M. Wu: ENEE631 Digital Image Processing (Spring'09) Wavelet Coding and Related Issues Spring ’09 Instructor: Min Wu Electrical and Computer Engineering.

M. Wu: ENEE631 Digital Image Processing (Spring'09)

Wavelet Coding and Related IssuesWavelet Coding and Related Issues

Spring ’09 Instructor: Min Wu

Electrical and Computer Engineering Department,

University of Maryland, College Park

bb.eng.umd.edu (select ENEE631 S’09) [email protected]

ENEE631 Spring’09ENEE631 Spring’09Lecture 13 (3/11/2009)Lecture 13 (3/11/2009)

M. Wu: ENEE631 Digital Image Processing (Spring'09) Lec13 – Wavelet Coding, etc. [2]

Overview and LogisticsOverview and Logistics

Last Time:

– JPEG Compression– Subband and Wavelet based compression

Subband decomposition; basic ideas of EZW encoding

Today– Continue on Subband and Wavelet based compression

Exploit the structures between coefficients for removing redundancy

More on wavelet transform, and consideration on filters

Assignment#3 is Due Friday noon March 13– JPEG part may be turned in after spring break

Project#1 Posted. Due Monday April 6

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Review: 2-D Subband/DWT Review: 2-D Subband/DWT DecompositionDecomposition

From Usevitch’s article in IEEE Sig.Proc. Mag. 9/01

Separable transform by successively applying 1-D DWT to rows and columns


Review: Key Concepts in EZW Review: Key Concepts in EZW

Exploit multi-resolution and self-similarity

Parent-children relation among coeff.– Each parent coeff at level k spatially correlates with

4 coeff at level (k-1) of same orientation – A coeff at lowest band correlates with 3 coeff.

Coding significance map via zero-tree– Encode “insignificance map” w/ zero-trees

Avoid large overhead by coding only significant coefficients

Successive approximation quantization– Send most-significant-bits first and gradually

refine coefficient value– “Embedded” nature of coded bit-stream

get higher fidelity image by adding extra refining bits

=> Quality Scalability with fine granularityFrom Usevitch (IEEE Sig.Proc. Mag. 9/01)

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Recall: EZW Algorithm and ExampleRecall: EZW Algorithm and Example

Initial threshold ~ 2 ^ floor(log2 xmax)– Put all coeff. in dominant list

Dominant Pass (“zig-zag” across bands)

– Assign symbol to each coeff. and entropy encode symbols ps – positive significance ns – negative significance iz – isolated zero ztr – zero-tree root

– Significant coefficients Move to subordinate list Set them to zero in dominant list

Subordinate Pass– Output one bit for subordinate list

According to position in up/low half of quantization interval

Repeat with half threshold, until bit budget achieved – E.g. first look for coeff. above 32; then focus on coeff. between 16

and 32

Fig. From Usevitch (IEEE Sig. Proc. Mag. 9/01)

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After 1After 1stst Pass PassFrom Usevitch (IEEE Sig.Proc. Mag. 9/01)

Divide quantization interval of [32,64) by half: [32,48) => 40 and [48, 64) => 56

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After 2After 2ndnd Pass Pass

From Usevitch (IEEE Sig.Proc. Mag. 9/01)

[16,24) => 20, [24, 32) => 28; [32,40) => 36, [40,48) => 44; [48, 56) => 52, [56, 64) => 60

2nd Subordinate Pass:

orig. 53 symbol 0 reconstruct 52 34 0 36

-22 0 -20

21 0 20

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EZW and BeyondEZW and Beyond

Can apply DWT to entire images or larger blocks than 8x8

Symbol sequence can be entropy encoded – e.g. arithmetic coding

Cons of EZW– Poor error resilience; Difficult for selective spatial decoding

SPIHT (Set Partitioning in Hierarchal Trees)– Further improvement over EZW to remove redundancy– Achieve equal/better performance than EZW w/o arithmetic coding

