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2.1 Explicit & Implicit Surfaces Hao Li - - PowerPoint PPT Presentation

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Spring 2015 CSCI 599: Digital Geometry Processing 2.1 Explicit & Implicit Surfaces Hao Li http://cs599.hao-li.com 1 Administrative Exercise 1 discussion: Next Tuesday Hao Li (Instructor) Office Hour: Tue 2:00 PM - 4:00

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SLIDE 1

CSCI 599: Digital Geometry Processing

Hao Li

http://cs599.hao-li.com

1

Spring 2015

2.1 Explicit & Implicit Surfaces

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SLIDE 2

Administrative

2

  • Exercise 1 discussion: Next Tuesday
  • Hao Li (Instructor)
  • Office Hour: Tue 2:00 PM - 4:00 PM, SAL 244
  • Kyle Olzsew (TA)
  • Office Hour: Still TBD :-)
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SLIDE 3

Last Time

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Polygonal meshes are

  • Effective representations
  • Flexible
  • Efficient, simple, enables unified processing
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SLIDE 4

Last Time

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Connection between Meshes and Graphs

  • Formalism (valence, connections, subgraph, embedding…)
  • Definitions (boundary, regular edge, singular edge, closed mesh)
  • triangulation → triangle mesh

A B C D E F G H I J K

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SLIDE 5

Last Time

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Topology

  • Genus, Euler characteristic
  • Euler Poincaré formula
  • Average valence of triangle mesh: 6
  • Triangles: F = 2V, E = 3V
  • Quads: F = V, E = 2V

V − E + F = 2(1 − g)

connected k=1 handle ≤2k edge loops

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SLIDE 6

Last Time

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2-Manifold Surface

  • Local Neighborhood is disk-shaped
  • Guarantees meaningful neighbor enumeration
  • Non-manifold

f(D✏[u, v]) = D[f(u, v)]

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SLIDE 7

Last Time

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Data Structures

  • Face-Based
  • Edge-Based, edges always have two faces
  • Halfedge-Based
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SLIDE 8

When is a Triangle Mesh a Manifold?

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  • Every Edge incident to 1 or 2 Triangles
  • Faces incident to a vertex form closed or open fan

closed fan

  • pen fan
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SLIDE 9

Outline

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  • Surface Representations
  • Explicit Surfaces
  • Implicit Surfaces
  • Conversion
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SLIDE 10

Explicit vs. Implicit

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Explicit:

  • Range of parameterization function

f(x) = (r cos(x), r sin(x))T

Implicit:

  • Kernel of implicit function

F(x, y) =

  • x2 + y2 − r

f([0, 2π]) F(x, y) < 0 F(x, y) > 0 F(x, y) = 0

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SLIDE 11

Explicit vs. Implicit

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Explicit:

  • Range of parameterization function
  • Piecewise approximation

F(x, y) =

  • x2 + y2 − r

Implicit:

  • Kernel of implicit function
  • Piecewise approximation

f(x) = (r cos(x), r sin(x))T

? ?

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SLIDE 12

Implicit:

  • Kernel of implicit function
  • Piecewise approximation
  • Scalar-valued 3D grid
  • Easy in/out test
  • Easy topology modification

Explicit vs. Implicit

12 12

Explicit:

  • Range of parameterization function
  • Piecewise approximation
  • Splines, triangle mesh, points
  • Easy enumeration
  • Easy geometry modification
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SLIDE 13

Examples: Fluid Simulation

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SLIDE 14

Examples: Collisions

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SLIDE 15

Examples: 3D Reconstruction

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Zippering Poisson Reconstruction

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SLIDE 16

Examples: Kinect Fusion

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http://msdn.microsoft.com/en-us/library/dn188670.aspx

  • 1. Capture
  • 2. Align
  • 3. Fuse
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SLIDE 17

Outline

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  • Surface Representations
  • Explicit Surfaces
  • Implicit Surfaces
  • Conversion
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SLIDE 18

Polynomial Approximation

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f(t) =

p

  • i=0

ci ti =

p

  • i=0

˜ ci φi(t) f(ti) = g(ti) , 0 ≤ t0 < · · · < tp ≤ h |f(t) − g(t)| ≤ 1 (p + 1)! max f (p+1)

p

⇤

i=0

(t − ti) = O

  • h(p+1)⇥

g(h) =

p

⇤

i=0

1 i! g(i)(0) hi + O

  • hp+1⇥

Polynomials are computable functions Taylor expansion up to degree Error for approximation by polynomial

p g f

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SLIDE 19

Spline Surfaces

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Piecewise polynomial approximation

f (u, v) =

n

  • i=0

m

  • j=0

cijN n

i (u) N m j (v)

