Image Space Reconstruction Algorithm Research Articles

Improvements in longitudinal phase space tomography at PITZ

Methodical studies to improve the existing e-beam Longitudinal Phase Space (LPS) tomography were performed at the Photo Injector Test facility at DESY in Zeuthen. Proof-of-principle simulations were done to address some core concerns e.g. booster phase range, space charge effects and noisy artefacts in results. Phase advance analysis was done with the help of an analytical model that determined the booster phase range and step size. A slit was introduced before the booster to truncate the beam and reduce space charge forces. The reconstruction method adopted was image space reconstruction algorithm owing to its assurance of non-negative solution. An initial scientific presumption of LPS from low energy momentum measurements was established to reduce artefacts in the phase space. This paper will explain the proof-of-principle simulations highlighting the key aspects to obtain accurate results. Reconstructed LPS for different experimental cases will be presented to demonstrate the diagnostic capability.

Alfa Laval AB, Sweden

The tomographic reconstruction of biological specimens requires robust algorithms, able to deal with low density contrast and low element concentrations. At the IST/ITN microprobe facility new GPU-accelerated reconstruction software, JPIXET, has been developed, which can significantly increase the speed of quantitative reconstruction of Proton Induced X-ray Emission Tomography (PIXE-T) data. It has a user-friendly graphical user interface for pre-processing, data analysis and reconstruction of PIXE-T and Scanning Transmission Ion Microscopy Tomography (STIM-T). The reconstruction of PIXE-T data is performed using either an algorithm based on a GPU-accelerated version of the Maximum Likelihood Expectation Maximisation (MLEM) method or a GPU-accelerated version of the Discrete Image Space Reconstruction Algorithm (DISRA) (Sakellariou (2001) [2]). The original DISRA, its accelerated version, and the MLEM algorithm, were compared for the reconstruction of a biological sample of Caenorhabditis elegans – a small worm. This sample was analysed at the microbeam line of the AIFIRA facility of CENBG, Bordeaux. A qualitative PIXE-T reconstruction was obtained using the CENBG software package TomoRebuild (Habchi et al. (2013) [6]). The effects of pre-processing and experimental conditions on the elemental concentrations are discussed.

Spatially Variant Resolution Modelling for Iterative List-Mode PET Reconstruction.

A spatially variant resolution modelling technique is presented which estimates the system matrix on-the-fly during iterative list-mode reconstruction. This is achieved by redistributing the endpoints of each list-mode event according to derived probability density functions describing the detector response function and photon acollinearity, at each iteration during the reconstruction. Positron range is modelled using an image-based convolution. When applying this technique it is shown that the maximum-likelihood expectation maximisation (MLEM) algorithm is not compatible with an obvious acceleration strategy. The image space reconstruction algorithm (ISRA), however, after being adapted to a list-mode based implementation, is well-suited to the implementation of the model. A comparison of ISRA and MLEM is made to confirm that ISRA is a suitable alternative to MLEM. We demonstrate that this model agrees with measured point spread functions and we present results showing an improvement in resolution recovery, particularly for off-centre objects, as compared to commercially available software, as well as the standard technique of using a stationary Gaussian convolution to model the resolution, for equal iterations and only slightly higher computation time.

ELF帯環境電磁界観測信号の背景信号推定における準L1ノルムに基づく非負値行列因子分解アルゴリズムの有用性

Our research group has been measuring Extremely Low Frequency (ELF) magnetic fields across Japan. The ELF measurements are mixtures of signals associated with various natural or artificial phenomena. When focus on specific factor, the signals related to other factors distort analysis result. In order to get specific information accurately, we should estimate desired signals or eliminate undesired signals. We found that Image Space Reconstruction Algorithm (ISRA), one of the Nonnegative Matrix Factorization (NMF) algorithm, works better than independent component analysis to estimate the ELF background signal. However, ISRA sometimes failed to estimate the weight vector for the background signal. We considered that ISRA has weakness for outliers and sparse signals because ISRA is based on minimizing L2 (Frobenius) norm between input matrix and projected matrix from estimated matrices. In order to improve robustness, we developed new methods based on minimizing quasi-L1 norm (QL1-NMF). In the experiment using generated signals and ELF observed signals which had outliers, the proposed method estimate background signal more accurately than ISRA and other L1 norm based algorithms.

$$\nu $$ ν acceleration of statistical iterative methods for image restoration

Image restoration problem is an important topic which appears in many different scientific areas. Several solving techniques are available, but generally in real applications, from which large-scale linear systems arise, the choice falls on iterative algorithms. In particular statistical methods (Lucy–Richardson method, image space reconstruction algorithm) have been extensively studied in the literature. Since they have low convergence rates, it is necessary to employ acceleration strategies. At this time, the most popular is the one introduced in 1997 by Biggs and Andrews, called automatic acceleration by the authors. In the present paper we describe \(\nu \) acceleration, that is the translation of the idea behind \(\nu \)-method, conceived for speeding up Landweber method, in the context of statistical methods. Computational results, which compare accelerated and classical methods, show the effectiveness of this strategy, which is able to get better performance than automatic acceleration.

