Invertibility of multi-energy X-ray transform.

PURPOSE: The goal is to provide a sufficient condition for the invertibility of a multi-energy (ME) X-ray transform. The energy-dependent X-ray attenuation profiles can be represented by a set of coefficients using the Alvarez-Macovski (AM) method. An ME X-ray transform is a mapping from N AM coefficients to N noise-free energy-weighted measurements, where N ≥ 2 . METHODS: We apply a general invertibility theorem to prove the equivalence of global and local invertibility for an ME X-ray transform. We explore the global invertibility through testing whether the Jacobian of the mapping J ( A ) has zero values over the support of the mapping. The Jacobian of an arbitrary ME X-ray transform is an integration over all spectral measurements. A sufficient condition for J ( A ) ≠ 0 for all A is that the integrand of J ( A ) is ≥ 0 (or ≤ 0 ) everywhere. Note that the trivial case of the integrand equals 0 everywhere is ignored. Using symmetry, we simplified the integrand of the Jacobian to three factors that are determined by the total attenuation, the basis functions, and the energy-weighting functions, respectively. The factor related to the total attenuation is always positive; hence, the invertibility of the X-ray transform can be determined by testing the signs of the other two factors. Furthermore, we use the Cramér-Rao lower bound (CRLB) to characterize the noise-induced estimation uncertainty and provide a maximum-likelihood (ML) estimator. RESULTS: The factor related to the basis functions is always negative when the photoelectric/Compton/Rayleigh basis functions are used and K-edge materials are not considered. The sign of the energy-weighting factor depends on the system source spectra and the detector response functions. For four special types of X-ray detectors, the sign of this factor stays the same over the integration range. Therefore, when these four types of detectors are used for imaging non-K-edge materials, the ME X-ray transform is globally invertible. The same framework can be used to study an arbitrary ME X-ray imaging system, for example, when K-edge materials are present. Furthermore, the ML estimator we presented is an unbiased, efficient estimator and can be used for a wide range of scenes. CONCLUSIONS: We have provided a framework to study the invertibility of an arbitrary ME X-ray transform and proved the global invertibility for four types of systems.

Correcting partial volume effects in biexponential T2 estimation of small lesions.

PURPOSE: T2 mapping provides a quantitative approach for focal liver lesion characterization. For small lesions, a biexponential model should be used to account for partial volume effects (PVE). However, conventional biexponential fitting suffers from large uncertainty of the fitted parameters when noise is present. The purpose of this work is to develop a more robust method to correct for PVE affecting small lesions. METHODS: We developed a region of interest-based joint biexponential fitting (JBF) algorithm to estimate the T2 of lesions affected by PVE. JBF takes advantage of the lesion fraction variation among voxels within a region of interest. JBF is compared to conventional approaches using Cramér-Rao lower bound analysis, numerical simulations, phantom, and in vivo data. RESULTS: JBF provides more accurate and precise T2 estimates in the presence of PVE. Furthermore, JBF is less sensitive to region of interest drawing. Phantom and in vivo results show that JBF can be combined with a reconstruction method for highly undersampled data, enabling the characterization of small abdominal lesions from data acquired in a single breath hold. CONCLUSION: The JBF algorithm provides more accurate and stable T2 estimates for small structures than conventional techniques when PVE is present. It should be particularly useful for the characterization of small abdominal lesions.

Molecular Imaging in the College of Optical Sciences - An Overview of Two Decades of Instrumentation Development.

During the past two decades, researchers at the University of Arizona's Center for Gamma-Ray Imaging (CGRI) have explored a variety of approaches to gamma-ray detection, including scintillation cameras, solid-state detectors, and hybrids such as the intensified Quantum Imaging Device (iQID) configuration where a scintillator is followed by optical gain and a fast CCD or CMOS camera. We have combined these detectors with a variety of collimation schemes, including single and multiple pinholes, parallel-hole collimators, synthetic apertures, and anamorphic crossed slits, to build a large number of preclinical molecular-imaging systems that perform Single-Photon Emission Computed Tomography (SPECT), Positron Emission Tomography (PET), and X-Ray Computed Tomography (CT). In this paper, we discuss the themes and methods we have developed over the years to record and fully use the information content carried by every detected gamma-ray photon.

Scanning linear estimation: improvements over region of interest (ROI) methods.

