OL-HeatMap: Effective Density Visualization of Multiple Overlapping Rectangles

Abstract

Visualization of the density of multiple overlapping axis-aligned objects is a challenging computational problem that can inform large-scale visual analytics, in diverse domains. For example, when dealing with crowd simulations, we care about constructing interaction maps, and in urban planning we care about city areas mostly frequented by people, to name a few. The primary objective of this research is, given a large set of axis-aligned two-dimensional (2D) objects, or simply rectangles, to devise efficient and effective data visualization methods that inform whether, where and how much these rectangles overlap. Currently, such visualizations rely on inefficient implementations of determining the size of the overlapping rectangles that do not scale well and are hard to accomplish. Approximate methods have also been proposed in the literature. To the contrary of these approaches, we aim to address this problem by exploiting state-of-the-art computational geometry methods based on the sweep line paradigm. These methods are fast and can determine the exact size of the overlap of multiple axis-aligned objects, therefore can effectively inform the visualization method. Towards that end, we present OL-HeatMap, a novel type of a heat-map visualization that can be used to represent and perceive density of overlapping rectangles. Our experimental evaluation demonstrates the effectiveness of the proposed method in terms of both accuracy and running time for synthetic and real-world data-sets.

Introduction

Density-based information visualization methods are commonly employed in big data visual analytics. They provide powerful abstract representations of large data sets that can help one to quickly perceive areas of interest due to a large concentration of data points (or their absence). Amongst a plethora of visualization techniques for density, such as scatter plots or treemaps, we focus on one of the most commonly used density-based visualization methods, the heat-map. A heat-map is a graphical representation of data where data values are represented as colors. These colors depict the characteristics of the data based on problem-specific requirements. Typically, darker colors depict regions with higher amounts or concentrations of data values present, while the opposite is true for lighter colors. Variants of heat-maps have been used to show the density or distribution of data on a given region of interest. This technique provides a general view of numerical data, and it can be customized to suit statistical and categorical data variants. It can also be employed to show the results of clustering algorithms. As the rendered graphic is easy to understand, it is typically used to check the expected results versus the actual results of an algorithm.

A common tool employed in the construction of a heat-map visualization is related to bounding volumes, and specifically bounding boxes. A bounding volume is a visual abstraction that is used to approximate complex objects and simplify the visualization process. Such visual abstractions introduce some flexibility to the problem, allowing for faster computation while avoiding significant losses in the information visualized. For different objects in real life, different bounding volumes such as rectangles, cuboids, spheres, and hyper-planes can be used. Furthermore, when the shapes used are rectangles or cuboids, they can be axis-aligned, meaning that their sides are parallel to the respective coordinate axis. In this work, we focus on 2-dimensional axis-aligned bounding boxes which we refer to as rectangles for ease of understanding. Previously, such rectangles have been used to approximate geographical objects [1], for the construction of spatial data structures [2], but also in VLSI design [3], to name a few.

We are interested in creating density-based visualizations that offer insights about the interactions (relationships) of these rectangles on a Cartesian plane. To that end, we need to identify and report the density value (i.e., the number of rectangles that overlap) of every point on the Cartesian plane. In addition, for each of these overlaps we want to determine the size of the overlap and its location in the Cartesian plane. There are a handful of approaches to address this problem, with one of the most common being grid-based [4]. According to grid-based methods, first a uniform grid is defined that would separate the observation space into equal size grid cells. Then, the method determines the overlap of each grid cell to the input rectangles using well-established orthogonal range query methods [5], such as R-trees [6]. However, grid-based methods inherit several limitations. Constructing a spatial grid-based data structure and performing range queries for each grid cell is computationally expensive. Furthermore, the accuracy of the visualization results would greatly depend on the size of the grid (grid granularity). This presents an interesting trade-off where a small grid will be computationally more efficient but less accurate, and a large grid will provide more accurate representation of the overlaps, but at the expense of running cost. An illustrative example of this trade-off is shown in Fig. 1. We further elaborate on this trade-off in the methodology and experimental evaluation sections.

A more desirable outcome would be to be able to identify the exact location, density and size of any overlap among the available rectangles in the data-set directly. The simplest, brute-force approach to accomplish this is to compare every rectangle with every other rectangle, pair-wise first, then proceed to compare the overlap of every pair with every other object to find triple overlaps, and so on. As is apparent, the computational cost of such a method is prohibitively high. Instead, an approach that is commonly used to answer such geometric object overlap problems efficiently is the algorithmic paradigm known as the sweep-line or plane sweep algorithm [7]. Algorithms belonging to this category utilize a conceptual line that sweeps across the plane and quickly identify overlapping objects.

In this work, we employ a recently proposed variation of the sweep-line algorithm that is able to determine the exact location, size and number of multiple overlaps of n-dimensional geometric objects [8]. That method is using a sweep-line to construct an auxiliary data structure known as a region intersection graph and has the potential to significantly reduce the computation required for the effective visualization of the density of overlapping rectangles. Specifically, the main contributions of our work are as follows:

• We present OL-HeatMap (OverLap HeatMap), a fast and exact density-based visualization method for effective representation of the overlaps of multiple axis-aligned rectangles, based on the sweep-line paradigm.

• We introduce an evaluation metric that can be used to determine the accuracy of grid-based heat-map visualizations.

• We conduct an extensive evaluation of the performance of OL-HeatMap which demonstrates that it significantly outperforms competitive grid-based methods, in terms of both running time and accuracy.

• We build an interactive visualization system that demonstrates the effectiveness of OL-HeatMap in practice.

• We make source code and data publicly available to encourage reproducibility of method and results.²

The remainder of this paper is organized as follows: Section 2 introduces notation and formally defines the problem of interest in this paper. Our proposed method, OL-HeatMap, along with the grid-based competitors and the overall computational framework are presented in Section 3. Section 4 presents a thorough experimental evaluation of the methods and algorithms. After reviewing the related work in Section 6, we conclude in Section 7.