  The anatomical structure of the brain can be observed via non-invasive
techniques such as diffusion imaging. However, these are imperfect because they
miss connections that are actually known to exist, especially long range
inter-hemispheric ones. In this paper we formulate the inverse problem of
inferring the structural connectivity of brain networks from experimentally
observed functional connectivity via functional Magnetic Resonance Imaging
(fMRI), by formulating it as a convex optimization problem. We show that
structural connectivity can be modeled as an optimal sparse representation
derived from the much denser functional connectivity in the human brain. Using
only the functional connectivity data as input, we present (a) an optimization
problem that models constraints based on known physiological observations, and
(b) an ADMM algorithm for solving it. The algorithm not only recovers the known
structural connectivity of the brain, but is also able to robustly predict the
long range inter-hemispheric connections missed by DSI or DTI, including a very
good match with experimentally observed quantitative distributions of the
weights/strength of anatomical connections. We demonstrate results on both
synthetic model data and a fine-scale 998 node cortical dataset, and discuss
applications to other complex network domains where retrieving effective
structure from functional signatures are important.
