Shearlets and Optimally Sparse Approximations

Multivariate functions are typically governed by anisotropic features such as edges in images or shock fronts in solutions of transport-dominated equations. One major goal both for the purpose of compression as well as for an efficient analysis is the provision of optimally sparse approximations of such functions. Recently, cartoon-like images were introduced in 2D and 3D as a suitable model class, and approximation properties were measured by considering the decay rate of the $L^2$ error of the best $N$-term approximation. Shearlet systems are to date the only representation system, which provide optimally sparse approximations of this model class in 2D as well as 3D. Even more, in contrast to all other directional representation systems, a theory for compactly supported shearlet frames was derived which moreover also satisfy this optimality benchmark. This chapter shall serve as an introduction to and a survey about sparse approximations of cartoon-like images by band-limited and also compactly supported shearlet frames as well as a reference for the state-of-the-art of this research field.Comment: in "Shearlets: Multiscale Analysis for Multivariate Data", Birkh\"auser-Springe

On the Doubly Sparse Compressed Sensing Problem

A new variant of the Compressed Sensing problem is investigated when the number of measurements corrupted by errors is upper bounded by some value l but there are no more restrictions on errors. We prove that in this case it is enough to make 2(t+l) measurements, where t is the sparsity of original data. Moreover for this case a rather simple recovery algorithm is proposed. An analog of the Singleton bound from coding theory is derived what proves optimality of the corresponding measurement matrices.Comment: 6 pages, IMACC2015 (accepted

Necessary and sufficient conditions of solution uniqueness in ℓ1\ell_1 minimization

This paper shows that the solutions to various convex $\ell_1$ minimization problems are \emph{unique} if and only if a common set of conditions are satisfied. This result applies broadly to the basis pursuit model, basis pursuit denoising model, Lasso model, as well as other $\ell_1$ models that either minimize $f(Ax-b)$ or impose the constraint $f(Ax-b)\leq\sigma$, where $f$ is a strictly convex function. For these models, this paper proves that, given a solution $x^*$ and defining I=\supp(x^*) and s=\sign(x^*_I), $x^*$ is the unique solution if and only if $A_I$ has full column rank and there exists $y$ such that $A_I^Ty=s$ and $|a_i^Ty|_\infty<1$ for $i\not\in I$. This condition is previously known to be sufficient for the basis pursuit model to have a unique solution supported on $I$. Indeed, it is also necessary, and applies to a variety of other $\ell_1$ models. The paper also discusses ways to recognize unique solutions and verify the uniqueness conditions numerically.Comment: 6 pages; revised version; submitte

Yet another breakdown point notion: EFSBP - illustrated at scale-shape models

The breakdown point in its different variants is one of the central notions to quantify the global robustness of a procedure. We propose a simple supplementary variant which is useful in situations where we have no obvious or only partial equivariance: Extending the Donoho and Huber(1983) Finite Sample Breakdown Point, we propose the Expected Finite Sample Breakdown Point to produce less configuration-dependent values while still preserving the finite sample aspect of the former definition. We apply this notion for joint estimation of scale and shape (with only scale-equivariance available), exemplified for generalized Pareto, generalized extreme value, Weibull, and Gamma distributions. In these settings, we are interested in highly-robust, easy-to-compute initial estimators; to this end we study Pickands-type and Location-Dispersion-type estimators and compute their respective breakdown points.Comment: 21 pages, 4 figure

Virtual Northern Analysis of the Human Genome

BACKGROUND: We applied the Virtual Northern technique to human brain mRNA to systematically measure human mRNA transcript lengths on a genome-wide scale. METHODOLOGY/PRINCIPAL FINDINGS: We used separation by gel electrophoresis followed by hybridization to cDNA microarrays to measure 8,774 mRNA transcript lengths representing at least 6,238 genes at high (>90%) confidence. By comparing these transcript lengths to the Refseq and H-Invitational full-length cDNA databases, we found that nearly half of our measurements appeared to represent novel transcript variants. Comparison of length measurements determined by hybridization to different cDNAs derived from the same gene identified clones that potentially correspond to alternative transcript variants. We observed a close linear relationship between ORF and mRNA lengths in human mRNAs, identical in form to the relationship we had previously identified in yeast. Some functional classes of protein are encoded by mRNAs whose untranslated regions (UTRs) tend to be longer or shorter than average; these functional classes were similar in both human and yeast. CONCLUSIONS/SIGNIFICANCE: Human transcript diversity is extensive and largely unannotated. Our length dataset can be used as a new criterion for judging the completeness of cDNAs and annotating mRNA sequences. Similar relationships between the lengths of the UTRs in human and yeast mRNAs and the functions of the proteins they encode suggest that UTR sequences serve an important regulatory role among eukaryotes

Minimizing Acquisition Maximizing Inference -- A demonstration on print error detection

Is it possible to detect a feature in an image without ever looking at it? Images are known to have sparser representation in Wavelets and other similar transforms. Compressed Sensing is a technique which proposes simultaneous acquisition and compression of any signal by taking very few random linear measurements (M). The quality of reconstruction directly relates with M, which should be above a certain threshold for a reliable recovery. Since these measurements can non-adaptively reconstruct the signal to a faithful extent using purely analytical methods like Basis Pursuit, Matching Pursuit, Iterative thresholding, etc., we can be assured that these compressed samples contain enough information about any relevant macro-level feature contained in the (image) signal. Thus if we choose to deliberately acquire an even lower number of measurements - in order to thwart the possibility of a comprehensible reconstruction, but high enough to infer whether a relevant feature exists in an image - we can achieve accurate image classification while preserving its privacy. Through the print error detection problem, it is demonstrated that such a novel system can be implemented in practise

