Finite Fourier Frame Approximation Using the Inverse Polynomial Reconstruction Method

In several applications, data are collected in the frequency (Fourier) domain non-uniformly, either by design or as a consequence of inexact measurements. The two major bottlenecks for image reconstruction from non-uniform Fourier data are (i) there is no obvious way to perform the numerical approximation, as the non-uniform Fourier data is not amenable to fast transform techniques and resampling the data first to uniform spacing is often neither accurate or robust; and (ii) the Gibbs phenomenon is apparent when the underlying function (image) is piecewise smooth, an occurrence in nearly every application. Recent investigations suggest that it may be useful to view the non-uniform Fourier samples as Fourier frame coefficients when designing reconstruction algorithms that attempt to mitigate either of these fundamental problems. The inverse polynomial reconstruction method (IPRM) was developed to resolve the Gibbs phenomenon in the reconstruction of piecewise analytic functions from spectral data, notably Fourier data. This paper demonstrates that the IPRM is also suitable for approximating the finite inverse Fourier frame operator as a projection onto the weighted \(L_2\) space of orthogonal polynomials. Moreover, the IPRM can also be used to remove the Gibbs phenomenon from the Fourier frame approximation when the underlying function is piecewise smooth. The one-dimensional numerical results presented here demonstrate that using the IPRM in this way yields a robust, stable, and accurate approximation from non-uniform Fourier data.     

A Hybrid Approach to Spectral Reconstruction of Piecewise Smooth Functions

Consider a piecewise smooth function for which the (pseudo-)spectral coefficients are given. It is well known that while spectral partial sums yield exponentially convergent approximations for smooth functions, the results for piecewise smooth functions are poor, with spurious oscillations developing near the discontinuities and a much reduced overall convergence rate. This behavior, known as the Gibbs phenomenon, is considered as one of the major drawbacks in the application of spectral methods. Various types of reconstruction methods developed for the recovery of piecewise smooth functions have met with varying degrees of success. The Gegenbauer reconstruction method, originally proposed by Gottlieb et al. has the particularly impressive ability to reconstruct piecewise analytic functions with exponential convergence up to the points of discontinuity. However, it has been sharply criticized for its high cost and susceptibility to round-off error. In this paper, a new approach to Gegenbauer reconstruction is considered, resulting in a reconstruction method that is less computationally intensive and costly, yet still enjoys superior convergence. The idea is to create a procedure that combines the well known exponential filtering method in smooth regions away from the discontinuities with the Gegenbauer reconstruction method in regions close to the discontinuities. This hybrid approach benefits from both the simplicity of exponential filtering and the high resolution properties of the Gegenbauer reconstruction method. Additionally, a new way of computing the Gegenbauer coefficients from Jacobian polynomial expansions is introduced that is both more cost effective and less prone to round-off errors.     

Inverse Polynomial Reconstruction of Two Dimensional Fourier Images

The Gibbs phenomenon is intrinsic to the Fourier representation for discontinous problems. The inverse polynomial reconstruction method (IPRM) was proposed for the resolution of the Gibbs phenomenon in previous papers [Shizgal, B. D., and Jung, J.-H. (2003) and Jung, J.-H., and Shizgal, B. D. (2004)] providing spectral convergence for one dimensional global and local reconstructions. The inverse method involves the expansion of the unknown function in polynomials such that the residue between the Fourier representations of the final representation and the unknown function is orthogonal to the Fourier or polynomial spaces. The main goal of this work is to show that the one dimensional inverse method can be applied successfully to reconstruct two dimensional Fourier images. The two dimensional reconstruction is implemented globally with high accuracy when the function is analytic inside the given domain. If the function is piecewise analytic and the local reconstruction is sought, the inverse method is applied slice by slice. That is, the one dimensional inverse method is applied to remove the Gibbs oscillations in one direction and then it is applied in the other direction to remove the remaining Gibbs oscillations. It is shown that the inverse method is exact if the two-dimensional function to be reconstructed is a piecewise polynomial. The two-dimensional Shepp–Logan phantom image of the human brain is used as a preliminary study of the inverse method for two dimensional Fourier image reconstruction. The image is reconstructed with high accuracy with the inverse method     

Recovery of High Order Accuracy in Radial Basis Function Approximations of Discontinuous Problems

Radial basis function (RBF) methods have been actively developed in the last decades. RBF methods are global methods which do not require the use of specialized points and that yield high order accuracy if the function is smooth enough. Like other global approximations, the accuracy of RBF approximations of discontinuous problems deteriorates due to the Gibbs phenomenon, even as more points are added. In this paper we show that it is possible to remove the Gibbs phenomenon from RBF approximations of discontinuous functions as well as from RBF solutions of some hyperbolic partial differential equations. Although the theory for the resolution of the Gibbs phenomenon by reprojection in Gegenbauer polynomials relies on the orthogonality of the basis functions, and the RBF basis is not orthogonal, we observe that the Gegenbauer polynomials recover high order convergence from the RBF approximations of discontinuous problems in a variety of numerical examples including the linear and nonlinear hyperbolic partial differential equations. Our numerical examples using multi-quadric RBFs suggest that the Gegenbauer polynomials are Gibbs complementary to the RBF multi-quadric basis.     

