Introduction to Wavelet Transform. Time Series are Ubiquitous Time Series are Ubiquitous! A random...

Introduction to Wavelet Transform

Time Series are UbiquitousTime Series are Ubiquitous!

A random sample of 4,000 graphics from 15 of the world’s newspapers published from 1974 to 1989 found that more than 75% of all graphics were time series (Tufte, 1983).

Why is Working With Time Series so Why is Working With Time Series so Difficult? Difficult?

The definition of similarity depends on the user, the domain and the task at hand. We need to be able to handle this subjectivity.

Answer:Answer: We are dealing with subjective notions of We are dealing with subjective notions of similarity. similarity.

Wavelet Transform - OverviewHistory

• Fourier (1807)

• Haar (1910)

• Math World

{ }i i i ii

f t / 2 2

22

nm

mn m

tt

Wavelet Transform - Overview

• What kind of Could be useful?– Impulse Function (Haar): Best time resolution

– Sinusoids (Fourier): Best frequency resolution

– We want both of the best resolutions

t

t

• Heisenberg (1930)– Uncertainty Principle

• There is a lower bound for

(An intuitive prove in [Mac91])

t

Wavelet Transform - Overview

• Gabor (1945)– Short Time Fourier Transform (STFT)

• Disadvantage: Fixed window size

Wavelet Transform - Overview

• Constructing Wavelets– Daubechies (1988)

• Compactly Supported Wavelets

• Computation of WT Coefficients– Mallat (1989)

• A fast algorithm using filter banks

1Aw( )ah k

( )dh k

2

2

( )ag k

( )dg k

2

2

S1Dw

S

Jean Fourier

1768-1830

0 20 40 60 80 100 120 140

0

1

2

3

X

X'

4

5

6

7

8

9

Discrete Fourier Discrete Fourier Transform ITransform I

Excellent free Fourier Primer

Hagit Shatkay, The Fourier Transform - a Primer'', Technical Report CS-95-37, Department of Computer Science, Brown University, 1995.

http://www.ncbi.nlm.nih.gov/CBBresearch/Postdocs/Shatkay/

Basic Idea: Represent the time series as a linear combination of sines and cosines, but keep only the first n/2 coefficients.

Why n/2 coefficients? Because each sine wave requires 2 numbers, for the phase (w) and amplitude (A,B).

n

kkkkk twBtwAtC

1

))2sin()2cos(()(

Discrete Fourier Discrete Fourier Transform IITransform II Pros and Cons of DFT as a time series Pros and Cons of DFT as a time series

representation.representation.

• Good ability to compress most natural signals.• Fast, off the shelf DFT algorithms exist. O(nlog(n)).• (Weakly) able to support time warped queries.

• Difficult to deal with sequences of different lengths.• Cannot support weighted distance measures.

0 20 40 60 80 100 120 140

0

1

2

3

X

X'

4

5

6

7

8

9

Note: The related transform DCT, uses only cosine basis functions. It does not seem to offer

any particular advantages over DFT.

History…

In late forties, stalwarts like John vonNeumann, Dennis Gabor, Leon BrillouxEugen Wigner addressed the shortcomingsof Fourier analysis and advocated for awindowed Fourier transform. In 1946[Theory of Communication J. IEEE93(1946), 429-457] introduced the conceptshort-time Fourier transform now known asWiondowed Fourier Transform or GaborTransform; for current development seeMallat [Wavelet Tour of Signal Process,Academic Press 1998]. The computationalproblems were realized by Gabor himselfand his coworkers von Neumann et al.

0 20 40 60 80 100 120 140

Haar 0

Haar 1

Haar 2

Haar 3

Haar 4

Haar 5

Haar 6

Haar 7

X

X'

DWT

Discrete Wavelet Discrete Wavelet Transform ITransform I

Alfred Haar

1885-1933

Excellent free Wavelets Primer

Stollnitz, E., DeRose, T., & Salesin, D. (1995). Wavelets for computer graphics A primer: IEEE Computer Graphics and Applications.

Basic Idea: Represent the time series as a linear combination of Wavelet basis functions, but keep only the first N coefficients.

Although there are many different types of wavelets, researchers in time series mining/indexing generally use Haar wavelets.

Haar wavelets seem to be as powerful as the other wavelets for most problems and are very easy to code.

22( ) (1 2 ) tt t e is a wavelet as it is thesecond derivative of the Gaussian

2 /2xe with minus sign. It is known as theMexican Hat Wavelet.

