Tree-based Filter Banks and based Filter Banks and...

ENEE630 P tENEE630 P t 11

TreeTree--based Filter Banks andbased Filter Banks and

ENEE630 PartENEE630 Part--11

TreeTree--based Filter Banks and based Filter Banks and MultiresolutionMultiresolution AnalysisAnalysis

CECE DepartmentUniv. of Maryland, College Park

Updated 10/2012 by Prof. Min Wu. bb eng md ed (select ENEE630) min @eng md ed

M. Wu: ENEE630 Advanced Signal Processing (Fall 2010)

bb.eng.umd.edu (select ENEE630); [email protected]

Min Wu

Text Box

Supplement

Dynamic Range of Original Dynamic Range of Original andand SubbandSubband SignalsSignalsand and SubbandSubband SignalsSignals

– Can assign more bits to represent coarse info

– Allocate remaining bits, if available, to finer details (via proper quantization)

M. Wu: ENEE630 Advanced Signal Processing [8]

Figures from Gonzalez/ Woods DIP 3/e book website.

Brief Note on Brief Note on SubbandSubband and Wavelet Codingand Wavelet Coding

The octave (“dyadic”) frequency partition can reflect the logarithmatic characteristics in human perceptiong p p

Wavelet coding and subband coding have many similarities (e.g. from filter bank perspectives)( g p p )– Traditionally subband coding uses filters that have little

overlap to isolate different bands

– Wavelet transform imposes smoothness conditions on the filters that usually represent a set of basis generated by shifting and scaling (“dilation”) of a “mother wavelet” functionshifting and scaling ( dilation ) of a mother wavelet function

– Wavelet can be motivated from overcoming the poor time-domain localization of short-time FT

Explore more in Proj#1. See PPV Book Chapter 11


Review and Examples of BasisReview and Examples of Basis

Standard basis vectors

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M. Wu: ENEE631 Digital Image Processing (Spring 2010) Lec10 – Unitary Transform [10]

TimeTime--Freq (or SpaceFreq (or Space--Freq) InterpretationsFreq) Interpretations– Inverse transf. represents a signal as a linear combination of basis vectors– Forward transf. determines combination coeff. by projecting signal onto basisE.g. Standard Basis (for data samples); Fourier Basis; Wavelet Basis


Figures from Gonzalez/ Woods DIP 2/e book website.

Recall: Matrix/Vector Form of DFT Recall: Matrix/Vector Form of DFT

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M. Wu: ENEE630 Advanced Signal Processing (Fall'09) 10/14/2009 [13]

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Example of 1Example of 1--D DCT: N = 8D DCT: N = 8From Ken Lam’s DCT talk 2001 (HK Polytech)

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Basis vectors Reconstructions

Haar TransformHaar Transform 1x

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filtering and downsampling

– Relatively poor energy compaction Equiv. filter response doesn’t have good

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=> Basis images of 2-D Haar transform

Compare Basis Images of DCT and HaarCompare Basis Images of DCT and Haar


See also: Jain’s Fig.5.2 pp136UMCP ENEE631 Slides (created by M.Wu © 2001)

Compressive SensingCompressive Sensing

Downsampling as a data compression tool– For bandlimited signals. Considered uniform sampling so farFor bandlimited signals. Considered uniform sampling so far

More general case of “sparsity” in some domainE.g. non-zero coeff. at a small # of frequencies but over a broad support

of frequency?

H l h i d d li ?– How to leverage such sparsity to get reduced average sampling rate?– Can we sample at non-equally spaced intervals?– How to deal with real-world issues e.g. approx. but not exactly sparse?g pp y p

Ref: IEEE Signal Processing Magazine: Lecture Notes on Compressive Sensing Ref: IEEE Signal Processing Magazine: Lecture Notes on Compressive Sensing (2007); Special Issue on Compressive Sensing (2008);(2007); Special Issue on Compressive Sensing (2008);

UMD ENEE630 Advanced Signal Processing (F'10) Discussions [20]

ENEE698A Fall 2008 Graduate Seminar: ENEE698A Fall 2008 Graduate Seminar: http://terpconnect.umd.edu/~dikpal/enee698a.htmlhttp://terpconnect.umd.edu/~dikpal/enee698a.html

