Applications of Surface Networks to Sampling Problems in Computer Graphics

Boundary volumes are arranged in a hierarchy based on the hierarchy of the surface network. We develop and study various partitioning mechanisms to control the sampling process for parametric functions.

Iconic Index of Figures

LJWLO

A triangle can be represented as a circular list of edges around the perimeter of the polygon.

Illustration of the merging process for two triangles . . . . . . . . . . . . . .

Nomenclature

Mij 2: mjx I Jij I ,

This representation allows the rapid integration of intensities within each isothetic rectangle in the image, and is useful for filtering texture maps.

NOMENCLATURE XXIX

Chapter 1 Introduction

Motivation
Surface Networks
Bounding Volumes for Parametric Functions
BOUNDING VOLUMES 3
Original Results
SUMMARY 5
Summary

Most standard techniques in computer graphics ignore the issue of the accuracy of surface approximations. The Lipschitz condition places an upper bound on the parametric rate of change of the function.

Figure 1.1: Parametric spike functions can be made arbitrarily sharp, such that their detection is extremely difficult

Chapter 2

Accurate Sampling of Deformed, Intersecting Surfaces t

Overview
Introduction

Motivation for Studying Surface Sampling Tech- n1ques
Background

INTRODUCTION 9
ACCURATE SAMPLING

Bounding Volumes for Parametric Surfaces
A Recursive Subdivision Mechanism
SUBDIVISION MECHANISM 17

Bicubic patches were rendered using recursive partitioning techniques in the parametric surface space ([Catmull 75]. A quadtree consists of pointers to regions arranged in a hierarchy that tessellate the parametric space of the surface.

Figure 2.1: Uniformly sampled chain, 60,000 polygons, sampling time 53 minutes

2.4. 7 Triangle Clipping

Recursive Subdivision Criteria
SUBDIVISION CRITERIA 23

Intersection Subdivision

Silhouette Boundaries
Proximity Subdivision
Efficient Combination of Subdivision Criteria

Imaging Results
IMAGING RESULTS 29

Puzzle
Bicycle Chainwheel
Nut and Bolt

Summary
Chapter 3

For toroids, the squares on the north edge of the quadtree are neighbors of squares on the south edge of the quadtree. It is possible for all the normal vectors to point in the same direction, but the tangent vectors can point in different directions. Tangent curvature subdivision eliminates the problem (Figure 2.17), improving the robustness of the curvature subdivision measure.

The flatness test of the boundary curve over the range uses the approximate tangent vectors of the boundary curve. Splitting stops when the curvature of the silhouette boundary in screen space is less than a threshold value or when the area is less than a pixel. The surface is divided until an intersection or local minimum is found in the inside-outside functions of other surfaces.

Some of the criteria use vision-dependent tests, since a vision transformation may be available at the time of sampling. The puzzle is modeled with six identical pieces that fill the volume of the interior of the puzzle.

Figure 2.15: Quadtree sampling with clipping at surface intersections.

Subdivision Mechanisms for

Adaptive Parametric Sampling

Uniform Sampling
Unrestricted Quadtrees
RESTRICTED QUADTREES 35
Restricted Quadtrees
Advantages of Triangular Surface Elements
ADVANTAGES OF TRIANGLES 37
Bintrees of Right Triangles
BINTREES OF RIGHT TRIANGLES 39
Planar Subdivisions with General Triangles
GENERAL TRIANGULAR SUBDIVISION 41

Triangular Subdivision with a Scaled Length Metric
Adjusting the Aspect Ratio of Triangular Subdivi-
Reorienting Triangular Subdivisions

It is difficult to construct a triangular subdivision from this configuration that avoids surface cracks, avoids too much oversampling across the square, and preserves the aspect ratio of the resulting triangles. If we use parametric quadrilaterals, we have no assurance that the image of the quadrilateral in the model space will remain convex. One way to avoid cracks in a surface is to form a planar subdivision of the parametric space.

A triangular subdivision of the plane can be constructed based on the limited-area quadtree (Figure 2.13). The length of each edge serves as the overall ordering function for the bintrees of right triangles, thus terminating the recursion. A simple extension of the right triangular subdivision algorithm is to use an edge length definition that is stretched in one direction, such as.

The statue silhouette of Figure 3.4 benefits from finer sampling in the u-direction in Figure 3.9, due to the vertical alignment of the edges. The technique in Figure 3.11 is slightly more successful at sampling the boundary of the figure.

