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Course 18.327 and 1.
Wavelets and Filter Banks
Wavelets and subdivision: nonuniform
grids; multiresolution for triangular
meshes; representation and
compression of surfaces.
2
Wavelets on Surfaces in R 3
Construction by Schröder and Sweldens
- uses lifting
- scaling functions are interpolating in most straightforward case
- typically work with triangular mesh generated by subdivision k (^1)
k (^2)
k (^4) k (^3)
k (^5)
k (^6)
m (^1)
m (^2)
m (^4) m (^3)
m (^5)
m (^6)
k
K(j) = {k, k 1 , k 2 , k 3 , k 4 , k 5 , k 6 , …} M(j) = {m 1 , m 2 , m 3 , m 4 , m 5 , m 6 , …}
mesh describing surface S
3
Notation: K(j) = all vertices at resolution j K(j + 1) = all vertices at resolution j + 1 M(j) = 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 1 if x = x (^) k k ∈∈∈∈ K(j)
0 if x = x (^) k′′′′ k′∈′∈′∈′∈ K(j) k′′′′ ≠≠≠≠ k
x = position vector of a point on S.
,
φφφφj,k(x) =
4
Simple interpolating scaling function: hat function
k (^1)
k (^2)
k 4 k^3
k (^5)
m (^1) m (^2) m (^3)
m (^5) m (^4)
k
φφφφj,k(x) Scaling functions at level j are all located at vertices in K(j)
Refinement equation φφ φφj,k(x) = φφφφj+1,k(x) + ½ ¦¦¦¦ φφφφj+1,m (x) In general, interpolating scaling functions will satisfy a refinement equation of the form φφ φφj,k(x) = φφφφj+1,k(x) + ¦¦¦¦ h 0 [k,m]φφφφj+1,m (x)
m (^6) m=m 1
m∈∈∈∈n(j,k)
j
m (^6)
k (^6)
7
To satisfy vanishing moment condition, choose ααααi = I j+1,m/2 I j,ki i = 1, 2 So the wavelet equation can be written as w (^) j,m (x) = φφφφj+i,m (x) - ¦¦¦¦ h 1 [k,m]φφφφj,k(x)
with
A(j,m) = two immediate neighbors in K(j) h 1 [k,m] = I j+1,m/2 I j,k
k∈∈∈∈A(j,m)
j
j
8
Wavelets on Surfaces in R^3
Synthesis scaling function
φφφφj,k(x) = φφφφj+1,k(x) + ¦¦¦¦ h 0 [k,m]φφφφj+1,m (x)
Linear interpolating functions: h 0 [k,m] = { ½^ m∈∈∈∈n(j,k)
k
k
k
k4 k
k
k
m
m m
m
m
m
m∈∈∈∈n(j,k)
j
n(j,k) = {m 1 , m 2 , m 3 , m 4 , m 5 , m 6 }
0 otherwise
j
Synthesis wavelet
w (^) j,m (x) = φφφφj+1,m (x) - ¦¦¦¦ h 1 [k,m] φφφφj,k(x) k∈∈∈∈A(j,m) k^1 m k (^2)
j
A(j,m) = {k 1 , k 2 }
9
What are the analysis functions?
Use alternating signs condition to get analysis filters, e.g. 1D interpolating filter
If F 0 (z) = {-z^3 + 0•z^2 + 9z + 16 + 9z -1^ + 0•z-2^ – z -3} then H 1 (z) = F 0 (-z) = {z^3 + 0.z^2 - 9z + 16 - 9z-1^ + 0.z-2^ + z- Change signs of all coefficients except center
So the analysis functions turn out to be
φφφφj,k(x) = φφφφj+1,k(x) + ¦¦¦¦ h 1 [k,m]w (^) j,m (x) a(j,k)={m:k∈∈∈∈A(j,m)}
w (^) j,m (x) = φφφφj+1,m (x) - ¦¦¦¦ h 0 [k,m]φφφφj+1,k(x) N(j,m)={k:m∈∈∈∈n(j,k)}
Exercise: verify that φφφφj,k(x), w (^) j,m (x), φφφφj,k(x), w (^) j,m (x) are biorthogonal.
16 16
m∈∈∈∈a(j,k)
~ ~ j ~
~ k∈∈∈∈N(j,m)
~ ~ j
10
Equations for the DWT:
Analysis (from analysis wavelet, refinement equations)
dj[m] = c j+1[m] - ¦¦¦¦ h 0 [k,m]c j+1[k] predict c j[k] = c j+1[k] + ¦¦¦¦ h 1 [k,m] d j[m] update
Synthesis (invert the lifting operations)
c j+1[k] = c j[k] - ¦¦¦¦ h 1 [k,m]dj[m]
c j+1[m] = dj[m] + ¦¦¦¦ h 0 [k,m]c j+1[k]
e.g.
m∈∈∈∈a(j,k)
j k∈∈∈∈N(j,m) (^) j
m∈∈∈∈a(j,k)
j
k∈∈∈∈N(j,m)
j
13
Loop Subdivision
8
1
8
1
8
3 8
3
8
5
16 −^1 16 −^1
16 −^1 16 −^1
16 − 1 16 −^1
Not an interpolating function
From: Zorin, Schroder and Sweldens, Interpolating subdivision for meshes with arbitrary topology,proceedings SIGGRAPH 1996. 14