Figure: Mexican Hat Wavelet

1

0 4¡ 4

Wavelet Series

0 20 40 60 80 100 120 140

Haar 0

Haar 1

Haar 2

Haar 3

Haar 4

Haar 5

Haar 6

Haar 7

X

X'

DWT

Discrete Wavelet Discrete Wavelet Transform IIITransform III

Pros and Cons of Wavelets as a time series Pros and Cons of Wavelets as a time series representation.representation.

• Good ability to compress stationary signals.• Fast linear time algorithms for DWT exist.• Able to support some interesting non-Euclidean similarity measures.

• Signals must have a length n = 2some_integer • Works best if N is = 2some_integer. Otherwise wavelets approximate the left side of signal at the expense of the right side.• Cannot support weighted distance measures.

0 20 40 60 80 100 120 140

X

X'

eigenwave 0

eigenwave 1

eigenwave 2

eigenwave 3

eigenwave 4

eigenwave 5

eigenwave 6

eigenwave 7

SVD

Singular Value Singular Value Decomposition IDecomposition I

Eugenio Beltrami

1835-1899

Camille Jordan (1838--1921)

James Joseph Sylvester 1814-1897

Basic Idea: Represent the time series as a linear combination of eigenwaves but keep only the first N coefficients.

SVD is similar to Fourier and Wavelet approaches, we represent the data in terms of a linear combination of shapes (in this case eigenwaves).

SVD differs in that the eigenwaves are data dependent.

SVD has been successfully used in the text processing community (where it is known as Latent Symantec Indexing ) for many years.

Good free SVD Primer

Singular Value Decomposition - A Primer. Sonia Leach

0 20 40 60 80 100 120 140

X

X'

eigenwave 0

eigenwave 1

eigenwave 2

eigenwave 3

eigenwave 4

eigenwave 5

eigenwave 6

eigenwave 7

SVD

Singular Value Singular Value Decomposition IIDecomposition II

How do we create the eigenwaves?

We have previously seen that we can regard time series as points in high dimensional space.

We can rotate the axes such that axis 1 is aligned with the direction of maximum variance, axis 2 is aligned with the direction of maximum variance orthogonal to axis 1 etc.

Since the first few eigenwaves contain most of the variance of the signal, the rest can be truncated with little loss.

TVUA This process can be achieved by factoring a M by n matrix of time series into 3 other matrices, and truncating the new matrices at size N.

0 20 40 60 80 100 120 140

X

X'

eigenwave 0

eigenwave 1

eigenwave 2

eigenwave 3

eigenwave 4

eigenwave 5

eigenwave 6

eigenwave 7

SVD

Singular Value Singular Value Decomposition IIIDecomposition III

Pros and Cons of SVD as a time series Pros and Cons of SVD as a time series representation.representation.

• Optimal linear dimensionality reduction technique .• The eigenvalues tell us something about the underlying structure of the data.

• Computationally very expensive.• Time: O(Mn2)• Space: O(Mn)

• An insertion into the database requires recomputing the SVD.• Cannot support weighted distance measures or non Euclidean measures.

Note: There has been some promising research into mitigating SVDs time and space complexity.

0 20 40 60 80 100 120 140

X

X'

Piecewise Linear Piecewise Linear Approximation IApproximation I

Basic Idea: Represent the time series as a sequence of straight lines.

Lines could be connected, in which case we are allowedN/2 lines

If lines are disconnected, we are allowed only N/3 lines

Personal experience on dozens of datasets suggest disconnected is better. Also only disconnected allows a lower bounding Euclidean approximation

Each line segment has • length • left_height (right_height can be inferred by looking at the next segment)

Each line segment has • length • left_height • right_height

Karl Friedrich Gauss

1777 - 1855

Problem with Fourier

A serious drawback in transforming to the frequencydomain, time information is lost. When looking at a Fouriertransform of a signal, it is impossible to tell when aparticular event took place.

Fourier analysis -- breaks down a signal into constituentsinusoids of different frequencies.

Function Representations

• sequence of samples (time domain)– finite difference method

• pyramid (hierarchical)• polynomial• sinusoids of various frequency (frequency domain)

– Fourier series

• piecewise polynomials (finite support)– finite element method, splines

• wavelet (hierarchical, finite support)– (time/frequency domain)

What Are Wavelets?

