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

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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 ...
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• Spring '03
• GilbertStrang
• Hungary, British K class submarine, Bucharest Metro, Bucharest Metro Lines, Motorway, Motorways in Hungary

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