Polychromatic X-ray CT Image Reconstruction and Mass-Attenuation Spectrum Estimation

We develop a method for sparse image reconstruction from polychromatic computed tomography (CT) measurements under the blind scenario where the material of the inspected object and the incident-energy spectrum are unknown. We obtain a parsimonious measurement-model parameterization by changing the integral variable from photon energy to mass attenuation, which allows us to combine the variations brought by the unknown incident spectrum and mass attenuation into a single unknown mass-attenuation spectrum function; the resulting measurement equation has the Laplace integral form. The mass-attenuation spectrum is then expanded into first order B-spline basis functions. We derive a block coordinate-descent algorithm for constrained minimization of a penalized negative log-likelihood (NLL) cost function, where penalty terms ensure nonnegativity of the spline coefficients and nonnegativity and sparsity of the density map. The image sparsity is imposed using total-variation (TV) and $\ell_1$ norms, applied to the density-map image and its discrete wavelet transform (DWT) coefficients, respectively. This algorithm alternates between Nesterov's proximal-gradient (NPG) and limited-memory Broyden-Fletcher-Goldfarb-Shanno with box constraints (L-BFGS-B) steps for updating the image and mass-attenuation spectrum parameters. To accelerate convergence of the density-map NPG step, we apply a step-size selection scheme that accounts for varying local Lipschitz constant of the NLL. We consider lognormal and Poisson noise models and establish conditions for biconvexity of the corresponding NLLs. We also prove the Kurdyka-{\L}ojasiewicz property of the objective function, which is important for establishing local convergence of the algorithm. Numerical experiments with simulated and real X-ray CT data demonstrate the performance of the proposed scheme

Convex-Set–Constrained Sparse Signal Recovery: Theory and Applications

Convex-set constrained sparse signal reconstruction facilitates flexible measurement model and accurate recovery. The objective function that we wish to minimize is a sum of a convex differentiable data-fidelity (negative log-likelihood (NLL)) term and a convex regularization term. We apply sparse signal regularization where the signal belongs to a closed convex set within the closure of the domain of the NLL. Signal sparsity is imposed using the l1-norm penalty on the signal\u27s linear transform coefficients. First, we present a projected Nesterov’s proximal-gradient (PNPG) approach that employs a projected Nesterov\u27s acceleration step with restart and a duality-based inner iteration to compute the proximal mapping. We propose an adaptive step-size selection scheme to obtain a good local majorizing function of the NLL and reduce the time spent backtracking. We present an integrated derivation of the momentum acceleration and proofs of O(k^(-2)) objective function convergence rate and convergence of the iterates, which account for adaptive step size, inexactness of the iterative proximal mapping, and the convex-set constraint. The tuning of PNPG is largely application independent. Tomographic and compressed-sensing reconstruction experiments with Poisson generalized linear and Gaussian linear measurement models demonstrate the performance of the proposed approach. We then address the problem of upper-bounding the regularization constant for the convex-set--constrained sparse signal recovery problem behind the PNPG framework. This bound defines the maximum influence the regularization term has to the signal recovery. We formulate an optimization problem for finding these bounds when the regularization term can be globally minimized and develop an alternating direction method of multipliers (ADMM) type method for their computation. Simulation examples show that the derived and empirical bounds match. Finally, we show application of the PNPG framework to X-ray computed tomography (CT) and outline a method for sparse image reconstruction from Poisson-distributed polychromatic X-ray CT measurements under the blind scenario where the material of the inspected object and the incident energy spectrum are unknown. To obtain a parsimonious mean measurement-model parameterization, we first rewrite the measurement equation by changing the integral variable from photon energy to mass attenuation, which allows us to combine the variations brought by the unknown incident spectrum and mass attenuation into a single unknown mass-attenuation spectrum function; the resulting measurement equation has the Laplace integral form. We apply a block coordinate-descent algorithm that alternates between an NPG image reconstruction step and a limited-memory BFGS with box constraints (L-BFGS-B) iteration for updating mass-attenuation spectrum parameters. Our NPG-BFGS algorithm is the first physical-model based image reconstruction method for simultaneous blind sparse image reconstruction and mass-attenuation spectrum estimation from polychromatic measurements. Real X-ray CT reconstruction examples demonstrate the performance of the proposed blind scheme

Blind polychromatic X-ray CT reconstruction from Poisson measurements

We develop a sparse image reconstruction method for Poisson distributed polychromatic X-ray computed tomography (CT) measurements under the blind scenario where the material of the inspected object and the incident energy spectrum are unknown. We employ our mass-attenuation spectrum parameterization of the noiseless measurements for single-material objects and express the mass-attenuation spectrum as a linear combination of B-spline basis functions of order one. A block coordinate descent algorithm is developed for constrained minimization of a penalized Poisson negative log-likelihood (NLL) cost function, where constraints and penalty terms ensure nonnegativity of the spline coefficients and nonnegativity and sparsity of the density-map image; the image sparsity is imposed using a convex total-variation (TV) norm penalty term. This algorithm alternates between a Nesterov’s proximal-gradient (NPG) step for estimating the density-map image and a limited-memory Broyden-Fletcher-Goldfarb-Shanno with box constraints (LBFGS- B) step for estimating the incident-spectrum parameters. We establish conditions for biconvexity of the penalized NLL objective function, which, if satisfied, ensures monotonicity of the NPG-BFGS iteration. We also show that the penalized NLL objective satisfies the Kurdyka-Łojasiewicz property, which is important for establishing local convergence of block-coordinate descent schemes in biconvex optimization problems. Simulation examples demonstrate the performance of the proposed scheme

