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## Lifting

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**Lifting**Part 2: Subdivision Ref: SIGGRAPH96**Subdivision Methods**On constructing more powerful predictors …**Subdivision methods**• Often referred to as the cascade algorithm • Systematic ways to build predictors • Concentrate on the P box • Types: • Interpolating subdivision • Average-interpolating • B-spline (more later) • …**Interpolating Subdivision**First proposed by Deslauriers-Dubic**Basic Ideas**• In general, use N (=2D) samples to build a polynomial of degree N-1 that interpolates the samples • Calculate the coefficient on the next finer level as the value of this polynomial • e.g., Lagrange polynomial (or Neville’s algorithm) • Order of the subdivision scheme is N • This can be extended to accommodate bounded interval and irregular sampling settings.**Math Review: Lagrange Polynomial**• The unique n-th degree polynomial that passes through (n+1) points can be expressed as follows:**Linear and Cubic Interpolation**Order = 2 Order = 4**…**… … … Numerical Example: Cubic Interpolation Stencil -1/16 9/16 9/16 -1/16**Scaling Functions**• All scaling functions at different levels are translates and dilates of one fixed function: • the fundamental solution (so named by the original inventor, Deslauriers-Dubuc) of the subdivision scheme • Obtained by cascade algorithm**0,1,0,0**0, .5, 1, .5, 0,0,0,0 0,0,0,0 .5, .5, 0,0 Cascading (linear interpolation)**0,1**0, .5,1, .5 0, .25, .5, .75, 1, .75, .5, .25 0,0 .5, .5 .25, .75, .75, .25**Compare with what we said before …**• From forward transform • Hi-wire: coarsened signal • Lo-wire: difference signal • Subdivision: Inverse transform with zero detail • Cascading: apply delta sequence to get impulse response (literally) • Hi-wire: scaling functions • Lo-wire: wavelets**Compact support**[-N+1, N-1] Interpolating Smoothness N large, smoother … Polynomial reproduction Polynomials up to degree N-1 can be expressed as linear combinations of scaling functions Properties of Scaling Functions**Refinability**Properties of Scaling Functions**upsampling**sj+1 Refinement Relations sj**sj**upsampling sj+1**Average-Interpolating Subdivision**Proposed by Donoho (1993)**Basic Ideas**Think of the signals as the intensity obtained from CCD**Meaning of Signal sj,k**p(x) area = Sj,k (width) Sj,k: the average signal in this interval CCD sensor**Which (constant) polynomial would have produced these**average? Subdivide according to the (implied) constant polynomial Averaging-interpolating subdivision (constant) Order = 1**defines the (implied) quadratic curve**produce the finer averages accordingly Average-interpolating subdivision (quadratic) Order = 3**Average-Interpolating (N=3)**The coefficient “2” is due to half width p(x) is the (implied) quadratic polynomial**3rd degree polynomial**Average-Interpolating (N=3) Define 4 conditions: P(x) can be determined**…**… 0 1 2 3 1.5 Numeric Example (N=3) 4.875 5.125 Solve for P(1.5) =5.4375 using Lagrange polynomial (next page)**Details**Lagrange Polynomial**Derive Weighting (N=3)**Check: If sj,k-1 = sj, k= sj,k+1 = x, P(1.5) = 1.5x = 24x/16**Consider in-place Computation**Problem: occupy the same piece of memory Solution 1 : compute sj+1,2k+1 first Not a good solution… dependent on execution sequence**Utilize inverse Haar transform !**Observe that …**Numerical Example (N=3)**2.75, 4.875, 4.25, 3.125 3, 5, 4, 3 0, 0, 0, 0 0.5, 0.25, –0.5, –0.25 3.25, 5.125, 3.75, 2.875 Merged Result: 2.75, 3.25, 4.875, 5.125, 4.25, 3.75, 3.125, 2.875**AI Scaling Function by Cascading (N=3)**-0.125, 1, 0.125, 0 0, 1, 0, 0 0, 0, 0, 0 0.125, 1, -0.125, 0 0.25, 0, –0.25, 0 Merged Result: -0.125, 0.125, 1, 1, 0.125, -0.125, 0, 0**Remark**• Recall inverse Haar preserves average … • Implying … • More about this later**Compact support**[-N+1, N] Average-interpolating Polynomial reproduction Up to degree N-1 Smoothness: continuous of order R(N) Refinability: Obtained similarly as in interpolating subdivision Properties of Scaling Functions**Types of Predictors:**Interpolating Average-interpolating B-spline So far, we only considered subdivision in inverse transform. How about its role in forward transform? Roles of Predictors In inverse transform Subdivision In forward transform: Predict results to generate the difference signal (low-wire) More … On constructing more powerful P boxes Define “power”!? Summary**MRA Properties**• Scaling functions at all levels are dilated and translated copies of a single function**Order of an MRA**• The order of MRA is N if every polynomial of degree < N can be written exactly as a linear combination of scaling functions of a given level • The order of MRA is the same as the order of the predictor used to build the scaling functions**Scaling functions: delta sequence on hi-wire**Wavelets: delta sequence on lo-wire Graphing by Cascading More on this later**Homeworks**• Derive the weights for cubic interpolation • Implement cascading to see scaling functions (and wavelets) at different levels • Use lifting to process audio data • Provide routines for read/write/plot data • denoising radio recordings (WAV)**Convention:**• Smaller index, smaller data set (coarser) • 2D lifting the same as classical?! • Lifting and biorthogonality!?**Filter coefficient**From lifting-2 Refinement relations follow from the fact that subdivision from level 0 with s0,k and level 1 with s1,k should be the same.