Surface Representations Leif Kobbelt RWTH Aachen University 1 - PowerPoint PPT Presentation

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Outline • (mathematical) geometry representations – parametric vs. implicit • approximation properties • types of operations – distance queries – evaluation – modification / deformation • data structures Leif Kobbelt RWTH Aachen University 2 2

Outline • (mathematical) geometry representations – parametric vs. implicit • approximation properties • types of operations – distance queries – evaluation – modification / deformation • data structures Leif Kobbelt RWTH Aachen University 3 3

Mathematical Representations • parametric – range of a function – surface patch f : R 2 → R 3 , S Ω = f (Ω) • implicit – kernel of a function – level set F : R 3 → R, S c = { p : F ( p ) = c } Leif Kobbelt RWTH Aachen University 4 4

2D-Example: Circle • parametric � r cos( t ) � f : t �→ S = f ([0 , 2 π ]) , r sin( t ) • implicit F ( x, y ) = x 2 + y 2 − r 2 S = { ( x, y ) : F ( x, y ) = 0 } Leif Kobbelt RWTH Aachen University 5 5

2D-Example: Island • parametric � r cos( t ) � ??? f : t �→ S = f ([0 , 2 π ]) , r sin( t ) ??? • implicit F ( x, y ) = x 2 + y 2 − r 2 ??? S = { ( x, y ) : F ( x, y ) = 0 } Leif Kobbelt RWTH Aachen University 6 6

Approximation Quality • piecewise parametric � r cos( t ) � ??? f : t �→ S = f ([0 , 2 π ]) , r sin( t ) ??? • piecewise implicit F ( x, y ) = x 2 + y 2 − r 2 ??? S = { ( x, y ) : F ( x, y ) = 0 } Leif Kobbelt RWTH Aachen University 7 7

Approximation Quality • piecewise parametric � r cos( t ) � ??? f : t �→ S = f ([0 , 2 π ]) , r sin( t ) ??? • piecewise implicit F ( x, y ) = x 2 + y 2 − r 2 ??? S = { ( x, y ) : F ( x, y ) = 0 } Leif Kobbelt RWTH Aachen University 8 8

Requirements / Properties • continuity f ( u i , v i ) ≈ p i – interpolation / approximation • topological consistency – manifold-ness • smoothness – C 0 , C 1 , C 2 , ... C k • fairness – curvature distribution Leif Kobbelt RWTH Aachen University 9 9

Requirements / Properties • continuity f ( u i , v i ) ≈ p i – interpolation / approximation • topological consistency – manifold-ness • smoothness – C 0 , C 1 , C 2 , ... C k • fairness – curvature distribution Leif Kobbelt RWTH Aachen University 10 10

Requirements / Properties • continuity f ( u i , v i ) ≈ p i – interpolation / approximation • topological consistency – manifold-ness • smoothness – C 0 , C 1 , C 2 , ... C k • fairness – curvature distribution Leif Kobbelt RWTH Aachen University 11 11

Requirements / Properties • continuity f ( u i , v i ) ≈ p i – interpolation / approximation • topological consistency – manifold-ness • smoothness – C 0 , C 1 , C 2 , ... C k • fairness – curvature distribution Leif Kobbelt RWTH Aachen University 12 12

Requirements / Properties • continuity f ( u i , v i ) ≈ p i – interpolation / approximation • topological consistency – manifold-ness • smoothness – C 0 , C 1 , C 2 , ... C k • fairness – curvature distribution Leif Kobbelt RWTH Aachen University 13 13

Topological Consistency Leif Kobbelt RWTH Aachen University 14 14

Topological Consistency Leif Kobbelt RWTH Aachen University 14 14

Topological Consistency Mesh Repair ... Leif Kobbelt RWTH Aachen University 14 14

Closed 2-Manifolds • parametric – disk-shaped neighborhoods f ( D ε [ u, v ]) = D δ [ f ( u, v )] – + injectivity • implicit – surface of a “physical” solid F ( x, y, z ) = c, �∇ F ( x, y, z ) � � = 0 – Leif Kobbelt RWTH Aachen University 15 15

Closed 2-Manifolds • parametric – disk-shaped neighborhoods f ( D ε [ u, v ]) = D δ [ f ( u, v )] – + injectivity • implicit – surface of a “physical” solid F ( x, y, z ) = c, �∇ F ( x, y, z ) � � = 0 – Leif Kobbelt RWTH Aachen University 16 16

Closed 2-Manifolds • parametric – disk-shaped neighborhoods f ( D ε [ u, v ]) = D δ [ f ( u, v )] – + injectivity • implicit – surface of a “physical” solid F ( x, y, z ) = c, �∇ F ( x, y, z ) � � = 0 – Leif Kobbelt RWTH Aachen University 17 17

