Image Reconstruction from Truncated Data in Single-Photon Emission Computed Tomomgraphy with Uniform Attenuation

International audienceWe present a mathematical analysis of the problem of image reconstruction from truncated data in two-dimensional (2D) single-photon emission computed tomography (SPECT). Recent results in classical tomography have shown that accurate reconstruction of some parts of the object is possible in the presence of truncation. We have investigated how these results extend to 2D parallel-beam SPECT, assuming that the attenuation map is known and constant in a convex region $\Omega$ that includes all activity sources. Our main result is a proof that, just like in classical tomography accurate SPECT reconstruction at a given location x ∈ $\Omega$,does not require the data on all lines passing through $\Omega$; some amount of truncation can be tolerated. Experimental reconstruction results based on computer-simulated data are given in support of the theory

Reconstruction from Uniformly Attenuated SPECT Projection Data Using the DBH Method

An algorithm was developed for the two-dimensional (2D) reconstruction of truncated and non-truncated uniformly attenuated data acquired from single photon emission computed tomography (SPECT). The algorithm is able to reconstruct data from half-scan (180o) and short-scan (180?+fan angle) acquisitions for parallel- and fan-beam geometries, respectively, as well as data from full-scan (360o) acquisitions. The algorithm is a derivative, backprojection, and Hilbert transform (DBH) method, which involves the backprojection of differentiated projection data followed by an inversion of the finite weighted Hilbert transform. The kernel of the inverse weighted Hilbert transform is solved numerically using matrix inversion. Numerical simulations confirm that the DBH method provides accurate reconstructions from half-scan and short-scan data, even when there is truncation. However, as the attenuation increases, finer data sampling is required

INVESTIGATION OF MYOCARDIAL SHORT-SCAN SPECT SCHEMES USING A REAL INHOMOGENEOUS ATTENUATING MEDIUM

In myocardial perfusion SPECT imaging the effect of photon attenuation may introduce artifacts in the reconstructed image due to the highly non-uniform distribution of tissue in the thorax region, potentially resulting in false-positive interpretations. It was the general consideration that the adequate compensation of photon attenuation requires, that the emission data be measured at projection angles over 2 π in the case the attenuation medium is inhomogeneous. The reduction of the scanning angle in SPECT imaging may be desirable because it can reduce scanning time and thereby minimize patient-motion and other artifacts. In SPECT myocardial imaging emission data is measured historically at projection angles over π from the right anterior oblique (RAO) to the left posterior oblique (LPO). This configuration results in better image contrast and, in some cases, better spatial resolution. However, in this case the reconstructed image may suffer more severely from geometric distortion than 2π angular sampling. It has been proven recently in analytical computer simulation studies that the data function over 2 π in SPECT with non-uniform attenuation contains redundant information; therefore the scanning angle theoretically can be reduced from 2π to π without loss of information. In this study our goal was to investigate how the various short-scan SPECT scheme configurations work in a real myocardial SPECT imaging system with highly inhomogeneous attenuating medium using attenuation correction. The measured projection images were reconstructed using the Maximum Likelihood Expectation Maximization algorithm with attenuation correction. The reconstructed slices of the various short-scan configurations and the full-scan slices were compared by a cardiac stress/rest software package

Mathematical Methods in Tomography

This is the seventh Oberwolfach conference on the mathematics of tomography, the first one taking place in 1980. Tomography is the most popular of a series of medical and scientific imaging techniques that have been developed since the mid seventies of the last century

Compensation for Nonuniform Resolution Using Penalized-Likelihood Reconstruction in Space-Variant Imaging Systems

Imaging systems that form estimates using a statistical approach generally yield images with nonuniform resolution properties. That is, the reconstructed images possess resolution properties marked by space-variant and/or anisotropic responses. We have previously developed a space-variant penalty for penalized-likelihood (PL) reconstruction that yields nearly uniform resolution properties . We demonstrated how to calculate this penalty efficiently and apply it to an idealized positron emission tomography (PET) system whose geometric response is space-invariant. In this paper, we demonstrate the efficient calculation and application of this penalty to space-variant systems. (The method is most appropriate when the system matrix has been precalculated.) We apply the penalty to a large field of view PET system where crystal penetration effects make the geometric response space-variant, and to a two-dimensional single photon emission computed tomography system whose detector responses are modeled by a depth-dependent Gaussian with linearly varying full-width at half-maximum. We perform a simulation study comparing reconstructions using our proposed PL approach with other reconstruction methods and demonstrate the relative resolution uniformity, and discuss tradeoffs among estimators that yield nearly uniform resolution. We observe similar noise performance for the PL and post-smoothed maximum-likelihood (ML) approaches with carefully matched resolution, so choosing one estimator over another should be made on other factors like computational complexity and convergence rates of the iterative reconstruction. Additionally, because the postsmoothed ML and the proposed PL approach can outperform one another in terms of resolution uniformity depending on the desired reconstruction resolution, we present and discuss a hybrid approach adopting both a penalty and post-smoothing.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85975/1/Fessler63.pd

