257 research outputs found

### A remark on the Restricted Isometry Property in Orthogonal Matching Pursuit

This paper demonstrates that if the restricted isometry constant
$\delta_{K+1}$ of the measurement matrix $A$ satisfies $\delta_{K+1} <
\frac{1}{\sqrt{K}+1},$ then a greedy algorithm called Orthogonal Matching
Pursuit (OMP) can recover every $K$--sparse signal $\mathbf{x}$ in $K$
iterations from A\x. By contrast, a matrix is also constructed with the
restricted isometry constant $\delta_{K+1} = \frac{1}{\sqrt{K}}$ such that
OMP can not recover some $K$-sparse signal $\mathbf{x}$ in $K$ iterations. This
result positively verifies the conjecture given by Dai and Milenkovic in 2009

### Compactly Supported Tensor Product Complex Tight Framelets with Directionality

Although tensor product real-valued wavelets have been successfully applied
to many high-dimensional problems, they can only capture well edge
singularities along the coordinate axis directions. As an alternative and
improvement of tensor product real-valued wavelets and dual tree complex
wavelet transform, recently tensor product complex tight framelets with
increasing directionality have been introduced in [8] and applied to image
denoising in [13]. Despite several desirable properties, the directional tensor
product complex tight framelets constructed in [8,13] are bandlimited and do
not have compact support in the space/time domain. Since compactly supported
wavelets and framelets are of great interest and importance in both theory and
application, it remains as an unsolved problem whether there exist compactly
supported tensor product complex tight framelets with directionality. In this
paper, we shall satisfactorily answer this question by proving a theoretical
result on directionality of tight framelets and by introducing an algorithm to
construct compactly supported complex tight framelets with directionality. Our
examples show that compactly supported complex tight framelets with
directionality can be easily derived from any given eligible low-pass filters
and refinable functions. Several examples of compactly supported tensor product
complex tight framelets with directionality have been presented

### A new proof of some polynomial inequalities related to pseudo-splines

AbstractPseudo-splines of type I were introduced in [I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (2003) 1â€“46] and [Selenick, Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10 (2000) 163â€“181] and type II were introduced in [B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. Anal. 22 (2007) 78â€“104]. Both types of pseudo-splines provide a rich family of refinable functions with B-splines, interpolatory refinable functions and refinable functions with orthonormal shifts as special examples. In [B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. Anal. 22 (2007) 78â€“104], Dong and Shen gave a regularity analysis of pseudo-splines of both types. The key to regularity analysis is Proposition 3.2 in [B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. Anal. 22 (2007) 78â€“104], which also appeared in [A. Cohen, J.P. Conze, RÃ©gularitÃ© des bases d'ondelettes et mesures ergodiques, Rev. Mat. Iberoamericana 8 (1992) 351â€“365] and [I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics, SIAM, Philadelphia, 1992] for the case l=Nâˆ’1. In this note, we will give a new insight into this proposition

### New bounds on the restricted isometry constant Î´2k

AbstractRestricted isometry constants play an important role in compressed sensing. In the literature, E.J. CandÃ¨s has proven that Î´2k<2âˆ’1â‰ˆ0.4142 is a sufficient condition for the l1 minimization problem having a k-sparse solution. Later, S. Foucart and M. Lai have improved the condition to Î´2k<0.4531 and S. Foucart has improved the bound to Î´2k<0.4652. In 2010, T. Cai, L. Wang and G. Xu have improved the condition to Î´2k<0.4721 for the cases such that k is a multiple of 4 or k is very large and S. Foucart has improved the bound to Î´2k<0.4734 for large values of k. In this paper, we have improved the sufficient condition to Î´2k<0.4931 for general k. Also, in some special cases, the sufficient condition can be improved to Î´2k<0.6569. These new bounds have several benefits on recovering compressible signals with noise

### WaveletQuant, an improved quantification software based on wavelet signal threshold de-noising for labeled quantitative proteomic analysis

<p>Abstract</p> <p>Background</p> <p>Quantitative proteomics technologies have been developed to comprehensively identify and quantify proteins in two or more complex samples. Quantitative proteomics based on differential stable isotope labeling is one of the proteomics quantification technologies. Mass spectrometric data generated for peptide quantification are often noisy, and peak detection and definition require various smoothing filters to remove noise in order to achieve accurate peptide quantification. Many traditional smoothing filters, such as the moving average filter, Savitzky-Golay filter and Gaussian filter, have been used to reduce noise in MS peaks. However, limitations of these filtering approaches often result in inaccurate peptide quantification. Here we present the WaveletQuant program, based on wavelet theory, for better or alternative MS-based proteomic quantification.</p> <p>Results</p> <p>We developed a novel discrete wavelet transform (DWT) and a 'Spatial Adaptive Algorithm' to remove noise and to identify true peaks. We programmed and compiled WaveletQuant using Visual C++ 2005 Express Edition. We then incorporated the WaveletQuant program in the <b>Trans-Proteomic Pipeline (TPP)</b>, a commonly used open source proteomics analysis pipeline.</p> <p>Conclusions</p> <p>We showed that WaveletQuant was able to quantify more proteins and to quantify them more accurately than the ASAPRatio, a program that performs quantification in the TPP pipeline, first using known mixed ratios of yeast extracts and then using a data set from ovarian cancer cell lysates. The program and its documentation can be downloaded from our website at <url>http://systemsbiozju.org/data/WaveletQuant</url>.</p

### Fast computation of observed cross section for $\psi^{\prime} \to PP$ decays

It has been conjectured that the relative phase between strong and
electromagnetic amplitudes is universally $-90^{\circ}$ in charmonium decays.
$\psi^{\prime}$ decaying into pseudoscalar pair provides a possibility to test
this conjecture. However, the experimentally observed cross section for such a
process is depicted by the two-fold integral which takes into account the
initial state radiative (ISR) correction and energy spread effect. Using the
generalized linear regression approach, a complex energy-dependent factor is
approximated by a linear function of energy. Taking advantage of this
simplification, the integration of ISR correction can be performed and an
analytical expression with accuracy at the level of 1% is obtained. Then, the
original two-fold integral is simplified into a one-fold integral, which
reduces the total computing time by two orders of magnitude. Such a simplified
expression for the observed cross section usually plays an indispensable role
in the optimization of scan data taking, the determination of systematic
uncertainty, and the analysis of data correlation.Comment: 8 pages, 5 figure

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