wavelet based image coding -...

ENEE631 Digital Image Processing (Spring'04)

Wavelet Based Image CodingWavelet Based Image Coding

Spring ’04 Instructor: Min Wu

ECE Department, Univ. of Maryland, College Park

� www.ajconline.umd.edu (select ENEE631 S’04)� [email protected]

Based on ENEE631 Spring’04Based on ENEE631 Spring’04

Section 11Section 11

ENEE631 Digital Image Processing (Spring'04) Lec14 –Wavelet Coding [2]

Overview and LogisticsOverview and Logistics

� Last Time:

– Transform coding

� Today:

– JPEG compression standard: Baseline block-DCT based algorithm

� lossy part: quantization with different step size for each coeff. Band

� lossless part: differential coding, run-length coding, Huffman

=> Continued with the class notes handed out in last lecture

– Subband and Wavelet based compression� Exploiting the structures between coefficients for removing redundancy

UMCP ENEE631 Slides (created by M.W

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2004)


Wavelet Transform for Image CompressionWavelet Transform for Image Compression

� ENEE631 emphasis

– Focus on conceptual aspects related to image compression

– Wavelet is also useful for denoising, enhancement, and image

analysis

– Build upon filterbank and subband coding from ENEE630

(For more in-depth info. on wavelet: wavelet course offered in Math Dept.)

� K-level 1-D wavelet/subband decomposition

– Successive lowpass/highpass filtering and downsampling� on different level: capture transitions of different frequency bands

� on the same level: capture transitions at different locations


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Successive Wavelet/Subband DecompositionSuccessive Wavelet/Subband Decomposition

Successive lowpass/highpass filtering and downsampling

� on different level: capture transitions of different frequency bands

� on the same level: capture transitions at different locations

Figure from Matlab Wavelet Toolbox Documentation


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Examples of 1Examples of 1--D Wavelet TransformD Wavelet Transform

From Matlab

Wavelet Toolbox

Documentation


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22--D Wavelet Transform via Separable FiltersD Wavelet Transform via Separable Filters

From Matlab Wavelet

Toolbox

Documentation


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22--D ExampleD Example

From Usevitch (IEEE

Sig.Proc. Mag. 9/01)


Subband Coding TechniquesSubband Coding Techniques

� General coding approach

– Allocate different bits for coeff. in different frequency bands

– Encode different bands separately

– Example: DCT-based JPEG and early wavelet coding

� Some difference between subband coding and early wavelet coding ~ Choices of filters

– Subband filters aims at (approx.) non-overlapping freq. response

– Wavelet filters has interpretations in terms of basis and typically designed for certain smoothness constraints

(=> will discuss more )

� Shortcomings of subband coding

– Difficult to determine optimal bit allocation for low bit rate applications

– Not easy to accommodate different bit rates with a single code stream

– Difficult to encode at an exact target rate


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Review: Filterbank & Multiresolution AnalysisReview: Filterbank & Multiresolution Analysis


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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


Embedded ZeroEmbedded Zero--Tree Wavelet Coding (EZW)Tree Wavelet Coding (EZW)

� “Modern” lossy wavelet coding exploits multi-resolution and self-similar nature of wavelet decomposition

– Energy is compacted into a small number of coeff.

– Significant coeff. tend to cluster at the same spatial location in

each frequency subband

� Two set of info. to code:

– Where are the significant

coefficients?

– What values are the significant

coefficients?

From Usevitch (IEEE



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Key Concepts in EZW Key Concepts in EZW

� 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 only high energy coefficients� Need to send location info. � large overhead

– Encode “insignificance map” w/ zero-trees

� Successive approximation quantization

– Send most-significant-bits first and

gradually refine coeff. value

– “Embedded” nature of coded bit-stream� get higher fidelity image by adding extra refining bits

From Usevitch (IEEE



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EZW Algorithm and ExampleEZW 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 coeff.� move to subordinate list� put 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

From Usevitch (IEEE



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After 1After 1stst PassPassFrom Usevitch (IEEE



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After 2After 2ndnd PassPass

From Usevitch (IEEE



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Beyond EZWBeyond EZW

� Cons of EZW

– Poor error resilience

– Difficult for selective spatial decoding

� SPIHT (Set Partitioning in Hierarchal Trees)

– Further improvement over EZW to remove redundancy

� 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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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 WaveletJPEG 2000: A Wavelet--Based New StandardBased 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

– David Taubman: “High Performance Scalable Image Compression

with EBCOT”, IEEE Trans. On Image Proc, vol.9(7), 7/2000.

– JPEG2000 Tutorial by Skrodras @ IEEE Sig. Proc Magazine 9/2001

– Taubman’s book on JPEG 2000 (on library reserve)

– Links and tutorials @ http://www.jpeg.org/JPEG2000.htm


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ExamplesExamples

JPEG2KJPEG2Kvs. 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

– 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 Tech.,v.9, no.5, 8/99, pp692-695.


