GVU Technical Report Series

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GVU Technical Report Series

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Now showing 1 - 10 of 65

SQUINT Fields, Maps, Patterns, and Lattices

(Georgia Institute of Technology, 2018-07-23) Rossignac, Jarek

The proposed Steady QUad INTerpolating (SQUINT) map is formulated in terms of a SQUINT Field of Similarities (FoS). It is controlled by four coplanar points. It maps the unit square onto a curved planar quad, R, which has these points as corners. Uniformly spaced, log-spiral isocurves decompose R into tiles that are similar to each other and, hence, each have equal angles at opposite corners. We provide closed-form expressions for computing the representation of the SQUINT map and for evaluating the map and its inverse. We discuss extensions and potential applications to texture maps and field warps and to the design, display, and constant-cost query of procedural models of arbitrarily complex lattices.
Permutation Classifier

(Georgia Institute of Technology, 2018-04-24) Zhou, Xinrui ; Guerra, Concettina ; Rossignac, Jarek ; Rossignac-Milon, Leo

We consider permutations of a given set of n different symbols. We are given two unordered training sets, T1 and T2, of such permutations that are each assumed to contain examples of permutations of the corresponding type, t1 and t2. Our goal is to train a classifier, C(q), by computing a statistical model from T1 and T2, which, when given a candidate permutation, q, decides whether q is of type t1 or t2. We discuss two versions of this problem. The ranking version focuses on the order of the symbols. Our Separation Average Distance Matrix (SADiM) solution expands on previously proposed ranking aggregation formulations. The grouping version focuses on contiguity of symbols and hierarchical grouping. We propose and compare two solutions: (1) The Population Augmentation Ratio (PAR) solution computes a PQ-tree for each training set and uses a novel measure of distance between these and q that is based on ratios of population counts (i.e., of numbers of permutations explained by specific PQ-trees). (2) The Difference of Positions (DoP) solution is computationally less expensive than PAR and is independent of the absolute population counts. Although DoP does not have the simple statistical grounding of PAR, our experiments show that it is consistently effective.
Optimized Blist Form (OBF)

(Georgia Institute of Technology, 2007-05-23) Rossignac, Jarek

Any Boolean expressions may be converted into positive-form, which has only union and intersection operators. Let E be a positive-form expression with n literals. Assume that the truth-values of the literals are read one at a time. The numbers s(n) of steps (operations) and b(n) of working memory bits (footprint) needed to evaluate E depend on E and on the evaluation technique. A recursive evaluation performs s(n)=n–1 steps but requires b(n)=log(n)+1 bits. Evaluating the disjunctive form of E uses only b(n)=2 bits, but may lead to an exponential growth of s(n). We propose a new Optimized Blist Form (OBF) that requires only s(n)=n steps and b(n)=⌈log2j⌉ bits, where j=⌈log2(2n/3+2)⌉. We provide a simple and linear cost algorithm for converting positive-form expressions to their OBF. We discuss three applications: (1) Direct CSG rendering, where a candidate surfel stored at a pixel is classified against an arbitrarily complex Boolean expression using a footprint of only 6 stencil bits; (2) the new Logic Matrix (LM), which evaluates any positive form logical expression of n literals in a single cycle and uses a matrix of at most n×j wire/line connections; and (3) the new Logic Pipe (LP), which uses n gates that are connected by a pipe of ⌈log2j⌉ lines and when receiving a staggered stream of input vectors produces a value of a logical expression at each cycle.
Multiple Object Selection in Pattern Hierarchies

(Georgia Institute of Technology, 2007) Jang, Justin ; Rossignac, Jarek

Hierarchies of patterns of features, of subassemblies, or of CSG sub-expressions are used in architectural and mechanical CAD to eliminate laborious repetitions from the design process. Yet, often the placement, shape, or even existence of a selection of the repeated occurrences in the pattern must be adjusted. The specification of a desired selection of occurrences in a hierarchy of patterns is often tedious (involving repetitive steps) or difficult (requiring interaction with an abstract representation of the hierarchy graph). The OCTOR system introduced here addresses these two drawbacks simultaneously, offering an effective and intuitive solution, which requires only two mouse-clicks to specify any one of a wide range of possible selections. It does not require expanding the graph or storing an explicit list of the selected occurrences and is simple to compute. It is hence well suited for a variety of CAD applications, including CSG, feature-based design, assembly mock-up, and animation. We discuss a novel representation of a selection, a technology that makes it possible to use only two mouse-clicks for each selection, and the persistence of these selections when the hierarchy of patterns is edited.
Ringing Js refinements