In general, a family of representations using:

• hierarchical (nested) basis functions

• finite (“compact”) support

• basis functions often orthogonal

• fast transforms, often linear-time

Function Representations –Desirable Properties

• generality – approximate anything well– discontinuities, nonperiodicity, ...

• adaptable to application– audio, pictures, flow field, terrain data, ...

• compact – approximate function with few coefficients– facilitates compression, storage, transmission

• fast to compute with– differential/integral operators are sparse in this basis– Convert n-sample function to representation in O(nlogn) or O(n)

time

Wavelet History, Part 1• 1805 Fourier analysis developed

• 1965 Fast Fourier Transform (FFT) algorithm

…

• 1980’s beginnings of wavelets in physics, vision, speech processing (ad hoc)

• … little theory … why/when do wavelets work?

• 1986 Mallat unified the above work

• 1985 Morlet & Grossman continuous wavelet transform … asking: how can you get perfect reconstruction without redundancy?

Wavelet History, Part 2

• 1985 Meyer tried to prove that no orthogonal wavelet other than Haar exists, found one by trial and error!

• 1987 Mallat developed multiresolution theory, DWT, wavelet construction techniques (but still noncompact)

• 1988 Daubechies added theory: found compact, orthogonal wavelets with arbitrary number of vanishing moments!

• 1990’s: wavelets took off, attracting both theoreticians and engineers

Time-Frequency Analysis

• For many applications, you want to analyze a function in both time and frequency

• Analogous to a musical score

• Fourier transforms give you frequency information, smearing time.

• Samples of a function give you temporal information, smearing frequency.

• Note: substitute “space” for “time” for pictures.

Comparison to Fourier Analysis• Fourier analysis

– Basis is global– Sinusoids with frequencies in arithmetic progression

• Short-time Fourier Transform (& Gabor filters)– Basis is local– Sinusoid times Gaussian– Fixed-width Gaussian “window”

• Wavelet– Basis is local– Frequencies in geometric progression– Basis has constant shape independent of scale

Wavelets are faster than ffts!

Fast Fourier Transform (FFT) (1966)Highly cited work: Cooley-TukeyComputation hazard: O[N log N)Fast Wavelet Transform:Computational Hazard: O(N).For details, see Mallat [Signal Poc. Book,Academic Press, 1998] or Neunzert andSiddiqi [Topics in Industrial Math – KluwerAcademic Publishers, 2000].

The results of the CWT are many wavelet coefficients, which are a function of scale and position

Gabor’s Proposal: Short Time Fourier Transform

Requirements:

Signal in time domain: require short time window to depict features of signal.

Signal in frequency domain: require short frequency window (long time window) to depict features of signal.

''2'' )]()([),( dtettgtxftSTFT ftjx

What are wavelets?What are wavelets?

Wavelets are functions defined over a finite interval and having an average value  of zero.

Haar wavelet

What is wavelet transform?What is wavelet transform?

The basic idea of the wavelet transform is to represent any arbitrary function ƒ(t) as a superposition of a set of such wavelets or basis functions.

These basis functions or baby wavelets are obtained from a single prototype wavelet called the mother wavelet, by dilations or contractions (scaling) and translations (shifts).

The wavelet transform is a tool for carving up functions, operators, or data into components of different frequency, allowing one to study each component separately.

The continuous wavelet transform (CWT)The continuous wavelet transform (CWT)

Fourier Transform

dtetfF tj

)()(

FT is the sum over all the time of signal f(t) multiplied by a complex exponential.

dtttfs s )()(),( *,

where * denotes complex conjugation. This equation shows how a function ƒ(t) is decomposed into a set of basis functions , called the wavelets. Z=r+iy, z*=r-iy

)(, ts

The variables s and are the new dimensions, scale and translation (position), after the wavelet transform.

Similarly, the Continuous Wavelet Transform (CWT) is defined as the sum over all time of the signal multiplied by scale , shifted version of the wavelet function :)(, ts

The wavelets are generated from a single basic wavelet , the so-called mother wavelet, by scaling and translation:

)(t

s

t

sts

1

)(,

s is the scale factor, is the translation factor and the factor s-1/2 is for energy normalization across the different scales.

It is important to note that in the above transforms the wavelet basis functions are not specified.

This is a difference between the wavelet transform and the Fourier transform, or other transforms.

Scale

Scaling a wavelet simply means stretching (or compressing) it.

Translation (shift)

Translating a wavelet simply means delaying (or hastening) its onset.

