Enhancement of a multi-sided Bzier surface representation Tams - - PowerPoint PPT Presentation

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Enhancement of a multi-sided Bzier surface representation Tams - - PowerPoint PPT Presentation

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Introduction Generalized Bzier (GB) patch Enhancements Examples Conclusion and future work Enhancement of a multi-sided Bzier surface representation Tams Vrady, Pter Salvi, Istvn Kovcs Budapest University of Technology and

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work

Enhancement of a multi-sided Bézier surface representation

Tamás Várady, Péter Salvi, István Kovács

Budapest University of Technology and Economics

CAGD 55, pp. 69–83, 2017. GMP 2018

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work

Outline

1

Introduction Motivation Previous work

2

Generalized Bézier (GB) patch Control structure Domain & parameterization Blending functions

3

Enhancements Problems New algorithms

4

Examples

5

Conclusion and future work

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Motivation

Applications of multi-sided patches

Curve network based design

Feature curves Automatic surface generation

Hole filling

E.g. vertex blends Cross-derivative constraints

3D point cloud approximation

Given boundary loops Smoothly connected patches

Representation?

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Motivation

Conventional representations

Trimmed/split tensor product surfaces

Detailed control in the interior CAD-compatible But: continuity problems

Recursive subdivision

Arbitrary topology Easy to design with But: hard to interpolate boundary cross-derivatives

Transfinite patches

Interpolates any number of sides Depends only on the boundary But: little interior control

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Previous work

Multi-sided surfaces with control networks

Loop and DeRose (1989)

S-patches – beautiful theory, difficult to use

Warren (1992)

Based on Bézier triangles, max. 6 sides

Zheng and Ball (1997)

High-degree expressions, max. 6 sides

Krasauskas (2002)

Toric patches – lattice-based, symmetry concerns

Várady et al. (2016)

Generalized Bézier patches Regular polygonal domain Symmetric control structure

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Control structure

Control net derivation from the quadrilateral case

Control grid → n ribbons Degree: d Layers: l = d+1

2

  • Control points:

C d,i

j,k

i = 1 . . . n j = 0 . . . d k = 0 . . . l −1

Weights: µi

j,k

1/4 1 1/2 1/2 1/2 1/2 1 1/2 1/2 1/2 1/2 1/2 1/2 1 1/2 1/2 1 1/2 1/2 1 1/2 1/2 1 1/2 1/2 1/4 α α 1 β β α α 1 β β α α 1 1 β β β β 1 1 α α 1/2 1/2

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Domain & parameterization

Domain

Regular domain in the (u, v) plane Side-based local parameterization functions si and hi

Based on Wachspress barycentric coordinates λi(u, v)

u v s4 h4 h2 s2 s3 h3 s1 h1 s1 h1 h2 s2 s3 h3 s4 h4 s5 h5

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Domain & parameterization

Local parameters

si =

λi λi−1+λi

hi = 1 − λi−1 − λi Barycentric coordinates λi λi ≥ 0 [positivity] n

i=1 λi = 1

[partition of unity] n

i=1 λi(u, v) · Pi = (u, v)

[reproduction] λi(Pj) = δij [Lagrange property]

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Blending functions

Bernstein functions with rational weights

C d,i

j,k : j-th control point on side i, layer k

Multiplied by µi

j,kBd j,k(si, hi) = µi j,kBd j (si)Bd k (hi)

µi

j,k is a rational function for 2 × 2 CPs in each corner

αi = hi−1/(hi−1 + hi), βi = hi+1/(hi+1 + hi)

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Blending functions

Central weight & patch equation

Weights do not add up to 1 Deficiency ⇒ weight of the central point: Bd

0 (u, v) = 1 − n

  • i=1

d

  • j=0

l−1

  • k=0

µi

j,kBd j,k(si, hi)

Patch equation: S(u, v) =

n

  • i=1

d

  • j=0

l−1

  • k=0

C d,i

j,k µi j,kBd j,k(si, hi) + C d 0 Bd 0 (u, v)

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work

Interpolation property

Definition A Bézier ribbon is a Bézier patch given by the first two layers (rows) of control points

  • n a given side.

Theorem The Generalized Bézier patch,

  • n its boundary, interpolates

the position and first cross- derivative of the Bézier ribbons of its respective sides.

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work

Overview

Fixed issues Weight deficiency

Increases with n and d ⇒ Influence of the central control point grows Strongly oscillates between even and odd degrees

Support for G 2 continuity between patches New/updated algorithms Degree elevation & reduction Fullness control Approximation of point clouds

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Problems

Weight deficiency

No central control point for odd-degree patches For even-degree patches: C d

l,l · n i=1 µi l,lBd l,l(si, hi)

Weight deficiency is distributed amongst the innermost blend functions

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Problems

New parameterization

Better isoline distribution Lower weight deficiency Constraints:

hi = 0.5 tangential to si−1 and si+1 Middle point

  • n a circular arc

Mi Mi+1 Vi-2 Vi+1 C Vi-1 Vi Mi-1 Vi-2 Vi+1 C Vi-1 Vi Mi+1 Mi-1 Mi

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Problems

G 2 continuity

Use squared terms in the rational weights: αi = h2

i−1/(h2 i−1 + h2 i )

βi = h2

i+1/(h2 i+1 + h2 i )

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work New algorithms

Degree elevation & reduction

Essentially the same as in the original paper Linear and bilinear combinations Modifies the surface (slightly) The control net is generated by reductions and elevations

Default positions for internal control points

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work New algorithms

Fullness control

Multi-resolution editing technique Edit a control point of a lower-degree patch

E.g. quartic central point

Its influence is propagated by degree elevation

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work New algorithms

Approximation

Least-squares fit of points Initial surface

Generated by the boundary constraints

Initial parameterization

Projection

Iteration:

Fit with smoothing Degree elevation Re-parameterization

Smoothing

Reduce oscillation

  • f the control points
  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Example 1

Torso

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Example 1

Torso – detail

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Example 2

Gamepad

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Example 2

Gamepad – detail

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Conclusion

Summary

Generalized Bézier patches Side-based interpretation All control points generated by the boundaries via degree elevation Interior control Enhancements Follow the quadrilateral patch more closely Central control point / weight deficiency fixed Better parameterization Curvature continuity Approximation algorithm

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Future work

Patches over concave domains

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation

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Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work

Any questions? Thank you for your attention.

  • T. Várady, P. Salvi, I. Kovács

BME Enhancement of a multi-sided Bézier surface representation