Signals on Pencils of Lines

Abstract

This paper proposes the "epipolar pencil transformation" (EPT). This is a tool for comparing the signals in different images, with no use of feature detection, yet taking advantage of the constraints given by epipolar geometry. The idea is to develop a descriptor for each point, summarizing the signals on the pencil of lines intersecting at that point. To compute the EPT, first find compact descriptors for each line, then combine these appropriately for each pencil. Given the EPT for two images, computing the epipolar geometry reduces to a closest pairs problem- select one pencil from each set such that the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> distance (in descriptor space) is minimized. By this reduction to a high dimensional closest pairs problem, recent advances in computational geometry can be used to efficiently identify the best global solution. This technique is robust, as each potential solution is evaluated by comparing the signals for all the lines passing through the two hypothesized epipoles. At the same time, as the closest pairs algorithm performs a global search, the solution is not distracted by local minima. The EPT is used here both for the problem of two-view rigid motion, and many-view place recognition.