EBCOT (Embedded Block Coding with Optimal Truncation)– Used in JPEG 2000– Address the shortcomings of EZW (random access, error resilience, …)– Embedded wavelet coding in each block + bit-allocations among blocks

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A Close Look at Wavelet TransformA Close Look at Wavelet Transform

Haar Transform – unitaryHaar Transform – unitary

Orthonormal Wavelet FiltersOrthonormal Wavelet Filters

Biorthogonal Wavelet FiltersBiorthogonal Wavelet Filters

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Construction of Haar FunctionsConstruction of Haar Functions

Unique decomposition of integer k (p, q)– k = 0, …, N-1 with N = 2n, 0 <= p <= n-1– 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) …

hk(x) = h p,q(x) for x [0,1]

k = 2p + q – 1“remainder”

power of 2

]1,0[other for 0

22for 2

1

22

1for 2

1

)()(

]1,0[for 1

)()(

21

2/

21

2/

,

0,00

x

qx

q-

N

q-x

q-

N

xhxh

xN

xhxh

ppp

ppp

qpk

1

x

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(0,0)

(0,1)

(1,1)

(1,2)

(2,1)


Haar TransformHaar Transform

Haar transform H

– Sample hk(x) at {m/N}for m = 0, …, N-1

– Real and orthogonal– Transition at each scale p is

localized according to q

Basis images of 2-D (separable) Haar transform– Outer product of two basis vectors

2200

0022

0000

0000

0000

0000

2 2 00

0 0 2 22222

0000

1111

1111

0000

2222

1111

1111

8

1

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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)


Time-Freq (or Space-Freq) InterpretationsTime-Freq (or Space-Freq) Interpretations

– Standard Basis (for data samples); Fourier Basis; Wavelet Basis


Recap on Haar TransformRecap on Haar Transform Two major sub-operations

– Scaling captures info. at different frequencies– Translation captures info. at different locations

Can be represented by filtering and downsampling

Relatively poor energy compaction– Equiv. filter response doesn’t have good cutoff & stopband attenuation

1

x

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Orthonormal FiltersOrthonormal Filters

Equiv. to projecting input signal to orthonormal basis

Energy preservation property– Convenient for quantizer design

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

Shortcomings: “coefficient expansion”– Linear filtering with N-element input & M-element filter

(N+M-1)-element output (N+M)/2 after downsample

– Length of output per stage grows ~ undesirable for compression

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

Symmetric filter

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Solutions to Coefficient ExpansionSolutions to Coefficient Expansion Circular convolution in place of linear convolution

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

Symmetric extension of input– Reduce border artifacts (note the signal length doubled with symmetry)– Problem: output at each stage may not be symmetric From Usevitch (IEEE

Sig.Proc. Mag. 9/01)

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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 artifacts– Symmetric filter (or asymmetric filter) => “linear phase filters”

(no phase distortion except by delays)

Problem

– Only one set of linear phase filters for real FIR orthogonal wavelets Haar filters: (1, 1) & (1,-1) do not give good energy compaction

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

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Biorthogonal WaveletsBiorthogonal Wavelets

“Biorthogonal”– Basis in forward and inverse transf.

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

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

filters’ performance

Advantage– Covers a much broader class of filters

including symmetric filters that eliminate coefficient expansion

Commonly used filters for compression– 9/7 biorthogonal symmetric filter– Efficient implementation: Lifting approach (ref: Swelden’s tutorial)

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)From Usevitch (IEEE Sig.Proc. Mag. 9/01)


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Smoothness Conditions on Wavelet FilterSmoothness Conditions on Wavelet Filter

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

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


JPEG 2000 Image Compression StandardJPEG 2000 Image Compression Standard

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From Christopoulos (IEEE Trans. on CE 11/00)

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JPEG 2000: A Wavelet-Based New StandardJPEG 2000: A Wavelet-Based New Standard