Reconstruction of Piecewise Smooth Functions from Non-uniform Grid Point Data

Spectral series expansions of piecewise smooth functions are known to yield poor results, with spurious oscillations forming near the jump discontinuities and reduced convergence throughout the interval of approximation. The spectral reprojection method, most notably the Gegenbauer reconstruction method, can restore exponential convergence to piecewise smooth function approximations from their (pseudo-)spectral coefficients. Difficulties may arise due to numerical robustness and ill-conditioning of the reprojection basis polynomials, however. This paper considers non-classical orthogonal polynomials as reprojection bases for a general order (finite or spectral) reconstruction of piecewise smooth functions. Furthermore, when the given data are discrete grid point values, the reprojection polynomials are constructed to be orthogonal in the discrete sense, rather than by the usual continuous inner product. No calculation of optimal quadrature points is therefore needed. This adaptation suggests a method to approximate piecewise smooth functions from discrete non-uniform data, and results in a one-dimensional approximation that is accurate and numerically robust.      

Parameter Optimization and Reduction of Round Off Error for the Gegenbauer Reconstruction Method

The Gegenbauer reconstruction method has been successfully implemented to reconstruct piecewise smooth functions by both reducing the effects of the Gibbs phenomenon and maintaining high resolution in its approximation. However, it has been noticed in some applications that the method fails to converge. This paper shows that the lack of convergence results from both poor choices of the parameters associated with the method, as well as numerical round off error. The Gegenbauer polynomials can have very large amplitudes, particularly near the endpoints x=±1, and hence the approximation requires that the corresponding computed Gegenbauer coefficients be extremely small to obtain spectral convergence. As is demonstrated here, numerical round off error interferes with the ability of the computed coefficients to decay properly, and hence affects the method's overall convergence. This paper addresses both parameter optimization and reduction of the round off error for the Gegenbauer reconstruction method, and constructs a viable black box method for choosing parameters that guarantee both theoretical and numerical convergence, even at the jump discontinuities. Validation of the Gegenbauer reconstruction method through a-posteriori estimates is also provided.     

基于惩罚最大似然优化模型的各向异性约束磁共振成像方法

传统磁共振(MR)傅里叶成像方法由于傅里叶不确定性,k空间扩展编码采样长度能提高图像空间分辨率,但是以降低图像信噪比为代价.提出基于最大似然优化模型的各向异性约束MR成像新方法,将离散傅里叶变换模型改进为惩罚约束函数的最优值搜索问题.利用医学结构的先验信息,将正则化惩罚运算细化至平滑区域、边界邻域、边界和边界的方向.实验结果表明,该方法不但能扩展k空间高频数据采样长度同时有效降低高斯噪声,而且能克服现有相关约束成像方法的二次模糊和Gibbs环状伪影.     

基于广义Gibbs先验的低剂量X-CT优质重建研究

为获取低剂量条件下X-CT的优质重建,提出基于广义Gibbs先验的低剂量X-CT重建算法。新算法首先对投影数据进行统计建模,其后采用Bayesian最大后验估计方法,将投影数据中非局部的先验信息加诸于该数据的恢复中,达到抑制噪声的效果,最后仍采用经典的滤波反投影方法对恢复后的投影数据进行解释CT重建。文中将非局部先验称为广义Gibbs先验,其原因在于该先验具有传统Gibbs先验形式的同时,可以通过选择较大邻域和自适应的加权方式充分利用投影数据的全局信息进行数据恢复。通过与已有算法的对比实验,表明该文提出的基于广义Gibbs先验的低剂量X-CT重建算法在降低噪声效果和保持边缘方面具有较好的表现。     

Image Reconstruction from Fourier Data Using Sparsity of Edges

Data of piecewise smooth images are sometimes acquired as Fourier samples. Standard reconstruction techniques yield the Gibbs phenomenon, causing spurious oscillations at jump discontinuities and an overall reduced rate of convergence to first order away from the jumps. Filtering is an inexpensive way to improve the rate of convergence away from the discontinuities, but it has the adverse side effect of blurring the approximation at the jump locations. On the flip side, high resolution post processing algorithms are often computationally cost prohibitive and also require explicit knowledge of all jump locations. Recent convex optimization algorithms using \(l^1\) regularization exploit the expected sparsity of some features of the image. Wavelets or finite differences are often used to generate the corresponding sparsifying transform and work well for piecewise constant images. They are less useful when there is more variation in the image, however. In this paper we develop a convex optimization algorithm that exploits the sparsity in the edges of the underlying image. We use the polynomial annihilation edge detection method to generate the corresponding sparsifying transform. Our method successfully reduces the Gibbs phenomenon with only minimal blurring at the discontinuities while retaining a high rate of convergence in smooth regions.     

