A tensor-based dictionary learning approach to tomographic image reconstruction

We consider tomographic reconstruction using priors in the form of a dictionary learned from training images. The reconstruction has two stages: first we construct a tensor dictionary prior from our training data, and then we pose the reconstruction problem in terms of recovering the expansion coefficients in that dictionary. Our approach differs from past approaches in that (a) we use a third-order tensor representation for our images and (b) we recast the reconstruction problem using the tensor formulation. The dictionary learning problem is presented as a non-negative tensor factorization problem with sparsity constraints. The reconstruction problem is formulated in a convex optimization framework by looking for a solution with a sparse representation in the tensor dictionary. Numerical results show that our tensor formulation leads to very sparse representations of both the training images and the reconstructions due to the ability of representing repeated features compactly in the dictionary.     

Global alignment of molecular sequences via ancestral state reconstruction

We consider the trace reconstruction problem on a tree (TRPT): a binary sequence is broadcast through a tree channel where we allow substitutions, deletions, and insertions; we seek to reconstruct the original sequence from the sequences received at the leaves. The TRPT is motivated by the multiple sequence alignment problem in computational biology. We give a simple recursive procedure giving strong reconstruction guarantees at low mutation rates. To our knowledge, this is the first rigorous trace reconstruction result on a tree in the presence of indels.     

On the stable sampling rate for binary measurements and wavelet reconstruction

This paper is concerned with the problem of reconstructing an infinite-dimensional signal from a limited number of linear measurements. In particular, we show that for binary measurements (modelled with Walsh functions and Hadamard matrices) and wavelet reconstruction the stable sampling rate is linear. This implies that binary measurements are as efficient as Fourier samples when using wavelets as the reconstruction space. Powerful techniques for reconstructions include generalized sampling and its compressed versions, as well as recent methods based on data assimilation. Common to these methods is that the reconstruction quality depends highly on the subspace angle between the sampling and the reconstruction space, which is dictated by the stable sampling rate. As a result of the theory provided in this paper, these methods can now easily use binary measurements and wavelet reconstruction bases.     

A variational proximal alternating linearized minimization in a given metric for limited-angle CT image reconstruction

Due to the restriction of computed tomography (CT) scanning environment, the acquired projection data may be incomplete for exact CT reconstruction. Though some convex optimization methods, such as total variation minimization based method, can be used for incomplete data reconstruction, the edge of reconstruction image may be partly distorted for limited-angle CT reconstruction. To promote the quality of reconstruction image for limited-angle CT imaging, in this paper, a nonconvex and nonsmooth optimization model was investigated. To solve the model, a variational proximal alternating linearized minimization (VPALM) method based on proximal mapping in a given metric was proposed. The proposed method can avoid computing the inverse of a huge system matrix thus can be used to deal with the larger-scale inverse problems. What’s more, we show that each bounded sequence generated by VPALM globally converges to a critical point based on the Kurdyka–Lojasiewicz property. Real data experiments are used to demonstrate the viability and effectiveness of VPALM method, and the results show that the proposed method outperforms two classical CT reconstruction methods.     

三维加权PFDK重建算法研究

基于滤波反投影的Feldkamp-Davis-Kress(FDK)算法,具有数学形式简单、容易实现和计算速度快等优点,在医疗和工业等领域得到了广泛的应用.平行重排(PF DK)算法是FDK算法的一种推广,针对PFDK算法重建出的图像受锥角的影响加大的问题,给出一种三维加权PFDK图像重建算法,并研究了重排过程中径向插值间隔对重建图像质量的影响,分别采用三种不同插值总数(插值间隔分别是1单位,0.5单位,0.25单位)重排数据.实验结果表明给出的三维加权PFDK算法可有效减少锥角的影响,且当采用2倍插值总数时重建结果较好.     

Optimal reconstruction systems for erasures and for the q-potential

In this paper we introduce the q-potential as an extension of the Benedetto-Fickus frame potential, defined on general reconstruction systems and show that protocols are the minimizers of this potential under certain restrictions. We extend recent results of B.G. Bodmann on the structure of optimal protocols with respect to 1 and 2 lost packets where the worst (normalized) reconstruction error is computed with respect to a compatible unitarily invariant norm. We finally describe necessary and sufficient (spectral) conditions, that we call q-fundamental inequalities, for the existence of protocols with prescribed properties by relating this problem to Klyachko’s and Fulton’s theory on sums of hermitian operators.     

Convolution, Average Sampling, and a Calderon Resolution of the Identity for Shift-Invariant Spaces

In this article, we study three interconnected inverse problems in shift invariant spaces: 1) the convolution/deconvolution problem; 2) the uniformly sampled convolution and the reconstruction problem; 3) the sampled convolution followed by sampling on irregular grid and the reconstruction problem. In all three cases, we study both the stable reconstruction as well as ill-posed reconstruction problems. We characterize the convolutors for stable deconvolution as well as those giving rise to ill-posed deconvolution. We also characterize the convolutors that allow stable reconstruction as well as those giving rise to ill-posed reconstruction from uniform sampling. The connection between stable deconvolution, and stable reconstruction from samples after convolution is subtle, as will be demonstrated by several examples and theorems that relate the two problems.     