Scale and Frequency

Low scale a Compressed wavelet Rapidly changing details

High frequency

High scale a stretched wavelet slowly changing details

low frequency

Haar wavelet

Discrete WaveletsDiscrete Wavelets

Discrete wavelet is written as

j

j

jkj s

skt

st

0

00

0

,

1)(

j and k are integers and s0 > 1 is a fixed dilation step. The translation factor depends on the dilation step. The effect of discretizing the wavelet is that the time-scale space is now sampled at discrete intervals. We usually choose s0 = 2

0

1)()( *

,, dttt nmkj If j=m and k=n

others

A band-pass filterA band-pass filter

The wavelet has a band-pass like spectrum

From Fourier theory we know that compression in time is equivalent to stretching the spectrum and shifting it upwards:

aF

aatfF

1)(

This means that a time compression of the wavelet by a factor of 2 will stretch the frequency spectrum of the wavelet by a factor of 2 and also shift all frequency components up by a factor of 2.

Suppose a=2

Subband codingSubband coding

If we regard the wavelet transform as a filter bank, then we can consider wavelet transforming a signal as passing the signal through this filter bank.

The outputs of the different filter stages are the wavelet- and scaling function transform coefficients.

That is called subband coding.

In general we will refer to this kind of analysis as a multiresolution.

Splitting the signal spectrum with an iterated filter bank.

f8B

fHPLP 4B 4B

fHPLP 4B

2B2B

Bf

HPLP 4B2B

B

Summarizing, if we implement the wavelet transform as an iterated filter bank, we do not have to specify the wavelets explicitly! This is a remarkable result.

The Discrete Wavelet TransformThe Discrete Wavelet Transform

Calculating wavelet coefficients at every possible scale is a fair amount of work, and it generates an awful lot of data. What if we choose only a subset of scales and positions at which to make our calculations?

It turns out, rather remarkably, that if we choose scales and positions based on powers of two -- so-called dyadic scales and positions -- then our analysis will be much more efficient and just as accurate. We obtain just such an analysis from the discrete wavelet transform (DWT).

Approximations and Details

The approximations are the high-scale, low-frequencycomponents of the signal. The details are the low-scale,high-frequency components. The filtering process, at itsmost basic level, looks like this:

The original signal, S, passes through two complementary filtersand emerges as two signals .

Downsampling

To correct this problem, we introduce the notion ofdownsampling. This simply means throwing away everysecond data point.

Unfortunately, if we actually perform this operation on areal digital signal, we wind up with twice as much data aswe started with. Suppose, for instance, that the originalsignal S consists of 1000 samples of data. Then theapproximation and the detail will each have 1000 samples,for a total of 2000.

An example:

Reconstructing Approximation and Details

Upsampling

Wavelet Decomposition

Multiple-Level Decomposition

The decomposition process can be iterated, with successive approximations being decomposed in turn, so that one signal is broken down into many lower-resolution components. This is called the wavelet decomposition tree.

Scaling function (two-scale relation)

)2()()2( 11 ktkht j

kj

j

Wavelet )2()()2( 11 ktkgt j

kj

j

The signal f(t) can be expresses as

)2()()2()()( 11

11 ktkktktf j

kj

j

kj

)()2()(1 mkmhk jm

j

)()2()(1 mkmgk jm

j

DWT

)()2()(1 mkmhk jm

j

)()2()(1 mkmgk jm

j

)()()(])2([)(1 kkhmmkhk jjm

j

)()()(])2([)(1 kkgmmkgk jjm

j

12

3

2

1

2

5

2

1)1()1()0()0()()()0( 11

1

012

jj

mjj hhmmh

42

3

2

1

2

5

2

1)1()1()0()0()()()0( 11

2

012

jj

mjj ggmmg

Wavelet Reconstruction (Synthesis)

Perfect reconstruction :

1'' HHGG

)()1()(

)()1()(1'

1'

ngnh

nhngn

n

1)0(2 j 4)0(2 j

(1,0)

(4,0)

x1

y1

0)2(

2/1)1()0()0()1()()1()1(

2/1)0()0()()()0(

)()()(

1

2'

2'

2'

1

2'

2'

1

2'

1

x

hhmmhx

hmmhx

mmkhkx

jjm

j

jm

j

mj

0)2(

2/4)1()0()0()1()()1()1(

2/4)0()0()()()0(

)()()(

1

2'

