14 May 2018 The inversion of highly singular linear equations with applications to imaging and spectral imaging
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Abstract
The use of linear algebra in optics has been shown to be an important tool in optical design and imaging system analysis for hundreds of years. More recently the use of matrix theory has allowed the development of image processing, particularly with the introduction of large scale computer processing. The ability to approximate matrices as Toplitz allowed matrix multiplies to be carried out with fast transforms and convolutions thus allowing much faster implementations for many image processing applications. There remain a large class of problems for which no Toplitz representation is feasible, particularly those requiring an inversion of a large matrix which is often ill-conditioned, or, formally singular. In this article we discuss a technique for providing an approximate solution for problems which are formally singular. We develop a method for solving problems with a high degree of singularity (those for which the number of equations is far less than the number of variables). By way of illuminating the utility of the overall technique, several examples are presented. The use of the method for solving small under-determined problems is presented as an introduction to the use and limitations of the solution. The technique is applied to digital zoom and the results compared with standard interpolation techniques. The development of multi spectral data cubes for tomographic type multi spectral imaging systems is shown with several simulated results based on real data.
Conference Presentation
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H. C. Schau, H. C. Schau, } "The inversion of highly singular linear equations with applications to imaging and spectral imaging", Proc. SPIE 10669, Computational Imaging III, 106690T (14 May 2018); doi: 10.1117/12.2301679; https://doi.org/10.1117/12.2301679
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