Jinhee Chun

Efficient Algorithms for Approximating a Multi-dimensional Voxel Terrain by a Unimodal Terrain

We consider the problem of approximating a function on a d-dimensional voxel grid by a unimodal function to minimize the L 2 approximation error with respect to a given measure distribution on the grid. The output unimodal function gives a layered structure on the voxel grid, and we give efficient algorithms for computing the optimal approximation under a reasonable assumption on the shape of each horizontal layer. Our main technique is a dominating cut algorithm for a graph.

Consistent digital rays

Given a fixed origin <i>o</i> in the <i>d</i>-dimensional grid, we give a novel definition of <i>digital rays dig(op)</i> from <i>o</i> to each grid point <i>p</i>. Each digital ray dig(<i>op</i>) approximates the Euclidean line segment <i>op</i> between <i>o</i> and <i>p</i>. The set of all digital rays satisfies a set of axioms analogous to the Euclidean axioms. We measure the approximation quality by the maximum Hausdorff distance between a digital ray and its Euclidean counterpart and establish an asymptotically tight Θ(log <i>n</i>) bound in the <i>n</i> x <i>n</i> grid. The proof of the bound is based on discrepancy theory and a simple construction algorithm. Without a monotonicity property for digital rays the bound is improved to <i>O</i>(1). Digital rays enable us to define the family of digital star-shaped regions centered at <i>o</i> which we use to design efficient algorithms for image processing problems.

Efficient Algorithms for Constructing a Pyramid from a Terrain

In [4], the following pyramid construction problem was proposed: Given nonnegative valued functions ρ and μ in d variables, we consider the optimal pyramid maximizing the total parametric gain of ρ against μ. The pyramid can be considered as the optimal unimodal approximation of ρ relative to μ, and can be applied to hierarchical data segmentation. In this paper, we give efficient algorithms for a couple of two-dimensional pyramid construction problems.

Linear Time Algorithm for Approximating a Curve by a Single-Peaked Curve

AbstractGiven a function y = f(x) in one variable, we consider the problem of computing the single-peaked (unimodal) curve y =φ(x) minimizing the L2-distance between them. If the input function f is a histogram with O(n) steps or a piecewise linear function with O(n) linear pieces, we design algorithms for computing φ in linear time. We also give an algorithm to approximate f with a function consisting of the minimum number of unimodal pieces under the condition that each unimodal piece is within a fixed L2-distance from the corresponding portion of f.

Algorithms for Computing the Maximum Weight Region Decomposable into Elementary Shapes

Motivated from the image segmentation problem, we consider the problem of finding the maximum weight region with a shape decomposable into elementary shapes in n ×n pixel grid where each pixel has a real valued weight. We give efficient algorithms for several interesting cases. This shows string constrast to the NP-hardness results to find the maximum weight union for the corresponding cases.

Distance interior ratio

A novel 2D shape signature named Distance Interior Ratio (DIR) is proposed.DIR represents an intersection pattern of line segments on a shape contour.Interval of each bin is defined using to the mean of the distribution method.Our method achieves higher retrieval rate than other distance distribution methods.DIR distance can differentiate the distance distributions of homometric pair objects. In this work, we propose a shape signature named Distance Interior Ratio (DIR) that utilizes intersection pattern of the distribution of line segments with the shape. To improve the efficiency of the histogram-based shape signature, we present a histogram alignment method for adjusting the interval of the histogram according to the distance distribution. The experimental result shows a 3.25% improvement using the proposed histogram alignment. When compared to other shape signatures, our experimental result gives a 77.69% retrieval rate using MPEG7 Part B dataset Latecki, etźal. (2000)14.

Distance Trisector of Segments and Zone Diagram of Segments in a Plane

Motivated by the work of Asano et al.[l], we consider the distance trisector problem and Zone diagram considering segments in the plane as the input geometric objects. As the most basic case, we first consider the pair of curves (distance trisector curves) trisecting the distance between a point and a line. This is a natural extension of the bisector curve (that is a parabola) of a point and a line. In this paper, we show that these trisector curves C1 and C2 exist and are unique. We then give a practical algorithm for computing the Zone diagram of a set of segments in a digital plane.

Gap-Planar Graphs

We introduce the family of k-gap-planar graphs for \(k \ge 0\), i.e., graphs that have a drawing in which each crossing is assigned to one of the two involved edges and each edge is assigned at most k of its crossings. This definition finds motivation in edge casing, as a \(k\)-gap-planar graph can be drawn crossing-free after introducing at most k local gaps per edge. We obtain results on the maximum density, drawability of complete graphs, complexity of the recognition problem, and relationships with other families of beyond-planar graphs.

EFFECT OF CORNER INFORMATION IN SIMULTANEOUS PLACEMENT OF k RECTANGLES AND TABLEAUX

We consider the optimization problem of finding k nonintersecting rectangles and tableaux in n × n pixel plane where each pixel has a real valued weight. We discuss existence of efficient algorithms if a corner point of each rectangle/tableau is specified.

Algorithms for computing the maximum weight region decomposable into elementary shapes

Motivated by the image segmentation problem, we consider the following geometric optimization problem: Given a weighted nxn pixel grid, find the maximum weight region whose shape is decomposable into a set of disjoint elementary shapes. We give efficient algorithms for several interesting shapes. This is in strong contrast to finding the maximum weight region that is the union of elementary shapes for the corresponding cases-a problem that we prove to be NP-hard. We implemented one of the algorithms and demonstrate its applicability for image segmentation.