Closed 2-Manifolds • parametric – disk-shaped neighborhoods f ( D ε [ u, v ]) = D δ [ f ( u, v )] – • implicit – surface of a “physical” solid F ( x, y, z ) = c, �∇ F ( x, y, z ) � � = 0 – Leif Kobbelt RWTH Aachen University 18 18

Closed 2-Manifolds • parametric – disk-shaped neighborhoods f ( D ε [ u, v ]) = D δ [ f ( u, v )] – • implicit – surface of a “physical” solid F ( x, y, z ) = c, �∇ F ( x, y, z ) � � = 0 – Leif Kobbelt RWTH Aachen University 19 19

Smoothness • position continuity : C 0 • tangent continuity : C 1 • curvature continuity : C 2 Leif Kobbelt RWTH Aachen University 20 20

Smoothness • position continuity : C 0 • tangent continuity : C 1 • curvature continuity : C 2 Leif Kobbelt RWTH Aachen University 21 21

Smoothness • position continuity : C 0 • tangent continuity : C 1 • curvature continuity : C 2 Leif Kobbelt RWTH Aachen University 22 22

Fairness • minimum surface area • minimum curvature • minimum curvature variation Leif Kobbelt RWTH Aachen University 23 23

Outline • (mathematical) geometry representations – parametric vs. implicit • approximation properties • types of operations – distance queries – evaluation – modification / deformation • data structures Leif Kobbelt RWTH Aachen University 24 24

Polynomials • computable functions p p c i t i = � � c � p ( t ) = i Φ i ( t ) i =0 i =0 • Taylor expansion p 1 i ! f ( i ) (0) h i + O ( h p +1 ) � f ( h ) = i =0 • interpolation error (mean value theorem) p ( t i ) = f ( t i ) , t i ∈ [0 , h ] p 1 � ( p + 1)! f ( p +1) ( t ∗ ) ( t − t i ) = O ( h ( p +1) ) � f ( t ) − p ( t ) � = i =0 Leif Kobbelt RWTH Aachen University 25 25

Polynomials • computable functions p p c i t i = � � c � p ( t ) = i Φ i ( t ) i =0 i =0 • Taylor expansion p 1 i ! f ( i ) (0) h i + O ( h p +1 ) � f ( h ) = i =0 • interpolation error (mean value theorem) p ( t i ) = f ( t i ) , t i ∈ [0 , h ] p 1 � ( p + 1)! f ( p +1) ( t ∗ ) ( t − t i ) = O ( h ( p +1) ) � f ( t ) − p ( t ) � = i =0 Leif Kobbelt RWTH Aachen University 26 26

Polynomials • computable functions p p c i t i = � � c � p ( t ) = i Φ i ( t ) i =0 i =0 • Taylor expansion p 1 i ! f ( i ) (0) h i + O ( h p +1 ) � f ( h ) = i =0 • interpolation error (mean value theorem) p ( t i ) = f ( t i ) , t i ∈ [0 , h ] p 1 ( p + 1)! f ( p +1) ( t ∗ ) � ( t − t i ) = O ( h ( p +1) ) � f ( t ) − p ( t ) � = i =0 Leif Kobbelt RWTH Aachen University 27 27

Implicit Polynomials • interpolation error of the function values � F ( x, y, z ) − P ( x, y, z ) � = O ( h ( p +1) ) • approximation error of the contour F ( p + � p ) − F ( p ) � p = λ ∇ F ( p ) ≈ �∇ F ( p ) � �� p � Leif Kobbelt RWTH Aachen University 28 28

Implicit Polynomials • interpolation error of the function values � F ( x, y, z ) − P ( x, y, z ) � = O ( h ( p +1) ) p p+ Δ p • approximation error of the contour F ( p + � p ) − F ( p ) � p = λ ∇ F ( p ) ≈ �∇ F ( p ) � �� p � Leif Kobbelt RWTH Aachen University 29 29

Implicit Polynomials • interpolation error of the function values � F ( x, y, z ) − P ( x, y, z ) � = O ( h ( p +1) ) p p+ Δ p • approximation error of the contour �� p � ≈ F ( p + � p ) − F ( p ) � p = λ ∇ F ( p ) �∇ F ( p ) � (gradient bounded from below) Leif Kobbelt RWTH Aachen University 30 30

Implicit Polynomials F(x,y,z) F(x,y,z) x,y,z x,y,z large small gradient gradient Leif Kobbelt RWTH Aachen University 31 31

Polynomial Approximation • approximation error is O(h p+1 ) • improve approximation quality by – increasing p ... higher order polynomials – decreasing h ... smaller / more segments • issues – smoothness of the target data ( max t f (p+1) (t) ) – handling higher order patches (e.g. boundary conditions) Leif Kobbelt RWTH Aachen University 32 32