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 PM, SAL 244 • Kyle Olzsew (TA) � • Office Hour: Still TBD :-) 2
Last Time Polygonal meshes are � • Effective representations • Flexible • Efficient, simple, enables unified processing 3
Last Time Connection between Meshes and Graphs � • Formalism (valence, connections, subgraph, embedding…) • Definitions (boundary, regular edge, singular edge, closed mesh) • triangulation → triangle mesh B A E F D C I H G K J 4
Last Time Topology � • Genus, Euler characteristic • Euler Poincaré formula V − E + F = 2(1 − g ) • Average valence of triangle mesh: 6 • Triangles: F = 2V, E = 3V • Quads: F = V, E = 2V connected k=1 handle ≤ 2k edge loops 5
Last Time 2-Manifold Surface � f ( D ✏ [ u, v ]) = D � [ f ( u, v )] • Local Neighborhood is disk-shaped • Guarantees meaningful neighbor enumeration • Non-manifold 6
Last Time Data Structures � • Face-Based • Edge-Based, edges always have two faces • Halfedge-Based 7
When is a Triangle Mesh a Manifold? • Every Edge incident to 1 or 2 Triangles � • Faces incident to a vertex form closed or open fan closed fan open fan 8
Outline • Surface Representations • Explicit Surfaces • Implicit Surfaces • Conversion 9
Explicit vs. Implicit f ( x ) = ( r cos( x ) , r sin( x )) T Explicit: � f ([0 , 2 π ]) • Range of parameterization function x 2 + y 2 − r � F ( x, y ) = Implicit: � F ( x, y ) < 0 • Kernel of implicit function F ( x, y ) = 0 F ( x, y ) > 0 10
Explicit vs. Implicit f ( x ) = ( r cos( x ) , r sin( x )) T Explicit: � ? • Range of parameterization function • Piecewise approximation x 2 + y 2 − r � F ( x, y ) = Implicit: � ? • Kernel of implicit function • Piecewise approximation 11
Explicit vs. Implicit Explicit: � • Range of parameterization function • Piecewise approximation • Splines, triangle mesh, points • Easy enumeration • Easy geometry modification Implicit: � • Kernel of implicit function • Piecewise approximation • Scalar-valued 3D grid • Easy in/out test • Easy topology modification 12 12
Examples: Fluid Simulation 13
Examples: Collisions 14
Examples: 3D Reconstruction Zippering Poisson Reconstruction 15
Examples: Kinect Fusion 1. Capture 2. Align 3. Fuse http://msdn.microsoft.com/en-us/library/dn188670.aspx 16
Outline • Surface Representations • Explicit Surfaces • Implicit Surfaces • Conversion 17
Polynomial Approximation Polynomials are computable functions p p c i t i = � � f ( t ) = c i φ i ( t ) ˜ i =0 i =0 Taylor expansion up to degree p p 1 i ! g ( i ) (0) h i + O ⇤ h p +1 ⇥ � g ( h ) = i =0 Error for approximation by polynomial f g f ( t i ) = g ( t i ) , 0 ≤ t 0 < · · · < t p ≤ h p 1 � h ( p +1) ⇥ ⇤ ( p + 1)! max f ( p +1) | f ( t ) − g ( t ) | ≤ ( t − t i ) = O i =0 18
Spline Surfaces Piecewise polynomial approximation n m � � c ij N n i ( u ) N m f ( u, v ) = j ( v ) i =0 j =0 19
Spline Surfaces Piecewise polynomial approximation Geometric constraints � • Large number of patches • Continuity between patches • Trimming Topological constraints � • Rectangular patches • Regular control mesh 20
Polygon Meshes Polygonal meshes are a good compromise � O ( h 2 ) • Piecewise linear approximation → error is • Error inversely proportional to #faces • Arbitrary topology surfaces • Piecewise smooth surfaces • Adaptive sampling • Efficient GPU-based rendering/processing 21
Triangle Meshes M = ( { v i } , { e j } , { f k } ) v i ∈ R 3 geometry e i , f i ⊂ R 3 topology 22
Outline • Surface Representations • Explicit Surfaces • Implicit Surfaces • Conversion 23
Implicit Representations Level set of 2D function defines 1D curve 24
Implicit Representations Level set of 3D function defines 2D curve 25
