Atlas F. Cook

Real-time density-based crowd simulation

Virtual characters in games and simulations often need to plan visually convincing paths through a crowded environment. This paper describes how crowd density information can be used to guide a large number of characters through a crowded environment. Crowd density information helps characters avoid congested routes that could lead to traffic jams. It also encourages characters to use a wide variety of routes to reach their destination. Our technique measures the desirability of a route by combining distance information with crowd density information. We start by building a navigation mesh for the walkable regions in a polygonal two‐dimensional (2‐D) or multilayered three‐dimensional (3‐D) environment. The skeleton of this navigation mesh is the medial axis. Each walkable region in the navigation mesh maintains an up‐to‐date density value. This density value is equal to the area occupied by all the characters inside a given region divided by the total area of this region. These density values are mapped onto the medial axis to form a weighted graph. An A* search on this graph yields a backbone path for each character, and forces are used to guide the characters through the weighted environment. The characters periodically replan their routes as the density values are updated. Our experiments show that we can compute congestion‐avoiding paths for tens of thousands of characters in real‐time. Copyright

Geodesic Fréchet distance inside a simple polygon

We present an alternative to parametric search that applies to both the nongeodesic and geodesic Fréchet optimization problems. This randomized approach is based on a variant of red-blue intersections and is appealing due to its elegance and practical efficiency when compared to parametric search. We introduce the first algorithm to compute the geodesic Fréchet distance between two polygonal curves <i>A</i> and <i>B</i> inside a simple bounding polygon <i>P</i>. The geodesic Fréchet <i>decision</i> problem is solved almost as fast as its nongeodesic sibling in <i>O</i>(<i>N</i>² log <i>k</i>) time and <i>O</i>(<i>k</i>+<i>N</i>) space after <i>O(k)</i> preprocessing, where <i>N</i> is the larger of the complexities of <i>A</i> and <i>B</i> and <i>k</i> is the complexity of <i>P</i>. The geodesic Fréchet <i>optimization</i> problem is solved by a randomized approach in <i>O</i>(<i>k</i>+<i>N</i>² log <i>kN</i> log <i>N</i>) expected time and <i>O</i>(<i>k</i>+<i>N</i>²) space. This runtime is only a logarithmic factor larger than the standard nongeodesic Fréchet algorithm [Alt and Godau 1995]. Results are also presented for the geodesic Fréchet distance in a polygonal domain with obstacles and the geodesic Hausdorff distance for sets of points or sets of line segments inside a simple polygon <i>P</i>.

Navigation meshes for realistic multi-layered environments

Virtual characters often need to plan visually convincing paths through a complicated environment. For example, a traveler may need to walk from an airport entrance to a staircase, descend the staircase, walk to a shuttle, ride the shuttle to a destination, ride an elevator back to the ground floor, and finally move on the ground floor again to reach the desired airplane. Most previous research only supports path planning in a single plane because the underlying data structures are two-dimensional. The goal of this paper is to permit visually convincing paths to be efficiently computed in a multi-layered environment such as an airport or a multi-storey building. We describe an algorithm to create a navigation mesh, and our implementation demonstrates the feasibility of the approach.

A navigation mesh for dynamic environments

Games and simulations frequently model scenarios where obstacles move, appear, and disappear in an environment. A city environment changes as new buildings and roads are constructed, and routes can become partially blocked by small obstacles many times in a typical day. This paper studies the effect of using local updates to repair only the affected regions of a navigation mesh in response to a change in the environment. The techniques are inspired by incremental methods for Voronoi diagrams. The main novelty of this paper is that we show how to maintain a 2D or 2.5D navigation mesh in an environment that contains dynamic polygonal obstacles. Experiments show that local updates are fast enough to permit real‐time updates of the navigation mesh. Copyright

