Using mathematical concepts from spectral graph theory, we provide algorithms that are not only robust to noisy experimental measurements, but also have running times orders of magnitude faster than existing solutions to two of the key problems in structural biology:
1) The reconstruction of molecular objects, such as proteins and viruses from noisy cryo electron microscopy images.
2) The distance geometry problem arising in atomic structure determination from NMR spectroscopy using NOE and other geometrical constraints.
We show that the two problems are closely related not only from the structural biology point of view, but also from the mathematical point of view. Our mathematical theory is applicable to both, though each problem has its unique underlying geometrical structure and geometrical constraints. Our algorithms are based on a novel construction of sparse linear operators and the computation of their eigenvectors. The tools and methods could also be useful in other imaging modalities, such as tomography of moving objects in medical imaging (X-ray, PET and Ultrasound), in other image and signal denoising problems, and in other research areas, such as sensor networks, numerical analysis and more.
Joint work with Ronald Coifman, Yoel Shkolnisky and Fred Sigworth.
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