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 … … … … 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

Cascading (linear interpolation) 0,1,0,0 0,0,0,0.5,.5, 0,0 0,.5, 1,.5, 0,0,0,0

0,1 0,0.5,.5 0,.5,1,.5.25,.75,.75,.25 0,.25,.5,.75, 1,.75,.5,.25

Cascading !

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

Interpolating Scaling Functions

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

Properties of Scaling Functions Refinability

Computing the filter coefficients N=2 N=4

Refinement Relations sjsj upsampling s j+1

sjsj upsampling s j+1

Average-Interpolating Subdivision Proposed by Donoho (1993)

Basic Ideas Think of the signals as the intensity obtained from CCD

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

Averaging-interpolating subdivision (constant) Which (constant) polynomial would have produced these average? Subdivide according to the (implied) constant polynomial Order = 1

Average-interpolating subdivision (quadratic) Order = 3 defines the (implied) quadratic curve produce the finer averages accordingly

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

Average-Interpolating (N=3) 3 rd degree polynomial Define 4 conditions: P(x) can be determined

Numeric Example (N=3) Solve for P(1.5) = using Lagrange polynomial (next page) … …

Lagrange Polynomial Details

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

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

Observe that … Utilize inverse Haar transform !

Closed form of quadratic P AI

Three-Stage Lifting

Numerical Example (N=3) 3, 5, 4, 3 0, 0, 0, 0 0.5, 0.25, –0.5, – , 4.875, 4.25, , 5.125, 3.75, Merged Result: 2.75, 3.25, 4.875, 5.125, 4.25, 3.75, 3.125, 2.875

AI Subdivision

AI Scaling Function by Cascading (N=3) 0, 1, 0, 0 0, 0, 0, , 0, –0.25, , 1, 0.125, , 1, , 0 Merged Result: , 0.125, 1, 1, 0.125, , 0, 0

Remark Recall inverse Haar preserves average … Implying … More about this later

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

Average-interpolating scaling functions

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”!?

MRA and Lifting (part I)

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

Graphing by Cascading Scaling functions: delta sequence on hi-wire Wavelets: delta sequence on lo-wire 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)

undecided

Convention: Smaller index, smaller data set (coarser) 2D lifting the same as classical?! Lifting and biorthogonality!?

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.