Acceleration of multiplicative iterative algorithms for image deblurring by duality maps in Banach spaces

An acceleration technique for multiplicative iterative methods, such as Lucy–Richardson and Image Space Reconstruction Algorithm, is presented. The technique is inspired by the Landweber method in Banach spaces and is based on the application of duality maps, which allow to compute the iterations in the dual space. We show the link between the proposed acceleration and the previously known Meinel acceleration, which consists in the introduction of an exponent in the basic iterative formulas. We prove that the new acceleration technique is more stable than the Meinel acceleration. This implies that, in the restoration process, the former is able to get better accuracy and higher speeding-up than the latter. In addition to the main focus of the paper, we propose a generalization of the Landweber method in Banach space, in order to overcome some drawbacks of this recent strategy when compared with classical (Hilbertian) Landweber method. Numerical results show the behavior and the features of the several techniques considered, highlighting the goodness of our proposals.

An Experimental Survey on Non-Negative Matrix Factorization for Single Channel Blind Source Separation

In applications such as speech and audio denoising, music transcription, music and audio based forensics, it is desirable to decompose a single-channel recording into its respective sources, commonly referred to as blind source separation (BSS). One of the techniques used in BSS is non-negative matrix factorization (NMF). In NMF both supervised and unsupervised mode of operations is used. Among them supervised mode outperforms well due to the use of pre-learned basis vectors corresponding to each underlying sources. In this paper NMF algorithms such as Lee Seung algorithms (Regularized Expectation Minimization Maximum Likelihood Algorithm (EMML) and Regularized Image Space Reconstruction Algorithm (ISRA)), Bregman Divergence algorithm (Itakura Saito NMF algorithm (IS-NMF)) and an extension to NMF, by incorporating sparsity, Sparse Non-Negative Matrix Factorization(SNMF) algorithm are used to evaluate the performance of BSS in which supervised mode is used. Here signal to distortion ratio (SDR), signal to interference ratio (SIR) and signal to artifact ratio (SAR) are measured for different speech and/or music mixtures and performance is evaluated for each combination.

A comparison of quantitative reconstruction techniques for PIXE-tomography analysis applied to biological samples

The tomographic reconstruction of biological specimens requires robust algorithms, able to deal with low density contrast and low element concentrations. At the IST/ITN microprobe facility new GPU-accelerated reconstruction software, JPIXET, has been developed, which can significantly increase the speed of quantitative reconstruction of Proton Induced X-ray Emission Tomography (PIXE-T) data. It has a user-friendly graphical user interface for pre-processing, data analysis and reconstruction of PIXE-T and Scanning Transmission Ion Microscopy Tomography (STIM-T). The reconstruction of PIXE-T data is performed using either an algorithm based on a GPU-accelerated version of the Maximum Likelihood Expectation Maximisation (MLEM) method or a GPU-accelerated version of the Discrete Image Space Reconstruction Algorithm (DISRA) (Sakellariou (2001) [2]). The original DISRA, its accelerated version, and the MLEM algorithm, were compared for the reconstruction of a biological sample of Caenorhabditis elegans – a small worm. This sample was analysed at the microbeam line of the AIFIRA facility of CENBG, Bordeaux. A qualitative PIXE-T reconstruction was obtained using the CENBG software package TomoRebuild (Habchi et al. (2013) [6]). The effects of pre-processing and experimental conditions on the elemental concentrations are discussed.

Preconditioners for image restoration by reblurring techniques

It is well known that iterative algorithms for image deblurring that involve the normal equations show usually a slow convergence. A variant of the normal equations which replaces the conjugate transpose AH of the system matrix A with a new matrix is proposed. This approach, which is linked with regularization preconditioning theory and reblurring processes, can be applied to a wide set of iterative methods; here we examine Landweber, Steepest descent, Richardson–Lucy and Image Space Reconstruction Algorithm. Several computational tests show that this strategy leads to a significant improvement of the convergence speed of the methods. Moreover it can be naturally combined with other widely used acceleration techniques.

Low-dose micro-CT imaging for vascular segmentation and analysis using sparse-view acquisitions.

