Quantifying admissible undersampling for sparsity-exploiting iterative image reconstruction in X-ray CT

Iterative image reconstruction (IIR) with sparsity-exploiting methods, such as total variation (TV) minimization, investigated in compressive sensing (CS) claim potentially large reductions in sampling requirements. Quantifying this claim for computed tomography (CT) is non-trivial, because both full sampling in the discrete-to-discrete imaging model and the reduction in sampling admitted by sparsity-exploiting methods are ill-defined. The present article proposes definitions of full sampling by introducing four sufficient-sampling conditions (SSCs). The SSCs are based on the condition number of the system matrix of a linear imaging model and address invertibility and stability. In the example application of breast CT, the SSCs are used as reference points of full sampling for quantifying the undersampling admitted by reconstruction through TV-minimization. In numerical simulations, factors affecting admissible undersampling are studied. Differences between few-view and few-detector bin reconstruction as well as a relation between object sparsity and admitted undersampling are quantified.Comment: Revised version that was submitted to IEEE Transactions on Medical Imaging on 8/16/201

A Compressed Sensing Algorithm for Sparse-View Pinhole Single Photon Emission Computed Tomography

Single Photon Emission Computed Tomography (SPECT) systems are being developed with multiple cameras and without gantry rotation to provide rapid dynamic acquisitions. However, the resulting data is angularly undersampled, due to the limited number of views. We propose a novel reconstruction algorithm for sparse-view SPECT based on Compressed Sensing (CS) theory. The algorithm models Poisson noise by modifying the Iterative Hard Thresholding algorithm to minimize the Kullback-Leibler (KL) distance by gradient descent. Because the underlying objects of SPECT images are expected to be smooth, a discrete wavelet transform (DWT) using an orthogonal spline wavelet kernel is used as the sparsifying transform. Preliminary feasibility of the algorithm was tested on simulated data of a phantom consisting of two Gaussian distributions. Single-pinhole projection data with Poisson noise were simulated at 128, 60, 15, 10, and 5 views over 360 degrees. Image quality was assessed using the coefficient of variation and the relative contrast between the two objects in the phantom. Overall, the results demonstrate preliminary feasibility of the proposed CS algorithm for sparse-view SPECT imaging

A Spectral CT Method to Directly Estimate Basis Material Maps From Experimental Photon-Counting Data

The proposed spectral CT method solves the constrained one-step spectral CT reconstruction (cOSSCIR) optimization problem to estimate basis material maps while modeling the nonlinear X-ray detection process and enforcing convex constraints on the basis map images. In order to apply the optimization-based reconstruction approach to experimental data, the presented method empirically estimates the effective energy-window spectra using a calibration procedure. The amplitudes of the estimated spectra were further optimized as part of the reconstruction process to reduce ring artifacts. A validation approach was developed to select constraint parameters. The proposed spectral CT method was evaluated through simulations and experiments with a photon-counting detector. Basis material map images were successfully reconstructed using the presented empirical spectral modeling and cOSSCIR optimization approach. In simulations, the cOSSCIR approach accurately reconstructed the basis map images

The Effects of Extending the Spectral Information Acquired by a Photon-counting Detector for Spectral CT

Photon-counting x-ray detectors with pulse-height analysis provide spectral information that may improve material decomposition and contrast-to-noise ratio (CNR) in CT images. The number of energy measurements that can be acquired simultaneously on a detector pixel is equal to the number of comparator channels. Some spectral CT designs have a limited number of comparator channels, due to the complexity of readout electronics. The spectral information could be extended by changing the comparator threshold levels over time, sub pixels, or view angle. However, acquiring more energy measurements than comparator channels increases the noise and/or dose, due to differences in noise correlations across energy measurements and decreased dose utilisation. This study experimentally quantified the effects of acquiring more energy measurements than comparator channels using a bench-top spectral CT system. An analytical and simulation study modeling an ideal detector investigated whether there was a net benefit for material decomposition or optimal energy weighting when acquiring more energy measurements than comparator channels. Experimental results demonstrated that in a two-threshold acquisition, acquiring the high-energy measurement independently from the low-energy measurement increased noise standard deviation in material-decomposition basis images by factors of 1.5–1.7 due to changes in covariance between energy measurements. CNR in energy-weighted images decreased by factors of 0.92–0.71. Noise standard deviation increased by an additional factor of due to reduced dose utilisation. The results demonstrated no benefit for two-material decomposition noise or energy-weighted CNR when acquiring more energy measurements than comparator channels. Understanding the noise penalty of acquiring more energy measurements than comparator channels is important for designing spectral detectors and for designing experiments and interpreting data from prototype systems with a limited number of comparator channels

Convergence for nonconvex ADMM, with applications to CT imaging

The alternating direction method of multipliers (ADMM) algorithm is a powerful and flexible tool for complex optimization problems of the form $\min\{f(x)+g(y) : Ax+By=c\}$. ADMM exhibits robust empirical performance across a range of challenging settings including nonsmoothness and nonconvexity of the objective functions $f$ and $g$, and provides a simple and natural approach to the inverse problem of image reconstruction for computed tomography (CT) imaging. From the theoretical point of view, existing results for convergence in the nonconvex setting generally assume smoothness in at least one of the component functions in the objective. In this work, our new theoretical results provide convergence guarantees under a restricted strong convexity assumption without requiring smoothness or differentiability, while still allowing differentiable terms to be treated approximately if needed. We validate these theoretical results empirically, with a simulated example where both $f$ and $g$ are nondifferentiable -- and thus outside the scope of existing theory -- as well as a simulated CT image reconstruction problem

How little data is enough? Phase-diagram analysis of sparsity-regularized X-ray CT

We introduce phase-diagram analysis, a standard tool in compressed sensing, to the X-ray CT community as a systematic method for determining how few projections suffice for accurate sparsity-regularized reconstruction. In compressed sensing a phase diagram is a convenient way to study and express certain theoretical relations between sparsity and sufficient sampling. We adapt phase-diagram analysis for empirical use in X-ray CT for which the same theoretical results do not hold. We demonstrate in three case studies the potential of phase-diagram analysis for providing quantitative answers to questions of undersampling: First we demonstrate that there are cases where X-ray CT empirically performs comparable with an optimal compressed sensing strategy, namely taking measurements with Gaussian sensing matrices. Second, we show that, in contrast to what might have been anticipated, taking randomized CT measurements does not lead to improved performance compared to standard structured sampling patterns. Finally, we show preliminary results of how well phase-diagram analysis can predict the sufficient number of projections for accurately reconstructing a large-scale image of a given sparsity by means of total-variation regularization.Comment: 24 pages, 13 figure

High resolution image reconstruction with constrained, total-variation minimization

This work is concerned with applying iterative image reconstruction, based on constrained total-variation minimization, to low-intensity X-ray CT systems that have a high sampling rate. Such systems pose a challenge for iterative image reconstruction, because a very fine image grid is needed to realize the resolution inherent in such scanners. These image arrays lead to under-determined imaging models whose inversion is unstable and can result in undesirable artifacts and noise patterns. There are many possibilities to stabilize the imaging model, and this work proposes a method which may have an advantage in terms of algorithm efficiency. The proposed method introduces additional constraints in the optimization problem; these constraints set to zero high spatial frequency components which are beyond the sensing capability of the detector. The method is demonstrated with an actual CT data set and compared with another method based on projection up-sampling.Comment: This manuscript appears in the proceedings of the 2010 IEEE medical imaging conferenc