基于压缩感知的自适应正则化磁共振图像重构

当前基于压缩传感理论的正则化磁共振(CS-MR)图像重构算法普遍采用全局正则化参数,不能很好地在保持边缘和平滑噪声方面做出平衡。为此,提出一种自适应的正则化CS-MRI重构算法。结合图像稀疏性和其局部光滑性的先验知识,采用非线性共轭梯度下降算法求取最优化问题,并在迭代过程中自适应地改变局部正则化参数。新的正则化参数可以更好地恢复图像边缘,并且有利于平滑噪声,使代价函数在定义域内具有凸性;同时先验信息包含于正则化参数中,以提高图像的高频成分。实验结果表明该算法能有效权衡恢复图像边缘和平滑噪声两者的关系。     

Sparsity Enforcing Edge Detection Method for Blurred and Noisy Fourier data

We present a new method for estimating the edges in a piecewise smooth function from blurred and noisy Fourier data. The proposed method is constructed by combining the so called concentration factor edge detection method, which uses a finite number of Fourier coefficients to approximate the jump function of a piecewise smooth function, with compressed sensing ideas. Due to the global nature of the concentration factor method, Gibbs oscillations feature prominently near the jump discontinuities. This can cause the misidentification of edges when simple thresholding techniques are used. In fact, the true jump function is sparse, i.e. zero almost everywhere with non-zero values only at the edge locations. Hence we adopt an idea from compressed sensing and propose a method that uses a regularized deconvolution to remove the artifacts. Our new method is fast, in the sense that it only needs the solution of a single l ₁ minimization. Numerical examples demonstrate the accuracy and robustness of the method in the presence of noise and blur.     

一种基于二阶和四阶PDEs的变分去噪方法

二阶偏微分方程图像去噪方法在去除噪声同时保护边缘效果较好,但在平滑区会产生阶梯效应,而四阶方法可以消除阶梯效应,基于此提出了一种将二阶与四阶偏微分方程相结合的变分去噪方法,给出了其欧拉方程和梯度下降流,并分析了数值离散化方法。通过去噪实验表明,该方法能有效去除噪声干扰,在一定程度上保护边缘,并且可以有效减弱阶梯效应,改善图像质量。     

Recovering Exponential Accuracy from Non-harmonic Fourier Data Through Spectral Reprojection

Spectral reprojection techniques make possible the recovery of exponential accuracy from the partial Fourier sum of a piecewise-analytic function, essentially conquering the Gibbs phenomenon for this class of functions. This paper extends this result to non-harmonic partial sums, proving that spectral reprojection can reduce the Gibbs phenomenon in non-harmonic reconstruction as well as remove reconstruction artifacts due to erratic sampling. We are particularly interested in the case where the Fourier samples form a frame. These techniques are motivated by a desire to improve the quality of images reconstructed from non-uniform Fourier data, such as magnetic resonance (MR) images.     

一种循环平移的Contourlet变换去噪新方法

与小波变换相比，Contourlet变换等多尺度几何分析方法，可以更好地逼近含线奇异的高维函数。针对Contourlet变换缺乏平移不变性的缺陷，提出了一种基于Contourlet变换以及循环平移的图像去噪方法，即MCT方法。由于阈值去噪会在重构的图像中产生虚假成分（视觉魇像），尤其是在奇异点附近交替出现较大的上下幅值振动。而循环平移的目的就是在给定范围内寻求最佳平移量（或平均平移量），通过改变图像的排列次序，从而改变奇异点在整个图像中的位置来达到减小或消除振荡幅度，进而改善由于伪Gibbs现象所导致的蚊状噪声。实验表明，与抽样小波去噪相比，该方法明显可以更好地保持图像边缘；同时也一定程度上改进了传统Contourlet变换去嗓方法所带来的视觉魇像的缺点，较好的保留了图像的细节部分，且峰值信噪比（PSNR）也较高。     

Edge Detection Free Postprocessing for Pseudospectral Approximations

Pseudospectral Methods based on global polynomial approximation yield exponential accuracy when the underlying function is analytic. The presence of discontinuities destroys the extreme accuracy of the methods and the well-known Gibbs phenomenon appears. Several types of postprocessing methods have been developed to lessen the effects of the Gibbs phenomenon or even to restore spectral accuracy. The most powerful of the methods require that the locations of the discontinuities be precisely known. In this work we discuss postprocessing algorithms that are applicable when it is impractical, or difficult, or undesirable to pinpoint all discontinuity locations.     