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SLIDE 20

Spline Surfaces

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Topological constraints

  • Rectangular patches
  • Regular control mesh

Geometric constraints

  • Large number of patches
  • Continuity between patches
  • Trimming

Piecewise polynomial approximation

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SLIDE 21

Polygon Meshes

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Polygonal meshes are a good compromise

  • Piecewise linear approximation → error is
  • Error inversely proportional to #faces
  • Arbitrary topology surfaces
  • Piecewise smooth surfaces
  • Adaptive sampling
  • Efficient GPU-based rendering/processing

O(h2)

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SLIDE 22

Triangle Meshes

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ei, fi ⊂ R3 topology vi ∈ R3

M = ({vi}, {ej}, { fk})

geometry

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SLIDE 23

Outline

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  • Surface Representations
  • Explicit Surfaces
  • Implicit Surfaces
  • Conversion
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SLIDE 24

Implicit Representations

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Level set of 2D function defines 1D curve

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SLIDE 25

Implicit Representations

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Level set of 3D function defines 2D curve

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SLIDE 26

Implicit Representations

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General implicit function:

  • Interior:
  • Exterior:
  • Surface:

Gradient is orthogonal to level set Special case

  • Signed distance function (SDF)
  • Gradient is unit surface normal

F(x, y) < 0 F(x, y) > 0 F(x, y) = 0

F(x, y, z) < 0 F(x, y, z) > 0 F(x, y, z) = 0 rF rF

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SLIDE 27

Signed Distance Function

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SDF of a circle?

  • General shapes
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SLIDE 28

SDF Discretization

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Regular cartesian 3D grid

  • Compute signed distance at nodes
  • Tri-linear interpolation within cells

F000 (1 − u) (1 − v) (1 − w) + F100 u (1 − v) (1 − w) + F010 (1 − u) v (1 − w) + F001 (1 − u) (1 − v) w + . . . F111 u v w

F000 F100 F010 F110 F111 F011 F101

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SLIDE 29

3-Color Octree

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1048576 cells 12040 cells interior,exterior,boundary

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SLIDE 30

Adaptively Sampled Distance Fields

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12040 cells 895 cells

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SLIDE 31

Binary Space Partitions

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895 cells 254 cells

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SLIDE 32

Regularity vs. Complexity

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Implicit surface discretizations

  • Uniform, regular voxel grids
  • Adaptive, 3-color octrees
  • Surface-adaptive refinement
  • Feature-adaptive refinement
  • Irregular hierarchies
  • Binary space partition (BSP)

O(h−3) O(h−2) O(h−1) O(h−1)

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SLIDE 33

Literature

33

  • Frisken et al., “Adaptively Sampled Distance Fields: A general

representation of shape for computer graphics”, SIGGRAPH 2000

  • Wu & Kobbel, “Piecewise Linear Approximation of Signed Distance

Fields”, VMV 2003

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SLIDE 34

Implicit Representations

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  • Natural representation for volumetric data: CT scans, density

fields, etc.

  • Advantageous when modeling shapes with complex and/or

changing topology (e.g., fluids)

  • Very suitable representation for Constructive Solid Geometry

(CSG)

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SLIDE 35

CSG Example

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Union Intersection Difference

FC∪S(·) = min {FC(·) , FS(·)} FC∩S(·) = max {FC(·) , FS(·)} FS\C(·) = max {−FC(·) , FS(·)}

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SLIDE 36

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CSG Example

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SLIDE 37

CSG Example: Milling

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SLIDE 38

Outline

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  • Surface Representations
  • Explicit Surfaces
  • Implicit Surfaces
  • Conversion
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SLIDE 39

Conversion

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Explicit to Implicit

  • Compute signed distance at grid points
  • Compute distance point-mesh
  • Fast marching

Implicit to Explicit

  • Extract zero-level iso-surface
  • Other iso-surfaces
  • Medical imaging, simulations, measurements, …

F(x, y, z) = 0 F(x, y, z) = C

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SLIDE 40

Signed Distance Computation

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Find closest mesh triangle

  • Use spatial hierarchies (octree, BSP tree)

Distance point-triangle

  • Distance to plane, edge, or vertx
  • http://www.geometrictools.com

Inside or outside?

