Journal of Applied and Numerical Optimization

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DOI: 10.23952/jano.6.2024.3.04
Volume 6, Issue 3, 1 December 2024, Pages 351-370

 

Abstract. Too much reliance has been placed on calculating single images meeting possibly arbitrary optimization criteria. By generating a dispersion of multiple images consistent with the data, we may be able to learn how wrong we could be. Our problem is to generate a way of seeing a representative sample of all of the solutions in the hyperplane of solutions, which would be an array of images, each of which is a solution to the equations. To accomplish this, we suggest that our sampling is in a space of image basis functions, rather than directly in the hyperplane. As the number of basis functions is large, we design a selection criterion for choosing a subset that reasonably spans the space of images. First, we try a random sampling, which gives high frequency or sequency samples. Then we turn to more systematic sampling, based on the methods developed for one-pixel imaging. Some numerical experiments demonstrate that the use of basis functions as starting images for ART-like iterative algorithms may suffice to span the hyperplane of solutions, allowing choices between solutions other than simple optimization of arbitrary criteria such as minimum norm or maximum entropy, or deconvolution of the point spread function of the algorithm.

 

How to Cite this Article:
C. Wang, R. Gordon, How wrong could we be? A new way to solve underdetermined linear equations, illustrated via computed tomography, J. Appl. Numer. Optim. 6 (2024), 351-370.