Targets and features

– Excellent low bit rate performance without sacrifice performance at higher bit rate

– Progressive decoding to allow from lossy to lossless– Region-of-interest (ROI) coding– Error resilience

For details

– JPEG2000 Tutorial by Skrodras @ IEEE Sig. Proc Magazine 9/2001 – David Taubman: “High Performance Scalable Image Compression

with EBCOT”, IEEE Trans. On Image Proc, vol.9(7), 7/2000.– Taubman’s book on JPEG 2000 (on library reserve)– Links and tutorials @ http://www.jpeg.org/JPEG2000.htm

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http://www.jpeg.org/JPEG2000.htm


ExamplesExamples

JPEG2KJPEG2K vs. vs. JPEGJPEG

From Christopoulos (IEEE Trans. on CE 11/00)

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DCT vs. Wavelet: Which is Better?DCT vs. Wavelet: Which is Better?

3dB improvement?

– Wavelet compression was claimed to have 3dB improvement over DCT-based compression

– Comparison is done on JPEG Baseline

Improvement not all due to transforms

– Main contribution from better rate allocation, advanced entropy coding, & smarter redundancy reduction via zero-tree

– Difference due to DCT vs. Wavelet transform is less than 1dB– DCT coder can be improved to decrease the gap

[Ref] "A comparative study of DCT- and wavelet-based image coding", Z. Xiong, K. Ramchandran, M. Orchard, Y-Q. Zhang, IEEE Trans. on Circuits and Systems for Video Technology, v.9, no.5, 8/99, pp692-695.

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Non-Dyadic Decomposition – Wavelet PacketsNon-Dyadic Decomposition – Wavelet Packets

Figures are from slides at Gonzalez/ Woods DIP book 2/e website (Chapter 7)


Bit Allocation in Image CodingBit Allocation in Image Coding

Focus on MSE-based optimization;Can further adjust based on HVS

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1

0

2 )()(

1min

N

kq kvkvE

ND

Here k2: variance of k-th coeff. v(k); rk is # bits allocated for v(k)

f(-): a func. relating bit rate to distortion based on a coeff’s p.d.f.

Bit AllocationBit Allocation

How many bits to be allocated for each coefficient?

– Related to each coeff’s variance (and probability distribution)– More bits for high variance k

2 to keep total MSE small

1

0

)(1 N

kkrf

N

1

0

.bits/coeff 1

subject toN

kk Rr

N

“Reverse Water-filling”– Try to keep same amount of error in each frequency band

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Rate/Distortion Allocation via Reverse Water-fillingRate/Distortion Allocation via Reverse Water-filling

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) Optimal solution for Gaussian distributed coefficients Idea: try to keep same amount of error in each freq. band; no need to

spend bits to represent coeff. w/ smaller variance than water level

– Results based on R-D function and via Lagrange-multiplier optimization

– Note the convex shape of R-D function for Gaussian => slope becomes milder; require more bits to further reduce distortion

D

RD

2

Given total rate R, determine and then D; or vice versa given D, determine R

Optimal Not Optimal


Details on Reverse Water-filling SolutionDetails on Reverse Water-filling Solution

RRRDi

i

n

iii

tosubj. )(min1

N

ii

N

iiiN RRRDRRJ

111 )(),...(

Construct func. using Lagrange multiplier

R and allfor 0 ii

i

i

i

i

RdR

dDi

dR

dD

R

J Necessary condition Keep the same marginal gain

0for 1

2

1 0,ln

2

1max

2

iii

i

i

i

ii

i RDdD

dR

DdD

d

dD

dR

Equal Slope Condition for bit allocation

D

RD

2

Resulting in equal distortions on Gaussian r.v’sas interpreted in “reverse water filling”


Key Result in Rate Allocation: Equal R-D SlopeKey Result in Rate Allocation: Equal R-D Slope

Keep the slope in R-D curve the same (not just for Gaussian r.v)

– Otherwise some bits can be applied to other r.v. for better “return” in reducing the overall distortion

If all r.v. are Gaussian, the same slope in R-D curves for these r.v. correspond to an identical amount of distortion.