The aim of this study is to investigate whether reliable and accurate 3D geometrical models of the murine aortic arch can be constructed from sparse-view data in vivo micro-CT acquisitions. This would considerably reduce acquisition time and X-ray dose. In vivo contrast-enhanced micro-CT datasets were reconstructed using a conventional filtered back projection algorithm (FDK), the image space reconstruction algorithm (ISRA) and total variation regularized ISRA (ISRA-TV). The reconstructed images were then semi-automatically segmented. Segmentations of high- and low-dose protocols were compared and evaluated based on voxel classification, 3D model diameters and centerline differences. FDK reconstruction does not lead to accurate segmentation in the case of low-view acquisitions. ISRA manages accurate segmentation with 1024 or more projection views. ISRA-TV needs a minimum of 256 views. These results indicate that accurate vascular models can be obtained from micro-CT scans with 8 times less X-ray dose and acquisition time, as long as regularized iterative reconstruction is used.

FPGA Implementation of Abundance Estimation for Spectral Unmixing of Hyperspectral Data Using the Image Space Reconstruction Algorithm

One of the most popular and widely used techniques for analyzing remotely sensed hyperspectral data is spectral unmixing, which relies on two stages: (i) identification of pure spectral signatures (endmembers) in the data, and (ii) estimation of the abundance of each endmember in each (possibly mixed) pixel. Due to the high dimensionality of the hyperspectral data, spectral unmixing is a very time-consuming task. With recent advances in reconfigurable computing, especially using field programmable gate arrays (FPGAs), hyperspectral image processing algorithms can now be accelerated for on-board exploitation using compact hardware components with small size and cost. Although in previous work several efforts have been directed towards FPGA implementation of endmember extraction algorithms, the abundance estimation step has received comparatively much less attention. In this work, we develop a parallel FPGA-based design of the image space reconstruction algorithm (ISRA), a technique for solving linear inverse problems with positive constraints that has been used to estimate the abundance of each endmember in each pixel of a hyperspectral image. It is an iterative algorithm that guarantees convergence (after a certain number of iterations) and positive values in the results of the abundances (an important consideration in unmixing applications). Our system includes a direct memory access (DMA) module and implements a pre-fetching technique to hide the latency of the input/output communications. The method has been implemented on a Virtex-4 XC4VFX60 FPGA (a model that is similar to radiation-hardened FPGAs certified for space operation) and tested using real hyperspectral data sets collected by the Airborne Visible Infra-Red Imaging Spectrometer (AVIRIS) over the Cuprite mining district in Nevada and the Jasper Ridge Biological Preserve in California. Experimental results demonstrate that our hardware version can significantly outperform an equivalent software version, thus being able to provide abundance estimation results in near real-time, which makes our reconfigurable system appealing for on-board hyperspectral data processing.

Generalized Alpha-Beta Divergences and Their Application to Robust Nonnegative Matrix Factorization

We propose a class of multiplicative algorithms for Nonnegative Matrix Factorization (NMF) which are robust with respect to noise and outliers. To achieve this, we formulate a new family generalized divergences referred to as the Alpha-Beta-divergences (AB-divergences), which are parameterized by the two tuning parameters, alpha and beta, and smoothly connect the fundamental Alpha-, Beta- and Gamma-divergences. By adjusting these tuning parameters, we show that a wide range of standard and new divergences can be obtained. The corresponding learning algorithms for NMF are shown to integrate and generalize many existing ones, including the Lee-Seung, ISRA (Image Space Reconstruction Algorithm), EMML (Expectation Maximization Maximum Likelihood), Alpha-NMF, and Beta-NMF. Owing to more degrees of freedom in tuning the parameters, the proposed family of AB-multiplicative NMF algorithms is shown to improve robustness with respect to noise and outliers. The analysis illuminates the links of between AB-divergence and other divergences, especially Gamma- and Itakura-Saito divergences.

ISWLS: Novel Algorithm for Image Reconstruction in PET

The purpose of this study is to introduce a novel empirical iterative algorithm for medical image reconstruction, under the short name ISWLS (image space weighted least squares), which is expected to have image space reconstruction algorithm (ISRA) properties in noise manipulation and weighted least-squares (WLS) acceleration of the reconstruction process. We used phantom data from a prototype small-animal positron emission tomography system and the methods presented here are applied to 2-D sinograms. Further, we assess the performance of the new algorithm by comparing it to the simultaneous version of algebraic reconstruction technique (ART), simultaneous algebraic reconstruction technique (SART), to expectation maximization maximum likelihood (EM-ML), ISRA, and WLS. All algorithms are compared in terms of cross-correlation coefficient, reconstruction time, and contrast-to-noise ratios (CNRs). As it turns out, ISWLS presents higher CNRs than EM-ML, ISRA, and SART for objects of different sizes. Also, ISWLS shows similar performance to WLS during the first iterations but it has better noise manipulation. Finally, ordered subsets ISWLS (OS-ISWLS), the OS version of ISWLS, shows its best performance between the first six-nine iterations. Its behavior seems to be a compromise between OS-ISRA and OS-WLS.