Real‐time path planning in heterogeneous environments

Modern virtual environments can contain a variety of characters and traversable regions. Each character may have different preferences for the traversable region types. Pedestrians may prefer to walk on sidewalks, but they may occasionally need to traverse roads and dirt paths. By contrast, wild animals might try to stay in forest areas, but they are able to leave their protective environment when necessary. This paper presents a novel path planning method named Modified Indicative Routes and Navigation (MIRAN) that takes a characters region preferences into account. Given an indicative route as a rough estimation of a characters preferred route, MIRAN efficiently computes a visually convincing path that is smooth, keeps clearance from obstacles, avoids unnecessary detours, and allows local changes to avoid other characters. To the best of our knowledge, MIRAN is the first path planning method that supports the aforementioned features while using an exact representation of the navigable space. Experiments show that with our approach, a wide range of different character behaviors can be simulated. It also overcomes problems that occur in previous path planning methods such as the Indicative Route Method. The resulting paths are well suited for real‐time simulations and gaming applications. Copyright

Fast Fréchet queries

Inspired by video analysis of team sports, we study the following problem. Let P be a polygonal path in the plane with n vertices. We want to preprocess P into a data structure that can quickly count the number of inclusion-minimal subpaths of P whose Frechet distance to a given query segment Q is at most some threshold value @e. We present a data structure that solves an approximate version of this problem: it counts all subpaths whose Frechet distance is at most @e, but this count may also include subpaths whose Frechet distance is up to (2+32)@e. For any parameter n@?s@?n^2, our data structure can be tuned such that it uses O(spolylogn) storage and has O((n/s)polylogn) query time.

Computing the Fréchet distance between folded polygons

Abstract Computing the Frechet distance for surfaces is a surprisingly hard problem and the only known polynomial-time algorithm is limited to computing it between flat surfaces. We study the problem of computing the Frechet distance for a class of non-flat surfaces called folded polygons. We present a fixed-parameter tractable algorithm for this problem. Next, we present a polynomial-time approximation algorithm. Finally, we present a restricted class of folded polygons for which we can compute the Frechet distance for L ∞ in polynomial time.

Shortest Path Problems on a Polyhedral Surface

We describe algorithms to compute edge sequences, a shortest path map, and the Fréchet distance for a convex polyhedral surface. Distances on the surface are measured by the length of a Euclidean shortest path. We describe how the star unfolding changes as a source point slides continuously along an edge of the convex polyhedral surface. We describe alternative algorithms to the edge sequence algorithm of Agarwal et al. (SIAM J. Comput. 26(6):1689–1713, 1997) for a convex polyhedral surface. Our approach uses persistent trees, star unfoldings, and kinetic Voronoi diagrams. We also show that the core of the star unfolding can overlap itself when the polyhedral surface is non-convex.

Link distance and shortest path problems in the plane

This paper describes algorithms to compute Voronoi diagrams, shortest path maps, the Hausdorff distance, and the Frechet distance in the plane with polygonal obstacles. The underlying distance measures for these algorithms are either shortest path distances or link distances. The link distance between a pair of points is the minimum number of edges needed to connect the two points with a polygonal path that avoids a set of obstacles. The motivation for minimizing the number of edges on a path comes from robotic motions and wireless communications because turns are more difficult in these settings than straight movements. Link-based Voronoi diagrams are different from traditional Voronoi diagrams because a query point in the interior of a Voronoi face can have multiple nearest sites. Our site-based Voronoi diagram ensures that all points in a face have the same set of nearest sites. Our distance-based Voronoi diagram ensures that all points in a face have the same distance to a nearest site. The shortest path maps in this paper support queries from any source point on a fixed line segment. This is a middle-ground approach because traditional shortest path maps typically support queries from either a fixed point or from all possible points in the plane. The Hausdorff distance and Frechet distance are fundamental similarity metrics for shape matching. This paper shows how to compute new variations of these metrics using shortest paths or link-based paths that avoid polygonal obstacles in the plane.

Visiting a sequence of points with a bevel-tip needle

Many surgical procedures could benefit from guiding a bevel-tip needle along circular arcs to multiple treatment points in a patient. At each treatment point, the needle can inject a radioactive pellet into a cancerous region or extract a tissue sample. Our main result is an algorithm to steer a bevel-tip needle through a sequence of treatment points in the plane while minimizing the number of times that the needle must be reoriented. This algorithm is related to [6] and takes quadratic time when consecutive points in the sequence are sufficiently separated. We can also guide a needle through an arbitrary sequence of points in the plane by accounting for a lack of optimal substructure.