2'

2'

1

2'

2'

1

2'

1

y

ggmmgy

gmmgy

mmkgky

jjm

j

jm

j

mj

2/3,2/51 j

2/3,2/31 j

2-D Discrete Wavelet Transform

A 2-D DWT can be done as follows:

Step 1: Replace each row with its 1-D DWT;

Step 2: Replace each column with its 1-D DWT;

Step 3: repeat steps (1) and (2) on the lowest subband for the next scale

Step 4: repeat steps (3) until as many scales as desired have been completed

original

L HLH HH

HLLL

LH HH

HL

One scale two scales

Image at different scales

Correlation between features at different scales

Wavelet construction – a simplified approach

• Traditional approaches to wavelets have used a filterbank interpretation

• Fourier techniques required to get synthesis (reconstruction) filters from analysis filters

• Not easy to generalize

3 steps

• Split

• Predict (P step)

• Update (U step)

Wavelet construction – lifting

Example – the Haar wavelet

• S step

Splits the signal into odd and even samples

ns 1ne

1no

even samples

odd samples

For the Haar wavelet, the prediction for the odd sample is the previous even sample :

lnln ss 2,12,ˆ

lns ,

l

Example – the Haar wavelet

• P step

Predict the odd samples from the even samples

lnlnln ssd 2,12,,1

Example – the Haar wavelet

lnd ,1

l

lns ,

l

Detail signal :

2/,12,,1 lnlnln dss

The signal average is maintained :

12

0

12

0,,1

1

2/1n n

l llnln ss

Example – the Haar wavelet

• U step

Update the even samples to produce the next coarser scale approximation

…..

-1

Summary of the Haar wavelet decomposition

lnlnln ssd 2,12,,1

2/,12,,1 lnlnln dss

Can be computed ‘in place’ :

lns 2, 12, lns12, lns

lnd ,1lns 2,1,1 lnd

…..-1

lnd ,11,1 lnd lns ,1

1/21/2

P step

U step

lnlnln dss ,12,12,

2/,1,12, lnlnln dss

Then merge even and odd samples

Mergelns 2,

12, lnslns ,

Inverse Haar wavelet

transform• Simply run the forward Haar wavelet transform

backwards!

General lifting stage of wavelet decomposition

-

Split P U

+

js

1js

1jd

1je

1jo

Multi-level wavelet decomposition

lift lift lift…ns

1ns

2nd

0s

1nd 0d

• We can produce a multi-level decomposition by cascading lifting stages

General lifting stage of inverse wavelet synthesis

-

MergePU

+

js

1je

1jo

1js

1jd

lift …...lift lift0s 1s 2s 1nsns

0d

1d 2d 1nd

• We can produce a multi-level inverse wavelet synthesis by cascading lifting stages

Multi-level inverse wavelet synthesis

Advantages of the lifting implementation

• Inverse transform Inverse transform is trivial – just run the code backwards No need for Fourier techniques

Generality The design of the transform is performed without reference

to particular forms for the predict and update operators Can even include non-linearities (for integer wavelets)

Example 2 – the linear spline wavelet

• A more sophisticated wavelet – uses slightly more complex P and U operators

• Uses linear prediction to determine odd samples from even samples

Linear prediction at odd samples Original signal

Detail signal (prediction error at odd samples)

The linear spline wavelet

• P-step – linear prediction

22,2,12,,1 2/1 lnlnlnln sssd

22,2,12, 2/1ˆ lnlnln sss

The linear spline wavelet

• The prediction for the odd samples is based on the two even samples either side :

)(4/1 ,11,12,,1 lnlnlnln ddss

The linear spline wavelet

• The U step – use current and previous detail signal sample

• Preserves signal average and first-order moment (signal position) :

The linear spline wavelet

12

0,

12

0,1 2/1

1 nn

lln

lln ss

12

0

12

0,,1

1

2/1n n

l llnln lsls

1/41/41/41/4

-1/2-1/2-1/2-1/2

lns 2, 12, lns 22, lns

lnd ,1lns 2, 22, lns

lns ,1 1,1 lnslnd ,1

P step

U step

The linear spline wavelet

• Can still implement ‘in place’

Summary of linear spline wavelet decomposition

22,2,12,,1 2/1 lnlnlnln sssd

)(4/1 ,11,12,,1 lnlnlnln ddss

Computing the inverse is trivial :

)(4/1 ,11,1,12, lnlnlnln ddss 22,2,,112, 2/1 lnlnlnln ssds

The even and odd samples are then merged as before

Wavelet decomposition applied to a 2D image

detail

detail

approx

lift

.

