A wavelet-based method for multiscale tomographic reconstruction

The authors represent the standard ramp filter operator of the filtered-back-projection (FBP) reconstruction in different bases composed of Haar and Daubechies compactly supported wavelets. The resulting multiscale representation of the ramp-filter matrix operator is approximately diagonal. The accuracy of this diagonal approximation becomes better as wavelets with larger numbers of vanishing moments are used. This wavelet-based representation enables the authors to formulate a multiscale tomographic reconstruction technique in which the object is reconstructed at multiple scales or resolutions. A complete reconstruction is obtained by combining the reconstructions at different scales. The authors' multiscale reconstruction technique has the same computational complexity as the FBP reconstruction method. It differs from other multiscale reconstruction techniques in that (1) the object is defined through a one-dimensional multiscale transformation of the projection domain, and (2) the authors explicitly account for noise in the projection data by calculating maximum a posteriori probability (MAP) multiscale reconstruction estimates based on a chosen fractal prior on the multiscale object coefficients. The computational complexity of this maximum a posteriori probability (MAP) solution is also the same as that of the FBP reconstruction. This result is in contrast to commonly used methods of statistical regularization, which result in computationally intensive optimization algorithms.     

基于稀疏编码的语音增强方法研究

本文利用带噪语音经特征基函数矩阵转换后所具有的稀疏特性,用最大似然估计方法对转换后得到的稀疏分量进行非线性压缩去噪,然后再经过反变换和重构恢复出原始语音信号的估计.特征基函数矩阵反映了语音数据本身的统计特性,因此具有很好的合理性和可取性.仿真结果表明利用稀疏编码方法能极大程度地抑制背景噪声,与小波消噪法相比优势明显.     

Sparsity induced convex nonnegative matrix factorization algorithm with manifold regularization

To address problems that the effectiveness of feature learned from real noisy data by classical nonnegative matrix factorization method,a novel sparsity induced manifold regularized convex nonnegative matrix factorization algorithm (SGCNMF) was proposed.Based on manifold regularization,the L_2,1norm was introduced to the basis matrix of low dimensional subspace as sparse constraint.The multiplicative update rules were given and the convergence of the algorithm was analyzed.Clustering experiment was designed to verify the effectiveness of learned features within various of noisy environments.The empirical study based on K-means clustering shows that the sparse constraint reduces the representation of noisy features and the new method is better than the 8 similar algorithms with stronger robustness to a variable extent.     

Kernel eigenmaps based multiscale sparse model for hyperspectral image classification

Hyperspectral imaging (HSI) is the emerging method that combines traditional imaging and spectroscopy to provide the image with both the spatial and spectral information of the object present in the image. The major challenges of the existing techniques for HSI classification are the high dimensionality of data and its complexity in classification. This paper devises a new technique to classify the HSI named Spatial–Spectral Schroedinger Eigen Maps based Multi-scale adaptive sparse representation (S²SEMASR). In this, two different phases are employed for the accurate classification of the HSI, namely, Schroedinger Eigen maps (SE) based spatial–spectral feature extraction and multi-scale adaptive sparse classification for the feature extracted image. SE makes use of spatial–spectral cluster potentials which allows the extraction of features that best describes the characteristics of different classes of HSI. The multiscale adaptive sparse representation (MASR) applied over the SE features provides the sparse coefficients that includes distinct scale level sparsity with same class level sparsity. With the obtained coefficients, the class label of each pixel is determined. The proposed HSI classifier well utilizes the spectral and spatial characteristics to exploit the within-class variability and thus reduces the misclassification of similar test pixels Experimental results demonstrated that the proposed S²SEMASR approach outperforms the traditional results both qualitatively and quantitatively with an overall accuracy of 98.3%.     

Convergence of the simultaneous algebraic reconstruction technique (SART)

Computed tomography (CT) has been extensively studied for years and widely used in the modern society. Although the filtered back-projection algorithm is the method of choice by manufacturers, efforts are being made to revisit iterative methods due to their unique advantages, such as superior performance with incomplete noisy data. In 1984, the simultaneous algebraic reconstruction technique (SART) was developed as a major refinement of the algebraic reconstruction technique (ART). However, the convergence of the SART has never been established since then. In this paper, the convergence is proved under the condition that coefficients of the linear imaging system are nonnegative. It is shown that from any initial guess the sequence generated by the SART converges to a weighted least square solution.     

