Gokul Varadhan

Accurate Minkowski sum approximation of polyhedral models

We present an algorithm to approximate the 3D Minkowski sum of polyhedral objects. Our algorithm decomposes the polyhedral objects into convex pieces, generates pairwise convex Minkowski sums and computes their union. We approximate the union by generating a voxel grid, computing signed distance on the grid points and performing isosurface extraction from the distance field. The accuracy of the algorithm is mainly governed by the resolution of the underlying volumetric grid. Insufficient resolution can result in unwanted handles or disconnected components in the approximation. We use an adaptive subdivision algorithm that overcomes these problems by generating a volumetric grid at an appropriate resolution. We guarantee that our approximation has the same topology as the exact Minkowski sum. We also provide a two-sided Hausdorff distance bound on the approximation. Our algorithm is relatively simple to implement and works well on complex models. We have used it for exact 3D translation motion planning, offset computation, mathematical morphological operations and bounded-error penetration depth estimation.

Fast swept volume approximation of complex polyhedral models

We present an efficient algorithm to approximate the swept volume (SV) of a complex polyhedron along a given trajectory. Given the boundary description of the polyhedron and a path specified as a parametric curve, our algorithm enumerates a superset of the boundary surfaces of SV. The superset consists of ruled and developable surface primitives, and the SV corresponds to the outer boundary of their arrangement. We approximate this boundary by using a five-stage pipeline. This includes computing a bounded-error approximation of each surface primitive, computing unsigned distance fields on a uniform grid, classifying all grid points using fast marching front propagation, iso-surface reconstruction, and topological refinement. We also present a novel and fast algorithm for computing the signed distance of surface primitives as well as a number of techniques based on surface culling, fast marching level-set methods and rasterization hardware to improve the performance of the overall algorithm. We analyze different sources of error in our approximation algorithm and highlight its performance on complex models composed of thousands of polygons. In practice, it is able to compute a bounded-error approximation in tens of seconds for models composed of thousands of polygons sweeping along a complex trajectory.

Topology preserving surface extraction using adaptive subdivision

We address the problem of computing a topology preserving isosurface from a volumetric grid using Marching Cubes for geometry processing applications. We present a novel topology preserving subdivision algorithm to generate an adaptive volumetric grid. Our algorithm ensures that every grid cell satisfies two local geometric criteria: a complex cell criterion and a star-shaped criterion. We show that these two criteria are sufficient to ensure that the surface extracted from the grid using Marching Cubes has the same genus and connectedness as that of the exact isosurface. We use our subdivision algorithm for accurate boundary evaluation of CSG combinations of polyhedra and low degree algebraic primitives, translational motion planning, model simplification and remeshing. The running time of our algorithm varies between a few seconds for simple models composed of a few thousand triangles to tens of seconds for complex polyhedral models represented using hundreds of thousands of triangles.

Out-of-core rendering of massive geometric environments

We present an external memory algorithm for fast display of very large and complex geometric environments. We represent the model using a scene graph and employ different culling techniques for rendering acceleration. Our algorithm uses a parallel approach to render the scene as well as fetch objects from the disk in a synchronous manner. We present a novel prioritized prefetching technique that takes into account LOD-switching and visibility-based events between successive frames. We have applied our algorithm to large gigabyte-sized environments that are composed of thousands of objects and tens of millions of polygons. The memory overhead of our algorithm is output sensitive and is typically tens of megabytes. In practice, our approach scales with the model sizes, and its rendering performance is comparable to that of an in-core algorithm.

Feature-sensitive subdivision and isosurface reconstruction

We present improved subdivision and isosurface reconstruction algorithms for polygonizing implicit surfaces and performing accurate geometric operations. Our improved reconstruction algorithm uses directed distance fields (Kobbelt et al., 2001) to detect multiple intersections along an edge, separates them into components and reconstructs an isosurface locally within each components using the dual contouring algorithm (Ju et al., 2002). It can reconstruct thin features without creating handles and results in improved surface extraction from volumetric data. Our subdivision algorithm takes into account sharp features that arise from intersecting surfaces or Boolean operations and generates an adaptive grid such that each voxel has at most one sharp feature. The subdivision algorithm is combined with our improved reconstruction algorithm to compute accurate polygonization of Boolean combinations or offsets of complex primitives that faithfully reconstruct the sharp features. We have applied these algorithms to polygonize complex CAD models designed using thousands of Boolean operations on curved primitives.