.

.

.

.lift

approx

approx

detaildetailapprox

approx

detailapproxdetail

lift lift lift lift

1,1 nni1

1,1 nnd 21,1 nnd 3

1,1 nnd

Wavelet decomposition applied to a 2D image

Why is wavelet-based compression effective?

• Allows for intra-scale prediction (like many other compression methods) – equivalently the wavelet transform is a decorrelating transform just like the DCT as used by JPEG

• Allows for inter-scale (coarse-fine scale) prediction

1 level Haar

1 level linear spline 2 level Haar

Original

Why is wavelet-based compression effective?

0

5000

10000

15000

20000

25000

-255 -205 -155 -105 -55 -5 45 95 145 195 245

Original

Haar wavelet

Why is wavelet-based compression effective?

• Wavelet coefficient histogram

  Entropy

Original image 7.22

1-level Haar wavelet 5.96

1-level linear spline wavelet 5.53

2-level Haar wavelet 5.02

2-level linear spline wavelet 4.57

Why is wavelet-based compression effective?

• Coefficient entropies

X

)(XP

Why is wavelet-based compression effective?

• Wavelet coefficient dependencies

))(|( SXPSXP

))(|( LXPLXP

Why is wavelet-based compression effective?

• Lets define sets S (small) and L (large) wavelet coefficients

• The following two probabilities describe interscale dependancies

2

#))(|(

N

SSXPSXP

2

#))(|(

N

LLXPLXP

Why is wavelet-based compression effective?

• Without interscale dependancies

0.886 0.529 0.781 0.219

))(|( SXPSXP 2

#

N

S))(|( LXPLXP

2

#

N

L

Why is wavelet-based compression effective?

• Measured dependancies from Lena

X

8

1

)(8

1

nnXcc

X1

X8 TcLX

TcSX

n

n

if

if

Why is wavelet-based compression effective?

• Intra-scale dependencies

0.912 0.623 0.781 0.219

)|( SXSXP n 2

#

N

S )|( LXLXP n 2

#

N

L

Why is wavelet-based compression effective?

• Measured dependancies from Lena

Why is wavelet-based compression effective?

• Have to use a causal neighbourhood for spatial prediction

• We will look at 3 state of the art algorithms

– Set partitioning in hierarchical sets (SPIHT)

– Significance linked connected components analysis (SLCCA)

– Embedded block coding with optimal truncation (EBCOT) which is the basis of JPEG2000

Example image compression algorithms

lsb

msb 1 1 0 0 0 0 0 0 0 0 0 0 0 0 …

x x 1 1 0 0 0 0 0 0 0 0 0 0 …

x x x x 1 1 1 0 0 0 0 0 0 …

x x x x x x x 1 1 1 1 1 1 0 …

x x x x x x x x x x x x x 1 …

x x x x x x x x x x x x x x …

Coeff. number 1 2 3 4 5 6 7 8 9 10 11 12 13 14…….

5

4

3

2

1

0

The SPIHT algorithm

• Coefficients transmitted in partial order

0

• 2 components to the algorithm – Sorting pass

– Sorting information is transmitted on the basis of the most significant bit-plane

• Refinement pass– Bits in bit-planes lower than the most significant bit

plane are transmitted

The SPIHT algorithm

N= msb of (max(abs(wavelet coefficient)))

for (bit-plane-counter)=N downto 1

transmit significance/insignificance wrt bit-plane counter

transmit refinement bits of all coefficients that

are already significant

The SPIHT algorithm

The SPIHT algorithm• Insignificant coefficients (with respect to current bitplane counter) organised into zerotrees

The SPIHT algorithm

• Groups of coefficients made into zerotrees by set paritioning

….1100101011100101100011………01011100010111011011101101….

bitstream

The SPIHT algorithm

• SPIHT produces an embedded bitstream

The SLCCA algorithm

Bit-plane encode

significant coefficients

Wavelet transform

Quantise coefficients

Cluster andtransmit

significance map

The SLCCA algorithm

• The significance map is grouped into clusters

Seed

Significant coeff

Insignificant coeff

The SLCCA algorithm

• Clusters grown out from a seed

Significance link

The SLCCA algorithm

• Significance link symbol

Image compression results

• Evaluation – Mean squared error

– Human visual-based metrics

– Subjective evaluation

21

0,2

),(ˆ),((1

N

cr

crIcrIN

mse

msedBPSNR

2

10

255log10)(

Usually expressed as peak-signal-to-noise (in dB)