Feasibility of half-data image reconstruction in 3-D reflectivity tomography with a spherical aperture

Reflectivity tomography is an imaging technique that seeks to reconstruct certain acoustic properties of a weakly scattering object. Besides being applicable to pure ultrasound imaging techniques, the reconstruction theory of reflectivity tomography is also pertinent to hybrid imaging techniques such as thermoacoustic tomography. In this work, assuming spherical scanning apertures, redundancies in the three-dimensional (3-D) reflectivity tomography data function are identified and formulated mathematically. These data redundancies are used to demonstrate that knowledge of the measured data function over half of its domain uniquely specifies the 3-D object function. This indicates that, in principle, exact image reconstruction can be performed using a "half-scan" data function, which corresponds to temporally untruncated measurements acquired on a hemi-spherical aperture, or using a "half-time" data function, which corresponds to temporally truncated measurements acquired on the entire spherical aperture. Both of these minimal scanning configurations have important biological imaging applications. An iterative reconstruction method is utilized for reconstruction of a simulated 3-D object from noiseless and noisy half-scan and half-time data functions.     

基于压缩感知稀疏信号重建的迭代硬阀值算法

Nyquist采样速率条件下的信号采样,采样系统表现良好并且信号可以被稀疏向量近似表示时,信号可以被有效而精确地重构。针对无噪声信号,利用确定的稀疏基和随机的观测矩阵,研究迭代硬阀值算法的有效性。若观测矩阵满足有限等距性质（RIP）,且稀疏基与随机观测矩阵不相干时,通过该算法,原始信号的稀疏投影可以被高概率重构。最后,利用哈达码正交矩阵作为稀疏基,高斯随机矩阵作为观测矩阵,对原始信号的稀疏投影进行重构,结果验证了该算法的有效性。     

Adaptive multiscale moment method (AMMM) for analysis of scatteringfrom perfectly conducting plates

Adaptive multiscale moment method (AMMM) is presented for the analysis of scattering from a thin perfectly conducting plate. This algorithm employs the conventional moment method and a special matrix transformation, which is derived from the tensor products of the two one-dimensional (1-D) multiscale triangular basis functions that are used for expansion and testing functions in the conventional moment method. The special feature of these new basis functions introduced through this transformation is that they are orthogonal at the same scale except at the initial scale and not between scales. From one scale to another scale, the initial estimate for the solution can be predicted using this multiscale technique. Hence, the compression is applied directly to the solution and the size of the linear equations to be solved is reduced, thereby improving the efficiency of the conventional moment method. The basic difference between this methodology and the other techniques that have been presented so far is that we apply the compression not to the impedance matrix, but to the solution itself directly using an iterative solution methodology. The extrapolated results at the higher scale thus provide a good initial guess for the iterative method. Typically, when the number of unknowns exceeds a few thousand unknowns, the matrix solution time exceeds generally the matrix fill time. Hence, the goal of this method is directed in solving electrically larger problems, where the matrix solution time is of concern. Two numerical results are presented, which demonstrate that the AMMM is a useful method to analyze scattering from perfectly conducting plates     

Multi-stage image denoising based on correlation coefficient matching and sparse dictionary pruning

We present a novel image denoising method based on multiscale sparse representations. In tackling the conflicting problems of structure extraction and artifact suppression, we introduce a correlation coefficient matching criterion for sparse coding so as to extract more meaningful structures from the noisy image. On the other hand, we propose a dictionary pruning method to suppress noise. Based on the above techniques, an effective dictionary training method is developed. To further improve the denoising performance, we propose a multi-stage sparse coding framework where sparse representations are obtained in different scales to capture multiscale image features for effective denoising. The multi-stage coding scheme not only reduces the computational burden of previous multiscale denoising approaches, but more importantly, it also contributes to artifact suppression. Experimental results show that the proposed method achieves a state-of-the-art denoising performance in terms of both objective and subjective quality and provides significant improvements over other methods at high noise levels.     

压缩感知中稀疏分解和重构精度改进的一种方法

稀疏分解、非相关观测和重构算法是压缩感知的三大要素,任何一个环节的设计优劣都对压缩感知的性能产生重大影响,稀疏分解是实现压缩感知的前提,现今使用的稀疏分解对大多数自然信号都不能做到理想的绝对稀疏,而是近似稀疏,这大大影响了压缩感知的重构性能。本文设计了一种可逆的阈值,并用其构造门限矩阵,从而门限矩阵可逆,将门限矩阵作用于信号经正交变换后的近似稀疏系数,可使系数更接近理想的绝对稀疏,而且门限矩阵对系数的处理过程是可逆的,即可由处理后的系数无损恢复原来的近似稀疏系数。重构算法采用贪婪算法中的OMP和CoSaMP,从理论上分析了在保证与CoSaMP同样的前提条件下,门限矩阵改进后的CoSaMP重构误差明显减小,仿真实验用门限矩阵对OMP和CoSaMP的改进前后进行对比,验证了门限矩阵对重构精度有进一步的提高。      

