MIT Media Laboratory Vision and Modeling Technical Report

Appears in Subband Coding edited by John Woods Kluwer Academic Press

Chapter

Subband Transforms

y z

Eero PSimoncelli and Edward H Adelson

Vision Science Group The Media Lab oratoryand

y

DepartmentofElectrical Engineering and Computer Science

z

Departmentof Brain and CognitiveScience

Massachusetts Institute of Technology

Cambridge Massachusetts

Linear transforms are the basis for manytechniques used in image pro

cessing image analysis and image co ding Subband transforms are a sub class

of linear transforms whichoer useful prop erties for these applications In

this chapter wediscuss a varietyofsubband decomp ositions and illustrate

their use in image co ding Traditionallycoders based on linear transforms

are divided into twocategories transform co ders and subband co ders This

distinction is due in part to the nature of the computational metho ds used for

the twotyp es of representation

Transform co ding techniques are usually based on orthogonal linear trans

forms The classic example of suchatransform is the discrete Fourier trans

form DFT whichdecomp oses a signal into sinusoidal frequency comp o

nents Twoother examples are the discrete cosine transform DCT and the

KarhunenLo evetransform KLT Conceptually these transforms are com

puted bytaking the inner pro duct of the nitelength signal with a set of basis

functions This pro duces a set of co ecients which are then passed on to the

This work was supp orted bycontracts with the IBM Corp oration agreementdated

and DARPA Rome Airforce FC and a grantfrom the NSF IRI

The opinions expressed are those of the authors and do not necessarily representthose of

the sp onsors

SUBBAND IMAGE CODING

quantization stage of the co der In practice manyofthesetransforms have

ecient implementations as cascades of buttery computations Further

more these transforms are usually applied indep endently to nonoverlapping

subblo cks of the signal

Subband transforms are generally computed by convolving the input sig

nal with a set of bandpass lters and decimating the results Eachdecimated

subband signal enco des a particular p ortion of the frequency sp ectrum corre

sp onding to information o ccurring at a particular spatial scale To reconstruct

the signal the subband signals are upsampled ltered and then combined ad

ditivelyFor purp oses of co ding subband transforms can b e used to control the

relative amounts of error in dierentparts of the frequency sp ectrum Most

lter designs for subband co ders attempt to minimize the resulting

from the subsampling pro cess In the spatial domain this aliasing app ears

as evidence of the sampling structure in the output image An ideal sub

band system incorp orates brickwall bandpass lters whichavoid aliasing

altogether Suchlters however pro duce ringing Gibbs phenomenon in the

spatial domain whichisperceptually undesirable

Although co ders are usually classied in one of these twocategories there

is a signicant amountofoverlap b etween the two In fact the latter part of

this chapter will fo cus on transforms whichmaybe classied under either cat

egoryAsan example consider the blo ckdiscrete cosine transform DCT in

which the signal image is divided into nonoverlapping blo cks and eachblock

is decomp osed into sinusoidal functions Several of these sinusoidal functions

are depicted in gure The basis functions are orthogonal since the DCT

is orthogonal and the blo cks are chosen so that they do not overlap Co ders

employing the blo ck DCT are typically classied as transform co ders

Wemayalso view the blo ck DCT as a subband transform Computing a

DCT on nonoverlapping blo cks is equivalenttoconvolving the image with each

of the blo ckDCT basis functions and then subsampling bya factor equal to

the blo ck spacing The of the basis functions also shown in

gure indicates that eachofthe DCT functions is selectivefora particular

frequency subband although it is clear that the subband lo calization is rather

poor Thus the DCT also qualies as a subband transform

Subband Transform Prop erties

Given the overlap b etween the categories of transform and subband co ders

what criteria should b e used in cho osing a linear transformation for co ding pur

poses We will consider a set of prop erties whichare relevanttothe problem

Chapter Subband Transforms

Figure Several of the p oint DCT basis functions left with their

corresp onding Fourier transforms right The Fourier transforms are plot

ted on a linear scale over the range from to

SUBBAND IMAGE CODING

of image co ding

Scale and Orientation

An explicit representation of scale is widely accepted as b eing imp ortantfor

eectiveimage representation Images contain ob jects and

features of manydierent sizes whichmaybeviewed over a large range of

distances and therefore a transformation should analyze the image simulta

neously and indep endently at dierentscales Several authors haveargued

that the correct partition in terms of scale is one in whichthe scales are re

lated bya xed constantofprop ortionality In the frequency domain this

corresp onds to a decomp osition into lo calized subbands with equal widths on

a logarithmic scale

For twodimensional signals a lo calized region in the frequency plane cor

resp onds spatially to a particular scale and orientation Orientation sp ecicity

allows the transform to extract higher order oriented structures typically found

in images suchasedges and lines Thus it is useful to construct transforma

tions whichpartition the input signal into lo calized patches in the frequency

domain

Spatial lo calization

In addition to lo calization in frequencyitisadvantageous for the basis func

tions to b e spatially lo calized that is the transform should enco de p ositional

