Implicit Surfaces Thomas Funkhouser Princeton University C0S 426, - - PowerPoint PPT Presentation

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Implicit Surfaces Thomas Funkhouser Princeton University C0S 426, - - PowerPoint PPT Presentation

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Implicit Surfaces Thomas Funkhouser Princeton University C0S 426, Fall 1999 Curved Surface Representations What makes a good surface representation? Accurate Concise Intuitive specification Local support Affine invariant

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Implicit Surfaces

Thomas Funkhouser Princeton University C0S 426, Fall 1999

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Curved Surface Representations

  • What makes a good surface representation?

Accurate Concise Intuitive specification Local support Affine invariant Arbitrary topology Guaranteed continuity Natural parameterization Efficient display Efficient intersections

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Curved Surface Representations

  • Polygonal meshes
  • Parametric surfaces
  • Subdivision surfaces
  • Implicit surfaces
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Implicit Surfaces

  • Represent surface with function defined over all

space

Kazhdan

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Implicit Surfaces

  • Surface defined implicitly by function:

f (x, y, z) = 0 (on surface) f (x, y, z) < 0 (inside) f (x, y, z) > 0 (outside)

f(x,y) = 0 on curve f(x,y) < 0 inside f(x,y) > 0 outside

Turk

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Implicit Surfaces

  • Normals defined by partial derivatives

normal(x, y, z) = (df/dx, df/dy, df/dz) Normals Tangents Curvatures

Bloomenthal

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Implicit Surfaces

Bourke

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Implicit Surface Properties

(1) Efficient check for whether point is inside

Evaluate f(x,y,z) to see if point is inside/outside/on 1

2 2 2

= −

  • +
  • +
  • z

y x

r z r y r x

H&B Figure 10.10

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Implicit Surface Properties

(2) Efficient surface intersections

Substitute to find intersections

Ray: P = P0 + tV Sphere: |P - O|2 - r 2 = 0 Substituting for P, we get: |P0 + tV - O|2 - r 2 = 0 Solve quadratic equation: at2 + bt + c = 0 where: a = 1 b = 2 V • (P0 - O) c = |P0 - C|2 - r 2 = 0

P0

V

O P

r

P’

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Implicit Surface Properties

(3) Efficient boolean operations (CSG)

Union, difference, intersect Union Difference

Bloomenthal

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Implicit Surface Properties

(4) Efficient topology changes

Surface is not represented explicitly!

Bourke

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Implicit Surface Properties

(4) Efficient topology changes

Surface is not represented explicitly!

Bloomenthal

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Comparison to Parametric Surfaces

  • Implicit

Efficient intersections & topology changes

  • Parametric

Efficient “marching” along surface & rendering

Bloomenthal

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Implicit Surface Representations

  • How do we define implicit function?

Algebraics Blobby models Skeletons Procedural Samples Variational

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Implicit Surface Representations

  • How do we define implicit function?

Algebraics Blobby models Skeletons Procedural Samples Variational

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Algebraic Surfaces

  • Implicit function is polynomial

f(x,y,z)=axd+byd+czd+dxd-1y+dxd-1z +dyd-1x+...

H&B Figure 10.10

1

2 2 2

= −

  • +
  • +
  • z

y x

r z r y r x

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Algebraic Surfaces

  • Most common form: quadrics

f(x,y,z)=ax2+by2+cz2+2dxy+2eyz+2fxz+2gx+2hy+2jz+k

  • Examples

Sphere Ellipsoid Torus Paraboloid Hyperboloid

Menon

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Algebraic Surfaces

  • Higher degree algebraics

Cubic Quartic Degree six

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Algebraic Surfaces

  • Function extends to infinity

Must trim to get desired patch (this is difficult!)

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Algebraic Surfaces

  • Equivalent parametric surface

Tensor product patch of degree m and n curves yields algebraic function with degree 2mn Bicubic patch has degree 18!

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Algebraic Surfaces

  • Intersection

Intersection of degree m and n algebraic surfaces yields curve with degree mn Intersection of bicubic patches has degree 324!

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Implicit Surface Representations

  • How do we define implicit function?

Algebraics Blobby models Skeletons Procedural Samples Variational

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Blobby Models

  • Implicit function is sum of spherical basis functions
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Blobby Models

  • Sum of two blobs

Turk

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Blobby Models

  • Sum of four blobs

Turk

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Blobby Models

  • Blobby molecules
  • Meta balls
  • Soft objects

2

) (

br

ae r D

−

=

  • ≤

≤ ≤ − ≤ ≤ − = r b b r b b r a b r b r a r D 3 / ) 1 ( 2 3 3 / ) 3 1 ( ) (

2 2 2

  • ≥

≤ − + − = b r b r b r b r b r a r D 9 22 9 17 9 4 1 ( ) (

2 2 4 4 6 6

Bourke

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Blobby Model of Face

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Blobby Model of Face

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Blobby Model of Face

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Blobby Model of Head

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Blobby Model of Head

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Blobby Model of Head

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Blobby Models

Objects resulting from CSG of implicit soft objects and other primitives Menon

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Implicit Surface Representations

  • How do we define implicit function?

Algebraics Blobby models Skeletons Procedural Samples Variational

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→

Skeletons

  • Bulge problem
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Skeletons

  • Bulge problem

→

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Skeletons

  • Convolution surfaces

Bloomenthal

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Implicit Surface Representations

  • How do we define implicit function?

Algebraics Blobby models Skeletons Procedural Samples Variational

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Procedural Implicits

  • f(x,y,z) is result of procedure

Example: Mandelbrot set

H&B Figure 10.100

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Implicit Surface Representations

  • How do we define implicit function?

Algebraics Blobby models Skeletons Procedural Samples Variational

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Sampled Functions

  • Most common example: voxels

Interpolate samples stored on regular grid Isosurface at f(x,y,z) =0 defines surface

2.3 1.2 0.3 0.2 1.7 0.4

  • 0.5
  • 0.9

0.9 0.1

  • 0.7
  • 1.7

0.2

  • 0.8
  • 1.4
  • 2.5
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Sampled Functions

  • Acquired from simulations or scans

Visible Human

(National Library of Medicine)

Airflow Inside a Thunderstorm

(Bob Wilhelmson, University of Illinois at Urbana-Champaign)

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Implicit Surface Representations

  • How do we define implicit function?

Algebraics Blobby models Skeletons Procedural Samples Variational

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Variational Implicit Surfaces

Bloomenthal

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Variational Implicit Surfaces

Bloomenthal

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Implicit Surface Summary

  • Advantages:

Easy to test if point is on surface Easy to compute intersections/unions/differences Easy to handle topological changes

  • Disadvantages:

Indirect specification of surface Hard to describe sharp features Hard to enumerate points on surface » Slow rendering

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Rendering Implicit Surfaces

  • How do we render images of implicit surfaces?

Polygonization Ray tracing Contours Floating particles

Turk

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Rendering with Polygons

  • Polygonization is not always easy

Marching Cubes

Lorensen Bloomenthal

Adaptive Polygonalization

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Rendering with Ray Tracing

Turk

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Rendering with Contours

Bloomenthal

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Rendering with Floating Particles

Turk

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Summary

Accurate No Yes Yes Yes Concise No Yes Yes Yes Intuitive specification No No Yes No Local support Yes No Yes Yes Affine invariant Yes Yes Yes Yes Arbitrary topology Yes Yes No Yes Guaranteed continuity No Yes Yes Yes Natural parameterization No No Yes No Efficient display Yes No Yes Yes Efficient intersections No Yes No No Polygonal Mesh Implicit Surface Parametric Surface Subdivision Surface Feature