Mathematical Methods of Physics.

Currently available tomographic image reconstruction schemes for optical tomography (OT) are mostly based on the limiting assumptions of small perturbations and a priori knowledge of the optical properties of a reference medium. Furthermore, these algorithms usually require the inversion of large, full, ill-conditioned Jacobian matrixes. In this work a gradient-based iterative image reconstruction (GIIR) method is presented that promises to overcome current limitations. The code consists of three major parts: (1) A finite-difference, time-resolved, diffusion forward model is used to predict detector readings based on the spatial distribution of optical properties; (2) An objective function that describes the difference between predicted and measured data; (3) An updating method that uses the gradient of the objective function in a line minimization scheme to provide subsequent guesses of the spatial distribution of the optical properties for the forward model. The reconstruction of these properties is completed, once a minimum of this objective function is found. After a presentation of the mathematical background, two- and three-dimensional reconstruction of simple heterogeneous media as well as the clinically relevant example of ventricular bleeding in the brain are discussed. Numerical studies suggest that intraventricular hemorrhages can be detected using the GIIR technique, even in the presence of a heterogeneous background.

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Gradient-based iterative image reconstruction scheme for time-resolved optical tomography

The problem of forming cross-sectional or tomographic images of the attenuation characteristics of objects arises in a variety of contexts, including medical x-ray computed tomography (CT) and nondestructive evaluation of objects in industrial inspection. In the context of emission imaging, such as positron emission tomography (PET) [1, 2], single photon emission computed tomography (SPECT) [3], and related methods used in the assay of containers of radioactive waste [4], it is useful to be able to form "attenuation maps," tomographic images of attenuation coefficients, from which one can compute attenuation correction factors for use in emission image reconstruction. One can measure the attenuating characteristics of an object by transmitting a collection of photons through the object along various paths or "rays" and observing the fraction that pass unabsorbed. From measurements collected over a large set of rays, one can reconstruct tomographic images of the object. Such image reconstruction is the subject of this chapter.

In all the above applications, the number of photons one can measure in a transmission scan is limited. In medical x-ray CT, source strength, patient motion, and absorbed dose considerations limit the total x-ray exposure. Implanted objects such as pacemakers also significantly reduce transmissivity and cause severe artifacts [5]. In industrial applications, source strength limitations, combined with the very large attenuation coefficients of metallic objects, often result in a small fraction of photons passing to the detector unabsorbed. In PET and SPECT imaging, the transmission scan only determines a "nuisance" parameter of secondary interest relative to the object's emission properties, so one would like to minimize the transmission scan duration. All the above considerations lead to "low-count" transmission scans. This chapter discusses algorithms for reconstructing attenuation images from low-count transmission scans. In this context, we define low-count to mean that the mean number of photons per ray is small enough that traditional filtered-backproject on (FBP) images, or even methods based on the Gaussian approximation to the distribution of the Poisson measurements (or logarithm thereof), are inadequate. We focus the presentation in the context of PET and SPECT transmission scans, but the methods are generally applicable to all low-count transmission studies. See [6] for an excellent survey of statistical approaches for the emission reconstruction problem.

Statistical Image Reconstruction Methods for Transmission Tomography

Positron emission tomography is a medical imaging modality for producing 3D images of the spatial distribution of biochemical tracers within the human body. The images are reconstructed from data formed through detection of radiation resulting from the emission of positrons from radioisotopes tagged onto the tracer of interest. These measurements are approximate line integrals from which the image can be reconstructed using analytical inversion formulae. However these direct methods do not allow accurate modeling either of the detector system or of the inherent statistical fluctuations in the data. Here we review recent progress in developing statistical approaches to image estimation that can overcome these limitations. We describe the various components of the physical model and review different formulations of the inverse problem. The wide range of numerical procedures for solving these problems are then reviewed. Finally, we describe recent work aimed at quantifying the quality of the resulting images, both in terms of classical measures of estimator bias and variance, and also using measures that are of more direct clinical relevance.

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Statistical approaches in quantitative positron emissiontomography

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 /sup 11/C-raclopride.

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Spatiotemporal reconstruction of list-mode PET data

The parameters of the prior, the hyperparameters, play an important role in Bayesian image estimation. Of particular importance for the case of Gibbs priors is the global hyperparameter, /spl beta/, which multiplies the Hamiltonian. Here we consider maximum likelihood (ML) estimation of /spl beta/ from incomplete data, i.e., problems in which the image, which is drawn from a Gibbs prior, is observed indirectly through some degradation or blurring process. Important applications include image restoration and image reconstruction from projections. Exact ML estimation of /spl beta/ from incomplete data is intractable for most image processing. Here we present an approximate ML estimator that is computed simultaneously with a maximum a posteriori (MAP) image estimate. The algorithm is based on a mean field approximation technique through which multidimensional Gibbs distributions are approximated by a separable function equal to a product of one-dimensional (1-D) densities. We show how this approach can be used to simplify the ML estimation problem. We also show how the Gibbs-Bogoliubov-Feynman (GBF) bound can be used to optimize the approximation for a restricted class of problems. We present the results of a Monte Carlo study that examines the bias and variance of this estimator when applied to image restoration.

Zhenyu Zhou

Papers

Approximate maximum likelihood hyperparameter estimation for Gibbs priors