D Wavelets for Different Sampling Grids and the Lifting Scheme Miroslav Vrankić University of Zagreb, Croatia Presented by: Atanas Gotchev
Lecture Outline 1D wavelets and FWT D separable wavelets D nonseparable wavelets different sampling grids Lifting scheme easy to construct filter banks
Two-Channel Filter Bank Analysis Synthesis x[n] H 0 x 0 [n] G 0 H 1 x 1 [n] G 1 x[n] ^ LP channel: H 0 and G 0 HP channel: H 1 and G 1 PR condition: xˆ [ n] = x[ n n0]
FWT: Analysis Filter Bank Fast wavelet transform enables efficient computation of DWT coefs. Iteration of the analysis FB on the low-pass channel DWT coefficients are computed recursively!
FWT: Analysis Filter Bank
FWT: Analysis Filter Bank
Synthesis Bank
Synthesis Bank
Complexity of FWT Number of operations proportional to: N size of data L length of filters in the filterbank (scaling and wavelet vectors)
Separable wavelet transforms products of 1D wavelet and scaling functions ϕ(x,y) = ϕ(x)ϕ(y) ψ Η (x,y) = ψ(x)ϕ(y) ψ V (x,y) = ϕ(x)ψ(y) ψ D (x,y) = ψ(x)ψ(y)
D separable FWT
Example: Symlets wavelets See functions symaux, dbaux in Wavelet Toolbox
Wavelet and the Scaling Function
D wavelets and scaling function
Sampling in D Image is split into several groups of pixels (phases) Not as straightforward as in 1D Many ways to split an image Separable Quincunx Hexagonal...
Quincunx Downsampling n n 1 Image is split into two phases (cosets) Simplest nonseparable sampling scheme
Subsampling Matrix n (1,1) Basis vectors form the unit cell Subsampling matrix (dilation matrix) defines the sampling operation (1,-1) n 1 D 1 1 = 1 1
Subsampling Matrix Defines the sampling grid For a D grid, D is a x matrix. There are M = det(d) image phases and also M samples in the unit cell. For the quincunx case, M =. Quincunx PR FB needs M = channels.
D Subsampling Operation D defines the sampling grid Take one coset of the image Renumber it to fit on the integer grid 1 1 D ( 1, ) ( 1, ), k n x n n = x k k where = k D n
Quincunx Subsampling Operation For the quincunx case: 1 1 D = 1 1 k1 1 1 n1 n1+ n k = = 1 1 n n n 1 xd ( n1, n ) = x ( n1+ n, n n1 )
Downsampling is actually... reading the image along the new axes. 45 rotation for the quincunx case n n (1,1) (0,1) n 1 (1,0) n 1 (1,-1)
To take the second phase... move the new axes by (1,0)... to the next element of the unit cell. n n (,1) (0,1) n 1 (1,0) n 1 (,-1)
Quincunx Polyphase Decomposition Phase 1 Phase Counterclockwise rotation
Separable Sampling n (0,) (,0) 4 elements of the unit cell Image is split into 4 phases Requires 4 channels of the PR filter bank n 1 0 D = 0
Hexagonal Sampling n (1,) (1,-) 4 elements of the unit cell Image is split into 4 phases Requires 4 channels of the PR filter bank n 1 1 1 D =
Voronoi cell Voronoi cell consists of points closer to the origin... than to any other point of the given lattice. Quincunx Voronoi cell n 1 1 n 1
Effects in the Frequency Domain Downsampling is defined with a D matrix 1 X X X π T D ( ω) = ( D) ( ω) = D ( ω k) det D T k N ( D ) where To avoid aliasing... signal should be bandlimited to Voronoi cell of the lattice defined by πd -T ω ω 1 = ω
Bandlimiting Properly bandlimited signal for quincunx downsampling ω ω π π π π ω 1 π π ω 1
Quincunx downsampling Input image has been properly bandlimited Spectrum support of the downsampled image ω ω π π π π π π ω 1 π π ω 1
Quincunx upsampling x U ( n) 1 x( D n) if n LAT( D) = 0 otherwise n n (0,1) (1,1) (1,0) n 1 n 1 (1,-1)
Upsampling effect on Z-transform ) ( ) ( ) ( ) ( ) ( 1 D Dk k n n n n z z k z n D z n z X x x x X U U = = = = 1 1 1 1 n n n n z z z z = = n z = = 1 1 11 1 1 11 1 1 1 d d d d d d d d z z z z z z D z k D Dk z z ) = ( Exercise: prove that
Frequency transformation ω z j e ω D D z T j d d j d d j d d d d e e e z z z z = = + + ) ( ) ( 1 1 1 1 1 1 11 1 1 11 ω ω ω ω ) ( ) ( ω D ω T U X X = Conclusion:
Quincunx upsampling X ( ω) XU T ( ω) = X( D ω) ω ω π π π π π π ω 1 π π ω 1
Iterated quincunx upsampling T XU ( ω ) = X ( D ω ) ω π ( T ) XU ( ω ) = X ( D ) ω π ω 1 ω π π ω 1 ω π ( 3 T ) XU ( ω ) = X ( D ) ω π ω 1
The Lifting Scheme Simple way to construct filter banks Easy to satisfy PR requirement Computationally efficient X(z) z -1 P(z) - + U(z) A(z) D(z) - U(z) P(z) + z -1 X(z) ^
The Lifting Scheme Basic structure: Polyphase decomposition Predict stage (dual lifting step) Update stage (primal lifting step) X(z) z -1 P(z) - + U(z) A(z) D(z) - U(z) P(z) + z -1 X(z) ^
Predict stage Prediction of the second phase sample...based on a number of samples from the first phase. Wavelet coefficients are obtained as... a prediction error. X(z) Smooth signal... gives small details. z -1 P(z) - D(z)
Update stage Input: detail coefs. Output is used to create approximation coefs. Average value of the input image must be retained. X(z) A(z) z -1 P(z) - + U(z) D(z)
Lifting Scheme in -D X(z 1,z ) D X e + A - D P(z 1,z ) U(z 1,z ) U(z 1,z ) P(z 1,z ) z 1-1 D X o - D + D z 1 X(z ^ 1,z ) similar structure as 1-D D polyphase decomposition D filters
Quincunx FB Example Lifting scheme based on quincunx interpolating filters J. Kovačević & W. Sweldens: Wavelet Families of Increasing Order in Arbitrary Dimensions. IEEE Trans. Image Proc., vol. 9, no. 3, pages 480-496, March 000.
Predict Filters Neville interpolating filters symmetric interpolation neighborhoods n example of a second order P filter: 1 1 1 P ( z1, z) = 0.5 + 0.5z1 + 0.5z + 0. 5z1 z n 1 1
Supports of the Prediction Filters
Update Filters updates the average value of the input image based on the corresponding predict filter 1 * UN( z1, z) = PN( z1, z)
Transfer Functions for P 4 and U Analysis LP Analysis HP Synthesis LP Synthesis HP
Wavelet and Scale for P 4 and U Analysis scale Analysis wavelet Synthesis scale Synthesis wavelet
Wavelet Decomposition Tree A J-1 A J- A J-3 DJ-1 DJ- DJ-3
Separable Versus Nonseparable Nonseparable higher complexity more freedom in FB design different directional properties Separable widely used simple realization based on 1D filter banks
Quincunx Wavelets Simplest nonseparable sampling grid Only two channels Double quincunx sampling = nonseparable sampling Less biased in horizontal and vertical directions Comparable results with separable wavelets