L1 vs. L2 OptimizationL1 vs. L2 Optimizationfor Sparse Signalfor Sparse Signalfor Sparse Signalfor Sparse Signal


( Fig. from Candes-Wakins SPM’08 article)

Example: Tomography problemExample: Tomography problem

Logan-Shepp phantomtest image

Sampling in the frequency planeAlong 22 radial lines with 512

samples on each

Minimum energy reconstruction

Reconstruction by minimizingtotal variation


Slide source: by Dikpal Reddy, ENEE698A, http://terpconnect.umd.edu/~dikpal/enee698a.html

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A Close Look at Wavelet TransformA Close Look at Wavelet Transform

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Orthonormal Wavelet FiltersOrthonormal Wavelet Filters

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Biorthogonal Wavelet FiltersBiorthogonal Wavelet FiltersU


Construction of Haar FunctionsConstruction of Haar Functions Unique decomposition of integer k (p, q)

– k = 0, …, N-1 with N = 2n, 0 <= p <= n-1q = 0 1 (for p=0); 1 <= q <= 2p (for p>0) k = 2p + q 1

power of 2

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– q = 0, 1 (for p=0); 1 <= q <= 2p (for p>0)

e.g., k=0 k=1 k=2 k=3 k=4 …(0,0) (0,1) (1,1) (1,2) (2,1) …

k = 2p + q – 1“remainder”

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More on Wavelets (1)More on Wavelets (1)20

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Linear expansion of a function via an expansion set

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– Form basis functions if the expansion is unique

Orthogonal basis

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Non-orthogonal basis – Coefficients are computed with

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Discrete Wavelet TransformWavelet expansion gives a setU – Wavelet expansion gives a set of 2-parameter basis functions and expansion coefficients:scale and translation


scale and translation

More on Wavelets (2)More on Wavelets (2)20

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1st generation wavelet systems:– Scaling and translation of a generating wavelet (“mother wavelet”)

Multiresolution conditions:

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– Use a set of basic expansion signals with half width and translated in half step size to represent a larger class of signals than the original expansion set (the “scaling function”)

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Orthonormal FiltersOrthonormal Filters Equiv. to projecting input signal to orthonormal basis

Energy preservation property2001

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Energy preservation property– Convenient for quantizer design

MSE by transform domain quantizer is same as reconstruction MSE in image domained

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– Length of output per stage grows ~ undesirable for compression

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Solutions to coefficient expansion– Symmetrically extended input (circular convolution) &

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Solutions to Coefficient ExpansionSolutions to Coefficient Expansion Circular convolution in place of linear con ol tion Circular convolution in place of linear convolution

– Periodic extension of input signal– Problem: artifacts by large discontinuity at borders

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Solutions to Coefficient Expansion (cont’d)Solutions to Coefficient Expansion (cont’d)

Symmetric extension + symmetric filters– No coefficient expansion and little artifacts20

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No coefficient expansion and little artifacts– Symmetric filter (or asymmetric filter) => “linear phase filters”

(no phase distortion except by delays)

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Only one set of linear phase filters for real FIR orthogonal wavelets Haar filters: (1, 1) & (1,-1)

do not give good energy compaction

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Ref: review ENEE630 discussions on FIR perfect reconstruction Qudrature Mirror Filters (QMF) for 2-channel filter banks.

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Biorthogonal WaveletsBiorthogonal WaveletsFrom Usevitch (IEEE Sig.Proc. Mag. 9/01)

“Biorthogonal”– Basis in forward and inverse transf.

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are not the same but give overall perfect reconstruction (PR) recall EE630 PR filterbank

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– No strict orthogonality for transf. filters so energy is not preserved But could be close to orthogonal

filters’ performance1 S

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including symmetric filters that eliminate coefficient expansion

Commonly used filters for compression– 9/7 biorthogonal symmetric filter

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g y– Efficient implementation: Lifting approach (ref: Swelden’s tutorial)

Smoothness Conditions on Wavelet FilterSmoothness Conditions on Wavelet Filter20

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– Ensure the low band coefficients obtained by recursive filtering can provide a smooth approximation of the original signal

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From M. Vetterli’s wavelet/filter-bank paper

Tree-based Filter Banks and based Filter Banks and...

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