In my dissertation, advanced theoretical approaches and algorithms are developed to investigate the two primary applications, (1) functional MRI (fMRI) and (2) structural virology. These new techniques provide novel insights in (i) brain functions and (ii) the heterogeneity of virus particles. Resting-state functional magnetic resonance imaging (rs-fMRI) examines the low frequency spontaneous fluctuations in blood oxygen level dependent (BOLD) signals. It is widely used to investigate the brain activity at resting state, which is the state consuming the majority brain energy metabolism. Over the past two decades, the established correlation method has been successful in delineating the large-scale brain networks. However, it cannot characterize the spatial and temporal causal relations in these networks. Other methods provide some information of the causal relations, but are not successful at detecting the underlying networks. To overcome these limitations, we introduce a new concept named prediction correlation (p-correlation) to replace the traditional methods for estimating brain networks from rs-fMRI. In particular, the correlation between two BOLD signals is replaced by a correlation between one BOLD signal and a prediction of this signal via a causal system driven by another BOLD signal. The advantages of the p-correlation approach include that (1) it is a generalization of correlation and is able to identify previously characterized large-scale brain networks, and (2) it determines more reliable patterns of directed network connectivity, and (3) it can estimate the duration of directed interactions in brain networks. By using p-correlation, for the first time, we are able to characterize the most and the least rapid information propagations in brain networks. For example, we found that the most rapid information flows are propagated from all other regions to the memory regions in human brain. This discovery agrees with the current understanding of brain functions at resting state, and provides novel insight into brain functions from a different dimension (e.g., the temporal scale). In addition, a realistic simulator for BOLD signals is developed with customized selection on mathematical models of brain activity and parameters. As a result, the new simulator simplified and enhanced the existing simulators by expanding customer options, especially allowing customers to integrate different components from different simulators into a single calculation. The structure of virus particles is central to understanding their function. Cryo electron microscopy (cryo EM) is a major experimental method for determining structure by measuring noisy randomly-oriented projection images, one image from each of 103-106 virus particles, which are then combined by reconstruction algorithms. Because each particle is individually imaged, unlike in x-ray crystallography, it is possible to gain information about the continuous heterogeneity, e.g., flexibility, of the virus particle. Symmetry is an important aspect of virus particles and our work has focused on particles with polyhedral symmetry which includes symmetry of tetrahedral, octahedral and icosahedral point groups. Traditional methods of characterizing heterogeneity assume that each particle has the full symmetry. This is probably not realistic. We have generalized this viewpoint so that individual particles lack symmetry, but the first and second order statistics of the particles have symmetry. In other words, the symmetry appears in the expectation and the covariance, rather than in each realization, of the random process. We represent the electron scattering intensity of the virus as a linear combination of basis functions where the coefficients are random variables. First, we derived and computed real-valued vector basis functions for square integrable functions on the surface of the sphere in 3-D where each vector basis function transforms as one of the irreducible representations of the point group symmetry. In particular, a new theorem in group representation theory was proven to guarantee the existence of such basis functions. Second, we derived conditions on the mean vector and covariance matrix of the coefficients that multiply the basis functions such that the linear combination of basis functions has the desired symmetry properties. Using this approach, we have been able to eliminate long-standing distortions in heterogeneity calculations associated with symmetry axes and demonstrate for the first time the space-varying anisotropy of the fluctuations in the virus' electron scattering intensity.