  • Based on interpolated surface normals
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SLIDE 41

Signed Distance Computation

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  • Closest point
  • Interpolated normal
  • Inside if

p = αpi + (1 − α)pj n = αni + (1 − α)nj (q − p)>n < 0 q n p pi pj nj

ni

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SLIDE 42

Fast Marching Techniques

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  • Initialize with exact distance in mesh’s vicinity
  • Fast-march outwards
  • Fast-march inwards
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SLIDE 43

Literature

43

  • Schneider, Eberly, “Geometric Tools for Computer Graphics”, Morgan

Kaufmann, 2002

  • Sethian, “Level Set and Fast Marching Methods”, Cambridge

University Press, 1999

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SLIDE 44

Conversion

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Explicit to Implicit

  • Compute signed distance at grid points
  • Compute distance point-mesh
  • Fast marching

Implicit to Explicit

  • Extract zero-level iso-surface
  • Other iso-surfaces
  • Medical imaging, simulations, measurements, …

F(x, y, z) = 0 F(x, y, z) = C

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SLIDE 45

2D: Marching Square

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  • 1. Classify grid nodes as inside/outside
  • Is or ?
  • 2. Classify cell: 24 configurations
  • In/out for each corner
  • 3. Compute intersection points
  • Linear interpolation along edges
  • 4. Connect them by edges
  • Look-up table for edge configuration

F(xi,j) > 0 < 0

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SLIDE 46

2D: Marching Square

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?

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SLIDE 47

3D: Marching Cubes

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  • 1. Classify grid nodes as inside/outside
  • Is or
  • 2. Classify cell: 28 configurations
  • In/out for each corner
  • 3. Compute intersection points
  • Linear interpolation along edges
  • 4. Connect them by edges
  • Look-up table for path configuration
  • Disambiguation by modified table [Montani ’94]

< 0 F(xi,j,k) > 0

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SLIDE 48

3D: Marching Cubes

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Classify grid nodes based on

  • Inside or outside

Classify all cubes based on

  • Inside, outside, or intersecting

Refined only intersected cells

  • 3-color adaptive octree
  • complexity

O(h−2) Fi,j,k xi,j,k Fi,j,k = F(xi,j,k)

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SLIDE 49

Intersection Points

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Linear interpolation along edges

xi,j,k · |Fi+1,j,k| + xi+1,j,k · |Fi,j,k| |Fi,j,k| + |Fi+1,j,k|

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SLIDE 50

Intersection Points

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Linear interpolation along edges

xi,j,k · |Fi,j+1,k| + xi,j+1,k · |Fi,j,k| |Fi,j,k| + |Fi,j+1,k|

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SLIDE 51

Intersection Points

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Linear interpolation along edges

xi,j,k · |Fi,j,k+1| + xi,j,k+1 · |Fi,j,k| |Fi,j,k| + |Fi,j,k+1|

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SLIDE 52

Intersection Points

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Linear interpolation along edges Lookup table for patch configuration

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SLIDE 53

Marching Cubes

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Look-up table with 28 entries

  • 15 representative cases shown
  • Others follow by symmetry
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SLIDE 54

Marching Cubes

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Algorithm for isosurface extraction from medical scans (CT, MRI)

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SLIDE 55

Marching Cubes

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Effect of grid size

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SLIDE 56

Marching Cubes

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Sample points restricted to edges of regular grid Alias artifacts at sharp features

65×65×65

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SLIDE 57

Increasing Resolution

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Does not remove alias problems!

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SLIDE 58

Extended Marching Cubes

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Locally extrapolate distance gradient Place samples on estimated features

65×65×65

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SLIDE 59

Extended Marching Cubes

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Feature detection

  • Based on angle between normals
  • Classify into edges / corners

ni

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SLIDE 60

Extended Marching Cubes

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Feature sampling

  • Intersect tangent planes
  • Over- or under-determined system
  • Solve by SVD pseudo-inverse

    . . . ni . . .     ·   x y z   =     . . . nT

i si

. . .     (si, ni)

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SLIDE 61

Extended Marching Cubes

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Feature sampling

  • Intersect tangent planes
  • Triangle fans centered at feature point

(si, ni)

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SLIDE 62

Extended Marching Cubes

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Feature Detection Feature Sampling Edge Flipping

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SLIDE 63

Milling Simulation

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257×257×257

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SLIDE 64

CSG Modeling

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65×65×65

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SLIDE 65

Marching Cubes

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+ Result is watertight, closed 2-manifold surface! + Easy to parallelize

  • Uniform (over-) sampling (→ mesh decimation)
  • Degenerate triangles (→ remeshing)
  • MC does not preserve features

+ EMC preserves features, but… about 10% more triangles 20-40% computational overhead

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SLIDE 66

Literature

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  • Lorensen & Cline, “Marching Cubes: A High Resolution 3D Surface

Construction Algorithm”, SIGGRAPH 1987

  • Montani et al., “A modified look-up table for implicit disambiguation
  • f Marching Cubes”, Visual Computer 1994
  • Kobbelt et al., “Feature Sensitive Surface Extraction from Volume

Data”, SIGGRAPH 2001

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SLIDE 67

Next Time

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Discrete Differential Geometry

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SLIDE 68

http://cs599.hao-li.com

Thanks!

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