Image compression results

• Mean-squared error 

25

27

29

31

33

35

37

39

41

43

0.2 0.4 0.6 0.8 1 1.2

bit-rate (bits/pixel)

PSNR(dB)

SPIHT

SLCCA

JPEG

Image compression results

25

27

29

31

33

35

37

39

41

43

0.2 0.4 0.6 0.8 1 1.2

bit-rate (bits/pixel)

PSNR(dB)

Haar

Linear spline

Daubechies 9-7

Image compression results

SPIHT 0.2 bits/pixel JPEG 0.2 bits/pixel

Image compression results

SPIHT JPEG

Image compression results

EBCOT, JPEG2000

• JPEG2000, based on embedded block coding and optimal truncation is the state-of-the-art compression standard

• Wavelet-based• It addresses the key issue of scalability

– SPIHT is distortion scalable as we have already seen– JPEG2000 introduces both resolution and spatial

scalability also• An excellent reference to JPEG2000 and compression

in general is “JPEG2000” by D.Taubman and M. Marcellin

• Resolution scalability is the ability to extract from the bitstream the sub-bands representing any resolution level

….1100101011100101100011………01011100010111011011101101….bitstream

EBCOT, JPEG2000

• Spatial scalability is the ability to extract from the bitstream the sub-bands representing specific regions in the image– Very useful if we want to selectively decompress

certain regions of massive images

….1100101011100101100011………01011100010111011011101101….bitstream

EBCOT, JPEG2000

Introduction to EBCOT

• JPEG2000 is able to implement this general scalability by implementing the EBCOT paradigm

• In EBCOT, the unit of compression is the codeblock which is a partition of a wavelet sub-band

• Typically, following the wavelet transform,each sub-band is partitioned into small blocks (typically 32x32)

Introduction to EBCOT

• Codeblocks – partitions of wavelet sub-bands

codeblock

Introduction to EBCOT

• A simple bit stream organisation could comprise concatenated code block bit streams

……0CB 1CB 2CB0L 1L 2L

Length of next code-block stream

Introduction to EBCOT

• This simple bit stream structure is resolution and spatially scalable but not distortion scalable

• Complete scalability is obtained by introducing quality layers– Each code block bitstream is individually (optimally)

truncated in each quality layer

– Loss of parent-child redundancy more than compensated by ability to individually optimise separate code block bitstreams

Introduction to EBCOT

• Each code block bit stream partitioned into a set of quality layers

0Q0QL

1QL 1Q …

00CB 0

1CB 02CB0

0L 01L 0

2L …

10CB 1

1CB 12CB1

0L 11L 1

2L …

EBCOT advantages

• Multiple scalability– Distortion, spatial and resolution scalability

• Efficient compression– This results from independent optimal truncation of

each code block bit stream

• Local processing– Independent processing of each code block allows for

efficient parallel implementations as well as hardware implementations

EBCOT advantages

• Error resilience– Again this results from independent code block

processing which limits the influence of errors

Performance comparison

• A performance comparison with other wavelet-based coders is not straightforward as it would depend on the target bit rates which the bit streams were truncated for– With SPIHT, we simply truncate the bit stream when

the target bit rate has been reached– However, we only have distortion scalability with

SPIHT

• Even so, we still get favourable PSNR (dB) results when comparing EBCOT (JPEG200) with SPIHT

Performance comparison

• We can understand this more fully by looking at graphs of distortion (D) against rate (R) (bitstream length)

R

D

R-D curve for continuously modulated quantisation step size

Truncation points

Performance comparison• Truncating the bit stream to some arbitrary

rate will yield sub-optimal performance

R

D

Performance comparison

25

27

29

31

33

35

37

39

41

43

0.0625 0.125 0.25 0.5 1

bit-rate (bits/pixel)

PSNR(dB)

Spiht

EBCOT/JPEG2000

Performance comparison

• Comparable PSNR (dB) results between EBCOT and SPIHT even though:– Results for EBCOT are for 5 quality layers (5

optimal bit rates)– Intermediate bit rates sub-optimal

• We have resolution, spatial, distortion scalability in EBCOT but only distortion scalability in SPIHT