Super-resolution far-field ghost imaging via compressive sampling

Much more image details can be resolved by improving the system's imaging resolution and enhancing the resolution beyond the system's Rayleigh diffraction limit is generally called super-resolution. By combining the sparse prior property of images with the ghost imaging method, we demonstrated experimentally that super-resolution imaging can be nonlocally achieved in the far field even without looking at the object. Physical explanation of super-resolution ghost imaging via compressive sampling and its potential applications are also discussed.Comment: 4pages,4figure

lp-Recovery of the Most Significant Subspace among Multiple Subspaces with Outliers

We assume data sampled from a mixture of d-dimensional linear subspaces with spherically symmetric distributions within each subspace and an additional outlier component with spherically symmetric distribution within the ambient space (for simplicity we may assume that all distributions are uniform on their corresponding unit spheres). We also assume mixture weights for the different components. We say that one of the underlying subspaces of the model is most significant if its mixture weight is higher than the sum of the mixture weights of all other subspaces. We study the recovery of the most significant subspace by minimizing the lp-averaged distances of data points from d-dimensional subspaces, where p>0. Unlike other lp minimization problems, this minimization is non-convex for all p>0 and thus requires different methods for its analysis. We show that if 0<p<=1, then for any fraction of outliers the most significant subspace can be recovered by lp minimization with overwhelming probability (which depends on the generating distribution and its parameters). We show that when adding small noise around the underlying subspaces the most significant subspace can be nearly recovered by lp minimization for any 0<p<=1 with an error proportional to the noise level. On the other hand, if p>1 and there is more than one underlying subspace, then with overwhelming probability the most significant subspace cannot be recovered or nearly recovered. This last result does not require spherically symmetric outliers.Comment: This is a revised version of the part of 1002.1994 that deals with single subspace recovery. V3: Improved estimates (in particular for Lemma 3.1 and for estimates relying on it), asymptotic dependence of probabilities and constants on D and d and further clarifications; for simplicity it assumes uniform distributions on spheres. V4: minor revision for the published versio

Wavelet Cycle Spinning Denoising of NDE Ultrasonic Signals Using a Random Selection of Shifts

Wavelets are a powerful tool for signal and image denoising. Most of the denoising applications in different fields were based on the thresholding of the discrete wavelet transform (DWT) coefficients. Nevertheless, DWT transform is not a time or shift invariant transform and results depend on the selected shift. Improvements on the denoising performance can be obtained using the stationary wavelet transform (SWT) (also called shift-invariant or undecimated wavelet transform). Denoising using SWT has previously shown a robust and usually better performance than denoising using DWT but with a higher computational cost. In this paper, wavelet shrinkage schemes are applied for reducing noise in synthetic and experimental non-destructive evaluation ultrasonic A-scans, using DWT and a cycle-spinning implementation of SWT. A new denoising procedure, which we call random partial cycle spinning (RPCS), is presented. It is based on a cycle-spinning over a limited number of shifts that are selected in a random way. Wavelet denoising based on DWT, SWT and RPCS have been applied to the same sets of ultrasonic A-scans and their performances in terms of SNR are compared. In all cases three well known threshold selection rules (Universal, Minimax and Sure), with decomposition level dependent selection, have been used. It is shown that the new procedure provides a good robust denoising performance, without the DWT fluctuating performance, and close to SWT but with a much lower computational cost.This work was partially supported by Spanish MCI Project DPI2011-22438San Emeterio Prieto, JL.; Rodríguez-Hernández, MA. (2015). Wavelet Cycle Spinning Denoising of NDE Ultrasonic Signals Using a Random Selection of Shifts. 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Automatic Detection of ECG Abnormalities by using an Ensemble of Deep Residual Networks with Attention

Heart disease is one of the most common diseases causing morbidity and mortality. Electrocardiogram (ECG) has been widely used for diagnosing heart diseases for its simplicity and non-invasive property. Automatic ECG analyzing technologies are expected to reduce human working load and increase diagnostic efficacy. However, there are still some challenges to be addressed for achieving this goal. In this study, we develop an algorithm to identify multiple abnormalities from 12-lead ECG recordings. In the algorithm pipeline, several preprocessing methods are firstly applied on the ECG data for denoising, augmentation and balancing recording numbers of variant classes. In consideration of efficiency and consistency of data length, the recordings are padded or truncated into a medium length, where the padding/truncating time windows are selected randomly to sup-press overfitting. Then, the ECGs are used to train deep neural network (DNN) models with a novel structure that combines a deep residual network with an attention mechanism. Finally, an ensemble model is built based on these trained models to make predictions on the test data set. Our method is evaluated based on the test set of the First China ECG Intelligent Competition dataset by using the F1 metric that is regarded as the harmonic mean between the precision and recall. The resultant overall F1 score of the algorithm is 0.875, showing a promising performance and potential for practical use.Comment: 8 pages, 2 figures, conferenc