Quantitative Object Reconstruction Using Abel Transform X-Ray Tomography and Mixed Variable Optimization

Abstract

Abstract. This paper introduces a new approach to the problem of quantitative reconstruction of an object from few radiographic views. A mixed variable programming problem is formulated in which the variables of interest are the number and types of materials and geometric parameters. To demonstrate the technique, we considered the problem of reconstructing cylindrically symmetric objects of multiple layers from a single radiograph. The mixed variable pattern search algorithm for linearly constrained problems was applied by means of the NOMADm MATLAB software package. Numerical results are presented for several test configurations and show that, while there are difficulties yet to be overcome, the method is promising for solving this class of problems.

Derivative-free optimization: a review of algorithms and comparison of software implementations

“…Derivative-free optimization is an area of long history and current rapid growth, fueled by a growing number of applications that range from science problems [4,42,52,143] to medical problems [90,103] to engineering design and facility location problems [2,10,15,48,49,54,57,91,92,98].…”

Quantitative Abel tomography robust to noisy, corrupted and missing data

“…The MVO approach (4) fixes the second problem, alleviates the first, and provides a solution framework robust to data noise. This is all accomplished by using an appropriate object space of significantly reduced dimension, 2k + 1, where k n, m. Abramson et al (2008) formulate the Abel inversion problem as…”

“…The procedural details are equivalent to those of the previous work (Abramson et al 2008). The results are shown in Table 2.…”

“…For a more in depth discussion on how to interpret differing material choices and orderings, we refer the reader to Abramson et al (2008).…”

“…, k. The number of object layers k is also a problem variable. Abramson et al (2008) solve this problem using the mixed variable pattern search method of Audet and Dennis (2000) for which they demonstrate convergence to appropriately defined stationary points (Lucidi et al 2005). The result is an unambiguous object description μ * .…”