information The necessityofspatial lo calization is particularly apparentin

machine vision systems where information ab out the lo cation of features in

the image is critical This lo calization should not however o ccur abruptly

as in the blo ckDCT example given earlier abrupt transitions lead to p o or

lo calization in the frequency domain

The concept of jointlocalization in the spatial and spatialfrequency do

mains maybe contrasted with the twomost common representations used for

the analysis of linear systems the sampled or discrete signal and its Fourier

transform The rst of these utilizes the standard basis set for discrete signals

consisting of impulses lo cated at eachsample lo cation These basis functions

are maximally lo calized in space but convey no information ab out scale On

the other hand the Fourier basis set is comp osed of even and o dd phase sinu

soidal sequences whose usefulness is primarily due to the fact that they are

the eigenfunctions of the class of linear shiftinvariantsystems Although they

are maximallylocalized in the frequency domain eachone covers the entire

spatial extentofthe signal

Chapter Subband Transforms

It is clear that representation in the space or frequency domains is ex

tremely useful for purp oses of system analysis but this do es not imply that

impulses or sinusoids are the b est waytoenco de signal information In a

numberof recentpapers the imp ortance of this issue is addressed and

related to a pap er by Dennis Gab or who showed that the class of

linear transformations maybe considered to span a range of jointlocalization

with the impulse basis set and the Fourier basis set at the twoextremes He

demonstrated that onedimensional signals can b e represented in terms of ba

sis functions whicharelocalized b oth in space and frequencyWewill return

to Gab ors basis set in section

Orthogonality

A nal prop ertytobeconsidered is orthogonality The justication usually

given for the orthogonality constraintisinterms of decorrelation Given a

signal with prescrib ed second order statistics ie a covariance matrix there

is an orthogonal transform the KarhunenLo eve transform whichwill decor

relate the signal ie diagonalize the covariance matrix In other words the

second order correlations of the transform co ecients will b e zero Orthogo

nalityisusually not discussed in the context of subband transforms although

manysuch transformas are orthogonal The examples in the next section will

demonstrate that although orthogonalityisnot strictly necessarya transform

that is strongly nonorthogonal maybeundesirable for co ding

Linear Transformations on Finite Images

The results presented in this chapter are based on analysis in b oth the spatial

and the frequency domains and thus rely on two separate notational frame

works the standard matrix notation used in linear algebra and the Fourier

domain representations commonly used in digital signal pro cessing In this

section wedescrib e the twotyp es of notation and makeexplicit the connec

tion b etween them For simplicitywewill restrict the discussion to analysis

of one dimensional systems although the notation maybe easily extended to

multiple dimensions

AnalysisSynthesis Filter Bank Formulation

We will b e interested in linear transformations on images of a nite size which

maybe expressed in terms of convolutions with nite impulse resp onse FIR

SUBBAND IMAGE CODING

lters The schematic diagram in gure depicts a convolutionbased sys

tem known as an analysissynthesis AS lter bank The notation in the

diagram is standard for digital signal pro cessing except that for the pur

poses of this pap er the b oxes H indicate circular convolution of a nite

i

input image of size N with a lter with impulse resp onse h nandFourier

i

transform

X

jn

H h ne

i i

n

Wedonot place a causalityconstraintonthe lter impulse resp onses since

they are meantfor application to images Wedohowever assume that the

k region of supp ort of the lter is smaller than the image size The b oxes

i

indicate that the sequence is subsampled byafactor of k where k is an integer

i i

for all i The b oxes k indicate that the sequence should b e upsampled by

i

inserting k zeros b etween each sample Wewill assume that the integers

i

k are divisors of N

i

The analysis section of the AS system takes an input sequence xn

of length N and p erforms a linear transformation to decomp ose it into M

sequences y noflength Nk The synthesis section p erforms the inverse

i i

op eration of the analysis transformation Here the M sequences y nare

i

upsampled and after ltering with lters g n are combined additively to give

i

an approximationx ntothe original sequence xn Note that although one

dimensional signals are indicated in the diagram the system is equally valid

for multidimensional signals if we replace o ccurences of the scalars n k

i

with vectors n and a matrix K resp ectively

i

The use of the AS formulation emphasizes the computation of the trans

form co ecients through convolution This is intuitively desirable since dier

ent regions of the image should b e pro cessed in the same manner Furthermore

the expression of the problem in the frequency domain allows us to easily sep

arate the error enx n xninto twoparts an aliasing comp onent

and a shiftinvariant comp onent To see this we write the contents of the

intermediate signals y ninthe frequency domain as

i

k 

X

j j

H X Y

i i

k k k k

k

j 

and the AS system output as

M 

X

Y kG X

i i

i

where wehaveusedwellknown facts ab out the eects of and

downsampling in the frequency domain Combining the twogives

Chapter Subband Transforms

Analysis section Synthesis section

y (n) ^ x(n) 0 x(n) H (∑) k £ k ‹ G (∑) 0 0 0 0

y (n) 1 H (∑) £ k ‹ G (∑) 1 k1 1 1

y (n) M H (∑) k £‹k G (∑)

