B-spline Wavelets Jyun-Ming Chen Spring 2001

Basic Ideas 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 {s 0,k }, subdivision tells us how to find a set of control points at the half integers which describe the same underlying B-spline curve cardinal cubic B-spline basis

B-spline Subdivision Upsampling then convolve with … … … … … … … …

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

Cascading

P odd P even /2 Details

P even /2 Details

B-spline Lifting

B-spline Wavelet Transform (inverse) U

B-spline Wavelet Transform (forward) P odd P even Split U

sum up to zero ! U

Design Update of Higher Order

B-spline Scaling Functions The Second Generation

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)

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: –P odd boxes remains as before –P even does not act on boundary; nor does the scaling operator

Examples

B-spline Wavelets

Numeric Example

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.