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## Lifting

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**Lifting**Part 3: Designing Update Operators Ref: SIGGRAPH 96**General Concepts**• We discussed several ways of realizing predictors in the context of an inverse transform with zero wavelet coefficients • Now we discuss how to design “Update” boxes • Design objective: • Ensure coarsened signal has the same average (and/or other properties) as the original (higher resolution) signal**Begin with m DOFs**2m DOFs after one step The extra m DOFs: the difference between Vj and Vj+1 in resolution Design update boxes is all about manipulating the inbetweeen DOFs So that the coarser signal (m DOFs) and finer signal (2m DOFs) are “similar” DOF Analysis of Inverse Transform**Recall Cascade Algorithm**• Scaling functions and wavelets**Interpolating Lifting (linear)**w/o U boxes !**Merge**PI Details (lo-wire)**Merge**Details**Observation**• These “wavelet” generated by the low wire do not have zero integral, a condition must be satisfied to ensure equality of the sample average • Manipulate the extra m DOF to achieve this goal • Design methodology: • Start from lazy wavelet and incorporate scaling functions of the same level to build a more performing wavelet Interlace splitting**…0,0,1,0,0…**Hi-Wire (after U) Lo-Wire (after P) Merged result should sum up to zero: A=1/4**even**odd Forward Linear Wavelet Transform**even**odd Inverse Linear Wavelet Transform**Wavelet Basis**• Cascading on the lower wire (with update) to get wavelet function**sj**upsampling Refinement Relations dj+1**0,-1/4, -1/4,0**0,0,0,0 0,-1/8,-1/4,3/4,-1/4,-1/8,0,0 0,1,0,0 -1/8,3/4,-1/8,0 Hi-wire (after U) Lo-wire (after P) Merged result Cascading of Linear Wavelet**0,0,0,0**0,1,0,0 Cascading to Reach Limit Function**More Powerful Update Operators**• Ensure not only the average but also the first moments of the sequences are preserved:**N = 4, = 4**0th moment (average): 2nd moment: Additionally, 1st and 3rd moment vanishes due to symmetry**Design Higher Order Update**• Note that symmetry is the key of this form • Solve A and B by requiring the vanishing of 0th and 2nd moment • Due to symmetry, 1st and 3rd moments are zero • Dual order is 4**sj-1**sj PAI UHaar PHaar Split dj-1 Average-Interpolating (AI) Lifting • Note that there is a slight difference for AI Lifting: • The lazy wavelet (interlace splitting) is followed by a Haar transform. Therefore, the forward transform looks like:**sj-1**sj PAI UHaar PHaar Split dj-1 Properties of AI Lifting • Use the update of Haar to preserve average • No need to further design new update operator (unless further vanishing moments requested) Same Average! (from Haar)**sj-1**sj dj-1 AI Lifting (cont) • Inverse Transform**Ex: AI Transform (forward)**sj+1,k After PHaar After UHaar After PAI sj,k: coarsened signal dj,k: difference signal**Ex: AI Transform (inverse)**3, 4, 1, 4 4, 3, 2, 3.5 3, 5, 4, 2, 1, 3, 4, 3 2.125, -1.5, 1.875, -1.5 2, -2, 2, -1 5, 2, 3, 3**0, -1/2, 0, 0**0, 0, 0, 0 0, 0, -1/2, 1/2, 0, 0, 0, 0 0, 1, 0, 0 0, 1/2, 0, 0 0, 1/16, -9/16, 7/16, 1/16, 0, 0, 0 0, -1/16, -7/16, 9/16, -1/16, 0, 0, 0 0, -1/8, 1/8, 1/8, -1/8, 0, 0, 0 AI Wavelets by Cascading (N=3) Merged result: 0, 0, 1/16, -1/16, -9/16, -7/16, 7/16, 9/16, 1/16, -1/16, 0, 0, 0**Function Projection**• Consider an initial signal, its coarser approximation: • Their difference lies in the space spanned by the wavelet functions**There are many things related to the definition of dn-1(x)**how dn-1,l is computed (the forward transform) The basis (wavelets) If the order of MRA is N, then the wavelet transform started from any polynomial sn(x) with degree less than N will yield zero dn-1,l That is, …**Dual Order of MRA**• We say the dual order is if the wavelets have vanishing moments: • All wavelets (translated and dilated) have the same vanishing moments • Recall: moment, time shifting, time scaling**Dual Order of MRA**• As a result, all detail functions dj(x), represented as linear combination of the wavelet basis, have the same vanishing moments • And, all coarser versions of sj(x) have the moments independent of j quantitatively stating what properties are preserved during coarsening**Homeworks**• derive update operator for cubic interpolation • Implement cascading for graphing wavelets • Verify biorthogonality of linear lifting • Compare interpolating and AI