Design and Model Verification of an Infrared Chromotomographic Imaging System

A prism chromotomographic hyperspectral imaging sensor is being developed to aid in the study of bomb phenomenology. Reliable chromotomographic reconstruction depends on accurate knowledge of the sensor specific point spread function over all wavelengths of interest. The purpose of this research is to generate the required point spread functions using wave optics techniques and a phase screen model of system aberrations. Phase screens are generated using the Richardson-Lucy algorithm for extracting point spread functions and Gerchberg-Saxton algorithm for phase retrieval. These phase screens are verified by comparing the modeled results of a blackbody source with measurements made using a chromotomographic sensor. The sensor itself is constructed as part of this research. Comparison between the measured and simulated results is based upon the noise statistics of the measured image. Four comparisons between measured and modeled data, each made at a different prism rotation angle, provide the basis for the conclusions of this research. Based on these results, the phase screen technique appears to be valid so long as constraints are placed on the field of view and spectral region over which the screens are applied

Markov random field image modelling

Includes bibliographical references.This work investigated some of the consequences of using a priori information in image processing using computer tomography (CT) as an example. Prior information is information about the solution that is known apart from measurement data. This information can be represented as a probability distribution. In order to define a probability density distribution in high dimensional problems like those found in image processing it becomes necessary to adopt some form of parametric model for the distribution. Markov random fields (MRFs) provide just such a vehicle for modelling the a priori distribution of labels found in images. In particular, this work investigated the suitability of MRF models for modelling a priori information about the distribution of attenuation coefficients found in CT scans

Partially Coherent Lab Based X-ray Micro Computed Tomography

X-ray micro computed tomography (CT) is a useful tool for imaging 3-D internal structures. It has many applications in geophysics, biology and materials science. Currently, micro-CT’s capability are limited due to validity of assumptions used in modelling the machines’ physical properties, such as penumbral blurring due to non-point source, and X-ray refraction. Therefore many CT research in algorithms and models are being carried out to overcome these limitations. This thesis presents methods to improve image resolution and noise, and to enable material property estimation of the micro-CT machine developed and in use at the ANU CTLab. This thesis is divided into five chapters as outlined below. The broad background topics of X-ray modelling and CT reconstruction are explored in Chapter 1, as required by later chapters. It describes each X-ray CT component, including the machines used at the ANU CTLab. The mathematical and statistical tools, and electromagnetic physical models are provided and used to characterise the scalar X-ray wave. This scalar wave equation is used to derive the projection operator through matter and free space, and basic reconstruction and phase retrieval algorithms. It quantifies the four types of X-ray interaction with matter for X-ray energy between 1 and 1000 keV, and presents common assumptions used for the modelling of lab based X-ray micro-CT. Chapter 2 is on X-ray source deblurring. The penumbral source blurring for X-ray micro-CT systems are limiting its resolution. This chapter starts with a geometrical framework to model the penumbral source blurring. I have simulated the effect of source blurring, assuming the geometry of the high-cone angle CT system, used at the ANU CTLab. Also, I have developed the Multislice Richardson-Lucy method that overcomes the computational complexity of the conjugate gradient method, while produces less artefacts compared to the standard Richardson-Lucy method. Its performance is demonstrated for both simulated and real experimental data. X-ray refraction, phase contrast and phase retrieval (PR) are investigated in Chapter 3. For weakly attenuating samples, intensity variation due to phase contrast is a significant fraction of the total signal. If phase contrast is incorrectly modelled, the reconstruction would not correctly account the phase contrast, therefore it would contribute to undesirable artefacts in the reconstruction volume. Here I present a novel Linear Iterative multi-energy PR algorithm. It enables material property estimation for the near field submicron X-ray CT system and reduces the noise and artefacts. This PR algorithm expands the validity range in comparison to the single material and data constrained modelling methods. I have also extended this novel PR algorithm to assume a polychromatic incident spectrum for a non-weakly absorbing object. Chapter 4 outlines the space filling X-ray source trajectory and reconstruction, on which I contributed in a minor capacity. This space filling trajectory reconstruction have improved the detector utilisation and reduced nonuniform resolution over the state-of-the-art 3-D Katsevich’s helical reconstruction, this patented work was done in collaboration with FEI Company. Chapter 5 concludes my PhD research work and provides future directions revealed by the present research