First results on ion micro-tomography at LIPSION

Abstract At the nanoprobe LIPSION ion micro-tomography can be used to determine the 3D distribution information of a sample’s mass density and elemental composition. For ion micro-tomography the two analytical techniques scanning transmission ion microscopy tomography (STIM-T) and particle induced X-ray emission tomography (PIXE-T) are combined. The required data are collected in two consecutive series of measurements, during which the sample is rotated by 180°/360° in small steps. Because all ions have to traverse the sample, the upper limit of the sample size is given by the range of the ions in the material. The tomogram is obtained using the discrete image space reconstruction algorithm (DISRA) by Sakellariou (1997) [1] . This algorithm iteratively corrects a sketchy initial tomogram estimated from the experimental reconstruction – obtained by backprojection of filtered projections (BFP) – and an a priori elemental composition. The necessary correction factors are calculated comparing the reconstruction of the experimental data with the reconstruction of simulated data. For the simulated data sets of STIM projections and PIXE maps are computed from the tomogram. These data sets are proceeded with the BFP algorithm to get simulated reconstruction data. Using the DISRA for ion micro-tomography, one can benefit from the high resolution of STIM-T by transferring it to the elemental distribution given by PIXE-T. This article presents first results of this technique applied on a phantom at the LIPSION facility.

Nonnegative least-squares image deblurring: improved gradient projection approaches

The least-squares approach to image deblurring leads to an ill-posed problem. The addition of the nonnegativity constraint, when appropriate, does not provide regularization, even if, as far as we know, a thorough investigation of the ill-posedness of the resulting constrained least-squares problem has still to be done. Iterative methods, converging to nonnegative least-squares solutions, have been proposed. Some of them have the ‘semi-convergence’ property, i.e. early stopping of the iteration provides ‘regularized’ solutions. In this paper we consider two of these methods: the projected Landweber (PL) method and the iterative image space reconstruction algorithm (ISRA). Even if they work well in many instances, they are not frequently used in practice because, in general, they require a large number of iterations before providing a sensible solution. Therefore, the main purpose of this paper is to refresh these methods by increasing their efficiency. Starting from the remark that PL and ISRA require only the computation of the gradient of the functional, we propose the application to these algorithms of special acceleration techniques that have been recently developed in the area of the gradient methods. In particular, we propose the application of efficient step-length selection rules and line-search strategies. Moreover, remarking that ISRA is a scaled gradient algorithm, we evaluate its behaviour in comparison with a recent scaled gradient projection (SGP) method for image deblurring. Numerical experiments demonstrate that the accelerated methods still exhibit the semi-convergence property, with a considerable gain both in the number of iterations and in the computational time; in particular, SGP appears definitely the most efficient one.

Retrieval of sub-pixel snow cover information in the Himalayan region using medium and coarse resolution remote sensing data

Numerous constrained and unconstrained algorithms have been used to retrieve sub-pixel snow-cover information quantitatively using medium and coarse spatial resolution multispectral images from the Advanced Wide Field Sensor (AWiFS) and Moderate Resolution Imaging Spectrometer (MODIS) sensors over the Himalayan region. Both the methods give slow convergence rates and inaccurate estimation of sub-pixel components analysed using root mean square (RMS) error and image deviation. Multiplicative iterative algorithms such as the Expectation Maximization Maximum Likelihood Method (EMML) and the Image Space Reconstruction Algorithm (ISRA) based on the minimization of least squares and Kullback–Leibler distances have been attempted to compute the endmembers' abundances in unmixing of satellite data. In this paper we discuss the eigenvalues of minimum noise fraction (MNF) transformation bands, data noise removal using MNF transformation and selection of pure endmembers using satellite images. The normalized difference snow index (NDSI) is also estimated using field spectral reflectance results and satellite images in green and shortwave infrared (SWIR) wavelength regions in order to carry out a comparative analysis for its variations with sub-pixel snow cover fractions. The present analysis shows the advantage of iterative over direct (constrained and unconstrained) methods; constraints are easily handled and allow better regularization of the solution for the ill-conditioned cases. Iterative methods are found to be faster compared to those of direct methods and can be used operationally for all resolution data for accurate estimation of sub-pixel snow cover.