Dynamic simplification and visualization of large maps

In this paper, we present an algorithm that performs simplification of large geographical maps through a novel use of graphics hardware. Given a map as a collection of non‐intersecting chains and a tolerance parameter for each chain, we produce a simplified map that resembles the original map, satisfying the condition that the distance between each point on the simplified chain and the original chain is within the given tolerance parameter, and that no two chains intersect. In conjunction with this, we also present an out‐of‐core system for interactive visualization of these maps. We represent the maps hierarchically and employ different pruning strategies to accelerate the rendering. Our algorithm uses a parallel approach to do rendering as well as fetching data from the disk in a synchronous manner. We have applied our algorithm to a gigabyte sized map dataset. The memory overhead of our algorithm (the amount of main memory it requires) is output sensitive and is typically tens of megabytes, much smaller than the actual data size.

Star-shaped roadmaps - A deterministic sampling approach for complete motion planning

We present a simple algorithm for complete motion planning using deterministic sampling. Our approach relies on computing a star-shaped roadmap of the free space. We partition the free space into star-shaped regions such that a single point called the guard can see every point in the starshaped region. The resulting set of guards capture the intraregion connectivity. We capture the inter-region connectivity by computing connectors that link guards of adjacent regions. We use the guards and connectors to construct a star-shaped roadmap of the free space. We present an efficient algorithm to compute the roadmap in a deterministic manner without computing an explicit representation of the free space. We show that the star-shaped roadmap captures the connectivity of the free space while providing sufficient information to perform complete motion planning. Our approach is relatively simple to implement for robots with translational and rotational degrees of freedom (dof). We highlight the performance of our algorithm on challenging scenarios with narrow passages or when there is no collision-free path for low-dof robots.

Generalized penetration depth computation

Penetration depth (PD) is a distance metric that is used to describe the extent of overlap between two intersecting objects. Most of the prior work in PD computation has been restricted to translational PD, which is defined as the minimal translational motion that one of the overlapping objects must undergo in order to make the two objects disjoint. In this paper, we extend the notion of PD to take into account both translational and rotational motion to separate the intersecting objects, namely generalized PD. When an object undergoes rigid transformation, some point on the object traces the longest trajectory. The generalized PD between two overlapping objects is defined as the minimum of the longest trajectories of one object under all possible rigid transformations to separate the overlapping objects.We present three new results to compute generalized PD between polyhedral models. First, we show that for two overlapping convex polytopes, the generalized PD is same as the translational PD. Second, when the complement of one of the objects is convex, we pose the generalized PD computation as a variant of the convex containment problem and compute an upper bound using optimization techniques. Finally, when both the objects are non-convex, we treat them as a combination of the above two cases, and present an algorithm that computes a lower and an upper bound on generalized PD. We highlight the performance of our algorithms on different models that undergo rigid motion in the 6-dimensional configuration space. Moreover, we utilize our algorithm for complete motion planning of polygonal robots undergoing translational and rotational motion in a plane. In particular, we use generalized PD computation for checking path non-existence.

A Simple Algorithm for Complete Motion Planning of Translating Polyhedral Robots

We present a new sampling-based algorithm for complete motion planning. Our algorithm relies on computing a star-shaped roadmap of the free space. We partition the free space into star-shaped regions such that a single point, called the guard, can see every point in the star-shaped region. The resulting set of guards capture the intra-region connectivity—the connectivity between points belonging to the same star-shaped region. We capture the inter-region connectivity by computing connectors that link guards of adjacent regions. The guards and connectors are combined to obtain the star-shaped roadmap. We present an efficient algorithm to compute the roadmap in a deterministic manner without explicit computation of the free space. We show that the star-shaped roadmap captures the connectivity of the free space, thereby enabling us to perform complete motion planning. Our approach is relatively simple to implement. We apply our approach to perform motion planning of robots with translational and rotational degrees of freedom (dof). We highlight its performance in challenging scenarios with narrow passages or when there is no collision-free path for robots with low degrees of freedom.

Efficient max-norm distance computation and reliable voxelization

We present techniques to efficiently compute the distance under max-norm between a point and a wide class of geometric primitives. We formulate the distance computation as an optimization problem and use this framework to design efficient algorithms for convex polytopes, algebraic primitives and triangulated models. We extend them to handle large models using bounding volume hierarchies, and use rasterization hardware followed by local refinement for higher-order primitives. We use the max-norm distance computation algorithm to design a reliable voxel-intersection test to determine whether the surface of a primitive intersects a voxel. We use this test to perform reliable voxelization of solids and generate adaptive distance fields that provides a Hausdorff distance guarantee between the boundary of the original primitives and the reconstructed surface.