Concepts and Algorithms of Scientific and Visual Computing - PowerPoint PPT Presentation

Concepts and Algorithms of Scientific and Visual Computing –Multiresolution Analysis– CS448J, Autumn 2015, Stanford University Dominik L. Michels

Multiresolution Analysis The multiresolution Analysis originally developed in [Mallat 1989] and [Meyer 1992] lays the theoretical foundation of the fast wavelet transform (FWT). Consider a signal f from a subspace V − 1 of L 2 ( R ), which we would like to decompose in its high frequency (rough) and its low frequency (smooth) part. The smooth part is described by an orthogonal projection P 0 f onto a smaller space V 0 containing the smooth functions from V − 1 . The orthogonal complement W 0 of V 0 in V − 1 contains the rough parts in V − 1 . Let P 0 denote the orthogonal projection onto W 0 , such that f = P 0 f + Q 0 f , V − 1 = V 0 ⊕ W 0 . Similarly, V 0 is described as the orthogonal sum of V 1 and W 1 . Let P 1 and Q 1 be the corresponding projections, such that P 0 f = P 1 P 0 f + Q 1 P 0 f , V 0 = V 1 ⊕ W 1 .

Multiresolution Analysis Because of P 1 P 0 f = P 1 f and Q 1 P 0 f = Q 1 f we obtain P 0 f = P 1 f + Q 1 f and therefore f = P 1 f + Q 1 f + Q 0 f . In the next step, P 1 f is decomposed in P 2 f and Q 2 f . Continuing this recursively leads to P 0 P 1 L 2 ( R ) ··· → − − → V 0 − − → V 1 ··· → { 0 } V − 1 ց Q 0 ց Q 1 ... W 0 W 1 ... In general, the projections P m f and Q m f describe the smooth and rough parts on the scale described by the space V m − 1 .

Multiresolution Analysis More precisely, a multiresolution analysis (MRA) of L 2 ( R ) contains a sequence ( V m ) m ∈ Z of closed subspaces V m ⊂ L 2 ( R ) and a scaling function ϕ ∈ V 0 with the following properties: 1 { 0 } ⊂ ··· ⊂ V 2 ⊂ V 1 ⊂ V 0 ⊂ V − 1 ⊂ V − 2 ⊂ ··· ⊂ L 2 ( R ), 2 ∪ m ∈ Z V m = L 2 ( R ) and ∩ m ∈ Z V m = { 0 } , 3 the spaces V m are scaled versions of V 0 , i.e. f ( · ) ∈ V m iff f (2 m · ) ∈ V 0 , 4 the set of the shifts ϕ ( · − k ), k ∈ Z is an orthonormal basis of V 0 . From (3) and (4) follows that ( ϕ m , k ) k ∈ Z with ϕ m , k ( x ) := 2 − m / 2 ϕ (2 − m x − k ) is a Hilbert basis of V m .

Filter Coefficients For a MRA (( V m ) m ∈ Z ,ϕ ), the sequence h ∈ ℓ 2 ( Z ) with h k = � ϕ | ϕ − 1 , k � fulfills the scaling equation � ϕ = h k ϕ − 1 , k k ∈ Z √ 2 � respectively ϕ ( x ) = k ∈ Z h k ϕ (2 x − k ). Furthermore the coefficients h k appear on every scale, i.e. � ϕ m , k = h j ϕ m − 1 , 2 k + j , j ∈ Z and fulfill the orthogonality relation � h k +2 j ¯ h k = δ 0 , j . k ∈ Z

Filter Coefficients Let us consider the Haar MRA with ϕ = χ [0 , 1) = 2 − 1 / 2 ( ϕ − 1 , 0 + ϕ − 1 , 1 ) , h 0 = h 1 = 2 − 1 / 2 and h k = 0 for all k ∈ Z \ { 0 , 1 } , and so-called wavelet coefficients g h 1 − k , i.e. g 0 = − g 1 = 2 − 1 / 2 and g k = 0 for all k ∈ Z \ { 0 , 1 } . with g k := ( − 1) k ¯ The corresponding frequency responses are given by H ( ω ) = 2 − 1 / 2 (1 + exp( − 2 π i ω )) , G ( ω ) = H ( ω + 1 / 2) . More precisely, it can be shown, that the convolution with h is a low-pass filter and the convolution with g is a high pass filter.

Filter Coefficients and Wavelets The coefficients of h allow for the construction of a mother wavelet ψ , whose shifted and scaled versions are Hilbert bases of the spaces W m . More precisely for a MRA (( V m ) m ∈ Z ,ϕ ) with scaling coefficients h ∈ ℓ 1 ( Z ) and wavelet coefficients g k := ( − 1) k ¯ h 1 − k we define ψ ∈ V − 1 with � ψ ( x ) := g k ϕ − 1 , k ( x ) . k ∈ Z For the functions ψ m , k ( x ) := 2 − m / 2 ψ (2 − m x − k ) , m , k ∈ Z , the following statements hold: 1 ψ m , k = � j ∈ Z g j ϕ m − 1 , 2 k + j , 2 ( ψ m , k ) k ∈ Z is a Hilbert basis of W m , 3 ( ψ m , k ) m , k ∈ Z is a Hilbert basis of L 2 ( R ) = � m ∈ Z W m , 4 ψ = ψ 0 , 0 is a wavelet.