R(D) = ½ [ log 2 – log D ] => dR/dD = – 1 / (2D)

Use operational/measured rate-distortion information for general r.v.

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) Rate allocation results:=> Equal slope(o.w. exist a better allocation strategy)


Summary of Today’s LectureSummary of Today’s Lecture Wavelet transform and coding

– Haar basis/transform; considerations for wavelet filters; JPEG 2000

Optimal bit allocation: “equal slope” in R-D curves– Lead to reverse water filling for Gaussian coefficients

Next lecture– Optimal transform

Readings: see wavelet coding references in Project #1– Gonzalez’s 3/e book Section 8.2.10; 7.1, 7.4-7.5; (7.2-7.3 on wavelet)– Transform coding articles in IEEE Sig. Proc. Magazine Sept. 2001

(special issue on transform coding)

– To explore more: Tutorial on rate-distortion by Ortega-Ramchandran in IEEE Sig. Proc. Magazine, Nov. 98, and Jain’s book Section 2.13;“Wavelets and the lifting scheme: A 5 minute tour” by Wim Sweldens [Zeitschrift für Angewandte Mathematik und Mechanik, 76 (2), pp. 41-44, 1996]

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Review: Correlation After a Linear TransformReview: Correlation After a Linear Transform

Consider an Nx1 zero-mean random vector x

– Covariance (or autocorrelation) matrix Rx = E[ x xH ] give us ideas of the correlation between elements Rx is a diagonal matrix for if all N r.v.’s are uncorrelated

Apply a linear transform to x: y = A x

What is the autocorrelation matrix for y ?

Ry = E[ y yH ] = E[ (Ax) (Ax)H ] = E[ A x xH AH ]

= A E[ x xH ] AH = A Rx AH

Decorrelation: try to search for A that can produce a decorrelated y (equiv. a diagonal correlation matrix Ry )

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Optimal TransformOptimal Transform

Recall: Why use transform in coding/compression?– Decorrelate the correlated data– Pack energy into a small number of coefficients

– Interested in unitary/orthogonal or approximate orthogonal transforms Energy preservation s.t. quantization effects can be better understood

and controlled

Unitary transforms we’ve dealt so far are data independent– Transform basis/filters are not depending on the signals we are processing

Can hard-wire them in encoder and decoder

What unitary transform gives the best energy compaction and decorrelation?– “Optimal” in a statistical sense to allow the codec works well with many

images => Signal statistics would play an important role

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Review: Correlation After a Linear TransformReview: Correlation After a Linear Transform

Consider an Nx1 zero-mean random vector x

– Covariance (or autocorrelation) matrix Rx = E[ x xH ] give ideas of correlation between elements Rx is a diagonal matrix for if all N r.v.’s are uncorrelated

Apply a linear transform to x: y = A x

What is the correlation matrix for y ?

Ry = E[ y yH ] = E[ (Ax) (Ax)H ] = E[ A x xH AH ]

= A E[ x xH ] AH = A Rx AH

Decorrelation: try to search for A that can produce a decorrelated y (equiv. a diagonal correlation matrix Ry )

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K-L Transform (Principal Component Analysis)K-L Transform (Principal Component Analysis)

Eigen decomposition of Rx: Rx uk = k uk

– Recall the properties of Rx

Hermitian (conjugate symmetric RH = R); Nonnegative definite (real non-negative eigen values)

Karhunen-Loeve Transform (KLT) y = UH x x = U y with U = [ u1, … uN ]

– KLT is a unitary transform with basis vectors in U being the orthonormalized eigenvectors of Rx

