Curve Modeling with Constrained B-spline Wavelets

1. Curve Modeling with Constrained B-spline Wavelets Denggao Li, Kaihuai Qin, Hanqiu Sun Computer Aided Geometric Design 22(2005) 45-56 Reporter: Huixia Xu Thursday, September 22, 2005

2. About Author • Denggao Li and Kaihuai Qin • Department of Computer Science and Technology, Tsinghua University • Hanqiu Sun • Associate Professor, Department of Computer Science and Engineering, The Chinese University of Hong Kong • Research interests: interactive animations, virtual & augmented reality, hypermedia, computer-assisted surgery, internet-based visualization/navigation, tele-medicine, realistic haptics simulation

3. Related Work • The development of B-spline wavelets • Forsey and Bartels [1988]: address the formulation of hierarchical B-splines, which form an over- representation for curves and surfaces. • Stollnitz et al. [1999] and Chui [1992]: present the wavelets which form a compact multiresolution representation. • Lyche et al. [2001]: a widely used class of B-spline wavelets is of mutually orthogonal wavelet spaces with minimally supported wavelet functions, which is the combination of B-splines and wavelets.

4. Related Work • B-spline wavelets’ research results on multiresolution techniques and applications • Finkelstein and Salesin [1994]: address the problems of multiresolution smoothing, editing and approximating of B-spline curves using uniform B-spline wavelets. • Kazinnik and Elber [1997]: utilize non-uniform B-spline wavelets to handle multiresolution editing of free-form curves and surfaces with non-uniform knots.

5. What to Do • In this paper, it presents a novel approach to construct B-spline wavelets under constraints, taking advantage of the lifting scheme

6. Application of The Method • Smoothing a curve while preserving user specified “feature points”. • Representing several segments of a single curve at different resolution levels, leaving no awkward “gaps”. • Multiresolution editing of B-spline curves under constraints. • etc.

7. Overview • B-spline wavelets • The philosophy of lifting • Applying constraints • Algorithm summary and computational complexity • Some examples • Conclusion

8. B-spline Wavelets

9. B-spline Wavelets • B-spline wavelet analysis starts from the nested sequence defined on a nested sequence . • : the linear space spanned by defined on knot vector . • : B-spline basis function, arranged as a row vector with dimension . • : the complement space of in . • : wavelet functions are a set of basis functions arranged as a row vector with dimension

10. Multiscale Relations • The multiscale relations state that: • : knot insertion matrix with dimension . • : wavelet coefficient matrix with dimension • The inverse basis transform can be expressed as

11. The Wavelet Decompostion and Reconstruction Process • Given a B-spline curve at resolution level : • : a column vector of control points. • The wavelet decompositon process: . • and are given by • The wavelet reconstruction process :

12. Semi-orthogonal B-spline wavelets • It requires and to be mutually orthogonal: • The notation stands for the inner product matrix of and , i.e. • This requirement can be further reduced into: • where,

13. The Philosophy of Lifting

14. Related Work • Sweldens [1996, 1997]:indicate that the lifting scheme is a promising framework for building customized wavelets. • SchrÖder and Sweldens [1995]:present spherical wavelets which can efficiently represent functions on the sphere.

15. Two Main Features • Our construction of constrained B-spline wavelets takes advantage of two main features of the lifting scheme: • The ability to construct customized (in our case, satisfying the constraints) wavelets. • In-place calculation, which leads to a fast wavelet transform.

16. The Lifting Scheme • Given a B-spline wavelet transform with synthesis filters: . • A new B-spline wavelet transform with synthesis filters can be obtained, following the lifting scheme: • : the lifting parameter matrix with dimension

17. Reconstruction Process • Under the new synthesis filters, the wavelet reconstruction process can be expressed as: • The above can be decomposed into two “in-place” calculation steps: • ; • .

18. Decomposition Process • The decomposition process can be obtained by simply perform the inverse operation of each step in the reverse order: • ; • .

19. Decomposition and Reconstruction Diagram

20. Applying Constraints

21. The Constraints • The constraints can be specified by a set of 2-tuples: • : the number of constraints. • : associates the jth constraint with position, tangent or high order derivatives. • : specifies the parameter value of the j th constraint.

22. Applying Constraints • : the B-spline curve at resolution level . • : the B-spline curve at th resolution level decomposed from using the new wavelets. • The constraints require that: • : the d th derivative of .

23. Applying Constraints • : the wavelet coefficient matrix of the new B-spline wavelets. • : the wavelet function of the new B-spline wavelets. • Applying the decomposition relation: • Then we have the following, which is a necessary property of constrained B-spline wavelet functions:

24. Applying Constraints Back

25. Solve the Lifting Parameter Matrix • Applying the multiscale relation, therefore the constraints can be expressed as: • The above equality gives us a way to solve , define the i th constraint matrix :

26. Solve the Lifting Parameter Matrix • One important property of is sparsity. • Using this new notation , the constraints can be rewritten as: • is a matrix with dimension and may not necessarily be square or of full rank. • Measures should be taken to deal with various under-constrained, over-constrained and rank-deficient conditions.

27. Under-Constrained • In this condition, it satisfies and rank( )= . • It’s the most usual case and we should add some optimization goals to make the solution unique. The most intuitive idea is to make as orthogonal as possible to for best approximation, as expressed by: for . • : of a vector.

28. Under-Constrained • Notice that: • means the l th column of matrix • This leads to the following approach: for , solve in the following linear equality-constrained least squares problem: • The method is straightforward but the computation too expensive.

29. Under-Constrained • By the variable substitution , we have a much cheaper approach as shown below: for , solve in the following underdetermined minimum norm problem: • Then form the lifting parameter matrix as

30. Over-Constrained • In this condition, it satisfies and rank( )= . • The remedy is to either discard some constraints and satisfy the others; or solve the over-constrained system to get an “as good as possible” solution, i.e., for , solve in the following linear least squares problem : • According to our observation, over-constraints do not work well in practice.

31. Rank-Deficient • In this condition, it satisfies • Our approach is as follows: By variable substitution , solve columnwise in the minimum norm least square problem: • Then form by • In fact, this approach is the generalized version of the former two conditions, therefore all cases can be solved uniformly in this way.

32. Algorithm Summary and Computational Complexity

33. Lemma for Algorithm Summary • Suppose matrix A has the LQ factorization: • : orthogonal matrix. • : lower triangular matrix. • The solution to the following underdetermined minimum norm problem : is given by:

34. Algorithm Summary • Using the Lemma and the former conclusion, we can solve the following problem: and get • “Left divide” means solving which yields the result . • “Right divide” means solving which yields the result

35. Decomposition Algorithm • Input: • Output: • Decompose into and by MBW; • Let • Solve the minimum norm problem: minimize

36. Reconstruction Algorithm • Input: • Output: • Let • Solve the minimum norm problem: minimize • Reconstruction from and by MBW.

37. Computational Complexity • The computational complexity (both time and storage) of MBW is linear in the number of control points for a given B-spline order. So the only computational expense we should take into consideration is in the lifting steps. • For a given B-spline order and the number of constraints, the total time and space requirements in the lifting steps are linear in

38. Some Examples

39. Some Examples Back

40. Some Examples Back

41. Some Examples Back

42. Conclusion • The constrained B-spline wavelets proposed in this paper have been proven to be a useful extension to current multiresolution analysis techniques based on B-spline wavelets. • When no constraints are imposed, the constrained B-spline wavelets will degenerate into traditional MBW. • The linear computational complexity of B-spline wavelets has been maintained in the constrained B-spline wavelets.

43. Thank you!