Reconstruction of 3D ion beam micro-tomography data for applications in Cell Biology

The DISRA (Discrete Image Space Reconstruction Algorithm) reconstruction code, created by A. Sakellariou, was conceived for the ideal case of complete three-dimensional (3D) PIXET (Particle Induced X-ray Emission Tomography) data. This implies two major difficulties for biological samples: first, the long duration of such experiments and second, the subsequent damage that occurs on such fragile specimens. For this reason, the DISRA code was extended at CENBG in order to probe isolated PIXET slices, taking into account the sample structure and mass density provided by 3D STIMT (Scanning Transmission Ion Microscopy Tomography) in the volume of interest. This modified version was tested on a phantom sample and first results on human cancer cells are also presented.

On the rate of convergence of the image space reconstruction algorithm

The Image Space Reconstruction Algorithm (ISRA) of Daube–Witherspoon and Muehllehner is a multiplicative algorithm for solving nonnegative least squares problems. Eggermont has proved the global convergence of this algorithm. In this paper, we analyze its rate of convergence. We show that if at the minimum the strict complementarity condition is satisfied and the reduced Hessian matrix is positive definite, then the ISRA algorithm which converges to it does so at a linear rate of convergence. If, however, the ISRA algorithm converges to a minimum which does not satisfy the strict complementarity condition, then the rate of convergence of the algorithm can degenerate to being sublinear. Our results here therefor hold under more general assumptions than in the work of Archer and Titterington who assume that at a minimum point all Lagrange multipliers are zero. We provide numerical examples to illustrate our rate of convergence results and to explain why the ISRA algorithm usually appears to converge slowly. Our work here heuristically justifies why the Lee–Seung algorithm for solving nonnegative matrix factorization problems has a slow rate of convergence. Mathematics subject classification (2000): 65F20, 65Y20, 15A48, 15A23.

Iterative Image Reconstruction for Clinical PET Using Ordered Subsets, Median Root Prior, and a Web-Based Interface

Purpose: The development, implementation and validation of simple, flexible and efficient iterative image reconstruction (IIR) methods for their take-up in routine clinical positron emission tomography (PET) static or dynamic studies. Procedures: The ordered subsets (OS) technique applied for the acceleration of the maximum likelihood expectation maximization (MLEM) IIR algorithm is here extended to include the weighted least-squares (WLS), image space reconstruction algorithm (ISRA) and the space alternating generalized EM (SAGE). The median root prior (MRP) has been successfully applied as a Bayesian regularization to control the noise level in the reconstructed images. All methods are implemented on distributed Pentium systems and tested using simulated PET data from a brain phantom. A Javascript is used for the initiation of the reconstruction. Results: Taking into consideration the image quality and the time required for the reconstruction, the MRP-OSEM (ordered subsets expectation maximization) seems to provide best results after four to eight iterations, with four subsets and a MRP coefficient of 0.2–0.4. Iterative reconstruction of the transmission images with OS-acceleration and MRP regularization with subsequent calculation of the attenuation correction factors (ACFs) is shown to effectively remove streak artifacts in the emission images, especially along paths of high attenuation. Conclusions: An efficient implementation using distributed processing principles and a web-based interface allows the reconstruction of one frame (with 63 cross-section slices) from a dynamic determination in few minutes. This work showed that regular PC systems can provide fast execution and produce results in clinically meaningful times. This eradicates the argument of the computational burden of the method that prevented the extensive use of IIR in today's modern PET systems. (Mol Imag Biol 2002;4:219–231)

SS3D-Fast fully 3D PET iterative reconstruction using stochastic sampling

We have developed a three-dimensional (3-D) reconstruction procedure for positron emission tomography (PET) using stochastic sampling techniques-the SS3D algorithm. A forward- and back-projector pair has been developed that eliminates the need to explicitly calculate the system response matrix (SRM). This is advantageous as it eliminates the memory requirements for storing this matrix. As compared to on-the-fly methods, our method is more accurate in its assumptions and can readily incorporate various physical effects of attenuation, scatter, and detector characteristics. This method can be efficiently used with simultaneous or row action algorithms like image space reconstruction algorithm (ISRA), expectation maximization (EM), and ordered subsets expectation maximization (OSEM). We demonstrate the successful implementation of this technique with the maximum likelihood expectation maximization (ML-EM), maximum a posteriori expectation maximization (MAP-EM), and maximum a posteriori ordered subsets expectation maximization (MAP-OS-EM) algorithms by evaluating the images on the basis of uniformity of noise, contrast recovery, signal-to-noise ratio (SNR), point-spread function, structural accuracy, and computational time.