1. SIAM Conference on Geometric Desing & Computing Approximation of spatial data with shape constraints Maria Lucia Sampoli University of Siena, Italy

2. 2 Overview  Methods for data approximation with shape control  Univariate case: variable degree polynomial spline spaces  Spatial data: shape information by zero moment analysis  Approximation by tensor product surfaces  Approximation by triangular surfaces  Examples and conclusion

3. 3 Related Publications 1997 P. Costantini, Variable degree polynomial splines, in Curves and Surfaces with Applications in CAGD, 85-94. 1997 P. Costantini, Variable degree polynomial splines, in Curves and Surfaces with Applications in CAGD, 85-94. 2000 P. Costantini, Curve and surface construction using variable degree polynomial splines, CAGD 17, 419-446. 2000 P. Costantini, Curve and surface construction using variable degree polynomial splines, CAGD 17, 419-446. 2000 P. Costantini and C. Manni, Interpolating polynomial macro- elements with tension properties, in Curves and Surface fitting: Saint-Malo 1999, 143-152. 2000 P. Costantini and C. Manni, Interpolating polynomial macro- elements with tension properties, in Curves and Surface fitting: Saint-Malo 1999, 143-152. 2001 P. Costantini and F. Pelosi, Shape preserving approximation by space curves, Num. Alg. 27, 237-264. 2001 P. Costantini and F. Pelosi, Shape preserving approximation by space curves, Num. Alg. 27, 237-264. 2003 P. Costantini, Properties and applications of new polynomial spaces, Tech.Rep. Univ. Siena, to appear in Inter. Jour. Wavel. Mult. Infor Proc. 2003 P. Costantini, Properties and applications of new polynomial spaces, Tech.Rep. Univ. Siena, to appear in Inter. Jour. Wavel. Mult. Infor Proc. 2004 P. Costantini and F. Pelosi, Shape preserving approximation of spatial data, Adv. Comp. Math. 20, 25-51. 2004 P. Costantini and F. Pelosi, Shape preserving approximation of spatial data, Adv. Comp. Math. 20, 25-51. 2005 P. Costantini and F. Pelosi, Shape preserving data approximation using new spline spaces, in Mathematical Methods for Curves and Surfaces: Tromsø 2004, 81-92. 2005 P. Costantini and F. Pelosi, Shape preserving data approximation using new spline spaces, in Mathematical Methods for Curves and Surfaces: Tromsø 2004, 81-92. 2005 I. Cravero and C. Manni, Detecting shape of spatial data via zero moments, in Mathematical Methods for Curves and Surfaces: Tromsø 2004, 93-102. 2005 I. Cravero and C. Manni, Detecting shape of spatial data via zero moments, in Mathematical Methods for Curves and Surfaces: Tromsø 2004, 93-102. 2005 I. Cravero and F. Pelosi, Zero moment analysis for bivariate data, preprint. 2005 I. Cravero and F. Pelosi, Zero moment analysis for bivariate data, preprint. 2005 P.Costantini and F. Pelosi, Tensor Product surface approximation, preprint. 2005 P.Costantini and F. Pelosi, Tensor Product surface approximation, preprint.

4. 4 Tension methods  Introduced for a control on the shape are shape parameters. For limit values of

5. 5 Variable degree polynomial splines  Costantini 1997  Costantini 2000  Given the knot vector { }, consider the polynomial space in  For limit values of shape parameters: the functions vanish in any    Equivalent to : “cubic like” variable degree polynomials

6. 6 Approximation with Variable Degree splines  Given select a sequence of significant knots  Construct the piecewise linear of best approximation λ on  Use λ to define the shape of the data  Find the degrees such that the spline curve of best approximation in has the same shape of λ  The structure of variable degree splines allows a geometric control of the shape constraints DRAWBACK: Good for interpolation problems but not always good for approximation

7. 7 The shape of many real objects from which the data are taken cannot be represented by linear curves or bilinear surfaces The shape of many real objects from which the data are taken cannot be represented by linear curves or bilinear surfaces ! IDEA: change the limit space 1998 Kaklis & Sapidis  admits a totally positive, normalized, “Bernstein- Bézier” basis  equivalence to   For limit values of shape parameters Costantini 2003 Costantini 2003 

8. 8 Approximation with new variable degree splines  Given select a sequence of significant knots  Different approach according to the new limit space  Different approach according to the new limit space  Construct σ in of best approximation to data on  Use σ to define the shape of the data  Find the degrees such that the spline curve of best approximation in has the same shape of σ  The structure of variable degree splines allows a geometric control of the shape constraints Costantini & Pelosi 2005 Costantini & Pelosi 2005

9. 9 admits a positive, normalized, “Bernstein-Bézier” basis  equivalence to Extension    For limit values of shape parameters  More flexibility due to more degrees of freedom

10. 10  Let be a parametric curve. For consider, a ball of radius ε with center. The zero moment of s in t is given by the barycenter Selection of significant knots  New approach based on Zero Moment analysis  automatic procedure  automatic procedure  stable with respect to noisy data  stable with respect to noisy data  easy to implement even for bivariate data  easy to implement even for bivariate data Clarenz,Rumpf &Telea 2004 Clarenz,Rumpf &Telea 2004 Cravero & Manni 2005 Cravero & Manni 2005

11. 11  The sign of vector gives information about concavity and convexity  The vector is called ε-normal, it is independent of translations and is such that can be used to select corners or sharp changes in data points (considering the zero-moment analysis to the piecewise line connecting the data   Local classification for curve features  Segmentation of discrete curves in regions with common shape

12. 12 Tensor-product surfaces  Assigned and  There exists a basis with similar properties of biquartic B-spline basis  Classical geometric construction  If for all indices - Given Costantini & Pelosi 2005 Costantini & Pelosi 2005

13. 13  The parameter values are given  The structure of data is supposed to be suitably approximated by a tensor product surface  The data have weak noise  they can be described by biquadratic patches Approximation with T-P variable degree splines Scheme 1. Extract significant knot sequences from the given parameter values 2. Compute σ in of best approximation to data 3. Define the shape of data as the shape of σ 4. Compute of best approximation to data in the least square sense 5. Find degrees such that has the same shape of σ 5. Find degrees such that s has the same shape of σ

14. 14 Example 1

15. 15 Example 2

16. 16 Resulting surface

17. 17 Drawbacks

18. 18  Equivalence to : bivariate polynomial space of degree 6 defined over a triangular domain Triangular patches  The parameter values are given  The structure of data is supposed to be suitably approximated by a triangulated surface  The data have weak noise  they can be described by quadratic triangular patches  variable degrees associated to the vertices 

19. 19 Free parameters  Triangular control net of degree 6

20. 20  The points are on a “quadratic net”  The points are on a net which is a degree elevated net from a proper quadratic net  Construction done in 3 steps -Generic patch of degree -Generic patch of degree

21. 21  The points are on a net which is a degree elevated net from a proper quadratic net  Construction done in 3 steps: general case -Generic patch of degree -Generic patch of degree -Consider the control net of -Consider the control net of

22. 22 Example

23. 23 Concluding remarks  “Quartic-like” variable degree polynomial splines can be used efficiently to approximate curves and tensor-product surfaces  Working with these spaces is computationally stable and not expensive (the computational cost does not depend on the variable degrees )  To define the shape of spatial data the zero moment method is considered  The extension to triangular patches is possible using a combination of a quartic-like and a degree 6-like variable degree polynomial spaces  New scheme to approximate scattered data with triangular splines of variable degree