discrète. La tomographie discrète sera abordée sous l'angle de ses liens avec la théorie de l'information, illustrés par l'application de la transformation Mojette et de la "Finite Radon Transform" au codage redondant d'information pour la transmission et le stockage distribué. Les distances discrètes seront exposées selon les points de vue théorique (avec une nouvelle classe de distances construites par des chemins à poids variables) et algorithmique (transformation en distance, axe médian, granulométrie) en particulier par des méthodes en un balayage d'image (en "streaming"). Le lien avec les séquences d'entiers non-décroissantes et l'inverse de Lambek-Moser sera mis en avant

Imaging applications of the sparse FFT

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 77-81).The sparse Fourier transform leverages the intrinsic sparsity of the frequency spectrum in many natural applications to compute the discrete Fourier Transform (DFT) in sub-linear time. Consequently, it has the potential to enable Big Data applications. In this thesis, we focus on extending the sparse Fourier transform (sparse FFT) to two imaging applications: 4D Light Field and Magnetic Resonance Spectroscopy. Directly applying sparse FFT to these applications however will not work. We need to extend the sparse FFT algorithm to address the following challenges: First, both applications are sample-intensive. It is time consuming, costly, and difficult to acquire samples. So, we need a new sparse FFT algorithm that minimizes the number of required input samples instead of purely focusing on the running time. Second, for these applications the spectra are not very sparse in the discrete Fourier domain. The sparsity is much greater in the continuous Fourier domain. Hence, we need a new sparse FFT algorithm that can leverage the sparsity in the continuous domain as opposed to the discrete domain. In this thesis, we design a sparse FFT algorithm suitable for our imaging applications. Our algorithm contains two phases: it first reconstructs a coarse discrete spectrum and then refines it using gradient descent in the continuous Fourier domain. In our experiments, we showed high-quality reconstruction of 4D light field with only 10% 20% of the samples, and a reduction of the MRS acquisition time by a factor of 3x 4x.by Lixin Shi.S.M

Seismic Full-Waveform Inversion of 3D Field Data – From the Near Surface to the Reservoir

The theory of FWI is well-established. However its practical application to 3D seismic datasets is still a subject of intense research. This technique has shown spectacular results in quantitatively extracting P-wave velocities in the shallow near surface at depths of less than 1 km, using wide-angle OBC datasets. This study deals with establishing a robust methodology for the application of FWI that can be routinely applied to analogous field datasets, both in the shallow near surface and at deeper reservoir depths. A practical strategy for anisotropic 3D acoustic FWI was developed and implemented. The stratergy is tested on a series of 3D datasets: (1) a synthetic Marmousi dataset, (2) an OBC field data and (3) a streamer data. A 3D synthetic Marmousi data is used to compare FWI implementations in both the time domain and the frequency domain. In both domains, it was possible to recover an almost ‘perfect’ model with complete data coverage, no noise, and few iterations. Both approaches were useful and competitive, and ideally both should be available within a comprehensive suite of inversion tools. The anisotropic time-domain FWI strategy was successfully implemented to complex OBC field data set with long offsets, full-azimuthal coverage and low frequencies. The FWI quantitatively recovered p-wave velocities in the shallow near surface, at intermediate depths where the sediments are gas bearing, and at deeper reservoir depths. The velocities are indeed realistic and are consistent with an independent reflection PSDM volume, well data and pressure data. The synthetic FWI data better match the field data, with the phase residuals between the two datasets significantly reduced to low values. The gathers are flatter and the depth-migrated images are more resolved and focused. The strategy was also successfully implemented to complex streamer field data set with short offsets, narrow-azimuthal coverage and reduced signal at the low frequencies. The FWI quantitatively recovered P-wave velocities down to depths of 750 m. A complex series of high and low velocity channels are recovered. These are consistent with an independent reflection PSTM volume. The synthetic FWI data better match the field data, with the phase residuals between the two datasets significantly reduced to low values. The depth-migrated images are more resolved and focused in the shallow section. Open Acces

Blind Retrospective Motion Correction of MR Images

Die Bewegung des Patienten während einer MRI Untersuchung kann die Bildqualität stark verringern. Eine Verschiebung des abzubildenden Objektes von nur ein Paar Millimetern ist genug um Bewegungsartefakte zu erzeugen und der Scan unbrauchbar für die medizinische Diagnostik zu machen. Obwohl in den letzten 20 Jahren mehrere Verfahren entwickelt wurden, ist die Bewegungskorrektur immer noch ein ungelöstes Problem. Wir schlagen einen neuen retrospektiven Bewegungskorrekturalgorithmus vor, mit dem man die Qualität von 3D MR Bildern verbessern kann. Mit diesem Verfahren ist es möglich sowohl starre als auch nicht starre Körperbewegungen zu korrigieren. Der wichtigste Aspekt unserer Algorithmen ist, dass keine Informationen über die Bewegungstrajektorie, z. B. von Kameras, nötig sind um die Bewegungskorrektur durchzuführen. Unsere Verfahren verwenden die RAW-Dateien von normalen MRT-Sequenzen und brauchen keinerlei Anderungen im Scanablauf. Wir benutzen Grafikprozessoren um die Bewegungskorrektur zu beschleunigen – im Fall von starren Körperbewegungen sind nur wenige Sekunden erforderlich, bei nicht starrer Körperbewegung nur einige Minuten Unser Bewegungskorrekturalgorithmus für starre Körper basiert auf der Minimierung einer Kostenfunktion, die die objektive Qualit ̈at des korrigierten Bildes abschätzt. Die Hauptidee ist, durch Optimierung eine Bewegungstrajektorie zu finden, die den kleinsten Betrag der Kostenfunktion liefert. Wir verwenden die Entropie der Bildgradienten als Bildqualitätsfunktion. Um nicht starre Körperbewegungen zu korrigieren, erweitern wir unser mathematisches Modell von Bewegungseffekten. Wir approximieren nicht starre Körperbewegungen als mehrere lokale starre Körperbewegungen. Um solche Bewegungen zu korrigieren, entwickeln wir ein neues annealing-basiert Optimierungsverfahren. Während der Optimierung wechseln wir zwei Schritte ab - die Kostenfunktionsminimierung durch Bild- und Bewegungsparameter. Wir haben mehrere Simulationen sowie in vivo Versuche am Menschen durchgeführt – beide lieferten wesentliche Bildqualitätsverbesserungen

Doctor of Philosophy

dissertationEach year in the United States, a quarter million cases of stroke are caused directly by atherosclerotic disease of the cervical carotid artery. This represents a significant portion of health care costs that could be avoided if high-risk carotid artery lesions could be detected early on in disease progression. There is mounting evidence that Magnetic Resonance Imaging of the carotid artery can better classify subjects who would benefit from interventions. Turbo Spin Echo sequences are a class of Magnetic Resonance Imaging sequences that provide a variety of tissue contrasts. While high resolution Turbo Spin Echo images have demonstrated important details of carotid artery morphology, it is evident that pulsatile blood and wall motion related to the cardiac cycle are still significant sources of image degradation. In addition, patient motion artifacts due to the relatively long scan times of Turbo Spin Echo sequences result in an unacceptable fraction of noninterpretable studies. This dissertation presents work done to detect and correct for types of voluntary and physiological patient motion