A point visibility graph is a graph induced by a set of points in the plane, where every vertex corresponds to a point, and two vertices are adjacent whenever the two corresponding points are visible from each other, that is, the open segment between them does not contain any other point of the set. We study the recognition problem for point visibility graphs: given a simple undirected graph, decide whether it is the visibility graph of some point set in the plane. We show that the problem is complete for the existential theory of the reals. Hence the problem is as hard as deciding the existence of a real solution to a system of polynomial inequalities. The proof involves simple substructures forcing collinearities in all realizations of some visibility graphs, which are applied to the algebraic universality constructions of Mnev and Richter-Gebert. This solves a longstanding open question and paves the way for the analysis of other classes of visibility graphs. Furthermore, as a corollary of one of our construction, we show that there exist point visibility graphs that do not admit any geometric realization with points having integer coordinates.

Recent development of Multi-detector Computerized Tomography (MDCT) holds the promise of a non-invasive diagnostic method to evaluate coronary artery disease as an alternative to standard angiography. A major impediment preventing its utilization in routine clinical practice is the presence of image "blooming" artifacts due to vascular calcium. For high-risk patients with serious coronary stenosis who often have significant calcified plaque, this problem becomes very acute. In this paper, we examine an approach to ameliorate the degradations due to the presence of calcium and thus increase the applicability of MDCT. We assume the blooming effect is well modeled as a linear convolutional blur. We then use this model as the basis to perform image restoration, thus deblurring the original image. The aim is to produce CT images with reduced calcium artifacts. We evaluate our method on simulated, ex vivo and clinical data and show that this method has the potential to improve coronary CT image and thus improve stenosis diagnosis.