Polygonization of Implicit Surfaces

1

• Polygonization of Implicit Surfaces
• Kenneth E. Hoff III
• Computational Geometry Presentation
• Sept 98

2
Overview

• Define implicit surfaces
• Why polygonize?
• Why are they more difficult to render than
  explicit surfaces?
• Why use them?
• General framework for polygonization
• 1D implicit surfaces (pts) root-finding
• 2D implicit surfaces (contour lines)
• Potential Problems
• Application of framework to 3D

3
What is an implicit surface?

• The set of points X that satisfies the implicit
  equation f(X)0
• Points that satisfy the property defined by the
  implicit function f
• sphere pts at a specific distance from a point
• map height contour pts at a specific height on
  a topographical map
• Also called iso-surfaces or contour surfaces
• The sign of the implicit function indicates
  whether a point is inside or outside the surface
  (we use to indicate inside)

4
Polygonization Explicit vs. Implicit Definitions

• Why polygonize?
• to obtain an efficient representation for
  rendering
• Explicit functions
• Simple, direct polygonization
• Parametric surfaces
• Vector-valued function so output is a surface
  point
• Just iterate through domain points to tesselate
  the surface
• Implicit functions
• More difficult to polygonize
• Reduces to solving the implicit equation -
  multiple-root finding
• Scalar-valued function so just gives a single
  value for any point in space.
• Zero-set (or root) defines the surface

5
Implicit Surfaces are difficult to render. Why
use them?

• Simpler collision queries
• Simpler CSG operations (union, difference,
  intersection)
• Allows simple blending between primitive surfaces
• only changes f (polygonization method need not be
  modified)
• Easy to apply free-form deformations
• only transforms input point to f (polygonization
  method need not be modified)
• have to invert the deformation M f(M-1x)0
• analytically deformed
• Blobby models, MetaBalls, etc.

6
1D Implicit Surfaces 1D root-finding

• Given a function f, we wish to find the set of x
  values (1D points) that satisfy f(x)0
• From calculus, the Intermediate Value Theorem
  states as x varies from a to b, the continuous
  function f takes on every value between f(a) and
  f(b)
• If f(a) and f(b) have opposite signs, then the
  root is said to be bracketed in the interval
  a,b
• BUT, how did we find a and b?

7
1D How to bracket the root(s)

• Uniformly subdivide 1D point domain into cells.
• Evaluate f for each cell boundary point
• Cells enclosed by two grid pts that evaluate to
  opposite signs contain at least one root
• Is a non-uniform sampling better? Maybe, but more
  difficult in higher dimensions.
• How much of the domain do we subdivide?

8
1D Finding the point domain subdivision
boundaries

• Small cells may bracket a root, but how much of
  the domain must be subdivided?
• Already know the extents of the implicit surface
  defined by f
• Conservatively guess the boundaries - inefficient
  and unreliable
• Use method that does not require domain
  subdivision

9
1D Finding the bracketed root

• If cell size is small enough, perhaps either
  boundary value may be close enough to the root
• OR use bisection search depends on the
  intermediate value theorem
• Recursively subdivide intervals bounded by grid
  pts evaluating to opposite signs
• Continue until cell is below some threshold size
• Boundaries or center value is considered close
  enough to the root of f
• Why not start the uniform subdivision at this
  threshold size? Excessive evaluation of grid pts.

10
2D Implicit Surfaces Finding the boundary
curve

• Given a surface defined by f(X)0 where X (x,y)
• Bracket roots by subdividing 2D point domain into
  rectangular cells
• Again, how much of the space should be subdivided
• Evaluate f at all grid pts
• Roots are bracketed in cells that have grid pt
  evaluations of opposite sign
• The surface passes through cells containing
  bracketed roots

11
2D Find the surface passing through a
bracketing cell

• Similar to 1D, if cell subdivision is considered
  small enough, simply find where the surface
  intersects the edges of the cell and connect the
  intersection points with a line.
• Only search for crossings along edges whose
  endpoints evaluate to opposite signs
• OR perform adaptive subdivision

12
2D Find the surface edge crossing

• Given an edge whose endpoints evaluate to
  opposite signs
• Use bisection search along the edge to find the
  intersection point

13
2D Adaptive-subdivision

• Recursively subdivide cells containing a surface
  crossing down to a threshold cell size
• Find surface crossings through smallest cells
  bisection search of crossed edges, connect
  intersections
• Since the adaptive subdivision is converged to a
  threshold cell size for all areas containing the
  boundary, adjacent cells containing the surface
  are all the same size - no cracks and no
  adjacency info is required just dump the line
  segments!
• Again, we perform adaptive subdivision rather
  than uniformly subdivide down to the threshold
  size to avoid excessive evaluations of the
  implicit function
• Can we do better? Evaluate only for the smallest,
  surface-crossing cells.