M M M M

Figure An analysissynthesis lter bank

M  k 

X X

j j

H X G X

i i

k k

k

i j 

M 

X

H G X

i i

k

i

M  k 

X X

j j

G H X

i i

k k

k

i j 

The rst sum corresp onds to a linear shiftinvariant system resp onse and the

second contains the system aliasing

Cascaded Systems

A further advantage of the AS system is that it allows explicit depiction and

analysis of hierarchically constructed transformations If weassume that we

are dealing with AS systems with p erfect resp onse that isx nxn then

anyintermediate signal y nofanAS system maybefurther decomp osed

i

byapplication of anyotherASsystem Tomake this notion more precise

an example is given in the diagram of gure in whichanAS system

has b een reapplied to its own intermediate signal y n If the original AS



system as shown in gure had a p erfect resp onse then it is clear that the

SUBBAND IMAGE CODING

twostage system shown in gure will also havea perfect resp onse If the

cascading is applied to eachofthe M intermediate signals y n wewillcallthe

i

system a uniform cascade system Otherwise it will b e termed a nonuniform

or pyramid cascade A system whichwe will discuss in section is based

on pyramid cascades of twoband AS systems Sucha cascade pro duces an

o ctavewidth subband decomp osition as illustrated in the idealized frequency

diagram in gure

Matrix Formulation

An alternativetothe frequency domain notation asso ciated with the AS lter

bank is the matrix notation of linear algebra An image of nite extentwhich

has b een sampled on a discrete lattice maybewritten as a nite length column

N

vector x which corresp onds to a p ointinR the set of all real N tuples The

value of each comp onentof x is simply the corresp onding sample value in

the image Multidimensional images are converted to this vector format by

imp osing an arbitrary but xed order on the lattice p ositions If weletN be

the length of the vector xa linear transformation on the image corresp onds

to multiplication of x by some matrix M with N columns

Since the analysis and synthesis stages of the system in gure eachcor

resp ond to linear transformations wemay represent the same transformations

using matrix notation Using the denition of convolution and assuming for

simplicity a onedimensional system wemay write

N 

X

y m xl h k m l

i i i

l

and

N



k

M 

i

X X

xn y mg n k m

i i i

m

i

where the lter and image sample lo cations k m l andn k m are com

i i

puted mo dulo N These expressions maybeformulated as matrixvector pro d

ucts

t

y H x

and

x Gy

or combining these two equations

Chapter Subband Transforms

y (n) 00 H (∑) k £ k ‹ G (∑) 0 0 0 0

y (n) 01 H (∑) £ k ‹ G (∑) 1 k1 1 1

y (n) 0M H (∑) k £‹k G (∑) M M M M

^ x(n) x(n) H (∑) k £ k ‹ G (∑) 0 0 0 0

y (n) 1 H (∑) £ k ‹ G (∑) 1 k1 1 1

y (n) M H (∑) k £‹k G (∑)

M M M M

Figure Anonuniformly cascaded analysissynthesis lter bank

SUBBAND IMAGE CODING

Figure Octave band splitting pro duced byafourlevel pyramid cas

cade of a twoband AS system The top picture represents the splitting

of the twoband AS system Each successive picture shows the eect of

reapplying the system to the lowpass subband indicated in grey of the

previous picture The b ottom picture gives the nal fourlevel partition of

the frequency domain All frequency axes cover the range from to

t

x GH x

where y and x are N vectors the sup erscript t indicates matrix transp osition

and

h h k h h k

     

h h k h h k

     

h h k h h k

     

h k h k H

   

h k h k

   

h h

 

h h

 

and

g g k g g k

     

g g k g g k

     

g g k g g k

     

g k g k

G

   

g k g k

   

g g

 

g g

 

