On the Convergence of Maximum Likelihood Expectation-maximization Algorithm for Iterative Tomographic Reconstruction...

It is desirable to iterate the maximum likelihood expectation-maximization (MLEM) algorithm to the point of convergence to obtain the optimal result. However, the impact of spatial frequency on the convergence rate of MLEM algorithm is not well investigated in a practical manner. Factors such as intensity of pixels and noise may also affect the algorithm's convergence. It is not also well-known that the algorithm converges uniformly or consistently across the field of view (FOV), termed as radial congruency or sensitivity to location and direction.

The purpose of the study was to evaluate the factors affecting the convergence of MLEM algorithm during iterative reconstruction including spatial frequency, pixel intensity, image contrast, and noise as well as the location or direction sensitivity or performance of algorithm in various parts of FOV.

Different sets of input test images were reconstructed by MLEM algorithm. To evaluate the role of spatial frequency on convergence rate and effect of direction, "zone plate test" image was used. Another set of images (six checkerboard images containing white and black boxes of different sizes) were used to examine the effect of location and direction in FOV. For the evaluation of image contrast and pixel intensity, a medium-sized checkerboard images was reconstructed. The original and reconstructed images were compared in each iteration using different performance metrics, error or discrepancy (mean squared error [MSE]), similarity (structural similarity index [SSIM]), and distance (Kullback-Leibler divergence). MSE curves were plotted for subregions (in FOV) and the whole image. Reconstruction was performed with 20, 50, 100, and 500 iterations. The Poisson noise was added to the sinograms, and the semiconvergence was assessed in reconstructed images.

Using "zone plate test" image, with 20 iterations, the low-frequency components located in the center were visible, but conversely, the middle- and high-frequency components were poorly reproduced. By increasing the iteration number to 50, 100, and 500, more details were visible in the periphery. In terms of the algorithm's performance in different locations, the center (includes lowest frequency components), the curve immediately starts to converge, however, for other three subregions (include low-, middle-, and high-frequency components, respectively), the convergence occurs later in the expected order. In the reconstructed images of checkerboard images, the central part was more blurred than its periphery. However, when more iterations were performed, the degree of blurring was less apparent. In the image with the smallest size of boxes (or "2"), the convergence occurred later (around 400 iterations). As the size of boxes increased, the convergence took place. The curves of the top and left side of the images were perfectly superimposed and converged earlier than the center. Image contrast and pixel intensity had no effect on the point of convergence. The optimal point in the reconstruction of noisy images, where the MSE was minimum, occurred very early, and it was directly related to the level of noise.

The convergence of the algorithm is slower in the center of FOV than periphery (location sensitivity), but no difference exists between the top and left subregions of FOV (no direction sensitivity). High-frequency details (sharp edges) required more iterations to be reproduced. Image contrast and pixel intensity have no discernible effect on the algorithm's rate of convergence.