14
2D Continuation methods

• Assume the the domain space is uniformly
  subdivided into cells at the threshold size, but
  do not perform any function evaluations
• Find a single starting seed point that lies on
  the surface (use ray-casting, etc)
• Find cell containing the seed point
• Grow the set of cells across the surface
• Evaluate adjacent grid pts to find next surface
  crossing cells (or use constrained dynamics)
• Do we even need a uniform subdivision (domain
  mesh)?

15
2D Using particle systems to avoid meshing

• Relaxation (Turk91)
• Distribute pts fairly evenly over surface
• Repeatedly iterate over all pts maintaining a
  certain neighbor distance
• Adaptive Repulsion (Witkin94)
• Find any pt on the surface (seed pt)
• Give pt a large sphere of influence
• Pts with spheres of influence over a threshold
  size are split - fissioning
• Split pts have smaller spheres of influence
• Pts within another pts sphere of influence are
  repelled across surface
• Pts with sphere of influence below another
  threshold size are destroyed

16
Potential Problems

• Bounding the Domain Space
• Discretization Error
• Disconnected Components
• Ambiguous Surface-Crossing Cells

17
Potential Problem 1 Bounding the Domain Space

• May not completely contain the object
• May miss disconnected components
• Will result in clipping

18
Potential Problem 2 Discretization Error

• Too large uniform cell size
• May not be able to converge (entirely miss the
  surface)
• Too large adaptive threshold cell size
• Misses small, completely-contained features
• Coarse resolution model
• Incorrect topology
• ambiguous cells
• connects or breaks components
• Too small cells sizes are inefficient and more
  susceptible to numerical error
• Particle systems are an attempt to avoid this
  problem - no meshing!

19
Potential Problem 3 Disconnected Components

• Uniform cell subdivision may find them, but
  inefficiently
• Particle system and continuation methods may miss
  them
• must have a seed pt on each component
• Related to discretization error
• Sometimes results from ambiguous cell polarity
  configurations

20
Potential Problem 4 Ambiguous Surface-Crossing
Cells

• Many cell vertex polarity configurations are
  ambiguous
• Possible to polygonize in different arrangements
• May even result in disjoint polygons
• Solutions
• Detect and recursively subdivide
• Choose a consistent convention (e.g. always join
  positive pts)
• Use simplices (triangulate)
• Also, related to discretization error

21
Review before going to 3D. . .

• Method 1 Uniform Spatial Subdivision
• Uniformly subdivide domain space into cells
• Evaluate implicit function at grid pts to find
  surface crossing cells
• Polygonize surface crossing cells
• Method 2 Adaptive Spatial Subdivision
• Start with Method 1
• Adaptively subdivide surface crossing cells to
  threshold size
• Polygonize surface crossing leaf cells
• Method 3 Continuation
• Find any pt on the surface (seed pt)
• Find cell containing seed pt at threshold size
• Grow surface crossing cells in directions
  determined by crossed cell boundaries
• Method 4 Particle Systems
• Distribute pts evenly and perform relaxation
  iterations
• OR perform adaptive repulsion and fissioning

22
3D Implicit Surface Polygonization using Spatial
Subdivision

• Conceptually identical to 1D and 2D version
• Given a surface defined by f(X)0 where X
  (x,y,z)
• Subdivide 3D space into uniform cells
• Evaluate implicit function for all grid points
• Find surface-crossing cells (check polarity of
  cell vertices)
• Subdivide surface-crossing cells to threshold
  size
• Search for surface-crossing along edges of
  opposite polarity
• Polygonize the cell based on the edge
  intersections - much more difficult in 3D

23
3D Polygonizing a Surface-Crossing Cell

• The most difficult part. . .
• Finding cell polygon vertices
• Algorithmic method
• Table lookup (based on vertex polarities)
• Resolving ambiguities
• Triangulating the cell polygon

24
3D Finding cell polygon face vertices

• Algorithmic method
• Begin with any edge-surface intersection point
  (1)
• Proceed to negative corner (white open pts) and
  then clockwise about the cube face (w.r.t
  outside) until another intersection point is
  found (2)
• Repeat for each subsequent face (3?4, 4?5, 5?1)

25
3D Finding cell polygon face vertices

• Table Lookup
• Cube table lookup 8 verts, 256 entries
• Tetrahedron table lookup 4 verts, 16 entries
• Requires tetrahedral decomposition of the cube
• Tetrahedral subdivision is more difficult