Tomographic reconstruction from experimentally obtained cone-beam projections

Direct reconstruction in three dimensions for two-dimensional projection data has been achieved by cone-beam reconstruction techniques. In this paper explicit formulas for a cone-beam convolution and back-projection reconstruction algorithm are given in a form which can be easily coded for a computer. The algorithm is justified by analyzing tomographic reconstructions of a uniformly attenuating sphere from simulated noisy projection data. A particular feature of this algorithm is the use of a one-dimensional rather than two-dimensional convolution function, greatly speeding up the reconstruction. The technique is applicable however large the cone angle of data capture and correctly reduces to the pure fan-beam reconstruction technique in the central section of the cone. The method has been applied to data captured on a cone-beam CT scanner designed for bone mineral densitometry.     

Multiscale autoregressive models and wavelets

The multiscale autoregressive (MAR) framework was introduced to support the development of optimal multiscale statistical signal processing. Its power resides in the fast and flexible algorithms to which it leads. While the MAR framework was originally motivated by wavelets, the link between these two worlds has been previously established only in the simple case of the Haar wavelet. The first contribution of this paper is to provide a unification of the MAR framework and all compactly supported wavelets as well as a new view of the multiscale stochastic realization problem. The second contribution of this paper is to develop wavelet-based approximate internal MAR models for stochastic processes. This will be done by incorporating a powerful synthesis algorithm for the detail coefficients which complements the usual wavelet reconstruction algorithm for the scaling coefficients. Taking advantage of the statistical machinery provided by the MAR framework, we will illustrate the application of our models to sample-path generation and estimation from noisy, irregular, and sparse measurements     

Adaptive multiscale moment method (AMMM) for analysis of scatteringfrom three-dimensional perfectly conducting structures

The adaptive multiscale moment method (AMMM) is presented for the analysis of scattering from three-dimensional (3D) perfectly conducting bodies. This algorithm employs the conventional moment method (MM) using the subsectional triangular patch basis functions and a special matrix transformation, which is derived from solving the Fredholm equation of the first kind by the multiscale technique. This methodology is more suitable for problems where the matrix solution time is much greater than the matrix fill time. The widely used triangular patch vector basis functions developed by Rao et al., (1982), are used for expansion and testing functions in the conventional MM. The objective here is to compress the unknowns in existing MM codes, which solves for the currents crossing the edges of the triangular patch basis functions. By use of a matrix transformation, the currents, source terms, and impedance matrix can be arranged in the form of different scales. From one scale to another scale, the initial guess for the solution can be predicted according to the properties of the multiscale technique. AMMM can reduce automatically the size of the linear equations so as to improve the efficiency of the conventional MM. The basic difference between this methodology and the other wavelet-based techniques that have been presented so far is that we apply the compression not to the impedance matrix but to the solution itself directly in an iterative fashion even though it is an unknown. Two numerical results are presented, which demonstrate that the AMMM is a useful method for analysis of electromagnetic scattering from arbitrary shaped 3D perfectly conducting bodies     

Robust Parametric Macromodeling Using Multivariate Orthonormal Vector Fitting

A robust multivariate extension of the orthonormal vector fitting technique is introduced for rational parametric macromodeling of highly dynamic responses in the frequency domain. The technique is applicable to data that is sparse or dense, deterministic or a bit noisy, and grid-based or scattered in the design space. For a specified geometrical parameter combination, a SPICE equivalent model can be calculated.      

A curve evolution approach to object-based tomographic reconstruction

We develop a new approach to tomographic reconstruction problems based on geometric curve evolution techniques. We use a small set of texture coefficients to represent the object and background inhomogeneities and a contour to represent the boundary of multiple connected or unconnected objects. Instead of reconstructing pixel values on a fixed rectangular grid, we then find a reconstruction by jointly estimating these unknown contours and texture coefficients of the object and background. By designing a new "tomographic flow", the resulting problem is recast into a curve evolution problem and an efficient algorithm based on level set techniques is developed. The performance of the curve evolution method is demonstrated using examples with noisy limited-view Radon transformed data and noisy ground-penetrating radar data. The reconstruction results and computational cost are compared with those of conventional, pixel-based regularization methods. The results indicate that the curve evolution methods achieve improved shape reconstruction and have potential computation and memory advantages over conventional regularized inversion methods.     

