The Lifting Scheme - Topics Reminder:approximations,details Haar wavelet transform Lifting scheme Update Higher order extensions

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  • The Lifting Scheme - Topics Reminder:approximations,details Haar wavelet transform Lifting scheme Update Higher order extensions
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  • One Stage Filtering Approximations and details: The low-frequency content is the most important part in many applications, and gives the signal its identity. This part is called Approximations The high-frequency gives the flavor, and is called Details
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  • Approximations and Details: Approximations:low-frequency components of the signal Details: high-frequency components Input Signal LPF HPF A D
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  • Decimation The former process produces twice the data To correct this, we Down sample (or: Decimate) the filter output by two. A complete one stage block : Input Signal LPF HPF A* D*
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  • Example*: * Wavelet used: db2
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  • Multi-level Decomposition Iterating the decomposition process, breaks the input signal into many lower-resolution components: Wavelet decomposition tree: Low pass filterhigh pass filter
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  • A Simple Example: The Haar Wavelet Consider two neighboring samples a and b of a sequence. - a and b have some correlation. A simple linear transform: High correlation - small |d|, fewer bits representation. (i.e. a=b,d=0)
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  • No loss of any information Reconstruction formula of a and b: The Haar Wavelet Con t The key behind Haar Wavelet Transform: these reconstruction formulas can be found by inverting a 2x2 matrix.
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  • The Haar Wavelet Con t Signal S n of 2 n sample values S n,l : S n = {S n,l | 0=< l odd j,i = even j,i + d n-1 Substituting this into the average (prev), we get : S n-1 = even j,i + even j,i + d n-1 2 S n-1 = even j,I + d n-1 = even + U(d n-1 ) 2
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  • The Lifting Scheme The averages (even elements) become the input for the next recursive step of the forward transform. SnSn S n-1 d n-1 d n-2 S n-2
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  • The Lifting Scheme In place computation: - (odd j-1,even j-1 ): = Split(S j ) - odd j-1 - = P(even j-1 ) - even j-1 + = U(odd j-1 )
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  • Inverse Lifting Scheme
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  • Inverse Lifting Scheme-3 steps Undo Update d n,S 0 given recover even samples by subtructing the update info: S n-1 = even j,i + U( d n-1 ) even j,i = S n-1 -U(d n-1 ) Haar: S n,2l = S n-1,l - d n-1,l/2
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  • Inverse Lifting Scheme - Undo Predict even n-1,d n-1 given recover odd samples by adding prediction info: d n-1 =odd n-1 - P(even n-1 ) odd n-1 = d n-1 + (even n-1 ) Haar: S n,2l+1 =d n-1,l +S n,2l
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  • Inverse Lifting Scheme- Merge zipper odd and even samples recover original signal-inverse Lazy wavelet: S n =Merge(even n-1,odd n-1 ) Inverse in place Even j-1 - = U(odd j-1 ) Odd j-1 + = P(even j-1 ) S j := Merge(odd j-1,even j-1 ) Inverse transform: reversing the order of the operations and flipping the signs
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  • Inverse Lifting Scheme Undo Update - Undo Predict Merge + Even values Odd values
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  • The Lifting Scheme-Example S n = X = [ 1 2 3 4 5 6 7 8 ] 2 n =8, n=3 Split: X e =[2 4 6 8] X o =[1 3 5 7] Pred: averaging neighboures (edges:simple subtruction of one neighbour-can fix by zero padding or wrap around or reflection) Pred{X e } = [2 3 5 7] d n-1 = d 2 = X o - Pred{X e } = [-1 0 0 0]
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  • The Lifting Scheme-Example Update: S n-1 = even j,I + d n-1,l 2 X e =[2 4 6 8] d n-1 = [-1 0 0 0] S 2 = even+d 2 /2 = [1.5 4 6 8] We repeat recursively: Split: S 2 is splitted simillarly: X e =[4 8] X o =[1.5 6] Pred{X e } = [4 6] d n-2 = d 1 = X o - Pred{X e } = [-2.5 0] S 1 = even+d 1 /2 = [2.75 8] X e =[8] X o =[2.75] Pred{X e } = [8] d n-3 = d 0 = X o - Pred{X e } = [-5.25] S 0 = even+d 0 /2 = [5.375]
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  • The Lifting Scheme-Example The pyramid will be: d 2 = [-1 0 0 0] d 1 = [-2.5 0] d 0 = [-5.25] S 0 = [5.375] Good result(most of elements are 0).
