Exploring the Influence of Initial Estimates on Iterative Maximum Likelihood Expectation–maximization in Tomographic Reconstruction

Abstract

Purpose: To explore the influence of initial guess or estimate (uniform as “ones” and “zeros” vs. filtered back projection [FBP] image) as an input image for maximum likelihood expectation–maximization (MLEM) tomographic reconstruction algorithm and provide the curves of error or convergence for each of these three initial estimates. Methods: Two phantoms, created as digital images, were utilized: one was a simple noiseless object and the other was a more complicated, noise-degraded object of the section of lower thorax in a matrix of 256 × 256 pixels. Both underwent radon transform or forward projection process and the corresponding sinograms were generated. For filtering during tomographic image reconstruction, ramp and Butterworth filters, as high-pass and low-pass ones, were applied to images. The second phantom (lower thorax) was radon-transformed and the resulting sinogram was degraded by noise. As initial guess or estimate images, in addition to FBP tomographic image, two uniform images, one with all pixels having a value of 1 (“ones”) and the other with all having zero (“zeros”), were created. The three initial estimates (FBP, ones, and zeros) were reconstructed with iterative MLEM tomographic reconstruction (with 1, 2, 4, 8, 16, 32, and 64 iterations). The difference between the object and the updated slice was calculated at the end of each iteration (as error matrix), and the mean squared error (MSE) was computed and plotted separately or in conjunction with the MSE curves of other initial estimates. All computations were implemented in MATLAB software. Results: The results of ones and zeros seemed strikingly similar. The curves of uniform ones and uniform zeros were so close to each other that overlap near-perfectly. However, in the FBP slice as an initial estimate, the resulting tomographic slice was similar with a much higher extent to the object even after 1 or 2 iterations. The pattern of convergence for all three curves was roughly similar. The normalized MSE decreased sharply up to 5 iterations and then, after 10 iterations, the curves reached a plateau until 32 iterations. For the phantom of the lower thorax section with its noise-degraded sinogram, similar to the pattern observed for simple disk-shaped phantom, the curves (normalized MSE) fell sharply up to 10 iterations and then rapidly converged thereafter until 64 iterations. Conclusion: Similar results are observed when choosing different initial guesses or estimates (uniform vs. FBP) as the starting point, based on the error calculation using MSE. The algorithm converges almost similarly for all initial estimates. Therefore, selecting a uniform initial guess image can be an appropriate choice and may be preferred over an FBP image. Reducing the processing time can be a valid reason for this choice.