Chapter Subband Transforms

The columns of Gcomposed of copies of the lter kernels shifted byincrements

of k and imb edded in vectors of length N are known as the basis functions

i

of the transformation and the columns of Hcomposedofcopiesofthetime

inverted lters h n shifted by increments of k arethe sampling functions

i i

of the transformation

From the discussion ab ove it is clear that we can express anylinear AS

system in matrix form The converse of this result is also true there is an AS

system corresp onding to the linear transformation and inverse transformation

dened byanyinvertible matrix MGiven a transformation matrix M with

l rows wetrivially create an analysis lter bank with k N for each i

i

containing l dierentlters eachdened byarowofthe matrix M

Inverse Transforms

A primary advantage of the matrix notation is the ease with whichitcan

express the conditions for transform invertibilityFrom equation we see

that in order for the AS system to p erfectly reconstruct the original signal

xn the corresp onding matrices must ob ey

t

GH I

where I is the identity matrix If H has rank N and is square wemaycho ose

asynthesis matrix

 t

G H

which will also b e square with rank N Thus transform inversion in the

spatial domain is a conceptually simple pro cedure and wewill nd it useful

in the analysis of AS systems Furthermore it should b e clear that H and

G maybeinterchanged thus using the basis functions as sampling functions

and vice versa

If the matrix H is of rank N but is not square that is the representa

tion is overcomplete wemay always build a p erfect reconstruction system by

cho osing G to b e the generalized inverse or pseudoinverse of H

t 

G HH H

If H is square equation reduces to the solution given in equation

Similarlyifwe start with a p ossibly nonsquare matrix G of rank N we

t 

maycho ose H GG G

SUBBAND IMAGE CODING

Orthogonal Transforms

As mentioned in the intro duction the issue of orthogonalityisusually not

considered when discussing subband lters It is however a prop ertywhich

is relevanttoimagecoding as we will discuss in the next section A matrix

M corresp onding to an orthogonal transformation is a square matrix with the

prop erty that

t t

MM M M I

In terms of the columns or basis functions of M this means that the inner

pro duct of anytwo distinct columns must b e zero and the inner pro duct of a

column with itself must b e unity

The orthogonality condition places a numberof restrictions on the corre

sp onding AS system Since the transformation matrix must b e square the

numberof samples in the transformed signal must b e equal to N thenumber

of samples in the original image For the AS system this means that

M 

X

k

i

i

where wehave assumed that N is divisible byallofthe k Suchasystem has

i

b een termed a maximal ly decimated or critical ly sampled lter bank

A second more imp ortantconstraintisplaced on the AS system by

orthogonalityCombining the p erfect reconstruction requirementin with

the orthogonality constraintin gives

G H

If weconsider the relationships b etween the AS lters h and g and the ma

trices H and G describ ed by equations and this means that the

lters must ob ey

g nh n for all i

i i

In other words the synthesis lters of an orthogonal transform are time

inverted versions of the analysis lters

Some Example Transforms

In this section wewill briey discuss three onedimensional transforms to

illustrate some of the p oints made in the previous sections Each transform will

demonstrate b oth advantageous and disadvantageous prop erties for co ding

Chapter Subband Transforms

The Gab or Transform

In the intro duction to this chapter weargued that the basis functions of a

useful decomp osition should b e lo calized in b oth the spatial and the spatial

frequency domains One solution to the problem of spatially lo calized subband

decomp osition is that prop osed by Dennis Gab or Gab or intro duced a

onedimensional transform in which the basis functions are sinusoids weighted

byGaussian windows The Gab or transform can b e considered to p erform

alocalized frequency decomp osition in a set of overlapping windows The

resulting basis functions are lo calized in b oth space and spatial frequency

in fact Gab or showed that this jointlocalization was optimal with resp ect

to a measure that he chose although Lerner later noted that altering

the measure of jointlocalization pro duces dierentoptimal functions The

rst ve basis functions of a Gab or transform are shown in gure along

with their frequency sp ectra Both the basis functions and their transforms

are smo oth and compact In two dimensions the Gab or basis functions are

directional sinusoids weighted bygaussian windows Daugman has

used twodimensional Gab or transforms for image compression

The primary dicultywith the Gab or transform is that it is strongly non

orthogonal ie the sampling functions are drastically dierent from the basis

functions The sampling functions corresp onding to the Gab or transform

computed byinverting the transformation matrix are depicted in gure

These functions are extremely p o orly b ehaved b oth in the spatial and spatial

frequency domains In a co ding application errors intro duced byquantization

of the co ecients will b e distributed throughout the spatial and frequency

domains even though the co ecientvalues are computed based on information

in lo calized spatial and frequency regions

It is interesting to note that the lo calization of the inverse Gab or functions

can b e substantially improved if one uses an overcomplete Gab or basis set

This can b e accomplished byspacing the Gaussian windows more closely than

is required or bydividing each windowinto more frequency bands This

results in an increase in the numberofcoecients however whichmaybe

disadvantageous for co ding systems The use of overcomplete Gab or sets for

co ding remains a topic for further research

Several authors havediscussed related overcomplete oriented transforms

for use in image co ding Kuntadvo cated the use of directional ie ori

entation sub division for image co ding and used an oriented decomp osition

for this purp ose Watson develop ed the Cortex transform an overcom

plete transform which decomp oses the image into oriented o ctavebandwidth subbands and used it to compress image data

SUBBAND IMAGE CODING

Figure Fiveofthe sixteen basis functions of a Gab or lter set with

their corresp onding Fourier transforms The transforms are plotted on a

linear scale over the range from to

Chapter Subband Transforms

Figure The inverse sampling functions of the Gab or lter set given in gure

SUBBAND IMAGE CODING

The DCT and LOT Transforms

The use of the discrete cosine transform DCT in image co ding systems is

often justied with the statement that it approximates the optimal transform

for a signal with rstorder GaussMarkovstatistics In practice the

transform is usually not computed globallybut is applied indep endently to

nonoverlapping subblo cks of the image As illustrated in gure the

resulting blo ckDCT basis functions constitute a subband transform but the

subbands are not very well lo calized Considered in the framework of the

AS system the subsampled subband images will contain severe amounts of

aliasing Since the transform is invertible in fact orthogonal it should b e

clear that this subband aliasing is cancelled in the synthesis stage However if

the transform co ecients are quantized or discarded eg in a co ding system

the aliasing no longer cancels and the errors app ear as blo ckedgeartifacts in

the reconstructed image

Recentwork byCassereau et al describ es an elegant technique for

reducing the aliasing of the blo ckDCT They p erform an orthogonal transfor

mation on the blo ckDCT co ecients which combines co ecients computed

from adjacent blo cks In the resulting transform whichtheyhavecalled a

LappedOrthogonal Transform LOT the basis functions from adjacentblocks

overlap eachother and their impulse resp onses are tap ered at the edges Mal

var has implementedan ecientversion of this transform known as the

fast LOT in which the additional orthogonal transformation is computed us

ing a buttery network of simple rotation transformations Several of the

evensymmetric basis functions of the fast LOT are shown in gure One

limitation whichapplies to b oth the DCT and the LOT is that the trans

forms are limited to equalsized subbands As discussed previouslyitmaybe

advantageous to sub divide the sp ectrum into equal logwidth subbands

The Laplacian Pyramid

One of the rst techniques for o ctavesubband decomp osition was develop ed by

Burt and applied to image co ding byBurtand Adelson They used a

pyramid cascade of small Gaussianlikelters to create an overcomplete sub

band representation whichtheycalled a Laplacian pyramid Asystem for

constructing one level of this pyramid in one dimension is illustrated in g

ure The signal is blurred with a lowpass lter B and then subsampled

to pro duce a lowpass subband W A highpass subband W is formed

 

byupsampling W convolving with an interp olation lter A and sub



tracting from the original signal The signal is reconstructed byupsampling

and ltering W withA and adding it to W This reconstruction is

 

Chapter Subband Transforms

Figure Fiveofthe eightevensymmetric basis functions of a LOT

The basis functions are illustrated on the left and their Fourier transforms

on the right The transforms are plotted on linear axes and cover the range

from to

SUBBAND IMAGE CODING

w (n) ^ x(n) 0 x(n) B(∑) 2 £ 2 ‹ A(∑)

2 ‹

A(∑) w (n)

- 1

Figure Signal pro cessing diagram depicting the standard construction

technique for one level of the Laplacian pyramid A full pyramid is built

by nonuniformly cascading this system This transformation mayalso b e

describ ed as an AS lter bank see text

exact regardless of the choice of the lters B andA The full pyramid

is constructed recursivelyby reapplying the system to the lowpass subband

Typicallythelters A and B are set to some common compact lowpass

lter although b etter co ding results are obtained bycho osing the twolters in

dep endentlySome example basis and sampling functions with A B

are plotted in gures and resp ectively

In addition to its suitabilityfordata compression the multiscale nature

of the pyramid makes it particularly useful for the task of progressivetransmis

sion Progressive transmission is a pro cess by whichanimage is sentthrough

alowcapacitychannel so that a lowresolution or blurred version of the im

age b ecomes available quicklyand higher resolution information is added in

a gradual manner In the case of a pyramid this is easily accomplished by

sending the transform co ecients in order from lowest to highest resolution

For comparison to other subband transforms wehavereformulated the

Laplacian pyramid scheme as a threeband AS system see diagram in g

ure by separating W into two subsignals Y contains the even

 

numbered samples and Y contains the o ddnumb ered samples The sub



sampling factors are k for all three AS branches thus pro ducing a repre

sentation that is overcomplete bya factor of The appropriate lters for

the AS system are dened in terms of the original lters A and B as

follows

H B G A

 

h i



H B A B A G

 



h i

j

e

j

H B A B A G e

  

