Raymond Lemieux

An expectation maximization reconstruction algorithm for emission tomography with non-uniform entropy prior

A Bayesian image reconstruction algorithm is proposed for emission tomography. It incorporates the Poisson nature of the noise in the projection data and uses a non-uniform entropy as an a priori probability distribution of the image in a maximum a posteriori (MAP) approach. The expectation maximization (EM) method was applied to find the MAP estimator. The Newton-Raphson numerical method whose convergence and positive solutions are proven, was used to solve the EM problem. The prior mean at iteration k was determined by smoothing the image obtained at iteration k-1. Comparisons between the ML and the MAP algorithm were carried out with a numerical phantom that contains a narrow valley region. The ML solution after 50 iterations was chosen as the initial solution for the MAP algorithm, since the global performance of the ML algorithm deteriorates with increasing number of iterations while its local performance in the valley region is always improving. The resulting algorithm is a compromise between ML who has the best local performance in the valley region and the MAP who has the best global performance.

Deriving the multiplicative algebraic reconstruction algorithm (MART) by the method of convex projection (POCS)

It is shown that the MART (multiplicative algebraic reconstruction technique) algorithm can be derived by POCS. This gives MART a new theoretical interpretation and a proof of convergence to a stable solution even when other convex constraints are introduced. However, MART, as a multiplicative algorithm, depends on the initial solution. It is noted that, far from being a flaw, this property can be used to introduce further a priori knowledge about the image to be reconstructed, to maximize the entropy, to keep the ratio between the regions of the original image constant, or to set to zero the area outside the reconstruction volume. MART should be preferred to MENT (a maximum entropy algorithm) for entropy maximization, for it performs as well but is much faster. ART is much less influenced by the initial solution than MART.<<ETX>>

Use of an optical flow algorithm to quantify and correct patient motion during tomographic acquisition

We have developed an automatic procedure for the detection, quantification and correction of translational motion during single photon emission computerized tomography (SPECT). The method computes the optical flow vector field between two successive views. The optical flow vector field assigns to each pixel of a tomographic view a two dimensional velocity that describes its motion across the image plane between two successive views. Motion can be corrected by shifting the moved views back by the amount of the measured motion distance. The method is applied to human studies where obvious motion was detected from visual inspection of a rotating cine display. Visually artifactual perfusion defects are eliminated after motion correction. The optical flow method can accurately detect the presence of motion, localize the camera angle at which motion occurred and measure the distance of motion.

Accelerated iterative reconstruction based on the maximum a posteriori expectation maximization

It is demonstrated that the ML-EM (maximum-likelihood expectation-maximization) algorithm is a particular case of the modified Newton method whose convergence is proved and can be optimally accelerated by an overrelaxation parameter. In order to overcome the checkerboard effect, this accelerated ML-EM algorithm can be penalized with a Gaussian a priori distribution in the framework of a MAP (maximum a posteriori) approach. The experimental results obtained here indicate that significant savings in computation time may be achieved using the accelerated MAP algorithm.<<ETX>>

Comparison between cross-correlation and optical flow methods for patient motion detection in SPECT

An automatic procedure for the detection and quantification of motion during single photon emission computerized tomography (SPECT) is proposed. The method computes the optical flow vector field between two successive views. The proposed method is also compared with the cross-correlation method.

Three-dimensional correction of experimentally determined gamma camera response

The response of a gamma camera depends on the source to-camera distance. It results in a two-dimensional blurring of the projection image and in an increasing loss in resolution with increasing distance from the face of the collimator. Here, the authors present a method for experimentally estimating a parametric model for the point source response of a gamma camera with parallel hole collimator. This model is used to incorporate the three-dimensional spatially varying camera response into the projection and back-projection operations of the tomographic reconstruction maximum likelihood algorithm which also compensates for the three-dimensional uniform attenuation. Using phantom experiments, the authors demonstrated a substantial improvement in the quality and the quantitative accuracy of the single photon emission computed tomography images.<<ETX>>

Comparison between ML-EM and modified Newton algorithms for SPECT image reconstruction

The expectation maximization method for maximum likelihood image reconstruction (ML- EM) is one of the most popular algorithms used in SPECT and PET, because it is based on the realistic assumption that photon emission and counts follow a Poisson process. Moreover, this method retains two important theoretical and practical properties namely nonnegativity and self-normalization of the reconstructed image. This latter property means that the number of emitted photons is equal to the number of counts. However, the major disadvantage of this method is the large amount of computation that is required, due to its slow rate of convergence. In this paper, we demonstrate that the ML-EM algorithm is a special case of the modified Newton method and can thus be accelerated by multiplying at each iteration the changes to the image, as calculated by the standard algorithm, by an overrelaxation parameter. This accelerated ML-EM algorithm can further be optimally accelerated, and converges to a good maximum likelihood estimator.

Iterative image reconstruction algorithms based on the theory of convex projections

Five iterative algorithms to reconstruct images From partial noisy data have been formulated and compared by using the theory of projection onto convex sets (POCS). This formulation clearly shows the dependence on the initial solution, allows the introduction of a prior distribution in the distance to be minimized and the use of additional convex constraints.

POCS: a uniform framework for iterative image reconstruction algorithms

The theory of projection onto convex sets (POCS) is very useful for comparing iterative reconstruction algorithms. Although originally developed with the Euclidian distance, it has been shown that POCS can be attended to pseudo-distances or can even use a different distance for each convex set. Five well known iterative algorithms that can be used to reconstruct images from partial noisy data have been formulated by POCS. Additional convex constraints and relaxation parameters can thus be introduced in these algorithms.

Patient motion quantification from tomographic data

The authors have developed an automatic procedure for the detection, quantification and correction of translational motion during tomographic acquisition. The method computes the optical flow vector field between two successive views. The optical flow vector field assigns to each pixel of a tomographic view a two dimensional velocity that describes its motion across the image plane between two successive views. The average optical flow of a region of interest is when computed to measure its interview global motion. Motion is corrected by shifting back the views where motion is detected by the amount of the predicted motion distance. The method is applied to simulated and patient tomographic raw data in single photon emission computed tomography (SPECT). Visually artifactual perfusion defects are eliminated. The optical flow method can accurately detect the presence of motion, localize the camera angle at which motion occurred and measure the distance of motion.