Computational Methods for Inverting the Soft X-Ray Transform

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The aim of this project is the mathematical development, computational implementation, testing and evaluation of procedures for inverting the soft x-ray transform, which has arisen recently from experimental cellular biology, but has not previously been subjected to a rigorous treatment from the computational mathematics point of view. The soft x-ray transform is the appropriate mathematical model for the physical process by which two-dimensional projections are acquired in soft x-ray microscopy (SXM), which has the unique capability of imaging whole cells in their native environment with high resolution. The proposednovel computational methods for inverting this accurate image formation model for SXM are designed to improve the resolution in the reconstructions of the three-dimensional (3D) structures from their SXM projections. It is conjectured that such a treatment will result in improved accuracy and usefulness as compared to the currently used heuristic procedures. In many areas of biology, the understanding of a 3D structure provides a way of understanding its function. Many structures, such as single molecules, viruses, cells, etc., are too small to be viewed with the naked eye. Microscopy has been providing images with information aboutdetails of these structures. However, unprocessed microscopic images present structures that are superimposed over each other, makingthem hard to interpret. Combining multiple images from different directions of the same structure by techniques of 3D reconstruction allows accurate visualization of such structures. The understanding of 3D shapes of structures is important in many biomedical areas, for example, in drug design. Of the many approaches to unraveling 3D biological structures, soft X-ray microscopy (SXM) has the unique capability of providing high-resolution details of subcellular 3D structures in their native environment, i.e., the whole cell. Such currently-unavailable structural information will be important for understanding many biological processes. An example is mitochondrial dysfunction as it occurs in human diseases, understanding of which is important for assessing cardiovascular and nervous system function in mitochondrial disorders.