Advanced Computer Graphics Computer Animation

1
Advanced Computer GraphicsComputer Animation

• Implicit Surfaces
• Spring 2002
• Professor Brogan

Many slides from Brian Wyvills online materials
at U. Calgary
2
Papers for Tuesday

• Spacetime Constraints, Witkin and Kass
• Siggraph
• Deep-water Animation and Rendering
• Gamasutra.com, Sept 26, 2001

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Implicit Surfaces

• Surfaces defined by points that satisfy
• f(P) 0 Implicit Function
• Example, a circle
• Parametric
• xr cos(a)
• yr sin(a)
• Implicit
• x2 y2 r2 0

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Implicit Surface Modeling

• Useful for modeling natural and smooth/organic
  synthetic phenomena
• Living forms, liquids, clouds
• Each primitive is represented by a skeletal
  element which contributes in defining a scalar
  field
• Every point in space is assigned a scalar value
  equal to shortest distance to a skeletal element

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Implicit Surface Modeling

• Simplest skeletal element is a point
• A (distance) contour of that point defines the
  (surface of the) model
• Ex two points approach and their contours blend

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Combining Skeletal Primitives
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Blending Skeletal Elements

• Define a simple surface as
• A central point, C
• A radius of influence, R
• A density function, f()
• A threshold value, T
• All points, P, for which dist (P, C) lt R
• Implicit surface f (dist(P, C)) T 0

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Blending Skeletal Elements

• Example metaballs
• f(dist)
• Surface drawn where f(dist) T 0

Surface drawnat this radius
Value of T
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Blending Skeletal Elements

• Example two metaballs
• Surface drawn where
• f(dist1) f(dist2) T 0

Do we drawsurface here?
r1
r2
r2
r1
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Blending Skeletal Elements

• Yes

http//www.lifl.fr/triquet/implicit/video/blend.m
pg
11
Blending Skeletal Elements

• In general, surface defined by multiple surface
  elements is

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Blending Lines

• Lines can bulge when their ends meet
• Usually not the desired effect
• Tinkering with density function and line
  separation can fix

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Versions of Density Functions


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Modeling a Dinosaur (Wyvill)
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Modeling Blood
http//www.lifl.fr/triquet/implicit/video/blood.m
pg
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Solving Implicit Functions

• 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)

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Solving Implicit Functions

• If f(a) and f(b) have opposite signs, the root is
  said to be bracketed in the interval a, b

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Bracketing the roots

• Uniformly subdividespace
• Evaluate f( ) for eachboundary
• Transition from to defines bracket

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Bisection search

• Iteratively subdivide to find exact zero

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2-D Implicit Surfaces

• Surface defined by f(X) 0 X (x, y)
• Uniformly subdivide
• Evaluate f( ) at all points
• Bracket roots

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2-D Implicit Surfaces

• Find intersection point along all grid lines with
  /- values at endpoints
• Edge defined by crossing points

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2-D Adaptive Subdivision

• Recursively subdivide cells containing a surface
  crossing down to a threshold size
• Calculate crossing values and connect dots to
  form polygon edge

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2-D Continuation Methods

• Overlay fine-grain uniform subdivision
• Find start (seed) point that lies on surface
• Find cell containing seed point
• Grow the set of cells across surface
• Evaluate adjacent cells to find next surface
  crossing cells

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Potential Problem 1 Bounding the Domain Space

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

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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-containedfeatures
• Coarse resolution model
• Incorrect topology
• ambiguous cells
• connects or breaks components
• Too small cells sizes are inefficient and more
  susceptible to numerical error

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Potential Problem 3 Disconnected Components

• Uniform cell subdivision may find them, but
  inefficiently
• Continuation methods may miss them
• must have a seed pt on each component
• Related to discretization error

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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)

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Moving on to 3-D

• Conceptually the same
• Discretize space and find crossing points on 12
  edges of cube
• This can be tricky to connect the crossing points
  into a polygon correctly

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3-D 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)

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3-D Tetrahedral Decomposition

• Tetrahedra reduce ambiguity and produce correct
  meshes, but many more polygons result
• Diagonal edges cut across cube faces
• Must make sure adjacent cubes have aligned
  tetrahedron edges to preserve topological
  correctness

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3-D Tetrahedral Decomposition

• Five
  Six

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3-D Implicit Surface Evaluation

• Particle Methods
• Turk, Hoppe, Szeliski, Witkin, Heckbert (cool
  paper)
• Constrain particles to exist on isosurface
• Allow particles to split and join
• Particle seeks to balance space between neighbors
• A constrained optimization problem

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Collision Detection

• Easy to detect colliding blobbies
• Evaluate sample points from one object in the
  implicit equation of another
• Deforming the blobbies is more complex
• Define collision boundary, F1(p) F2(p)
• Create addition term, G(p) that is added to
  density function, F(p) to prevent passing through
  collision boundary
• Add terms to G(p) so blobbies preserve volume as
  they are retarded by collision boundary

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Spacetime Constraints

• Find min of f(x) y
• Find zeros (roots) of derivative function, f(x)
• Newton Raphson Method does this
• Find min of f(X) Y X and Y vectors
• Same method, but find where Jacobian is equal to
  zero

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Spacetime Constraints

• Null space of a function
• The inputs of a function for which the function
  evaluates to zero
• Finite differences
• Approximate derivatives (inverse of Euler
  integration)
• Solving for all forces in time simultaneously