Chapter Subband Transforms

Figure Five example basis functions of a four level Laplacian pyramid

SUBBAND IMAGE CODING

Figure Fiveexample inverse sampling functions of the Laplacian

pyramid

Chapter Subband Transforms

Notice that when A and B arelowpass lters the sampling functions

are bandpass and the basis functions are broadband Since the resulting AS

system violates the constraintinequation the transform is clearly not

orthogonal In two dimensions Burt and Adelson constructed Laplacian pyra

mids using the same algorithm but with a separable twodimensional blurring

lter The twodimensional Laplacian pyramid maybereformulated as a

band AS system with eachbandsubsampled bya factor of twobothhori

zontally and vertically

The Laplacian pyramid has certain disadvantages for image co ding The

most serious of these is the fact that quantization errors from highpass sub

bands do not remain in these subbands Instead they app ear in the recon

structed image as broadband noise As with the Gab or transform the non

orthogonalityofthe transform is the source of the diculty Furthermore

the basis set is overcomplete requiring an increase in twodimensions bya



factor of in the number of sample points over the original image Finally



the twodimensional basis functions are not oriented and thus will not ex

tract the oriented structural redundancy typically found in natural images

Despite these disadvantages for stillimage co ding the Laplacian pyramid has

b een eectively used for motioncomp ensated video co ding where its overcom

pleteness makes it robust in to motioncomp ensation errors

Quadrature Mirror Filters

In the previous section wedescrib ed three example transforms eachdemon

strating useful prop erties for co ding Nowwe consider a transform which

captures the advantages of the previous examples while avoiding the disad

vantages

As was illustrated with the Laplacian pyramid an o ctavesubband trans

form maybe constructed by cascading a twoband AS system in a non

uniform manner A useful twoband subband transform whichwas develop ed

for sp eechcoding is based on banks of quadrature mirror lters QMF de

velop ed by Croiser et al They discovered a class of nonideal FIR

bandpass lters that could b e used in an AS system while still avoid aliasing

in the overall system output Although they did not describ e them as such

these lters form an orthogonal subband transform as was discussed by Adel

son et al and Mallat Mallat related QMFs to the mathematical

theory of wavelets Vetterli was the rst to suggest the the application of

QMFs to twodimensional images In this section wegivea brief review of

QMFs in one dimension A more thorough review maybefound in or

SUBBAND IMAGE CODING

The original QMF problem was formulated as a twoband critically sam

pled analysissynthesis lter bank problem The overall system resp onse of

the lter bank is given byequation with subsampling factor on each

branch set to k

h i

X H G H G X

   

h i

H G H G X

   

The rst term is a linear shiftinvariant LSI system resp onse and the second

is the system aliasing

The term QMF refers to a particular choice of lters that are related by

spatial shifting and frequency mo dulation We dene

H G F

 

j

H G e F

 

for F anarbitrary function of This is a more general denition than that

originally provided byCroisier et al and makes explicit the orthogonality

of the transform see section In particular the analysis and synthesis

lters satisfy the relationship in equation and the relationship b etween

the lters in the two branches ie H and H ensures that the corresp onding

 

basis functions are orthogonal

With the choice of lters given in equation b ecomes

h i

X H H H H X

h i

j

H H e H H X

The second aliasing term cancels and the remaining LSI system resp onse is

h i

X H H H H X

Note that the aliasing cancellation is exact indep endentofthechoice of the

function F Weshouldemphasize however that it is the overall system

aliasing that cancels the individual subbands do contain aliasing

QMF Design

The design problem is nowreduced to nding a lowpass lter with Fourier

transform H that satises the constraint

h i

H H H H

Chapter Subband Transforms

or

 