Reconstruction From Limited-Angle Projections Based on $delta-u$ Spectrum Analysis

This paper proposes a sparse representation of an image using discrete ?-u functions. A ?-u function is defined as the product of a Kronecker delta function and a step function. Based on the sparse representation, we have developed a novel and effective method for reconstructing an image from limited-angle projections. The method first estimates the parameters of the sparse representation from the incomplete projection data, and then directly calculates the image to be reconstructed. Experiments have shown that the proposed method can effectively recover the missing data and reconstruct images more accurately than the total-variation (TV) regularized reconstruction method.     

Platelets: a multiscale approach for recovering edges and surfaces in photon-limited medical imaging

The nonparametric multiscale platelet algorithms presented in this paper, unlike traditional wavelet-based methods, are both well suited to photon-limited medical imaging applications involving Poisson data and capable of better approximating edge contours. This paper introduces platelets, localized functions at various scales, locations, and orientations that produce piece-wise linear image approximations, and a new multiscale image decomposition based on these functions. Platelets are well suited for approximating images consisting of smooth regions separated by smooth boundaries. For smoothness measured in certain H?lder classes, it is shown that the error of m-term platelet approximations can decay significantly faster than that of m-term approximations in terms of sinusoids, wavelets, or wedgelets. This suggests that platelets may outperform existing techniques for image denoising and reconstruction. Fast, platelet-based, maximum penalized likelihood methods for photon-limited image denoising, deblurring and tomographic reconstruction problems are developed. Because platelet decompositions of Poisson distributed images are tractable and computationally efficient, existing image reconstruction methods based on expectation-maximization type algorithms can be easily enhanced with platelet techniques. Experimental results suggest that platelet-based methods can outperform standard reconstruction methods currently in use in confocal microscopy, image restoration, and emission tomography.     

Sparse‐View CT Image Recovery Using Two‐Step Iterative Shrinkage‐Thresholding Algorithm

We investigate an image recovery method for sparse‐view computed tomography (CT) using an iterative shrinkage algorithm based on a second‐order approach. The two‐step iterative shrinkage‐thresholding (TwIST) algorithm including a total variation regularization technique is elucidated to be more robust than other first‐order methods; it enables a perfect restoration of an original image even if given only a few projection views of a parallel‐beam geometry. We find that the incoherency of a projection system matrix in CT geometry sufficiently satisfies the exact reconstruction principle even when the matrix itself has a large condition number. Image reconstruction from fan‐beam CT can be well carried out, but the retrieval performance is very low when compared to a parallel‐beam geometry. This is considered to be due to the matrix complexity of the projection geometry. We also evaluate the image retrieval performance of the TwIST algorithm using measured projection data.     

二维压缩感知多投影矩阵特征融合的SAR目标识别方法

针对合成孔径雷达（synthetic aperture radar,SAR）目标识别问题,提出联合多层次二维压缩感知投影特征的方法。采用二维压缩感知投影作为基础特征提取算法,具有不依赖训练样本、效率高等显著优势。构建多个二维压缩感知投影矩阵提取原始SAR图像的多层次特征。不同投影矩阵获得的特征具有差异性,从不同方面描述SAR图像的灰度分布特性;同时,这些特征源自相同的输入图像,因此也具有一定的内在关联性。采用联合稀疏表示对提取的多个特征矢量进行表征分析,在内在关联性约束下考察不同特征的独立鉴别能力,从而提升每一类特征的稀疏表示精度。最终,根据求解的稀疏表示系数,分别在各个训练类别上对测试样本的多类特征进行重构,获得重构误差。根据最小误差的准则,判定测试样本所属目标类别。通过综合运用多层次二维压缩感知特征提取和联合稀疏表示分类,提高SAR目标识别的整体性能。利用MSTAR数据集中的多类目标SAR图像对方法进行测试验证,结果反映其在标准操作条件（standard operating condition,SOC）和扩展操作条件（extended operating condition,EOC）均可保持可靠的识别性能。     

Compressed Sensing and Redundant Dictionaries

This paper extends the concept of compressed sensing to signals that are not sparse in an orthonormal basis but rather in a redundant dictionary. It is shown that a matrix, which is a composition of a random matrix of certain type and a deterministic dictionary, has small restricted isometry constants. Thus, signals that are sparse with respect to the dictionary can be recovered via basis pursuit (BP) from a small number of random measurements. Further, thresholding is investigated as recovery algorithm for compressed sensing, and conditions are provided that guarantee reconstruction with high probability. The different schemes are compared by numerical experiments.