Vanishing Point

pages 34-3

Sweeping an oval to a vanishing point

Given a convex region in the plane, and a sweep-line as a tool, what is best way to reduce the region to a single point by a sequence of sweeps? The problem of sweeping points by orthogonal sweeps was first studied in [2]. Here we consider the following \emph{slanted} variant of sweeping recently introduced in [1]: In a single sweep, the sweep-line is placed at a start position somewhere in the plane, then moved continuously according to a sweep vector $\vec v$ (not necessarily orthogonal to the sweep-line) to another parallel end position, and then lifted from the plane. The cost of a sequence of sweeps is the sum of the lengths of the sweep vectors. The (optimal) sweeping cost of a region is the infimum of the costs over all finite sweeping sequences for that region. An optimal sweeping sequence for a region is one with a minimum total cost, if it exists. Another parameter of interest is the number of sweeps. We show that there exist convex regions for which the optimal sweeping cost cannot be attained by two sweeps. This disproves a conjecture of Bousany, Karker, O'Rourke, and Sparaco stating that two sweeps (with vectors along the two adjacent sides of a minimum-perimeter enclosing parallelogram) always suffice [1]. Moreover, we conjecture that for some convex regions, no finite sweeping sequence is optimal. On the other hand, we show that both the 2-sweep algorithm based on minimum-perimeter enclosing rectangle and the 2-sweep algorithm based on minimum-perimeter enclosing parallelogram achieve a $4/\pi \approx 1.27$ approximation in this sweeping model.Comment: 9 pages, 4 figure

Fixation prediction with a combined model of bottom-up saliency and vanishing point

By predicting where humans look in natural scenes, we can understand how they perceive complex natural scenes and prioritize information for further high-level visual processing. Several models have been proposed for this purpose, yet there is a gap between best existing saliency models and human performance. While many researchers have developed purely computational models for fixation prediction, less attempts have been made to discover cognitive factors that guide gaze. Here, we study the effect of a particular type of scene structural information, known as the vanishing point, and show that human gaze is attracted to the vanishing point regions. We record eye movements of 10 observers over 532 images, out of which 319 have vanishing points. We then construct a combined model of traditional saliency and a vanishing point channel and show that our model outperforms state of the art saliency models using three scores on our dataset.Comment: arXiv admin note: text overlap with arXiv:1512.0172