H H

Several authors have studied the design and implementation of these lters

Johnston designed a set of widely used evenlength lters

by minimizing an error function containing a shiftinvarianterror term and

aweighted stopband ripple term for a xed numberoflter taps Jain and

Cro chiere used a similar error criterion in the time domain and for

mulated an iterativedesign scheme whichinwhicheach iteration required the

constrained minimization of a quadratic function

A technique for design of p erfect reconstruction lter sets is given by Smith

and Barnwell in They rst design a lowpass product lter F whichis

factorable as



F H

and whichsatises

n

f n n

The resulting F isfactored to get hn the lowpass lter Wackersreuther

indep endently arrived at a similar design metho d in the time domain The

problem with these design metho ds is the somewhat arbitrary choice of the

pro duct lter

Simoncelli prop osed an exploratory design metho d utilizing an itera

tive matrix averaging technique and designed a set of o ddlength lters using a

frequencysampling metho d with error criteria similar to Johnston The design

constraints for QMFs do not necessitate sharp transitions and thus frequency

sampling designs p erform quite well Furthermore it was found that o dd

length lters could b e made smaller for a given transition band width The

basis functions for a fourlevel QMF pyramid based on a tap kernel are shown

in gure A set of example QMF kernels and a more detailed description

of this design technique are given in the app endix to this chapter

QMFs are typically applied to images in a separable manner In order

to compute a multiscale pyramid the transform is applied recursively to the

lowpass subimage Sucha cascaded transformation partitions the frequency

domain into o ctavespaced oriented subbands as illustrated in the idealized

frequency diagram of gure Thus the QMF pyramid satises the prop

erties describ ed in the intro duction to this chapter it is multiscale and ori

ented it is spatially lo calized and it is an orthogonal transformation and so

constrains quantization errors to remain within subbands One unfortunate

asp ect of the transform is that the orientation decomp osition is incomplete

SUBBAND IMAGE CODING

Figure Fiveofbasis functions of a tap QMF pyramid transform

Chapter Subband Transforms

The two diagonal orientations are lump ed together in a single subband We

will address this problem in section

An Asymmetrical System

Thus far wehave ignored the issue of computational eciency For many

applications this is relatively unimp ortantdue to the steady increase in the

sp eed of signal pro cessing hardware There are however situations where

it is desirable to quickly deco de or enco de a co ded image using conventional

or generalpurp ose hardware For example an image data base that will b e

accessed by millions of users with p ersonal computers should b e quickly deco d

able on standard hardware the cost of enco ding these images is of relatively

minor imp ortance as long as the deco ding is simple Atthe other extreme a

remotely piloted vehicle demands a very simple enco ding scheme in order to

minimizeweight and p ower requirements

For these situations it is advantageous to develop asymmetric co ding tech

niques in whichsimplicityisemphasized at one end at the exp ense of com

plexityatthe other end For a QMF transform the computational complexity

is directly prop ortional to the size of the lters employed Thus we wish

to relax the orthogonalityconstraintwhichforces the synthesis lters to b e

timereversed copies of the analysis lters Consider the situation in whichwe

require ecient deco ding The increase in eciency can b e accomplished by

using a very compact lter pair in the synthesis stage of an AS system

In particular one can cho ose the tap lowpass lter g n with a



highpass counterpart g n Convolutions with these lters may



be performed using only arithmetic shifting and addition op erations

j

The relationship G e G as in equation ensures

 

that the linear subspaces spanned bythe basis functions corresp onding to each

lter will b e orthogonal Conceptuallyaset of inverse lters h nisfound

i

by forming a square matrix of the g nasinequation and inverting it

i

The size of the matrix determines the size of the resulting inverse lters In

practice a b etter design technique is to minimizean error function for a given

kernel size Wehave designed a set of inverse lters given in the app endex

by minimizing the maximal reconstruction error for a step edge input signal

These kernels are given in the app endix

Another highly ecientAS system was prop osed by LeGall He derived

the following set of simple lters for use in an AS lter bank

H A G B

 

j j

H e B G e A

 

SUBBAND IMAGE CODING

ωy

ωx

ωy ωy ωy ωy

ωx ωx ωx ωx

ωy ωy ωy ωy

ωx ωx ωx ωx

....

Figure Idealized diagram of the partition of the frequency plane

resulting from a level pyramid cascade of separable band lters The

top plot represents the frequency sp ectrum of the original image with axes

ranging from  to This is divided into four subbands at the next

level On eachsubsequent level the lowpass subband outlined in b old is sub divided further

Chapter Subband Transforms

where the lter kernels impulse resp onses corresp onding to A and B

are

an

bn

Note that the denitions of anand bnmaybeinterchanged These lters

allow ecientenco ding and deco ding and provide exact reconstruction with

p erfect aliasing cancellation

Nonseparable QMF Transforms

In the previous section wedescrib ed the separable QMF pyramid transform

Most twodimensional work with QMFs has employed separable lters or non

oriented nonseparable lters As discussed in the previous section sepa

rable application of onedimensional QMFs pro duces a representation in which

one of the subbands contains a mixture of two orientations This problem is

inherentintherectangular sampling scheme Rectangular sampling of a sig

nal in the spatial domain corresp onds to summing aliased or mo dulated copies

of the sp ectrum in the frequency domain Thus the frequency resp onse of

anyrectangularly sampled function has the same value at the p oints

and ie this p oint corresp onds to twoopp osing

orientations Splitting the frequencies in the neighborhood of this p ointinto

dierent orientation bands requires the use of very large lters In general the

highfrequency diagonal regions of the sp ectra of natural images are relatively

insignicant But if the lter bank is cascaded to form a pyramid then the

lower frequency diagonals where there is signicantpower will also b e mixed

Hexagonal Systems

In this section wewill discuss the use of hexagonal sampling systems and l

ters We will show that the mixed orientation problem discussed ab ovecanbe

avoided byusing hexagonally symmetriclters This nonseparable extension

of the QMF concept was rst describ ed byAdelson et al and improved

and generalized in later work Other authors havealsoexplored the

use of hexagonal sampling systems for image representation Crettez and Si

mon and Watson describ e decomp ositions on hexagonal lattices using

nonoverlapping basis functions The blo cked nature of these functions sug

gests that they are unlikely to oer ecient image compression

SUBBAND IMAGE CODING

Figure shows a hexagonal sampling lattice and its Fourier transform

The sampling lattice is dened byapairofsampling vectors in the plane

p

v v

 

The lo cations of the lattice p oints consist of all linear combinations of these

vectors with integer co ecients In the frequency domain the eect of this

sampling is to convolve the original frequency sp ectrum of the image with a

modulation or reciprocal lattice whichis the Fourier transform of the sampling

lattice The mo dulation lattice is dened byapair of modulation vectors in

the frequency plane

p p

v v

 

Thus if F is the Fourier transform of a hexagonally sampled signal image

then it is invariantto translations bymultiples of the vectors v

i

F F n v n v

   

for n and n anytwointegers

 

In general the relationship b etween the sampling vectors and mo dulation

vectors is easily describ ed in terms of matrices If weconsider the

sampling matrix V with columns containing the vectors v and the modulation

i

matrix V with columns containing the vectors v thenthetwo matrices are

i

related bytheequation

 t

V V

Note that we know V is invertible since weassumethat the sampling vectors

span the space ie they are linearly indep endent

As stated in section the AS system depicted in gure is valid

for twodimensional signals but the ltering and subsampling is done in two

dimensions is nowatwodimensional vector and the subsampling is pa

rameterized byanonsingular twobytwo subsampling matrix Kwithinteger

entries Figure illustrates twodimensional subsampling in b oth the spa

tial and frequency domains

In order to write a general expression for the output of a multidimensional

analysissynthesis system weneed a frequencydomain equation analogous to

that given in relating the subsampled signal to the sampled signal We

also need an equation relating an upsampled signal to the original sampled

signal For rectangular sampling lattices in d dimensions the relationship is

simple The sampling matrix K generates a sublattice dened by

d

fn n Km m Z g

Chapter Subband Transforms

Figure Relationship b etween hex sampling lattices in the spatial and

spatialfrequency domains On the left is the lattice dened by the sampling

vectors On the rightis the Fourier transform of this lattice dened by the

mo dulation vectors

Figure Illustration of subsampling on a hexagonal lattice The

points in the diagram on the left represent the original sampling lattice

and the circles represent the subsampled lattice p oints The picture on the

right shows the Fourier transform of the lattice p oints and the Fourier transform of the subsampled lattice circles

