Bspline Wavelets

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B-spline Wavelets

• Jyun-Ming Chen
• Spring 2001

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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 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

cardinal cubic B-spline basis
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B-spline Subdivision

• Upsampling then convolve with

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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
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Cascading
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Details
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Details
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B-spline Lifting
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B-spline Wavelet Transform (inverse)
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B-spline Wavelet Transform (forward)
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sum up to zero !
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Design Update of Higher Order
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B-spline Scaling Functions

• The Second Generation

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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)

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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

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Examples
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B-spline Wavelets
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Numeric Example
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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.