Fast Wavelet Transform (FWT) Instead of computing the wavelet coefficients � f | ψ s , t � by approximating the integral, a discrete version of the signal f is low-pass filtered with h and high-pass filtered with g in the fast wavelet transform (FWT). Let (( V m ) m ∈ Z ,ϕ ) be a MRA with scaling coefficients ϕ = � k ∈ Z h k ϕ − 1 , k . Consider f ∈ V 0 . Since ( ϕ 0 , k ) k ∈ Z is a Hilbert basis of V 0 , there exists a uniquely determined sequence ( v 0 k ∈ Z v 0 k ) k ∈ Z ∈ ℓ 2 ( Z ) with Fourier series representation f = � k ϕ 0 , k . Let ψ k ∈ Z g k ϕ − 1 , k with g k := ( − 1) k ¯ be the wavelet corresponding to ϕ , i.e. ψ = � h 1 − k . Then ( ψ m , k ) m , k ∈ Z with ψ m , k ( x ) = 2 − m / 2 ψ (2 − m x − k ) is a Hilbert basis of L 2 ( R ) for which reason we only have to evaluate the CTWT ˜ f ( s , t ) at the specific positions ( s , t ) ∈ { (2 m , 2 m k ) | m , k ∈ Z } . Because of � V 0 = W m , m > 0 is it sufficient to compute ˜ f (2 m , 2 m k ) = � f | ψ m , k � for all m ∈ Z > 0 and k ∈ Z .

Fast Wavelet Transform (FWT) We define ( w m k ) k ∈ Z ∈ ℓ 2 ( Z ) with w m k := � f | ψ m , k � and ( v m k ) k ∈ Z ∈ ℓ 2 ( Z ) with v m k := � f | ϕ m , k � , so that we obtain from ψ m , k = � l ∈ Z g l ϕ m − 1 , 2 k + l and ϕ m , k = � l ∈ Z h l ϕ m − 1 , 2 k + l , � � g l − 2 k v m − 1 h l − 2 k v m − 1 ¯ w m v m k = ¯ , k = l l l ∈ Z l ∈ Z due to the substitution l ← 2 k + l . According to this, we define the operators � � ¯ ( H v ) k := h l − 2 k v l , ( G v ) k := ¯ g l − 2 k v l . l ∈ Z l ∈ Z These operators can be seen as convolutions with filter coefficients ˜ h respectively ˜ g with ˜ h k = ¯ h − k , ˜ g k = ¯ g − k for k ∈ Z , followed by a downsampling operator ( ↓ 2)( x )( n ) := x (2 n ), i.e. w m = ( ↓ 2)(˜ v m = ( ↓ 2)(˜ g ∗ v m − 1 ) , h ∗ v m − 1 ) .

Fast Wavelet Transform (FWT) Fast wavelet transform for input Fourier coefficients v 0 = ( v 0 k ) k ∈ Z . function FWT( v 0 , M (number of scales)) begin for m ← 1 to M do w m ← G v m − 1 v m ← H v m − 1 end return ( w 1 ,..., w M , v M ) end The mapping v 0 �→ ( w 1 ,..., w M , v M ) given by the FWT is based on the decomposition  M  �   V 0 =  ⊕ V M .  W m         m =1 In practice, samples f ( k ) = � f | δ 0 , k � ≈ � f | ϕ 0 , k � = v 0 k are used.

Inverse Fast Wavelet Transform (IFWT) The mapping v 0 �→ ( w 1 ,..., w M , v M ) is an isomorphism. Its inverse can be described using the adjunct operators � � ( H ∗ v ) k := ( G ∗ v ) k := h k − 2 l v l , g k − 2 l v l , l ∈ Z l ∈ Z describing an upsampling step with ( ↑ 2)( x )( n ) := x ( n / 2) for even inputs x and ( ↑ 2)( x )( n ) := 0 for odd inputs x , followed by convolutions with filter coefficients h respectively g , so that a reconstruction step v m − 1 = H ∗ v m + G ∗ w m from scale m to scale m − 1 by can be realized by v m − 1 = h ∗ ( ↑ 2)( v m ) + g ∗ ( ↑ 2)( w m ) .

Inverse Fast Wavelet Transform (IFWT) Inverse Fast wavelet transform for input decomposition ( w 1 ,..., w M , v M ). function FWT( w 1 ,..., w M , v M )) begin for m ← M down to 1 do v m − 1 ← H ∗ v m + G ∗ w m end return v 0 end The complexities of both transform, FWT and IFWT, are in O ( | h | N ), in which N denotes the length of the input signal and | h | the length of the filter coefficients h and g . Since | h | ≪ N , it can be considered as a program constant leading to linear complexity O ( N ) .