Robust Dual Reconstruction Systems and Fusion Frames

We study the duality of reconstruction systems, which are g-frames in a finite dimensional setting. These systems allow redundant linear encoding-decoding schemes implemented by the so-called dual reconstruction systems. We are particularly interested in the projective reconstruction systems that are the analogue of fusion frames in this context. Thus, we focus on dual systems of a fixed projective system that are optimal with respect to erasures of the reconstruction system coefficients involved in the decoding process. We consider two different measures of the reconstruction error in a blind reconstruction algorithm. We also study the projective reconstruction system that best approximate an arbitrary reconstruction system, based on some well known results in matrix theory. Finally, we present a family of examples in which the problem of existence of a dual projective system of a reconstruction system of this type is considered.     

基于截断加权基追踪模型的迭代支撑探测算法

迭代支撑探测算法是基于截断的基追踪(Basis Pursuit,BP)模型的一种l_1最小化信号重构算法,它可以实现信号的快速重构并且所需要的观测值比经典的L1算法以及迭代加权L1算法更少.本文针对非零元具有快速退化分布性质的稀疏信号,提出了一种改进算法一一基于截断的加权BP模型的迭代支撑探测算法.在迭代的过程中,改进的算法探测原信号支撑集中元素的同时调整重构模型的权值,使得重构模型更有利于实现信号的精确重构.根据所考虑的信号的非零元具有快速退化分布性质这样的先验信息,利用阈值法则探测原信号支撑集中的元素.最后通过Matlab数值实验实现了算法,验证了基于截断的加权BP模型的迭代支撑探测算法比迭代加权L1算法需要的观测值更少,并且比迭代加权L1算法以及传统的迭代支撑探测算法需要更少的重构时间就可以实现信号的精确重构.     

On the error behaviour of the filtered back projection

The filtered back projection (FBP) formula allows us to reconstruct bivariate functions from given Radon samples. However, the FBP formula is numerically unstable and low-pass filters with finite bandwidth and a compactly supported window function are employed to make the reconstruction by FBP less sensitive to noise. In this paper we analyse the inherent reconstruction error which is incurred by the application of a low-pass filter with finite bandwidth. We present L²-error estimates on Sobolev spaces of fractional order along with asymptotic convergence rates, where the filter's bandwidth goes to infinity. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quantization of compressive samples with stable and robust recovery

In this paper we study the quantization stage that is implicit in any compressed sensing signal acquisition paradigm. We propose using Sigma–Delta (ΣΔ) quantization and a subsequent reconstruction scheme based on convex optimization. We prove that the reconstruction error due to quantization decays polynomially in the number of measurements. Our results apply to arbitrary signals, including compressible ones, and account for measurement noise. Additionally, they hold for sub-Gaussian (including Gaussian and Bernoulli) random compressed sensing measurements, as well as for both high bit-depth and coarse quantizers, and they extend to 1-bit quantization. In the noise-free case, when the signal is strictly sparse we prove that by optimizing the order of the quantization scheme one can obtain root-exponential decay in the reconstruction error due to quantization.     

Uniqueness and reconstruction theorems for pseudodifferential operators with a bandlimited Kohn-Nirenberg symbol

Motivated by the problem of channel estimation in wireless communications, we derive a reconstruction formula for pseudodifferential operators with a bandlimited symbol. This reconstruction formula uses the diagonal entries of the matrix of the pseudodifferential operator with respect to a Gabor system. In addition, we prove several other uniqueness theorems that shed light on the relation between a pseudodifferential operator and its matrix with respect to a Gabor system.     

An integrated solution and analysis of bioluminescence tomography and diffuse optical tomography

While diffuse optical tomography (DOT) has been studied for years, bioluminescence tomography (BLT) is emerging as a promising optical molecular imaging tool. These two modalities have different goals. DOT is for reconstruction of optical parameters of a medium such as a breast from surface measurements induced by external sources. BLT is for reconstruction of a bioluminescent source distribution in a medium such as a mouse from surface measurements induced by internal bioluminescent sources. However, an important pre-requisite for BLT reconstruction is the knowledge on the distribution of optical parameters within the medium, which is the output of DOT. In this paper, we propose a mathematical model integrating BLT and DOT at the fundamental level; that is, performing the two types of reconstructions simultaneously instead of doing them sequentially. The model is introduced through minimizing the difference between predicted quantities and boundary measurements, as well as incorporating regularization terms. Then, we show the solution existence, introduce numerical schemes and prove convergence of the numerical solution. We also present numerical results to illustrate the utility of our approach.     

