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B-spline Wavelets

B-spline Wavelets. Jyun-Ming Chen Spring 2001. Here refers to cubic B-spline most commonly used in CG Assume cardinal cubic B-spline for now No boundary effects

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B-spline Wavelets

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  1. B-spline Wavelets Jyun-Ming Chen Spring 2001

  2. Here refers to cubic B-spline most commonly used in CG Assume cardinal cubic B-spline for now No boundary effects Given a set of cubic B-spline control points at integers {s0,k}, subdivision tells us how to find a set of control points at the half integers which describe the same underlying B-spline curve Basic Ideas cardinal cubic B-spline basis

  3. … … … … … … … … B-spline Subdivision • Upsampling then convolve with

  4. … … … … … … … … Consider In-place Computation 9.5, 18, 15.5, 9 4,10,8,4 4.75, 9, 7.75, 4.5 7,9,6,4

  5. Cascading

  6. Podd Peven /2 Details

  7. Peven /2 Details

  8. B-spline Lifting

  9. U B-spline Wavelet Transform (inverse)

  10. Split Peven Podd U B-spline Wavelet Transform (forward)

  11. 0 0 0 0 0 0 0 1 0 0 U sum up to zero !

  12. Design Update of Higher Order

  13. B-spline Scaling Functions The Second Generation

  14. Remarks • The first generation refers to • regular sampling in interpolating and AI wavelets • In B-spline, the regularity refers to uniform knot sequence (all piecewise polynomial components of the curve are regular in parametric space) • The second generation B-spline must consider the boundary effects (near the two end points) • Such that the curve passes through the two end points (desirable for geometric design consideration)

  15. B-spline Scaling Functions • Chui and Quak • Use knot insertion • Does not fit into the lifting framework of inserting new points between old ones • (in fact, the control points are not even distributed !) • Here, use a different treatment: • Podd boxes remains as before • Peven does not act on boundary; nor does the scaling operator

  16. Examples

  17. B-spline Wavelets

  18. Numeric Example

  19. Homework • Given 32 control points in 2D. Sketch the B-spline curve (by subdivision) • Derive the corresponding multiresolution curve of 16-, 8-, 4- control points. Sketch each curve by subdivision and plot the control points. • Do it for cardinal and end-point interpolating B-splines.

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