(Georgia Institute of Technology, 2007) Schaefer, Scott ; Rossignac, Jarek

Both the four-point and the uniform cubic B-spline refinement (i.e. subdivision) schemes double the number of vertices of a closed-loop polygonal curve jP and respectively produce sequences of vertices fk and bk. The Js refinement proposed here produces vertices vk=(1-s)fk+sbk. Iterative applications of Js yield a family of curves parameterized by s. It includes the four-point curve (J0), the uniform cubic B-spline (J8/8), and the quintic B-spline (J12/8). Iterating Js converges to a C2 curve for 0[less than]s[less than]1, to a C3 curve for 1[less than or equal to]s[less than]3/2, and to a C4 curve for s=S3/2. J3/8 tends to reduce the error between consecutive refinements and is useful to reduce popping when switching levels-of-detail in multi-resolution rendering. J4/8 produces the Jarek curve, which, in 2D, encloses a surface area that is usually very close to the area enclosed by the original control polygon 0P. We propose model-dependent and model-independent optimizations for these parameter values. As other refinement schemes, the Js approach extends trivially to open curves, animations, and surfaces. To reduce memory requirements when evaluating the final refined curve, surface, or animation, we introduce a new evaluation technique, called Ringing. It requires a footprint of only 5 points per subdivision level for each curve and does not introduce any redundant calculations.
Simulation of Bubbles in Foam With The Volume Control Method

(Georgia Institute of Technology, 2007) Kim, Byungmoon ; Liu, Yingjie ; Llamas, Ignacio ; Jiao, Xiangmin ; Rossignac, Jarek

Liquid and gas interactions often produce bubbles that stay for a long time without bursting on the surface, making a dry foam structure. Such long lasting bubbles simulated by the level set method can suffer from a small but steady volume error that accumulates to a visible amount of volume change. We propose to address this problem by using the volume control method. We track the volume change of each connected region, and apply a carefully computed divergence that compensates undesired volume changes. To compute the divergence, we construct a mathematical model of the volume change, choose control strategies that regulate the modeled volume error, and establish methods to compute the control gains that provide robust and fast reduction of the volume error, and (if desired) the control of how the volume changes over time.
Pressing: Smooth Isosurfaces with Flats from Binary Grids

(Georgia Institute of Technology, 2007) Chica, Antonio ; Williams, Jason ; Andujar, Carlos ; Brunet, Pere ; Navazo, Isabel ; Rossignac, Jarek ; Vinacua, Alvar

We explore the automatic recovery of solids from their binary volumetric discretizations. In particular, we propose an approach, called Pressing, for smoothing isosurfaces extracted from binary volumes while recovering their large planar regions (flats). Pressing yields a surface that is guaranteed to contain the samples of the volume classified as interior and exclude those classified as exterior. It uses global optimization to identify flats and constrained bilaplacian smoothing to eliminate sharp features and high-frequencies from the rest of the isosurface. It recovers sharp edges between flat regions and between flat and smooth regions. Hence, the resulting isosurface is usually a very accurate approximation of the original solid. Furthermore, the segmentation of the isosurface into flat and curved faces and the sharp/smooth labelling of their edges may be valuable for shape recognition, simplification, compression, and various reverse engineering and manufacturing applications.
Simulation of Bubbles and Liquid Films

(Georgia Institute of Technology, 2006) Kim, Byungmoon ; Liu, Yingjie ; Llamas, Ignacio ; Rossignac, Jarek

Liquid and gas interactions often contain bubbles surrounded by thin liquid films. Simulation of these liquid films is challenging since they quickly become thinner than the grid resolution, which leads to premature bursting or merging of the bubbles. We prevent this thinning process by applying a disjoining force to the film, obtaining bubbles that last much longer without bursting or merging. The surface tension on the liquid film is the next diffuculty. Since the level set is not differentiable at the center of the thin liquid film, the curvature computed from the level set gradient is noisy, and the thin liquid film ruptures quickly. To prevent this, we compute the surface tension from the local isosurface, obtaining long-lasting liquid films. However, since bubbles stay longer without bursting or merging, the volume loss of each bubble is noticeable. To solve this problem, we modify the pressure projection to produce a velocity field whose divergence is controlled by the proportional and integral feedback. This allows us to preserve the volume or, if desired, to inflate or deflate the bubbles. In addition to premature bursting and volume change, another difficulty is the complicated liquid surface, which increases memory and computational costs. To reduce storage requirement, we collocate the velocity and pressure to simplify the octree mesh. To reduce the computational complexity of the pressure projection, we use a multigrid method.
Boundary of the Volume Swept By a Free-form Solid in Screw Motion

(Georgia Institute of Technology, 2006) Rossignac, Jarek ; Kim, J. J. ; Song, S. C. ; Suh, K. C. ; Joung, C. B.

The swept volume of a moving solid provides an excellent aid for path and accessibility planning in robotics and for simulating various manufacturing operations. To compute the boundary of the swept volume, we approximate the motion by a polyscrew (continuous, piecewise-screw), generate candidate faces, compute the two-cells of their arrangement, and use a new point-in-sweep test to select the correct cells whose union forms the boundary of the swept volume.
Advections with Significantly Reduced Dissipation and Diffusion

(Georgia Institute of Technology, 2006) Kim, Byungmoon ; Liu, Yingjie ; Llamas, Ignacio ; Rossignac, Jarek

Back and Forth Error Compensation and Correction (BFECC) can be applied to reduce dissipation and diffusion in advection steps, such as velocity, smoke density, and image advections. It can be implemented trivially as a small modification of the first-order upwind or semi-Lagrangian integration of advection equations. It provides second-order accuracy in both space and time and reduces volume loss significantly. We demonstrate its benefits on the simulation of smoke, bubbles, and interaction between water, a solid, and air.