Shock capturing by the spectral viscosity method

A main disadvantage of using spectral methods for nonlinear conservation laws lies in the formation of Gibbs phenomenon, once spontaneous shock discontinuities appear in the solution. The global nature of spectral methods then pollutes the unstable Gibbs oscillations over all the computational domain, and the lack of entropy dissipation prevents convergences in these cases. In this paper, we discuss the spectral viscosity method, which is based on high frequency-dependent vanishing viscosity regularization of the classical spectral methods. We show that this method enforces the convergence of nonlinear spectral approximations without sacrificing their overall spectral accuracy.     

Gibbs artifact reduction for POCS super-resolution image reconstruction

The topic of super-resolution image reconstruction has recently received considerable attention among the research community. Super-resolution image reconstruction methods attempt to create a single high-resolution image from a number of low-resolution images (or a video sequence). The method of projections onto convex sets (POCS) for super-resolution image reconstruction attracts many researchers’ attention. In this paper, we propose an improvement to reduce the amount of Gibbs artifacts presenting on the edges of the high-resolution image reconstructed by the POCS method. The proposed method weights the blur PSF centered at an edge pixel with an exponential function, and consequently decreases the coefficients of the PSF in the direction orthogonal to the edge. Experiment results show that the modification reduces effectively the visibility of Gibbs artifacts on edges and improves obviously the quality of the reconstructed high-resolution image.     

Gibbs artifact reduction for POCS super-resolution image reconstruction

The topic of super-resolution image reconstruction has recently received considerable attention among the research community. Super-resolution image reconstruction methods attempt to create a single high-resolution image from a number of low-resolution images (or a video sequence). The method of projections onto convex sets (POCS) for super-resolution image reconstruction attracts many researchers’ attention. In this paper, we propose an improvement to reduce the amount of Gibbs artifacts presenting on the edges of the high-resolution image reconstructed by the POCS method. The proposed method weights the blur PSF centered at an edge pixel with an exponential function, and consequently decreases the coefficients of the PSF in the direction orthogonal to the edge. Experiment results show that the modification reduces effectively the visibility of Gibbs artifacts on edges and improves obviously the quality of the reconstructed high-resolution image.     

优化加权TV的复合正则化压缩感知图像重建

目的：压缩感知理论突破了传统的Shanon-Nyquist采样定理的限制,能够以较少的采样值来进行原信号的恢复。针对压缩感知图像重建问题,本文提出了一种基于优化加权全变差（Total Variation, TV）的复合正则化压缩感知图像重建模型。方法：提出的重建模型是以TV正则化模型为基础的。首先,为克服传统TV正则化会导致重建图像的边缘和纹理细节部分模糊或丢失的缺点,本文引入图像的梯度信息估计权重,构建加权TV的重建模型。其次,利用全变差去噪（Rudin–Osher–Fatemi,ROF）模型对权重进行优化估计,从而减少计算权重时受噪声的影响。再次,本文将非局部结构相似性先验和局部自回归性先验引入提出的加权TV模型,得到优化加权TV的复合正则化重建模型。最后,结合投影法和算子分裂法对优化模型求解。结果：针对自然图像的不同特性,本文使用复合正则化先验进行建模,实验表明上述重建问题通过我们的方法得到了很好的解决,加权TV正则化先验使得图像的平坦区域和强边重建较好,而非局部结构相似性先验和局部自回归性先验能够保证图像的精细结构部分的重建效果。结论：本文提出了一种新的复合正则化压缩感知重建模型。与其它基于TV正则化的重建模型相比,实验结果表明本文模型的重建性能无论是在视觉效果还是在客观评价指标上都有明显的提高。     

A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model

We propose a new multiphase level set framework for image segmentation using the Mumford and Shah model, for piecewise constant and piecewise smooth optimal approximations. The proposed method is also a generalization of an active contour model without edges based 2-phase segmentation, developed by the authors earlier in T. Chan and L. Vese (1999. In Scale-Space'99, M. Nilsen et al. (Eds.), LNCS, vol. 1682, pp. 141–151) and T. Chan and L. Vese (2001. IEEE-IP, 10(2):266–277). The multiphase level set formulation is new and of interest on its own: by construction, it automatically avoids the problems of vacuum and overlap; it needs only log n level set functions for n phases in the piecewise constant case; it can represent boundaries with complex topologies, including triple junctions; in the piecewise smooth case, only two level set functions formally suffice to represent any partition, based on The Four-Color Theorem. Finally, we validate the proposed models by numerical results for signal and image denoising and segmentation, implemented using the Osher and Sethian level set method.