– Rx U = [ 1u1, … N uN ] = U diag{1, 2, … , N}

=> UH Rx U = diag{1, 2, … , N} i.e. KLT performs decorrelation Often order {ui} so that 1 2 … N

– Also known as Hotelling transform or Principle Component Analysis (PCA)

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Properties of K-L TransformProperties of K-L Transform

Decorrelation

– E[ y yH ]= E[ (UH x) (UH x)H ]= UH E[ x xH ] U = diag{1, 2, … , N}

– Note: Other matrices (unitary or nonunitary) may also decorrelate the transformed sequence [Jain’s e.g.5.7 pp166]

Minimizing MSE under basis restriction

– If only allow to keep m coefficients for any 1 m N, what’s the best way to minimize reconstruction error?

Keep the coefficients w.r.t. the eigenvectors of the first m largest eigen values

Reference: Theorem 5.1 and Proof in Jain’s Book (pp166)

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KLT Basis RestrictionKLT Basis Restriction Basis restriction

– Keep only a subset of m transform coefficients and then perform inverse transform (1 m N)

– Basis restriction error: MSE between original & new sequences

Goal: to find the forward and backward transform matrices to minimize the restriction error for each and every m– The minimum is achieved by KLT arranged according to the

decreasing order of the eigenvalues of R

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K-L Transform for ImagesK-L Transform for Images

Work with 2-D autocorrelation function

– R(m,n; m’,n’)= E[ x(m, n) x(m’, n’) ] for all 0 m, m’, n, n’ N-1– K-L Basis images is the orthonormalized eigen functions of R

Rewrite images into vector form (N2x1)

– Need solve the eigen problem for N2xN2 matrix! ~ O(N 6)

Reduced computation for separable R

– R(m,n; m’,n’)= r1(m,m’) r2(n,n’)

– Only need solve the eigen problem for two NxN matrices ~ O(N3)– KLT can now be performed separably on rows and columns

Reducing the transform complexity from O(N4) to O(N3)

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Pros and Cons of K-L TransformPros and Cons of K-L Transform

Optimality

– Decorrelation and MMSE for the same# of partial transf coeff.

Data dependent

– Have to estimate the 2nd-order statistics to determine the transform– Can we get data-independent transform with similar performance?

DCT

Applications

– (non-universal) compression– Pattern recognition: e.g., eigen faces– Analyze the principal (“dominating”) components and reduce

feature dimensions

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Energy Compaction of DCT vs. KLTEnergy Compaction of DCT vs. KLT

DCT has excellent energy compaction for highly correlated data

DCT is a good replacement for K-L– Close to optimal for highly correlated data– Not depend on specific data like K-L does– Fast algorithm available

[ref and statistics: Jain’s pp153, 168-175]

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Energy Compaction of DCT vs. KLT (cont’d)Energy Compaction of DCT vs. KLT (cont’d)

Preliminaries– The matrices R, R-1, and R-1 share the same set of eigen vectors

– DCT basis vectors are eigenvectors of a symmetric tri-diagonal matrix Qc

– Covariance matrix R of 1st-order stationary Markov sequence with has an inverse in the form of symmetric tri-diagonal matrix

DCT is close to KLT on 1st-order stationary Markov

– For highly correlated sequence, a scaled version of R-1 approx. Qc

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More on Rate-Distortion Based More on Rate-Distortion Based

Bit Allocation in Image CodingBit Allocation in Image Coding

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Basic Steps in R-D OptimizationBasic Steps in R-D Optimization

Determine source’s statistics– Simplified models: Gaussian, Laplacian, Generalized Gaussian– Obtain from training samples

Obtain operational R-D points/functions/curves– E.g., compute distortion for each candidate quantizer

Determine objective function and constraints

Search for optimal operational R-D points– Lagrange multiplier approach

select only operating points on the convex hull of overall R-D characteristics (Fig.19 of Ortega’s tutorial)

may handle different coding unit independently

– Dynamic programming approach not constrained by convex hull but has larger search space

~ higher complexity => See Ortega’s SPM’98 tutorial for details


Bridging the Theory and the Practice Bridging the Theory and the Practice

Operational R-D curves– Directly achievable R-D pairs with practical implementation

Tradeoff! – How close are the R-D operational points to info. theory bounds?