SUBBAND IMAGE CODING

d

where Z is the set of all ddimensional vectors with integer comp onents The

sublattice has jKj distinct cosets eachcosetbeing a copyofthe sublattice

translated byaninteger vector and the union of the cosets is the original

sampling lattice Consider twosignals related by subsampling sn

r Kn Then their Fourier transforms are related by the expression

jKj

X

 t

S R K k

i

jKj

i

where S and R arethe Fourier transforms of sn and r n resp ectively

and the k are a set of p olyphase shift vectors corresp onding to eachofthe

i

jKj sublattice cosets Asimple example of a set of shift vectors is the

following

n o

 t d d

k K k k Z

d

where is the halfop en unit interval in d dimensions

The corresp onding expression for nonrectangular sampling lattices is ob

tained bymapping from the rectangular case The result of subsampling in

the analysissynthesis system may then b e written as a convolution of the

sampled sp ectrum with a set of subsampling modulation vectors

j

jKj

X

 t  t

H K X K Y

i j j i

jKj

j 

where the are dened as

j

o n

j jKj

j

o n

 t  t d d

V K n K n n Z

The eect of upsampling in the frequency domain is the same as for the rect

angular case Combining equation with the frequency domain up

sampling relationship gives an expression for the overall lter bank resp onse

jKj

X

t

X G Y K

i i

jKj

i

jKj jKj

X X

G H X

i i j j

jKj

i j 

jKj jKj

X X

X G H

j i i j

jKj

j  i

Chapter Subband Transforms

Figure Illustration of the mo dulating eect of subsampling in the

frequency domain Assume that the sampled image has a sp ectrum ban

dlimited to the grayregion in the upp er left frequency diagram Subsam

pling will mo dulate the sp ectrum to the gray regions in the other three

diagrams The resulting sp ectrum will b e the sum of the four sp ectra

As in equation the rst term of the sum j corresp onds to the LSI

system resp onse and the remaining terms are the system aliasing

Returning nowto the sp ecic case of the hexagonal sampling lattice we

describ e a system obtained byusing a sp ecic sampling matrix KSince we

wanttobeable to apply the transform recursivelywecho ose a subsampling

scheme whichpreserves the geometry of the original sampling lattice

K

On the hexagonal sampling lattice with this subsampling scheme the denition

given in pro duces the following mo dulation vectors

p p p

   

Figure oers an idealized picture of this mo dulation

Analogous to the onedimensional case wecancho ose the lters to elim

inate the aliasing terms in equation

H G F F

 

SUBBAND IMAGE CODING

j s

H G e F

  

j s

F H G e

  

j s

H G e F

  

where H is a function that is invariantunder negation of its argument the

s are a set of spatial shift vectors dened in the next paragraph and the

i

expressions s indicates an inner pro duct of the twovectors As in equa

i

tion the lters are related byspatial shifting and frequency mo dulation

For the subsampling matrix weare using here there are four sublattice cosets

and therefore four distinct shifting vectors including the zero vector Two

assignments of the s lead to system aliasing cancellation and these twoas

i

signments are related by reection through the origin So without loss of

generalitywecho ose the shifting vectors to b e

p p

s s s

  

After cancellation of the aliasing terms in equation the remaining

LSI system resp onse is



X

X X G H

i i

i



X

F F X

i i

i



X



jF j X

i

i

As in one dimension the aliasing cancellation is exact indep endentofthe

choice of F and the design problem is reduced to nding a lter with

Fourier transform F satisfying the constraint



X



jF j

i

i

This is analogous to the onedimensional equation Again a lowpass

solution will pro duce a bandsplitting system whichmaybecascaded hier

archically to pro duce an o ctavebandwidth decomp osition in two dimensions

An idealized illustration of this is given in gure Finer frequency and ori

entation sub divisions maybeachieved by recursively applying the lter bank

to some of the high frequency subbands as illustrated in gure

Filters maybe designed using the metho ds describ ed in section

Several example lter sets are given in the app endix to this chapter The

Chapter Subband Transforms

ωy

ωx

ωy ωy ωy ωy

ωx ωx ωx ωx

ωy ωy ωy ωy

ωx ωx ωx ωx

.....

Figure Idealized diagram of the partition of the frequency plane

resulting from a fourlevel pyramid cascade of hexagonal lters The top

plot represents the frequency sp ectrum of the original image This is divided

into four subbands at the next level On each subsequentlevel the lowpass subband outlined in b old is subdivided further

