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

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

Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Outline Introduction 1 Motivation Previous work Generalized Bézier (GB) patch 2 Control structure Domain & parameterization Blending functions Enhancements 3 Problems New algorithms Examples 4 Conclusion and future work 5 T. Várady, P. Salvi, I. Kovács BME Enhancement of a multi-sided Bézier surface representation

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

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

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

Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Control structure Control net derivation from the quadrilateral case Control grid → n ribbons 1/4 1/2 1/2 Degree: d 1/2 1/2 1 1/2 1/2 1/2 1/2 1 1 1/2 1/2 Layers: 1/2 1/2 1 1/2 1/2 1/2 1/2 1 1 1/2 1/2 � d + 1 � l = 2 Control points: C d , i j , k i = 1 . . . n j = 0 . . . d 1/4 k = 0 . . . l − 1 1/2 1/2 Weights: µ i α α 1 β β j , k α α 1 1 β β α α 1 β β α α 1 1 β β T. Várady, P. Salvi, I. Kovács BME Enhancement of a multi-sided Bézier surface representation

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 s i and h i Based on Wachspress barycentric coordinates λ i ( u , v ) s 4 s 3 s 3 h 4 h 3 h 3 h 4 s 4 s 2 h 5 h 2 s 5 s 2 h 2 h 1 s 1 h 1 v s 1 u T. Várady, P. Salvi, I. Kovács BME Enhancement of a multi-sided Bézier surface representation

Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Domain & parameterization Local parameters λ i s i = λ i − 1 + λ i h i = 1 − λ i − 1 − λ i Barycentric coordinates λ i λ i ≥ 0 [positivity] � n i = 1 λ i = 1 [partition of unity] � n i = 1 λ i ( u , v ) · P i = ( u , v ) [reproduction] λ i ( P j ) = δ ij [Lagrange property] T. Várady, P. Salvi, I. Kovács BME Enhancement of a multi-sided Bézier surface representation

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 µ i j , k B d j , k ( s i , h i ) = µ i j , k B d j ( s i ) B d k ( h i ) µ i j , k is a rational function for 2 × 2 CPs in each corner α i = h i − 1 / ( h i − 1 + h i ) , β i = h i + 1 / ( h i + 1 + h i ) T. Várady, P. Salvi, I. Kovács BME Enhancement of a multi-sided Bézier surface representation

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: n d l − 1 � � � B d µ i j , k B d 0 ( u , v ) = 1 − j , k ( s i , h i ) i = 1 j = 0 k = 0 Patch equation: n d l − 1 � � � C d , i j , k µ i j , k B d j , k ( s i , h i ) + C d 0 B d S ( u , v ) = 0 ( u , v ) i = 1 j = 0 k = 0 T. Várady, P. Salvi, I. Kovács BME Enhancement of a multi-sided Bézier surface representation

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 on a given side. Theorem The Generalized Bézier patch, on 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

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

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: l , l · � n C d i = 1 µ i l , l B d l , l ( s i , h i ) Weight deficiency is distributed amongst the innermost blend functions T. Várady, P. Salvi, I. Kovács BME Enhancement of a multi-sided Bézier surface representation

Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Problems New parameterization Better isoline distribution Lower weight deficiency V i-2 V i+1 V i-2 V i+1 Constraints: C h i = 0 . 5 tangential M i-1 M i+1 M i-1 M i+1 C to s i − 1 and s i + 1 Middle point V i-1 V i V i-1 V i M i M i on a circular arc T. Várady, P. Salvi, I. Kovács BME Enhancement of a multi-sided Bézier surface representation

Introduction Generalized Bézier (GB) patch Enhancements Examples Conclusion and future work Problems G 2 continuity Use squared terms in the rational weights: α i = h 2 i − 1 / ( h 2 i − 1 + h 2 β i = h 2 i + 1 / ( h 2 i + 1 + h 2 i ) i ) T. Várady, P. Salvi, I. Kovács BME Enhancement of a multi-sided Bézier surface representation

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

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

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 of the control points T. Várady, P. Salvi, I. Kovács BME Enhancement of a multi-sided Bézier surface representation

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

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

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