In tomographic medical imaging, a signal activity is typically estimated by summing voxels from a reconstructed image. We introduce an alternative estimation scheme that operates on the raw projection data and offers a substantial improvement, as measured by the ensemble mean-square error (EMSE), when compared to using voxel values from a maximum-likelihood expectation-maximization (MLEM) reconstruction. The scanning-linear (SL) estimator operates on the raw projection data and is derived as a special case of maximum-likelihood estimation with a series of approximations to make the calculation tractable. The approximated likelihood accounts for background randomness, measurement noise and variability in the parameters to be estimated. When signal size and location are known, the SL estimate of signal activity is unbiased, i.e. the average estimate equals the true value. By contrast, unpredictable bias arising from the null functions of the imaging system affect standard algorithms that operate on reconstructed data. The SL method is demonstrated for two different tasks: (1) simultaneously estimating a signal's size, location and activity; (2) for a fixed signal size and location, estimating activity. Noisy projection data are realistically simulated using measured calibration data from the multi-module multi-resolution small-animal SPECT imaging system. For both tasks, the same set of images is reconstructed using the MLEM algorithm (80 iterations), and the average and maximum values within the region of interest (ROI) are calculated for comparison. This comparison shows dramatic improvements in EMSE for the SL estimates. To show that the bias in ROI estimates affects not only absolute values but also relative differences, such as those used to monitor the response to therapy, the activity estimation task is repeated for three different signal sizes.

Image Science with Photon-Processing Detectors.

We introduce and discuss photon-processing detectors and we compare them with photon-counting detectors. By estimating a relatively small number of attributes for each collected photon, photon-processing detectors may help understand and solve a fundamental theoretical problem of any imaging system based on photon-counting detectors, namely null functions. We argue that photon-processing detectors can improve task performance by estimating position, energy, and time of arrival for each collected photon. We consider a continuous-to-continuous linear operator to relate the object being imaged to the collected data, and discuss how this operator can be analyzed to derive properties of the imaging system. Finally, we derive an expression for the characteristic functional of an imaging system that produces list-mode data.

T2 mapping from highly undersampled data by reconstruction of principal component coefficient maps using compressed sensing.

Recently, there has been an increased interest in quantitative MR parameters to improve diagnosis and treatment. Parameter mapping requires multiple images acquired with different timings usually resulting in long acquisition times. While acquisition time can be reduced by acquiring undersampled data, obtaining accurate estimates of parameters from undersampled data is a challenging problem, in particular for structures with high spatial frequency content. In this work, principal component analysis is combined with a model-based algorithm to reconstruct maps of selected principal component coefficients from highly undersampled radial MRI data. This novel approach linearizes the cost function of the optimization problem yielding a more accurate and reliable estimation of MR parameter maps. The proposed algorithm--reconstruction of principal component coefficient maps using compressed sensing--is demonstrated in phantoms and in vivo and compared with two other algorithms previously developed for undersampled data.

Design and Validation of an Adaptive SPECT System: AdaptiSPECT.

In order to obtain optimal image quality with respect to a particular task, adaptive imaging systems automatically change their acquisition parameters in response to preliminary data being recorded from the object under study. Currently, the adaptive aspect in Single Photon Emission Computed Tomography (SPECT) is limited to a manual collimator interchange and the choice of detector rotation radius. Furthermore, there is often no optimization of any kind with respect to a certain task. There is thus a need for more versatile SPECT systems that autonomously optimize their acquisition geometry for every task and every patient. Here we describe a pinhole SPECT imager, AdaptiSPECT, which is being developed at the Center for Gamma Ray Imaging (CGRI) to enable adaptive SPECT imaging in a pre-clinical context. Furthermore, ideas for an autonomous adaptation procedure are discussed and some preliminary results are reported upon.

Singular Value Decomposition of Pinhole SPECT Systems.

A single photon emission computed tomography (SPECT) imaging system can be modeled by a linear operator H that maps from object space to detector pixels in image space. The singular vectors and singular-value spectra of H provide useful tools for assessing system performance. The number of voxels used to discretize object space and the number of collection angles and pixels used to measure image space make the matrix dimensions H large. As a result, H must be stored sparsely which renders several conventional singular value decomposition (SVD) methods impractical. We used an iterative power methods SVD algorithm (Lanczos) designed to operate on very large sparsely stored matrices to calculate the singular vectors and singular-value spectra for two small animal pinhole SPECT imaging systems: FastSPECT II and M(3)R. The FastSPECT II system consisted of two rings of eight scintillation cameras each. The resulting dimensions of H were 68921 voxels by 97344 detector pixels. The M(3)R system is a four camera system that was reconfigured to measure image space using a single scintillation camera. The resulting dimensions of H were 50864 voxels by 6241 detector pixels. In this paper we present results of the SVD of each system and discuss calculation of the measurement and null space for each system.

Processing of radial fast spin-echo data for obtaining T2 estimates from a single k-space data set.

Radially acquired fast spin-echo data can be processed to obtain T2-weighted images and a T2 map from a single k-space data set. The general approach is to use data at a specific TE (or narrow TE range) in the center of k-space and data at other TE values in the outer part of k-space. With this method high-resolution T2-weighted images and T2 maps are obtained in a time efficient manner. The mixing of TE data, however, introduces errors in the T2-weighted images and T2 maps that affect the accuracy of the T2 estimates. In this work, various k-space data processing methods for reconstructing T2-weighted images and T2 maps from a single radial fast spin-echo k-space data set are analyzed in terms of the accuracy of T2 estimates. The analysis is focused on the effect of image artifacts, object dependency, and noise on the T2 estimates. Results are presented in computer-generated phantoms and in vivo.