Slides17

Slides17 - Course 18.327 and 1.130 Course Wavelets and...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Course 18.327 and 1.130 Course Wavelets and Filter Banks Wavelets and subdivision: nonuniform Wavelets grids; multiresolution for triangular meshes; representation and compression of surfaces. Wavelets on Surfaces in R3 Wavelets Construction by Schröder and Sweldens Construction • uses lifting uses • scaling functions are interpolating in most scaling straightforward case • typically work with triangular mesh generated by typically subdivision k6 m6 k5 K(j) = {k, k1, k2, k3, k4, k5, k6, …} K(j M(j) = {m1, m2, m3, m4, m5, m6, …} k1 m1 km 2 k2 m5 m4 m3 k4 k3 mesh describing surface S 2 Notation: Notation: K(j) = K(j + 1) = M(j) = all vertices at resolution j all vertices at resolution j + 1 vertices obtained by subdividing the resolution j mesh to produce the resolution j + 1 mesh So K(j + 1) = K(j) \ M(j) { Interpolating property means that scalings functions satisfy k ∈ K(j) 1 if x = xk K(j) φj,k(x) = k′∈ K(j) 0 if x = xk′ K(j) k′ ≠ k x = position vector of a point on S. position 3 Simple interpolating scaling function: hat function Simple φj,k(x) k5 m5 m4 k4 k6 m6 k m3 Scaling functions at level j are all located at vertices in K(j) k1 m1 m2 k2 k3 Refinement equation m6 φj,k(x) = φj+1,k(x) + ½ ∑ φj+1,m(x) m=m1 m=m In general, interpolating scaling functions will satisfy a refinement equation of the form φj,k(x) = φj+1,k(x) + ∑ h0[k,m]φj+1,m(x) m∈n(j,k) j 4 n(j,k) = vertices in the neighborhood of vertex k that n(j,k) contribute to the refinement equation. Because of interpolating property, n(j,k) can only consist of vertices in M(j). How to construct the wavelet? Start with Wavelets at level j are all wj,m(x) = φj+1,m(x) located at vertices in M(j) Then use the lifting idea to impose vanishing moment. 5 m k1 k2 Consider a wavelet of the form Consider wj,m(x) = φj+1,m(x) - α1φj,k1(x) - α2φj,k2(x) For the zeroth moment to vanish moment 0 = Ij+1,m - α1 Ij,k1 - α2Ij,k2 where where Ij,k = ∫ φj,k(x)dS s 6 To satisfy vanishing moment condition, choose To i = 1, 2 αi = Ij+1,m/2Ij,ki 1, So the wavelet equation can be written as wj,m(x) = φj+i,m(x) - ∑ with k∈A(j,m) h1[k,m]φj,k(x) j A(j,m) = two immediate neighbors in K(j) j h1[k,m] = Ij+1,m/2Ij,k 7 Wavelets on Surfaces in R3 Wavelets Synthesis scaling function Synthesis j φj,k(x) = φj+1,k(x) + ∑ h0[k,m]φj+1,m(x) m∈n(j,k) k6 k6 k5 k5 k1 k1 m6 m6 m1 m1 k m5 m5 m4 m4 m2 m2 k2 k2 m3 m3 k3 k3 k4 Linear interpolating functions: Linear j h0[k,m] m∈n(j,k = { ½ otherwise ) 0 n(j,k) = {m1, m2, m3, m4, m5, m6} n(j,k) Synthesis wavelet Synthesis j wj,m(x) = φj+1,m(x) - ∑ h1[k,m] φj,k(x) k∈A(j,m) k1 m k2 A(j,m) = {k1, k2} A(j,m) 8 What are the analysis functions? What Use alternating signs condition to get analysis filters, e.g. 1D interpolating filter If F0(z) = 6 {-z3 + 0•z2 + 9z + 16 + 9z-1 + 0•z-2 – z-3} 9z then H1(z) = F0(-z) = 6 {z3 + 0.z2 - 9z + 16 - 9z-1 + 0.z-2 + z-3 0.z 0.z ⇒Change signs of all coefficients except center So the analysis functions turn out to be j ~ ~ ~ φj,k(x) = φj+1,k(x) + ∑ h1[k,m]wj,m(x) a(j,k)={m:k∈A(j,m)} m∈a(j,k) ~ j ~ ~ wj,m(x) = φj+1,m(x) - ∑ h0[k,m]φj+1,k(x) N(j,m)={k:m∈n(j,k)} k∈N(j,m) ~ φ Exercise: verify that φj,k(x), wj,m(x), ~j,k(x), wj,m(x) are biorthogonal. 9 Equations for the DWT: Equations Analysis (from analysis wavelet, refinement equations) dj[m] = j - ∑ h0[k,m]cj+1[k] k∈N(j,m) j +∑ h1[k,m] dj[m] m∈a(j,k) cj+1[m] cj[k] = cj+1[k] Synthesis (invert the lifting operations) cj+1[k] = cj[k] - ∑ e.g. update j h1[k,m]dj[m] m∈a(j,k) cj+1[m] = dj[m] + ∑ predict j h0[k,m]cj+1[k] k∈N(j,m) 10 10 Cubic Interpolating Scaling Function Cubic φ(x) = ∑ h0[k]φ(2x – k) h0[k] = k 11 11 Butterfly Subdivision Butterfly − 1 8 1 16 Also an interpolating function 1 16 1 16 1 2 1 2 − − 1 8 − 1 16 12 12 Loop SubdivisionS Loop Subdivision 1 8 3 8 3 8 1 8 − − − 1 16 5 8 1 16 − − Not an interpolating function 1 16 1 16 − 1 16 1 16 13 13 From: Zorin, Schroder and Sweldens, Interpolating subdivision for meshes with arbitrary topology, proceedings SIGGRAPH 1996. 14 14 ...
View Full Document

This note was uploaded on 12/04/2011 for the course ESD 18.327 taught by Professor Gilbertstrang during the Spring '03 term at MIT.

Ask a homework question - tutors are online