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  • The Lifting Scheme- Inverse Example We will recover the original signal from the pyramid: The pyramid : d 2 = [-1 0 0 0], d 1 = [-2.5 0], d 0 = [-5.25], S 0 = [5.375] Inverse transform: X e =S 0 d 0 /2 = [8] Pred{X e } = [8] X o =d 0 +Pred{Xe} = [2.75] S 1 = [2.75 8]
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  • The Lifting Scheme- Inverse Example X e =S 1 d 1 /2 = [4 8] Pred{X e } = 4 6 X o =d 1 + Pred{X e } = [1.5 6] S 2 = [1.5 4 6 8] Xe=S 2 d 2 /2 = [2 4 6 8] Pred{X e } = 2 3 5 7 X o =d 0 + Pred{X e } = [1 3 5 7] S 3 = [1 2 3 4 5 6 7 8]
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  • The Lifting Scheme Suppose Predict and Update are linear. description of their operation as matrices P and U: X o new =X o -PXe X e new =X e +UX o new =X e +UX o -UPX e or: X o new I -P X o X e new U I-UP X e
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  • The Lifting Scheme - Example 2 Assume: P=I and U=0.5I X o new I -p Xo I -I Xo X e new U I-UP Xe 0.5I 0.5I Xe Odd elements Even elements differences average of neighbouring pairs
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  • An example: linear wavelet transform Haar - simple and fast wavelet transform Limitations - not smooth enough: blocky Erasing Haar Coefficients: Fourier analysis not always applicable
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  • The Lifting Scheme Lifting Build more powerful versions An example linear wavelet transform
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  • linear wavelet transform Prediction Linear Prediction - Use even on either side - Predictor for odd sample S n,2l+1 : average of neighboring samples:on left S n,2l, on right S n,2l+2
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  • linear wavelet transform - Prediction
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  • linear wavelet transform Even values are subsampled
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  • linear wavelet transform - Update Sets S n-1 to be the average of even and odd elements 2 n elements => 2 n-1 even elements/averages Example: 2 n =8
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  • linear wavelet transform - Update Ex. Example: 2 n =8 -
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  • linear wavelet transform - Update Ex. sum of the s j-1 elements is equal to the sum of the s j elements, divided by two:
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  • linear wavelet transform - Update Signal average is not preserved Update S n-1,l using previously compute detail signals d n-1,l. Using Neighboring wavelet coefficients: S n-1,l =S n,2l +A(d n-1,l-1 +d n-1,l )
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  • Update S n-1,l = S n,2l +A(d n-1,l-1 +d n-1,l ) A=1/4 to maintain the average Inverse-easy to compute: S n,2l = S n-1,l - 1/4(d n-1,l-1 +d n-1,l ) to recover even S n,2l+1 =d n,l +1/2(S n,2l +S n,2l+2 ) to recover odd samples
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  • linear wavelet transform Original signal
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  • linear wavelet transform
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  • Extend - build higher polynomial order predictors - Use more (D) neighbors on left and right
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  • Example: - Cubic polynom interpolating 4 values - Bilinear Interpolation: the assigned value is an intermediate value between the 4 nearest pixels : aweighted sum of the 4 nearest pixels - Each weight is proportional to the distance from each existing pixel. effective weights: -1/16 9/16 9/16 -1/16
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  • Higher Order Prediction
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  • Lifting Scheme - Predict Linear average of neighboring samples:on d n-1 = S n,2l+1 -1/2(S 2n +S 2n+2 ) Bilinear intermediate value between the 4 nearest pixels Haar The even sample is the prediction for the odd sample. d n-1 = S n,2l+1 - S n,2l
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  • Summary Lifting Scheme - construction of transforms - Haar example - rewriting Haar in place Three steps - split - Predict - Update
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  • Summary Predict - detail coefficient is failure of prediction Update - smooth coefficient to preserve average - B spline C2-ensure smoothness Higher order extensions - increase order of prediction and update
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  • linear wavelet transform - Update (why A=1/4) =1/2Sj[n]