Figure 3.1: An unrestricted quadtree of a circular region showing squares with a large number of neighbors

3. 7 General Tetrahedral Subdivision for Para- metric Functions of Three Variables

Non-planar Surface Elements

An interesting potential future extension of this work would be to consider using non-planar surface elements instead of triangles to approximate parametric surfaces. The problem with this approach is that a polygon is the most complex primitive that many rendering systems render directly. One option is to consider a method that generates simple curved patches and then generate triangles from the curved patches.

These are uneven triangular patches without turning points in their interior (Figure 3.13). Given three arbitrary corner points, a set of four Steiner patches can be sufficient to ensure continuity of C1 across the surface. Since Steiner patches are triangular, it is attractive to use bindrees of right triangles to create surface patterns.

Another approach is the use of biquadratic surface elements to model surface curvature as well as orientation ([Barr 86, p. 294]). Another possible approach to using nonplanar surface elements comes from image analysis research in the segmentation of an image into smooth subelements ([Besl and Jain 88]).

SUMMARY 45

Summary

Collision Determination for Parametric Surfaces

Introduction

Problem Statement
Problems with Arbitrary Surfaces

INTRODUCTION 49

Solution for Surfaces with Lipschitz Conditions

PREVIOUS WORK 51
Previous Work
Finding Surface Intersections for Station- ary Parametric Surfaces

k-d Trees in Parametric Space

SURFACE INTERSECTIONS 53

Spherical Bounds in Modeling Space for Parametric Subregions
Intersection Computation

Intersection Algorithm in Common Lisp

Extension to Variable L Values over a Surface
Sphere Example

Bounding Volumes for Moving Parametric Surfaces

Bounding Spheres Derived from the Rate Condition

MOVING PARAMETRIC SURFACES 59

Bounding Volumes Based on the Jacobian of the Parametric Function

Finding a narrow peak becomes arbitrarily difficult as the parametric width of the peak approaches zero. Knowing the maximum speed of two surfaces, we can find the first collision of the surfaces. It is possible to develop a similar constraint on the temporal aspects of the collision determination problem.

In this case, L sets an upper limit for the parametric surface velocity as well as for other parametric leads. We can automatically find the lower bound of the separation distance between objects based on the upper bounds of the parametric derivatives of the function. The aspect ratio of parametric fields can be controlled by splitting the fields multiple times in the same direction.

Compare the sum of the two radii to the distance between the points in the pattern space. Near the polar regions of the parametric sphere, the bounding volumes become significantly smaller for the functional determination of Lu(R).

Figure 4.1: The collision-determination problem for arbitrary surfaces: to find the time and location of the first collision between two moving parametric surfaces

COLLISION ALGORITHM 61
Algorithm for Collision Determination

Collision Algorithm Approach
Common Lisp Implementation

Termination Condition
Complexity Analysis for Colliding Spheres

COLLISION ALGORITHM

Experimental Results for Colliding Spheres
Experimental Results for Other Objects

PARAMETRIC COLLISIONS

Determining Constraints on the Jacobian of a Parametric Function
JACOBIAN DETERMINATION 69
JACOBIAN DETERMINATION 71

Chapter 5

This section deals with calculating collisions from the bounding box information of the previous section. The task is to calculate whether two objects collide, as determined by the separation loss of the two parametric surfaces. For the parametric k-d tree hierarchy, each halving of the partition distance requires a constant number of additional levels of subdivision.

The total computation time is a function of the minimum separation distance between the two objects. With this information, we can solve difficult collision problems using a simple application of the collision algorithm of Figure 4.7. 4.6 Determination of constraints on the Jacobian of a Parametric Function A Parametric Function. For the collision technique to be useful, we need to determine constraints on the Jacobian of the parametric functions.

We can take the process a step further and simply calculate the maxima for each component of the Jacobian over the entire surface. Perhaps the most general and flexible way to compute constraints on the Jacobi matrix is to create a special function that computes the maxima of the parametric derivatives of each parametric function.