Sparse matrices in frame theory

Frame theory is closely intertwined with signal processing through a canon of methodologies for the analysis of signals using (redundant) linear measurements. The canonical dual frame associated with a frame provides a means for reconstruction by a least squares approach, but other dual frames yield alternative reconstruction procedures. The novel paradigm of sparsity has recently entered the area of frame theory in various ways. Of those different sparsity perspectives, we will focus on the situations where frames and (not necessarily canonical) dual frames can be written as sparse matrices. The objective for this approach is to ensure not only low-complexity computations, but also high compressibility. We will discuss both existence results and explicit constructions.     

Local Sampling and Reconstruction in Shift-Invariant Spaces and Their Applications in Spline Subspaces

The local reconstruction from samples is one of the most desirable properties for many applications in signal processing. Local sampling is practically useful since we need only to consider a signal on a bounded interval and computer can only process finite samples. However, the local sampling and reconstruction problem has not been given as much attention. Most of known results concern global sampling and reconstruction. There are only a few results about local sampling and reconstruction in spline subspaces. In this article, we study local sampling and reconstruction in general shift-invariant spaces and multiple generated shift-invariant spaces with compactly supported generators. Then we give several applications in spline subspaces and multiple generated spline subspaces.     

Continuous-time image reconstruction for binary tomography

Binary tomography is the process of reconstructing a binary image from a finite number of projections. We present a novel method for solving binary tomographic inverse problems using a continuous-time image reconstruction (CIR) system described by nonlinear differential equations based on the minimization of a double Kullback–Leibler divergence. We prove theoretically that the divergence measure monotonically decreases in time. Moreover, we demonstrate numerically that the quality of the reconstructed images of the nonlinear CIR system is better than those from an iterative reconstruction method.     

The Multichannel Deconvolution Problem: A Discrete Analysis

We study the multichannel deconvolution problem (MDP) in a discrete setting by developing the theory for converting the method used in the continuous setting in [36]. We give a method for solving the MDP when the convolvers are characteristic functions, derive the explicit form of the linear system, and obtain an upper bound on the condition number of the system in a particular case. We compare the Schiske reconstruction [28] to our solution in the discrete setting, and give an explicit formula for the corresponding error. We then give the algorithm for solving the general MDP and discuss in detail the local reconstruction aspects of the problem. Finally, we describe a method for improving the reconstruction by regularization and give some explicit estimates on error bounds in the presence of noise.     

Unique reconstruction of band-limited signals by a Mallat-Zhong wavelet transform algorithm

We show that uniqueness and existence for signal reconstruction from multiscale edges in the Mallat and Zhong algorithm become possible if we restrict our signals to Paley-Wiener space, band-limit our wavelets, and irregularly sample at the wavelet transform (absolute) maxima—the edges—while possibly including (enough) extra points at each level. We do this in a setting that closely resembles the numerical analysis setting of Mallat and Zhong and that seems to capture something of the essence of their (practical) reconstruction method. Our work builds on a uniqueness result for reconstructing an L² signal from irregular sampling of its wavelet transform of Grochenig and the related work of Benedetto, Heller, Mallat, and Zhong. We show that the rate of convergence for this reconstruction algorithm is geometric and computable in advance. Finally, we consider the effect on the rate of convergence of not sampling enough local maxima.     

Continuous-time image reconstruction using differential equations for computed tomography

An approach for reconstructing tomographic images based on the idea of continuous dynamical methods is presented. The method consists of a continuous-time image reconstruction (CIR) system described by differential equations for solving linear inverse problems. We theoretically demonstrate that the trajectories converge to a least squares solution to the linear inverse problem. An implementation of its equivalent electronic circuit is significantly faster than conventional discrete-time image reconstruction (DIR) systems executed in a digital computer. Moreover, the merits of our CIR are demonstrated on a tomographic inverse problem where simulated noisy projection data are generated from a known phantom. Here, we numerically demonstrate that the CIR system does not produce unphysical negative pixel values if one starts out with positive initial values. Besides, CIR also recovers the phantom with almost the same quality as DIR images.     

Unique Reconstruction of Band-limited Signals by a Mallat-Zhong Wavelet Transform Algorithm

We show that uniqueness and existence for signal reconstruction from multiscale edges in the Mallat and Zhong algorithm become possible if we restrict our signals to Paley-Wiener space, band-limit our wavelets, and irregularly sample at the wavelet transform (absolute) maxima—the edges—while possibly including (enough) extra points at each level. We do this in a setting that closely resembles the numerical analysis setting of Mallat and Zhong and that seems to capture something of the essence of their (practical) reconstruction method. Our work builds on a uniqueness result for reconstructing an L² signal from irregular sampling of its wavelet transform of Gröchenig and the related work of Benedetto, Heller, Mallat, and Zhong. We show that the rate of convergence for this reconstruction algorithm is geometric and computable in advance. Finally, we consider the effect on the rate of convergence of not sampling enough local maxima.