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Published byVictoria Lang Modified over 9 years ago
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Lifting Part 2: Subdivision Ref: SIGGRAPH96
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Subdivision Methods On constructing more powerful predictors …
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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) –…
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Interpolating Subdivision First proposed by Deslauriers-Dubic
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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.
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Math Review: Lagrange Polynomial The unique n-th degree polynomial that passes through (n+1) points can be expressed as follows:
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Linear and Cubic Interpolation Order = 2 Order = 4
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Numerical Example: Cubic Interpolation Stencil … … … … 9/16 -1/16
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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
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Cascading (linear interpolation) 0,1,0,0 0,0,0,0.5,.5, 0,0 0,.5, 1,.5, 0,0,0,0
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0,1 0,0.5,.5 0,.5,1,.5.25,.75,.75,.25 0,.25,.5,.75, 1,.75,.5,.25
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Cascading !
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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
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Interpolating Scaling Functions
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Properties of Scaling Functions 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
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Properties of Scaling Functions Refinability
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Computing the filter coefficients N=2 N=4
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Refinement Relations sjsj upsampling s j+1
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sjsj upsampling s j+1
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Average-Interpolating Subdivision Proposed by Donoho (1993)
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Basic Ideas Think of the signals as the intensity obtained from CCD
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Meaning of Signal s j,k area = S j,k (width) S j,k : the average signal in this interval p(x)p(x) CCD sensor
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Averaging-interpolating subdivision (constant) Which (constant) polynomial would have produced these average? Subdivide according to the (implied) constant polynomial Order = 1
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Average-interpolating subdivision (quadratic) Order = 3 defines the (implied) quadratic curve produce the finer averages accordingly
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Average-Interpolating (N=3) p(x) is the (implied) quadratic polynomial p(x) is the (implied) quadratic polynomial The coefficient “2” is due to half width
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Average-Interpolating (N=3) 3 rd degree polynomial Define 4 conditions: P(x) can be determined
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Numeric Example (N=3) Solve for P(1.5) =5.4375 using Lagrange polynomial (next page) … 0123 1.5 … 4.875 5.125
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Lagrange Polynomial Details
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Derive Weighting (N=3) Check: If s j,k-1 = s j, k = s j,k+1 = x, P(1.5) = 1.5x = 24x/16
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Consider in-place Computation Solution 1 : compute s j+1,2k+1 first Not a good solution… dependent on execution sequence Problem: occupy the same piece of memory
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Observe that … Utilize inverse Haar transform !
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Closed form of quadratic P AI
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Three-Stage Lifting
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Numerical Example (N=3) 3, 5, 4, 3 0, 0, 0, 0 0.5, 0.25, –0.5, –0.25 2.75, 4.875, 4.25, 3.125 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
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AI Subdivision
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AI Scaling Function by Cascading (N=3) 0, 1, 0, 0 0, 0, 0, 0 0.25, 0, –0.25, 0 -0.125, 1, 0.125, 0 0.125, 1, -0.125, 0 Merged Result: -0.125, 0.125, 1, 1, 0.125, -0.125, 0, 0
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Remark Recall inverse Haar preserves average … Implying … More about this later
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Properties of Scaling Functions 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
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Average-interpolating scaling functions
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Summary 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”!?
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MRA and Lifting (part I)
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MRA Properties Scaling functions at all levels are dilated and translated copies of a single function
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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
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Graphing by Cascading Scaling functions: delta sequence on hi-wire Wavelets: delta sequence on lo-wire More on this later
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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)
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undecided
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Convention: Smaller index, smaller data set (coarser) 2D lifting the same as classical?! Lifting and biorthogonality!?
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From lifting-2 Filter coefficient Refinement relations follow from the fact that subdivision from level 0 with s 0,k and level 1 with s 1,k should be the same.
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