Large Scale Inverse Problems

This book is thesecond volume of a three volume series recording the "Radon Special Semester 2011 on Multiscale Simulation &amp Analysis in Energy and the Environment" that took placein Linz, Austria, October 3-7, 2011. This volume addresses the common ground in the mathematical and computational procedures required for large-scale inverse problems and data assimilation in forefront applications. The solution of inverse problems is fundamental to a wide variety of applications such as weather forecasting, medical tomography, and oil exploration. Regularisation techniques are needed to ensure solutions of sufficient quality to be useful, and soundly theoretically based. This book addresses the common techniques required for all the applications, and is thus truly interdisciplinary. This collection of survey articles focusses on the large inverse problems commonly arising in simulation and forecasting in the earth sciences

Non-uniform resolution and partial volume recovery in tomographic image reconstruction methods

Acquired data in tomographic imaging systems are subject to physical or detector based image degrading effects. These effects need to be considered and modeled in order to optimize resolution recovery. However, accurate modeling of the physics of data and acquisition processes still lead to an ill-posed reconstruction problem, because real data is incomplete and noisy. Real images are always a compromise between resolution and noise; therefore, noise processes also need to be fully considered for optimum bias variance trade off. Image degrading effects and noise are generally modeled in the reconstruction methods, while, statistical iterative methods can better model these effects, with noise processes, as compared to the analytical methods. Regularization is used to condition the problem and explicit regularization methods are considered better to model various noise processes with an extended control over the reconstructed image quality. Emission physics through object distribution properties are modeled in form of a prior function. Smoothing and edge-preserving priors have been investigated in detail and it has been shown that smoothing priors over-smooth images in high count areas and result in spatially non-uniform and nonlinear resolution response. Uniform resolution response is desirable for image comparison and other image processing tasks, such as segmentation and registration. This work proposes methods, based on MRPs in MAP estimators, to obtain images with almost uniform and linear resolution characteristics, using nonlinearity of MRPs as a correction tool. Results indicate that MRPs perform better in terms of response linearity, spatial uniformity and parameter sensitivity, as compared to QPs and TV priors. Hybrid priors, comprised of MRPs and QPs, have been developed and analyzed for their activity recovery performance in two popular PVC methods and for an analysis of list-mode data reconstruction methods showing that MPRs perform better than QPs in different situations

Single photon emission computed tomography: performance assessment, development and clinical applications

This is a general investigation of the SPECT imaging process. The primary aim is to determine the manner in which the SPECT studies should be performed in order to maximise the relevant clinical information given the characteristics and limitations of the particular gamma camera imaging system used. Thus the first part of this thesis is concerned with an assessment of the performance characteristics of the SPECT system itself. This involves the measurement of the fundamental planar imaging properties of the camera, their stability with rotation, the ability of the camera to rotate in a perfect circle and the accuracy of the transfer of the information from the camera to the computing system. Following this the performance of the SPECT system as a whole is optimised. This is achieved by examining the fundamental aspects of the SPECT imaging process and by optimising the selection of the parameters chosen for the acquisition and reconstruction of the data. As an aid to this a novel mathematical construct is introduced. By taking the logarithm of the power spectrum of the normalised projection profile data the relationship between the signal power and the noise power in the detected data can be visualised. From a theoretical consideration of the available options the Butterworth filter is chosen for use because it provides the best combination of spatial frequency transfer characteristics and flexibility. The flexibility of the Butterworth filter is an important feature because it means that the form of the actual function used in the reconstruction of a transaxial section can be chosen with regard to the relationship between the signal and the noise in the data. A novel method is developed to match the filter to the projection data. This consists of the construction of a mean angular power spectrum from the set of projection profiles required for the reconstruction of the particular transaxial section in question. From this the spatial frequency at which the the signal becomes dominated by the noise is identified. The value which the Butterworth filter should take at this point can then be determined with regard to the requirements of the particular clinical investigation to be performed. The filter matching procedure can be extended to two dimensions in a practical manner by operating on the projection data after it has been filtered in the y direction. The efficacy of several methods to correct for the effects of scatter, attenuation and camera non-uniformity are also investigated. Having developed the optimised methodology for the acquisition and reconstruction of the SPECT data the results which are obtained are applied in the investigation of some specific clinical problems. The assessment of intractable epilepsy using 99mTc-HMPAO is performed followed by the investigation of ischaemic heart disease using 99mTc-MIBI and finally, the diagnosis of avascular necrosis of the femoral head using 99mTc-MDP is studied. The SPECT studies described in this thesis make a significant contribution to patient management