Nonlinear three-dimensional signal reconstruction from spherically averaged Fourier transform magnitudes: Determining virus structures from x-ray solution scattering

3-D reconstruction from spherically averaged Fourier transform magnitudes arise in the structural biology of viruses based on x-ray solution scattering experiments. Such problems can be computed by solving a nonlinear least squares problem. Solving a least squares problem often cannot include all biological knowledge about the virus because such knowledge cannot be incorporated into the model and least squares cost. In addition, incorporation of this additional knowledge decreases the effect on the solution of noise in the data. To address the two problems, regularizers are proposed based on the spherically averaged electron scattering density autocorrelation and based on 2-D and 3-D measures of curvature. A second issue is the limited amount of data from a solution scattering experiment. To address this issue experiments are proposed using particles labeled by a strong scatterer(e.g., nano gold cluster). Models and reconstruction algorithms are developed for the problem. Labels have the desirable effect of removing certain non-uniqueness in the solution which is due to symmetry