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Polygonization of Implicit Surfaces

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Recursively subdivide cells containing a surface crossing down to a threshold cell size ... pts to find next surface crossing cells (or use constrained dynamics) ... – PowerPoint PPT presentation

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Title: 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.
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