Dyadic Wavelets Energy Zero-Crossings

An important problem in signal analysis is to define a general purpose signal representation which is well adapted for developing pattern recognition algorithms. In this paper we will show that such a representation can be defined from the position of the zero-crossings and the local energy values of a dyadic wavelet decomposition. This representation is experimentally complete and admits a simple distance for pattern matching applications. It provides a multiscale decomposition of the signal and at each scale characterizes the locations of abrupt changes in the signal. We have developed a stereo matching algorithm to illustrate the application of this representation to pattern matching

Complex Wavelet Bases, Steerability, and the Marr-Like Pyramid

Our aim in this paper is to tighten the link between wavelets, some classical image-processing operators, and David Marr's theory of early vision. The cornerstone of our approach is a new complex wavelet basis that behaves like a smoothed version of the Gradient-Laplace operator. Starting from first principles, we show that a single-generator wavelet can be defined analytically and that it yields a semi-orthogonal complex basis of L-2 (R-2), irrespective of the dilation matrix used. We also provide an efficient FFT-based filterbank implementation. We then propose a slightly redundant version of the transform that is nearly translation -invariant and that is optimized for better steerability (Gaussian-like smoothing kernel). We call it the Marr-like wavelet pyramid because it essentially replicates the processing steps in Marr's theory of early vision. We use it to derive a primal wavelet sketch which is a compact description of the image by a multiscale, subsampled edge map. Finally, we provide an efficient iterative algorithm for the reconstruction of an image from its primal wavelet sketch

Investigation of Different Sparsity Transforms for the PICCS Algorithm in Small- Animal Respiratory Gated CT

Data Availability Statement: All relevant data are available from the Zenodo database, under the DOI: http://dx.doi.org/10.5281/zenodo.15685.Respiratory gating helps to overcome the problem of breathing motion in cardiothoracic small-animal imaging by acquiring multiple images for each projection angle and then assigning projections to different phases. When this approach is used with a dose similar to that of a static acquisition, a low number of noisy projections are available for the reconstruction of each respiratory phase, thus leading to streak artifacts in the reconstructed images. This problem can be alleviated using a prior image constrained compressed sensing (PICCS) algorithm, which enables accurate reconstruction of highly undersampled data when a prior image is available. We compared variants of the PICCS algorithm with different transforms in the prior penalty function: gradient, unitary, and wavelet transform. In all cases the problem was solved using the Split Bregman approach, which is efficient for convex constrained optimization. The algorithms were evaluated using simulations generated from data previously acquired on a micro-CT scanner following a high-dose protocol (four times the dose of a standard static protocol). The resulting data were used to simulate scenarios with different dose levels and numbers of projections. All compressed sensing methods performed very similarly in terms of noise, spatiotemporal resolution, and streak reduction, and filtered back-projection was greatly improved. Nevertheless, the wavelet domain was found to be less prone to patchy cartoon-like artifacts than the commonly used gradient domain.This work was partially funded by the RICRETIC network (RD12/0042/0057) from the Ministerio de Economía y Competitividad (www.mineco.gob.es/) and projects TEC2010-21619-C04-01 and PI11/00616 from Ministerio de Ciencia e Innovación (www.micinn.es/). The research leading to these results was supported by funding from the Innovative Medicines Initiative (www.imi.europa.eu) Joint Undertaking under grant agreement n°115337, the resources of which comprise financial contributions from the European Union's Seventh Framework Programme (FP7/2007-2013) and EFPIA companies ("in kind contribution"). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript

Non local spatial and angular matching : enabling higher spatial resolution diffusion MRI datasets through adaptive denoising

Diffusion magnetic resonance imaging (MRI) datasets suffer from low Signal-to-Noise Ratio (SNR), especially at high b-values. Acquiring data at high b-values contains relevant information and is now of great interest for microstructural and connectomics studies. High noise levels bias the measurements due to the non-Gaussian nature of the noise, which in turn can lead to a false and biased estimation of the diffusion parameters. Additionally, the usage of in-plane acceleration techniques during the acquisition leads to a spatially varying noise distribution, which depends on the parallel acceleration method implemented on the scanner. This paper proposes a novel diffusion MRI denoising technique that can be used on all existing data, without adding to the scanning time. We first apply a statistical framework to convert both stationary and non stationary Rician and non central Chi distributed noise to Gaussian distributed noise, effectively removing the bias. We then introduce a spatially and angular adaptive denoising technique, the Non Local Spatial and Angular Matching (NLSAM) algorithm. Each volume is first decomposed in small 4D overlapping patches, thus capturing the spatial and angular structure of the diffusion data, and a dictionary of atoms is learned on those patches. A local sparse decomposition is then found by bounding the reconstruction error with the local noise variance. We compare against three other state-of-the-art denoising methods and show quantitative local and connectivity results on a synthetic phantom and on an in-vivo high resolution dataset. Overall, our method restores perceptual information, removes the noise bias in common diffusion metrics, restores the extracted peaks coherence and improves reproducibility of tractography on the synthetic dataset. On the 1.2 mm high resolution in-vivo dataset, our denoising improves the visual quality of the data and reduces the number of spurious tracts when compared to the noisy acquisition. Our work paves the way for higher spatial resolution acquisition of diffusion MRI datasets, which could in turn reveal new anatomical details that are not discernible at the spatial resolution currently used by the diffusion MRI community

