arXiv subjects

Haizhao Yang

Publications and source records attributed to Haizhao Yang.

134 records · Page 8Linked to original sources

arXiv2016

Statistical Analysis of Synchrosqueezed Transforms

Synchrosqueezed transforms are non-linear processes for a sharpened time-frequency representation of wave-like components. They are efficient tools for identifying and analyzing wave-like components from their superposition. This paper is concerned with the statistical properties of compactly supported synchrosqueezed transforms for wave-like components embedded in a generalized Gaussian random process in multidimensional spaces. Guided by the theoretical analysis of these properties, new numerical implementations are proposed to reduce the noise fluctuations of these transforms on noisy data. A MATLAB package SynLab together with several heavily noisy examples is provided to support these theoretical claims.

math.ST↗

arXiv2015

Multidimensional Butterfly Factorization

This paper introduces the multidimensional butterfly factorization as a data-sparse representation of multidimensional kernel matrices that satisfy the complementary low-rank property. This factorization approximates such a kernel matrix of size $N\times N$ with a product of $Ø(\log N)$ sparse matrices, each of which contains $Ø(N)$ nonzero entries. We also propose efficient algorithms for constructing this factorization when either (i) a fast algorithm for applying the kernel matrix and its adjoint is available or (ii) every entry of the kernel matrix can be evaluated in $Ø(1)$ operations. For the kernel matrices of multidimensional Fourier integral operators, for which the complementary low-rank property is not satisfied due to a singularity at the origin, we extend this factorization by combining it with either a polar coordinate transformation or a multiscale decomposition of the integration domain to overcome the singularity. Numerical results are provided to demonstrate the efficiency of the proposed algorithms.

math.NA↗

arXiv2015

Crystal image analysis using $2D$ synchrosqueezed transforms

We propose efficient algorithms based on a band-limited version of 2D synchrosqueezed transforms to extract mesoscopic and microscopic information from atomic crystal images. The methods analyze atomic crystal images as an assemblage of non-overlapping segments of 2D general intrinsic mode type functions, which are superpositions of non-linear wave-like components. In particular, crystal defects are interpreted as the irregularity of local energy; crystal rotations are described as the angle deviation of local wave vectors from their references; the gradient of a crystal elastic deformation can be obtained by a linear system generated by local wave vectors. Several numerical examples of synthetic and real crystal images are provided to illustrate the efficiency, robustness, and reliability of our methods.

math.NA↗

arXiv2015

Combining $2D$ synchrosqueezed wave packet transform with optimization for crystal image analysis

We develop a variational optimization method for crystal analysis in atomic resolution images, which uses information from a 2D synchrosqueezed transform (SST) as input. The synchrosqueezed transform is applied to extract initial information from atomic crystal images: crystal defects, rotations and the gradient of elastic deformation. The deformation gradient estimate is then improved outside the identified defect region via a variational approach, to obtain more robust results agreeing better with the physical constraints. The variational model is optimized by a nonlinear projected conjugate gradient method. Both examples of images from computer simulations and imaging experiments are analyzed, with results demonstrating the effectiveness of the proposed method.

cond-mat.mtrl-sci↗

arXiv2015

Butterfly Factorization

The paper introduces the butterfly factorization as a data-sparse approximation for the matrices that satisfy a complementary low-rank property. The factorization can be constructed efficiently if either fast algorithms for applying the matrix and its adjoint are available or the entries of the matrix can be sampled individually. For an $N \times N$ matrix, the resulting factorization is a product of $O(\log N)$ sparse matrices, each with $O(N)$ non-zero entries. Hence, it can be applied rapidly in $O(N\log N)$ operations. Numerical results are provided to demonstrate the effectiveness of the butterfly factorization and its construction algorithms.

math.NA↗

arXiv2015

A Multiscale Butterfly Algorithm for Multidimensional Fourier Integral Operators

This paper presents an efficient multiscale butterfly algorithm for computing Fourier integral operators (FIOs) of the form $(\mathcal{L} f)(x) = \int_{R^d}a(x,ξ) e^{2πıΦ(x,ξ)}\hat{f}(ξ) dξ$, where $Φ(x,ξ)$ is a phase function, $a(x,ξ)$ is an amplitude function, and $f(x)$ is a given input. The frequency domain is hierarchically decomposed into a union of Cartesian coronas. The integral kernel $a(x,ξ) e^{2πıΦ(x,ξ)}$ in each corona satisfies a special low-rank property that enables the application of a butterfly algorithm on the Cartesian phase-space grid. This leads to an algorithm with quasi-linear operation complexity and linear memory complexity. Different from previous butterfly methods for the FIOs, this new approach is simple and reduces the computational cost by avoiding extra coordinate transformations. Numerical examples in two and three dimensions are provided to demonstrate the practical advantages of the new algorithm.

math.NA↗

arXiv2014

Synchrosqueezed Wave Packet Transforms and Diffeomorphism Based Spectral Analysis for 1D General Mode Decompositions

This paper develops new theory and algorithms for 1D general mode decompositions. First, we introduce the 1D synchrosqueezed wave packet transform and prove that it is able to estimate the instantaneous information of well-separated modes from their superposition accurately. The synchrosqueezed wave packet transform has a better resolution than the synchrosqueezed wavelet transform in the time-frequency domain for separating high frequency modes. Second, we present a new approach based on diffeomorphisms for the spectral analysis of general shape functions. These two methods lead to a framework for general mode decompositions under a weak well-separation condition and a well different condition. Numerical examples of synthetic and real data are provided to demonstrate the fruitful applications of these methods.

math.NA↗

arXiv2013

Synchrosqueezed Curvelet Transform for 2D Mode Decomposition

This paper introduces the synchrosqueezed curvelet transform as an optimal tool for 2D mode decomposition of wavefronts or banded wave-like components. The synchrosqueezed curvelet transform consists of a generalized curvelet transform with application dependent geometric scaling parameters, and a synchrosqueezing technique for a sharpened phase space representation. In the case of a superposition of banded wave-like components with well-separated wave-vectors, it is proved that the synchrosqueezed curvelet transform is capable of recognizing each component and precisely estimating local wave-vectors. A discrete analogue of the continuous transform and several clustering models for decomposition are proposed in detail. Some numerical examples with synthetic and real data are provided to demonstrate the above properties of the proposed transform.

math.NA↗