– Practical considerations: cost of memory, computation, delay, etc.

Often narrowing down to an optimization problem:~ optimize an objective function subject to a set of constraints

Problem of model mismatch

– How good are our assumptions on source modeling?– How well does a coding algorithm tolerate model mismatch?


Ref: convex hullRef: convex hull

Formally

– Smallest convex set containing all points For R-D problem, we concerned of the lower boundary of this

set that close to the info. theoretical bound (the upper/right end are open)

– Convex set if X1,..,Xn are in the set, then the linear combinations of

them (with non-negative combination factors) are still in the set

Intuitively

– Use a rubber band to wrap the outermost points so that the band encloses all points


1

0

2 )()(

1min

N

kq kvkvE

ND

where k2 -- variance of k-th coeff. v(k); nk -- # bits allocated for v(k)

f(-) – quantization distortion func.

Bit AllocationBit Allocation How many bits to be allocated for each coeff.?

– Determined by the variance of coeff.– More bits for high variance k

2 to keep total MSE small

1

0

2 )(1 N

kkk nf

N

1

0

.bits/coeff 1

subject toN

kk Bn

N

“Inverse Water-filling” (from info. theory)

– Try to keep same amount of error in each freq. band

See Jain’s pp501 for detailed bit-allocation algorithm


Details on Reverse Water-filling Solution (cont’d)Details on Reverse Water-filling Solution (cont’d)

DDD i

i

in

i

tosubj. ,lnmin2

121

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Lagrangian Opt. for Indep. Budget ConstraintLagrangian Opt. for Indep. Budget Constraint

Previous: fix distortion, minimize total rate

Alternative: fix total rate (bit budget), minimize distortion

(Discrete) Lagrangian optimization on general source

– “Constant slope optimization” (e.g. in Box 6 Fig. 17 of Ortega’s tutorial)– Need to determine the quantizer q(i) for each coding unit i– Lagrangian cost for each coding unit

use a line with slope - to intersect with each operating point (Fig.14)

– For a given operating quality , the minimum can be computed independently for each coding unit

=> Find operating quality satisfying the rate constraint

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(from Box 6 Fig. 17 of Ortega’s tutorial)(from Fig. 14 of Ortega’s tutorial)


Bridging the Theory and Ad-hoc Practice Bridging the Theory and Ad-hoc Practice

Operational R-D curves– Directly achievable with practical implementation

Tradeoff! Tradeoff! Tradeoff! – Rate vs. Distortion

how close are the operational points to info. theory bounds?– Also tradeoff among practical considerations

Cost of memory, computation, delay, etc.

Often narrowing down to an optimization problem:~ optimize an objective func. subj. to a set of constraints

Model mismatch problem– How good are our assumptions on source modeling?– How well does a coding algorithm tolerate model mismatch?


Basic Steps in R-D OptimizationBasic Steps in R-D Optimization

Determine source’s statistics– Simplified models: Gaussian, Laplacian, Generalized Gaussian– Obtain from training samples

Obtain operational R-D points/functions/curves– E.g., compute distortion for each candidate quantizer

Determine objective func. and constraints

Search for optimal operational R-D points– Lagrange multiplier approach

select only operating points on the convex hull of overall R-D characteristics (Fig.19 of Ortega’s tutorial)

may handle different coding unit independently

– Dynamic programming approach not constrained by convex hull but has larger search space

(higher complexity)

M. Wu: ENEE631 Digital Image Processing (Spring'09) Wavelet Coding and Related Issues Spring ’09 Instructor: Min Wu Electrical and Computer Engineering.

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