Implicit Representations F ( x, y ) > 0 General implicit function: � F ( x, y, z ) < 0 • Interior: F ( x, y ) < 0 F ( x, y, z ) > 0 • Exterior: F ( x, y, z ) = 0 • Surface: F ( x, y ) = 0 Gradient is orthogonal to level set r F Special case � • Signed distance function (SDF) • Gradient is unit surface normal r F 26
Signed Distance Function SDF of a circle? � � � � General shapes 27
SDF Discretization Regular cartesian 3D grid � • Compute signed distance at nodes • Tri-linear interpolation within cells F 011 F 111 F 000 (1 − u ) (1 − v ) (1 − w ) + u (1 − v ) (1 − w ) + F 100 F 010 F 101 F 010 (1 − u ) v (1 − w ) + F 001 (1 − u ) (1 − v ) w + . . . F 110 F 111 u v w F 000 F 100 28
3-Color Octree interior,exterior,boundary 1048576 cells 12040 cells 29
Adaptively Sampled Distance Fields 12040 cells 895 cells 30
Binary Space Partitions 895 cells 254 cells 31
Regularity vs. Complexity Implicit surface discretizations � O ( h − 3 ) • Uniform, regular voxel grids • Adaptive, 3-color octrees O ( h − 2 ) • Surface-adaptive refinement O ( h − 1 ) • Feature-adaptive refinement • Irregular hierarchies O ( h − 1 ) • Binary space partition (BSP) 32
Literature • 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 33
Implicit Representations • 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) 34
CSG Example Union F C ∪ S ( · ) = min { F C ( · ) , F S ( · ) } Intersection F C ∩ S ( · ) = max { F C ( · ) , F S ( · ) } Difference F S \ C ( · ) = max { − F C ( · ) , F S ( · ) } 35
CSG Example 36
CSG Example: Milling 37
Outline • Surface Representations • Explicit Surfaces • Implicit Surfaces • Conversion 38
Conversion Explicit to Implicit � • Compute signed distance at grid points • Compute distance point-mesh • Fast marching Implicit to Explicit F ( x, y, z ) = 0 • Extract zero-level iso-surface F ( x, y, z ) = C • Other iso-surfaces • Medical imaging, simulations, measurements, … 39
Signed Distance Computation 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 40
Signed Distance Computation p = α p i + (1 − α ) p j • Closest point n = α n i + (1 − α ) n j • Interpolated normal ( q − p ) > n < 0 • Inside if n j q p j n p n i p i 41
Fast Marching Techniques • Initialize with exact distance in mesh’s vicinity • Fast-march outwards • Fast-march inwards 42
Literature • Schneider, Eberly, “Geometric Tools for Computer Graphics”, Morgan Kaufmann, 2002 • Sethian, “Level Set and Fast Marching Methods”, Cambridge University Press, 1999 43
Conversion Explicit to Implicit • Compute signed distance at grid points • Compute distance point-mesh • Fast marching Implicit to Explicit � F ( x, y, z ) = 0 • Extract zero-level iso-surface F ( x, y, z ) = C • Other iso-surfaces • Medical imaging, simulations, measurements, … 44
2D: Marching Square 1. Classify grid nodes as inside/outside � • Is or ? F ( x i,j ) > 0 < 0 2. Classify cell: 2 4 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 45
2D: Marching Square ? 46
3D: Marching Cubes 1. Classify grid nodes as inside/outside � • Is or F ( x i,j,k ) > 0 < 0 2. Classify cell: 2 8 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] 47
3D: Marching Cubes F i,j,k = F ( x i,j,k ) Classify grid nodes based on � x i,j,k • Inside or outside Classify all cubes based on � F i,j,k • Inside, outside, or intersecting Refined only intersected cells � • 3-color adaptive octree O ( h − 2 ) • complexity 48
Intersection Points Linear interpolation along edges x i,j,k · | F i +1 ,j,k | + x i +1 ,j,k · | F i,j,k | | F i,j,k | + | F i +1 ,j,k | 49
Intersection Points Linear interpolation along edges x i,j,k · | F i,j +1 ,k | + x i,j +1 ,k · | F i,j,k | | F i,j,k | + | F i,j +1 ,k | 50
Intersection Points Linear interpolation along edges x i,j,k · | F i,j,k +1 | + x i,j,k +1 · | F i,j,k | | F i,j,k | + | F i,j,k +1 | 51
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