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A Close Look at Wavelet TransformA Close Look at Wavelet Transform


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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“reminder”

power of 2

∈

<≤−

<≤

==

∈==

]1,0[other for 0

22for 2

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22

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1

)()(

]1,0[for 1

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21

2/

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2/

,

0,00

x

qx

q-

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q-x

q-

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xhxh

xN

xhxh

pp

p

pp

p

qpk

1

x


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Haar TransformHaar Transform

� Haar transform H

– Sample hk(x) at {m/N}� 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 2

2222

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)


Summary on Haar TransformSummary 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

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

� 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



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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


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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 EE624 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 (see Swelden’s tuotial)


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


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


u ©

2004) � Linear expansion of a

function via an expansion set

– Basis function – if the

expansion is unique

� Orthogonal basis

� Non-orthogonal basis

– Coefficients are computed with

a set of dual-basis

� Discrete Wavelet Transform

– Wavelet expansion gives a set

of 2-parameter basis functions

and expansion coefficients:

scale and translation


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


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� 1st-generation wavelet systems:

– Scaling and translation of a generating wavelet (“mother wavelet”)

� Multiresolution conditions:

– 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”)

� Represent a signal by combining scaling functions and wavelets


Summary of Compression ToolsSummary of Compression Tools

� Lossless compression tools

– Entropy coding� Huffman, Arithmetic, Lempel-Ziv

� run-length

– Predictive coding� reduce the dynamic range to code

– Transform� enhance energy compaction

� Lossy compression tools

– Discarding and thresholding

– Quantization� ( previous lectures ) Scalar quantizer

� Quantize more than one component => vector quantization


Summary of Today’s LectureSummary of Today’s Lecture

� DCT-based coding and JPEG

� Wavelet-based coding– Exploring tree-based structure in Wavelet coefficients: EZW– JPEG 2000

� Next lecture (after spring break):– More on wavelet transform/filters and JPEG 2000– Bit allocation techniques in image compression

� Readings on Wavelet

– Gonzalez book: 8.5.3, 8.6.2

– “A Tutorial on Modern Lossy Wavelet Image Compression” by Usevitch @ IEEE Sig. Proc Magazine 9/2001

– See also reference list provided with project#1


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AssignmentAssignment

� Required readings

– [Wavelet coding] � Usevitch’s tutorial in IEEE Sig. Proc. Magazine 9/01 � Xiong’s paper on DCT vs. Wavelet on IEEE Trans. CSVT (Univ. Library’s E-journal)� MATLAB 2-D Wavelet Demos

– [JPEG2000] Christopoluos’ tutorial in IEEE Trans. CE 11/00� http://etro.vub.ac.be/~chchrist/paper_ieee_ce_jpeg2000_Nov2000.pdf

– [VQ] Bovik’s Handbook Sec.5.3

� Recommended readings

– www.jpeg.org

– Gray’s tutorial paper on VQ

– Bovik’s Handbook Sec.5.4



� Wavelet-based coding

– Exploring tree-based structure in Wavelet coefficients: EZW

– JPEG 2000

– Wavelet transform/filters

� Next lecture :

– More on wavelet filters

– Bit allocation techniques in image compression

� Readings on Wavelet

– Gonzalez book: 7.1.1-7.1.2, 8.5.3, 8.6.2

– “A Tutorial on Modern Lossy Wavelet Image Compression” by Usevitch @ IEEE Sig. Proc Magazine 9/2001

– Taubman: “JPEG 2000”, Proc. of IEEE,

– See also reference list provided with project#1


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� Wavelet filters

� Bit allocation techniques in image compression

– Rate-distortion functions and reverse water-filling solution

� Next Lecture:

– Optimal transform

� Readings

– Wavelets and JPEG2000: reference list provided in project#1� See also Gonzalez’s book Section 7.1, (7.2-7.3), 8.5.3, 8.6.2

– “Wavelets and the lifting scheme: A 5 minute tour” by Wim Sweldens

[Zeitschrift für Angewandte Mathematik und Mechanik, 76 (2), pp. 41-44, 1996]

– Results on rate-distortion theory: Jain’s book Section 2.13

� Further exploration – Tutorial on rate-distortion by Ortega-Ramchandran in IEEE Sig. Proc. Magazine, Nov. 98


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Question for Today: Bring in Motion Question for Today: Bring in Motion �� VideoVideo

� Capturing video

– Frame by frame => image sequence

– Image sequence: A 3-D signal

� 2 spatial dimensions & time dimension

� continuous I( x, y, t ) => discrete I( m, n, tk )

� Encode digital video

– Simplest way ~ compress each frame image individually

� e.g., “motion-JPEG”

� only spatial redundancy is explored and reduced

– How about temporal redundancy?� Will differential coding work well? U

MCP ENEE408G Slides (created by M.W

u &

R.Liu © 2002)

wavelet based image coding -...

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