26
3D Finding cell polygon face vertices

• Tetrahedral decomposition of the cube
• Different ways to decompose 5, 6, or 12
  tetrahedra
• Adjacency problems 5 must alternate, 6 and 12 is
  ok, but more expensive
• Even 24 is possible to avoid bias

27
3D Finding cell polygon face vertices

• Resolving ambiguities
• many ambiguous polarity configurations
• resolve by subdividing, using tetrahedral
  decomposition, or by being consistent (e.g.
  always join positive pts)

28
3D Triangulating the cell polygon

• non-coplanar
• preserve aspect ratio
• may insert new points

29
3D Continuation Methods

• Same as in 2D
• Start with a seed pt and its containing cell
• Visit adjacent cells through surface crossing
  cube faces (faces that have opposite vertex
  polarities)
• Grow until surface is covered
• Polygonize each cell

30
3D Particle Systems

• Similar to 2D
• Use relaxation if we have a uniform distribution
• more expensive starting conditions
• Use adaptive repulsion if we wish to grow
  across the surface
• starts with a single point
• better for interactive applications

31
ASIDE Ray Intersection with Implicit Surface

• Particle system methods require ability to push
  particles around on the surface
• Usually a random direction is chosen in the
  tangent plane
• Requires gradient of implicit function
• Jumping in random direction causes new position
  to be off of the surface
• We know the gradient at the new position, so we
  have a ray (start is at new pt, dir is along
  gradient towards the surface)
• RAY CASTING!
• Ray defined by r(t) S Dt where S is ray
  start, D is ray direction, t is step in units of
  length of D along ray
• Implicit equation is defined as f(X)0 where X is
  a point on the surface if f(X)0
• Point along ray is defined by r(t), so substitute
  into f(X) f(r(t))f(SDt)0
• This reduces to the 1D implicit equation f(t)0
• Solution(s) t of this equation correspond to the
  distance along the ray to a point that lies on
  the surface. . .

32
ASIDE Ray Intersection with Implicit Surface

• We need only find the smallest positive root t of
  the 1D implicit equation f(t)0
• We have the gradient vector-valued function
  fx(X), fy(X), that gives the direction (in
  the domain) of maximum change in value of the
  implicit function for a particular point (in the
  domain)
• The domain of f(t) is 0,? since we are only
  concerned with intersections in front of the ray
  start pos
• We can try a variety of methods to find the root
  uniform step size starting at t0, adaptive step
  size, Newton iterations (use directional
  derivative)
• To use Newtons methods, we need the derivative
  at particular t values f(t), but we only have
  the gradient for pts in the implicit function
  domain use the directional derivative in the ray
  direction to get f(t)
• Given a t value for which we want f(t)
• Find pt in implicit function domain r(t)SDt
• Evaluate gradient at r(t) G(t) fx(r(t)),
  fy(r(t)),
• f(t) directional derivative in ray dir w.r.t.
  G(t) D?G(t)

33
References

• Bloomenthal, Jules. Introduction to Implicit
  Surfaces. Morgan Kaufmann Publishers, Inc. San
  Francisco, CA. 1997.
• Bloomenthal, Jules. Polygonization of Implicit
  Surfaces. Computer Aided Geometric Design, Nov.
  1988, vol 5, p 341-355.
• Witken, Andrew P. and Paul S. Heckbert. Using
  Particles to Sample and Control Implicit
  Surfaces. SIGGRAPH 94, 1994, p 269-277.
• Bloomenthal, Jules. Interactive Techniques for
  Implicit Modeling. Computer Graphics 24, 2 (Mar.
  1990), p 109-116.
• Sclaroff, Stan and Alex Pentland. Generalized
  Implicit Functions for Computer Graphics.
  SIGGRAPH 91, July 1991, p 247-250.
• Stander Barton T. and John C. Hart. Guaranteeing
  the Topology of Implicit Surface Polygonization
  for Interactive Modeling. SIGGRAPH 97, 1997, p
  279-286.
• Lorenson, William E. and Harvey E. Cline.
  Marching Cubes A High Resolution 3D Surface
  Construction Algorithm. SIGGRAPH 87, July 1987,
  p 163-169.
• Ning, Paul and Jules Bloomenthal. An Evaluation
  of Implicit Surface Tilers. IEEE Computer
  Graphics and Applications. November 1993, p
  33-41.
• Turk, Greg. Generating Textures on Arbitrary
  Surfaces Using Reaction-Diffusion. SIGGRAPH 91,
  July 1991, p 289-298.
• Szeliski, Richard and David Tonnesen. Surface
  Modeling with Oriented Particle Systems. SIGGRAPH
  92, July 1992, p 185-194.