We examine the spatial resolution and variance properties of PET images reconstructed using maximum a posteriori (MAP) or penalized-likelihood methods. Resolution is characterized by the contrast recovery coefficient (CRC) of the local impulse response. Simplified approximate expressions are derived for the local impulse response CRCs and variances for each voxel. Using these results we propose a practical scheme for selecting spatially variant smoothing parameters to optimize lesion detectability through maximization of the local CRC-to-noise ratio in the reconstructed image. I. INTRODUCTION PET image reconstruction algorithms based on maximum likelihood (ML) or maximum a posteriori (MAP) principles can produce improved spatial resolution and noise properties in comparison to conventional filtered backprojection (FBP) methods. It is often important to be able to quantify this improvement in terms of the resolution (or bias) and variance of the resulting images. These measures can be...

Tomographic image reconstruction using statistical methods can provide more accurate system modeling, statistical models, and physical constraints than the conventional filtered backprojection (FBP) method. Because of the ill-posedness of the reconstruction problem, a roughness penalty is often imposed on the solution to control noise. To avoid smoothing of edges, which are important image attributes, various edge-preserving regularization methods have been proposed. Most of these schemes rely on information from local neighborhoods to determine the presence of edges. In this paper, we propose a cost function that incorporates nonlocal boundary information into the regularization method. We use an alternating minimization algorithm with deterministic annealing to minimize the proposed cost function, jointly estimating region boundaries and object pixel values. We apply variational techniques implemented using level-sets methods to update the boundary estimates; then, using the most recent boundary estimate, we minimize a space-variant quadratic cost function to update the image estimate. For the PET transmission reconstruction application, we compare the bias-variance tradeoff of this method with that of a “conventional” penalized-likelihood algorithm with local Huber roughness penalty.

Abstract—We describe a method for computing a continuous time estimate of tracer density using list-mode positron emission tomography data. The rate function in each voxel is modeled as an inhomogeneous Poisson process whose rate function can be represented using a cubic B-spline basis. The rate functions are estimated by maximizing the likelihood of the arrival times of detected photon pairs over the control vertices of the spline, modified by quadratic spatial and temporal smoothness penalties and a penalty term to enforce nonnegativity. Randoms rate functions are estimated by assuming independence between the spatial and temporal randoms distributions. Similarly, scatter rate functions are estimated by assuming spatiotemporal independence and that the temporal distribution of the scatter is proportional to the temporal distribution of the trues. A quantitative evaluation was performed using simulated data and the method is also demonstrated in a human study using IIC-raclopride. Index Terms—Calibration, conjugate gradient methods, estimation, Poisson processes, smoothing methods, spline functions. I.

Abstract. In X-ray tomography, the structure of a three dimensional body is reconstructed from a collection of projection images of the body. Medical CT imaging does this using an extensive set of projections from all around the body. However, in many practical imaging situations only a small number of truncated projections is available from a limited angle of view. Three dimensional imaging using such data is complicated for two reasons: (i) Typically, sparse projection data does not contain sufficient information to completely describe the 3-D body, and (ii) Traditional CT reconstruction algorithms, such as filtered backprojection, do not work well when applied to few irregularly spaced projections. Concerning (i), existing results about the information content of sparse projection data are reviewed and discussed. Concerning (ii), it is shown how Bayesian inversion methods can be used to incorporate a priori information into the reconstruction method, leading to improved image quality over traditional methods. Based on the discussion, a low-dose three-dimensional X-ray imaging modality is described. Submitted to: Phys. Med. Biol. 1.

Positron Emission Tomography (PET) is a diagnostic imaging tool that provides images of radioactive substances injected into the body to trace biological functions. The radioactive substance emits a positron which annihilates with an electron to produce two 511 keV photons traveling in approximately opposite directions to be coincidentally detected by two detectors. Many photons are absorbed or scattered, reducing the number of detected emission events. Attenuation correction is crucial for quantitatively accurate PET reconstructions. PET transmission scans are performed to estimate attenuation parameters which are in turn used to correct the emission scans for attenuation effects. The noise in estimating the attenuation parameters propagates to the emission images affecting their quality and quantitative correctness. Thus, attenuation image reconstruction is extremely important in PET. Conventional methods of attenuation correction are suboptimal and ignore the Poisson nature of the data. We propose to use penalized likelihood image reconstruction techniques for transmission scans. Current algorithms for transmission tomography have two important problems: 1) they are not guaranteed to converge, 2) if they converge, the convergence is slow. We develop new fast and monotonic optimization algorithms for penalized likelihood image reconstruction based on a novel paraboloidal surrogates principle. We present results showing the speed of the new optimization algorithms as compared to previous ones. We apply the algorithms to PET data obtained from an anthropomorphic thorax phantom and real patient data. A transmission scan per...

Bayesian MAP (maximum a posteriori) methods for SPECT reconstruction can both stabilize reconstructions and lead to better bias and variance relative to ML methods. In previous work [1], a nonquadratic prior (the weak plate) that imposed piecewise smoothness on the first derivative of the solution led to much improved bias/variance behavior relative to results obtained using a more conventional nonquadratic prior (the weak membrane) that imposed piecewise smoothness of the zeroth derivative. By relaxing the requirement of imposing spatial discontinuities and using instead a quadratic (no discontinuities) smoothing prior, algorithms become easier to analyze, solutions easier to compute, and hyperparameter calculation becomes less of a problem. In this work, we investigated whether the advantages of weak plate relative to weak membrane are retained when non-piecewise quadratic versions - the thin plate and membrane priors - are used. We compared, with three different phantoms, the bias/v...

When interpolating incomplete data, one can choose a parametric model, or opt for a more general approach and use a non-parametric model which allows a very large class of interpolants. A popular non-parametric model for interpolating various types of data is based on regularization, which looks for an interpolant that is both close to the data and also "smooth" in some sense. Formally, this interpolant is obtained by minimizing an error functional which is the weighted sum of a "fidelity term" and a "smoothness term".

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. We introduce a new prior---the weak plate---to Bayesian tomographic reconstruction. The weak plate captures the piecewise ramplike spatial structure evident in primate autoradiograph source distributions. The weak plate is a part of a family of "mechanical" models---weak membrane (1st order), weak plate (2nd order), and weak quadric (3rd order)---in which a class of smoothness constraints derived from properties of ideal physical materials are used as models in the associated reconstruction problem. Since "weak" priors generate local minima in MAP estimation, we have designed novel Generalized Expectation--Maximization deterministic annealing algorithms to alleviate this problem. Our simulation studies qualitatively demonstrate the improvements over the weak membrane and maximum likelihood reconstructions. Key words: tomographic reconstruction, weak plate, piecewise smooth, deterministic annealing, expectation--maximization, free energy, saddle point. 1. Introduction Maximum likel...

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