Figure 4.7: An algorithm and data structure for testing for the collision of two para- para-metric surfaces moving as a function of time

Applications of Adaptive

Sampling with Surface Networks

Triangulation of Texture Maps

Conversion of Texture Maps into Polygon Tilings

TEXTURE TRIANGULATION 75

Previous Work
Subdivision Mechanism Using Right Triangular Sub- divisions of the Plane

Subdivision Criterion
Merging Step

Polygons Represented as Circular Lists

The right image shows a triangulation of the same texture map using the tails of right triangles. A particular representation is chosen based on the desired spatial frequency components of the map. A third approach is discussed in [Besl and Jain 88], in which the local Gaussian and the mean curvature of the intensity values of the texture map are computed at each pixel.

Since each polygon is shaded linearly over its surface, this limits the spatial frequencies of the texture map. The middle figure demonstrates the splitting process, where subdivision is only allowed by splitting the hypotenuse of the triangles. We choose a representation for polygons as circular lists of edges around the perimeter of the polygon to facilitate the merging operation.

Once we've simplified the expression, we have a new list of edges that make up the perimeter of the joined polygon. As a final step, the redundant elements of the path are removed, in the figure on the right, leaving path (A,B,D,E).

Figure 5.1: A texture map of a playing card. The left image shows the original texture map of the jack of hearts

5.1. 7 Imaging Results and Applications

Potential E-Collision Applications

Robotic Path Verification
Collision Prediction for Aircraft

We review some potential applications of collision theory in robotics and air traffic control. The problem can be posed as follows: Given a parametric description of a robot, its motion as a function of time, and a dynamic surrounding environment, determine whether the robot collides with its environment. The robot and the environment can consist of a large set of parametric surfaces.

Recent articles have spoken of the possibility of grid-lock in the nation's air transportation system ([New York Times, June due to aging airport facilities and the vast expansion of air traffic over the past decade. Air transportation currently has a $57) billion market per year, or 1.5 percent of the gross national product of the United States. Airlines also provide 92 percent of public transportation between cities in the United States.

One of the major problems with increased air traffic is the coordination of arriving and departing traffic in large metropolitan areas. In general, the separation diameter of the volume is a function of the airspeed and altitude of the aircraft.

Figure 5.12: Path of a robot navigating a cluttered environment. We must determine if the robot will collide with any objects

E-COLLISION APPLICATIONS 87

Appendix A

Theorems for E-Collisions

I Finding E-Collisions

FINDING (,-COLLISIONS 91
Making Bounding Boxes
MAKING BOUNDING BOXES 93
Finding E-Collisions Efficiently
Termination of the E-Collision Algorithm
INSOLUBLE COLLISIONS 95
Example of an Insoluble Collision Without the Lipschitz Condition

In other words, when the central time difference of the two fields exceeds the half duration of the two fields, there can be no time overlap. By sorting the E-collisions according to max(tcf - /j,tf, tc9 -/j,t9) we can find the earliest possible collision time and the location of the two surfaces. This determines the parametric dimension that contributes most to the size of the bounding box in the modeling space.

So it is sufficient to divide until the sum of the radii of the bounding box is less than 8/2. The recursive procedure divides the maximum of Du, Dv, Dt in half, to obtain a smaller bounding box. If the terms differ by more than a factor of two, the sum is reduced by more than a factor of two.

This ensures that the set of fields with width less than b will be finite; and the rank matrix M ensures that each bounding box will bound its own parametric subregion. In this way, it is possible to hierarchically inspect the subregions of the two surfaces and quickly determine whether an l-collision occurs.

Figure A.1: Partitioning of the €-collision theorem into three cases.

Appendix B

Derivations for Adaptive Sampling

Derivation of Upper Bounds on Parametric Derivatives for a Sphere
Derivation of the Jacobian Matrix for a Parametric Sphere
NONDIFFERENTIABLE SURFACES 99
Non-differentiable Surfaces with Lipschitz Constants
Binary Right Triangles vs. Restricted Quad- trees

For this example, the bounded quadtree requires exactly 3.5 times as many triangles as the bintree of right triangles in the case of infinite recursion.

Figure B.1: Plot of f(u) = limh_,ohlu/hj, for a finite value of h.

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Lane in Carpenter 79] Lane, Jeff in Loren Carpenter, "A Generalized Scan Line Algorithm for the Computer Display of Parametrically Defined Surfaces," Computer Graphics and Image Processing. Riesenfeld, "Teoretični razvoj za računalniško generiranje in prikaz kosovnih polinomskih površin," IEEE Transactions on Pattern Analysis and Machine Intelligence 2, l, januar. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences, Macmillan Publishing Co., Inc., New York, 1974, str.

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104 REFERENCES