Model-based probing strategies for convex polygons

AbstractWe prove that n+4 finger probes are sufficient to determine the shape of a convex n-gon from a finite collection of models, improving the previous result of 2n+1. Further, we show that n−1 are necessary, proving this is optimal to within an additive constant. For line probes, we show that 2n+4 probes are sufficient and 2n−3 necessary. The difference between these results is particularly interesting in light of the duality relationship between finger and line probes

Study of modularity in images

Modularity is an important concept in Biology, among other areas. In script complexity, there are some theories stating that the symbols of a writing system are built from smaller, common components, that could be thought of as modules. We introduce a representation of black and white images as sequences of symbols, which allows us to apply sequence based motif finding techniques on images. We present a modification of the algorithm in [27] to search for motifs in a variable number of sequences, instead of in a fixed number, while guaranteeing the same properties about the statistical significance of the found motifs of the original paper. Finally, we apply the proposed algorithm to sequences describing images of latin letters

Study of modularity in images

Modularity is an important concept in Biology, among other areas. In script complexity, there are some theories stating that the symbols of a writing system are built from smaller, common components, that could be thought of as modules. We introduce a representation of black and white images as sequences of symbols, which allows us to apply sequence based motif finding techniques on images. We present a modification of the algorithm in [27] to search for motifs in a variable number of sequences, instead of in a fixed number, while guaranteeing the same properties about the statistical significance of the found motifs of the original paper. Finally, we apply the proposed algorithm to sequences describing images of latin letters

Probing Convex Polygons with a Wedge

Minimizing the number of probes is one of the main challenges in reconstructing geometric objects with probing devices. In this paper, we investigate the problem of using an $\omega$-wedge probing tool to determine the exact shape and orientation of a convex polygon. An $\omega$-wedge consists of two rays emanating from a point called the apex of the wedge and the two rays forming an angle $\omega$. To probe with an $\omega$-wedge, we set the direction that the apex of the probe has to follow, the line $\overrightarrow L$, and the initial orientation of the two rays. A valid $\omega$-probe of a convex polygon $O$ contains $O$ within the $\omega$-wedge and its outcome consists of the coordinates of the apex, the orientation of both rays and the coordinates of the closest (to the apex) points of contact between $O$ and each of the rays. We present algorithms minimizing the number of probes and prove their optimality. In particular, we show how to reconstruct a convex $n$-gon (with all internal angles of size larger than $\omega$) using $2n-2$ $\omega$-probes; if $\omega = \pi/2$, the reconstruction uses $2n-3$ $\omega$-probes. We show that both results are optimal. Let $N_B$ be the number of vertices of $O$ whose internal angle is at most $\omega$, (we show that $0 \leq N_B \leq 3$). We determine the shape and orientation of a general convex $n$-gon with $N_B=1$ (respectively $N_B=2$, $N_B=3$) using $2n-1$ (respectively $2n+3$, $2n+5$) $\omega$-probes. We prove optimality for the first case. Assuming the algorithm knows the value of $N_B$ in advance, the reconstruction of $O$ with $N_B=2$ or $N_B=3$ can be achieved with $2n+2$ probes,- which is optimal.Comment: 31 pages, 27 figure

Probing the arrangement of hyperplanes

AbstractIn this paper we investigate the combinatorial complexity of an algorithm to determine the geometry and the topology related to an arrangement of hyperplanes in multi-dimensional Euclidean space from the “probing” on the arrangement. The “probing” by a flat means the operation from which we can obtain the intersection of the flat and the arrangement. For a finite set H of hyperplanes in Ed, we obtain the worst-case number of fixed direction line probes and that of flat probes to determine a generic line of H and H itself. We also mention the bound for the computational complexity of these algorithms based on the efficient line probing algorithm which uses the dual transform to compute a generic line of H.We also consider the problem to approximate arrangements by extending the point probing model, which have connections with computational learning theory such as learning a network of threshold functions, and introduce the vertical probing model and the level probing model. It is shown that the former is closely related to the finger probing for a polyhedron and that the latter depends on the dual graph of the arrangement.The probing for an arrangement can be used to obtain the solution for a given system of algebraic equations by decomposing the μ-resultant into linear factors. It also has interesting applications in robotics such as a motion planning using an ultrasonic device that can detect the distances to obstacles along a specified direction

On the Area of Overlap of Translated Polygons

(Also cross-referenced as CAR-TR-699) Given two simple polygons P and Q in the plane and a translation vector t E R2, the area-oJ-overlap function of P and Q is the function Ar(t) = Area(P n (t + Q)), where t + Q denotes Q translated by t. This function has a number of applications in areas such as motion planning and object recognition. We present a number of mathematical results regarding this function. We also provide efficient algorithms for computing a representation of this function, and for tracing contour curves of constant area of o verlap