Numerical simulation of shock wave/turbulent boundary layer interactions in over-expanded nozzles

The performances of first-stage liquid rocket engines are highly dependent on the fluid dynamic behaviour of the expansion nozzle and for launch-trajectory optimisation purposes, large values of the ratio between the exit and throat areas are desirable. The maximum limit to this ratio is imposed by the need to avoid internal flow separation, since at sea level the flow is highly overexpanded. However, during the start-up phase the chamber pressure is below the design pressure and the flow separates from the nozzle wall. This condition is characterised by complex physical features, including the formation of a shock-wave system that adapts the exhaust flow to the higher ambient pressure, shock-wave/boundary-layer interactions (SWBLI), and a turbulent recirculation zone. As a global effect, the nozzle experiences non-axial forces, known as side-loads, which can be of sufficient strength to cause structural damage to the engine. Despite several studies in the last decades, a clear physical understanding of the driving factors of the unsteadiness is still lacking. The experiments on axi-symmetric nozzles suffer from the lack of flow measurements inside the nozzle itself, due to the challenging flow conditions and absence of optical access. Therefore, numerical simulations represent an important complementary tool to gain a more complete insight into the physics of separated rocket nozzle flows, giving the opportunity to address important open questions. The present thesis investigates shock wave induced flow separation in over-expanded rocket nozzles by means of large-scale high-fidelity numerical computations based on the delayed detached eddy simulation (DDES) methodology, a hybrid RANS/LES method that allows the simulation of high-Reynolds number flows involving massive flow separation. In this approach, attached boundary layers are treated in RANS mode, lowering the computational requirements, while the most energetic turbulent scales of separated shear layers and turbulent recirculating zones are directly treated by the LES mode of the method. The potential of DDES has been first tested on a simple planar nozzle configuration for which experimental and numerical studies are available, with the main aim of highlighting the strengths and weaknesses of the approach. The results indicate that the DDES is able to capture the shock oscillations and that the computed characteristic frequency is close to that reported in literature for the same test case. The study then focuses on the investigation of the unsteadiness in a truncated ideal contoured (TIC) nozzle, a configuration for which experimental data are available. The numerical data agree well with the experimental results in terms of mean and fluctuating wall pressure statistics. The frequency spectra are characterised by the presence of a large bump in the low-frequency range associated to an axi-symmetric (piston-like) motion of the shock system and a broad and high amplitude peak at high frequencies generated by the turbulent activity of the detached shear layer. Moreover, a distinct peak at an intermediate frequency (f « 1 kHz) is observed in the wall-pressure spectra downstream of the separation shock. A Fourier-based spectral analysis performed in both time and azimuthal wave number space, reveals that this peak is associated with the first (non- symmetrical) pressure mode and is thus related to the generation of the aerodynamic side loads. Furthermore, it is found that the unsteady Mach disk is characterised by an intense vortex shedding activity that, together with the vortical structures of the annular shear layer, contributes to the sustainment of an aeroacoustic feedback loop occurring within the nozzle

Wavelet and Multiscale Methods

Various scientific models demand finer and finer resolutions of relevant features. Paradoxically, increasing computational power serves to even heighten this demand. Namely, the wealth of available data itself becomes a major obstruction. Extracting essential information from complex structures and developing rigorous models to quantify the quality of information leads to tasks that are not tractable by standard numerical techniques. The last decade has seen the emergence of several new computational methodologies to address this situation. Their common features are the nonlinearity of the solution methods as well as the ability of separating solution characteristics living on different length scales. Perhaps the most prominent examples lie in multigrid methods and adaptive grid solvers for partial differential equations. These have substantially advanced the frontiers of computability for certain problem classes in numerical analysis. Other highly visible examples are: regression techniques in nonparametric statistical estimation, the design of universal estimators in the context of mathematical learning theory and machine learning; the investigation of greedy algorithms in complexity theory, compression techniques and encoding in signal and image processing; the solution of global operator equations through the compression of fully populated matrices arising from boundary integral equations with the aid of multipole expansions and hierarchical matrices; attacking problems in high spatial dimensions by sparse grid or hyperbolic wavelet concepts. This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computation and to promote the exchange of ideas emerging in various disciplines