SUBBAND IMAGE CODING

ωy

ωx

Figure An example of a frequencydomain partition whichcould

be computed using a nonpyramid cascade of the hexagonal lter bank

transform describ ed in the text The shaded region indicates the frequency

region asso ciated with one of the transform subbands

power sp ectra of an example set of lters the ring lters are plotted in

gure These lters are extremely compact requiring only nine multipli

cations p er convolution p ointassuming one takes advantage of the twelvefold

hexagonal symmetry Figure shows the results of applying this bank of

lters recursively to an image of a disk Examples of images co ded using these

lters will b e given in section

RhombicDodecahedral Systems

The extension of the concepts develop ed in the previous section to three

dimensional signal pro cessing is fairly straightforward Suchsystemsare use

ful for applications suchascompression of medical images or video motion

sequences Analogous to the twodimensional hexagonal case one can cho ose

aperio dic sampling lattice which corresp onds to the densest packing of spheres

Chapter Subband Transforms

Figure The p ower sp ectra for the ring set of hexagonal QMF

lters The lter kernels are given in the app endix

SUBBAND IMAGE CODING

a b c

Figure Results of applying a hexagonal QMF bank to an image of

adisk a The original image b The result after one application of the

analysis section of the lter bank The image has b een decomp osed into a

lowpass and three oriented highpass images at densitycThe result

of applying the lter bank recursively to the lowpass image to pro duce a

twolevel pyramid decomp osition

in three dimensions This packing corresp onds to the crystal structure of gar

net Wecho ose as a band limiting region the Voronoi region of this lattice

a rhombic do decahedron whichisillustrated in gure The sampling

matrix for the lattice is

V

p

Using equation the mo dulation matrix is then

V

p p

To preservethe geometry of the original sampling lattice wecho ose an

eightband AS system with subsampling matrix

K

This pro duces the following subsampling mo dulation p oints as determined by

equation

B C C B C B B C

A A A A

   

p p

Chapter Subband Transforms

Figure Arhombic do decahedron This is the shap e of the bandlim

iting frequency region for the garnet lter

B B B C C C C B

A A A A

  

p p p p

When applied to video motion sequences these mo dulation vectors corresp ond

to a decomp osition into the following subbands lowpass stationary vertical

stationary horizontal motion upright motion upleft motion downright

motion downleft and combined stationary diagonals and fulleld icker

Unfortunately there seems to b e no waytoavoid the last lter which contains

mixed orientations The overall system resp onse of the lter bank is

X X

X G H X

i i j j

i j 

where the rst term is the LSI system resp onse and the remaining terms are

aliasing terms

Once again we can cho ose lters related byshifts and mo dulations that

will cancel the system aliasing terms

H G F F

 

j s

i

H G e F i f g

i i i

where the shift vectors s are dened as

i

B B C C C B B C

s s s s

A A A A

   

p p p

SUBBAND IMAGE CODING

B C B C C B C B

s s s s

A A A A

  

p p

Note that as in the hexagonal case the choice of shift vectors is not unique

With the choice of lters given ab ove the aliasing terms in equation

cancel and the remaining LSI system resp onse is

X



X X jF j

i

i

indep endentofthechoice of the function F The design constraintequation

is now

X



jF j

i

i

To illustrate the use of the garnet lter weapplyit to an image sequence

of a sinusoidal pinwheel rotating in a counterclo ckwise direction One frame

of the sequence is shown in gure a The squared resp onses of the four

dierentmotionselective lters lters H through H are shown in

 

gure be

Image Co ding Examples

Several authors have used QMFs for purp oses of image co ding Woods and

ONeill were the rst to implementanimage co ding system using QMFs

They constructed a separable sixteenband decomp osition using a uniform

cascade of and tap lters designed by Johnston and then co ded

the bands using adaptive DPCM Gharavi and Tabatabai used a pyramid

of separable lters and in applied it to color images Tran et al used

an extension of Chen and Pratts combined Human and runlength co ding

scheme to co de QMF pyramids Adelson et al haveused b oth separable and

hexagonal QMF pyramids for image co ding Mallat used lters

derived from wavelet theory to co de images Westerink et al haveused

vector quantization for subband co ding of images

In gures and wegiveexamples of data compression of the

Lena image using a separable tap QMF bank a tap asymmetric

lter bank describ ed in section and a hexagonal ring QMF bank In

all cases a fourlevel pyramid transform was computed byrecursiveapplication

of the analysis p ortion of the AS system to the lowpass image The total bit

Chapter Subband Transforms

Figure Atthe top is one image from a rotating pinwheel image

sequence The four lower images are the squared result of convolving the

sequence with four of the garnet lters describ ed in the text Each lter

resp onds preferentially to one direction of motion

SUBBAND IMAGE CODING

rate R was xed and the bit rates assigned to the co ecients of the transform

were determined using the standard optimal allo cation formula



k

R R log

k



N

N 

Y



j

j 



where as b efore is the variance of the k th co ecientin the transform

k

Negativevalues of R were set to zero and the other bit rates raised to maintain

k

the correct overall bit rate R



are the same for Note that if weassume stationary image statistics the

k

all co ecients b elonging to the same subimage of the transform It has b een

shown that the optimal quantizer for entropycoding is nearly uniform

for bit rates which are high enough that the image probabilitydistribution is

approximately constantover eachbinEven though the examples shown were

compressed to relatively low bit rates uniform quantization was used due to

its simplicity Eachsubimage was quantized with the bin size chosen to give

a rst order entropyequal to the optimal bit rate R for that subimage

k

For the hexagonal pyramid additional pre and p ostpro cessing was neces

sary to resample the image on a hexagonal grid Before building the pyramid

weresampled the original image vertically byafactorofusingsinc inter

n n

x y

p olation We then multipliedbythe function f n This

metho d whichissimilar to one suggested in gives a reasonable geomet

ric approximation to a hexagonal sampling lattice After resynthesizing the

image weinterp olated the zerovalued pixels and vertically resampled bya

factor of

The hexagonal QMF system generally oers co ding p erformance p ercep

tually sup erior to that of the separable system p erhaps b ecause the aliasing

errors are not as visually disturbing as those of separable QMFs Of course

the hexagonal system has the disadvantage of b eing more inconvenienttouse

in conjunction with standard hardware

Conclusion

Wehave discussed the prop erties of linear transforms that are relevanttothe

task of image compression In particular wehavesuggested that the basis and

sampling functions of the transform should b e lo calized in b oth the spatial and

the spatialfrequency domains Wehave also suggested that it is desirable for the transform to b e orthogonal

Chapter Subband Transforms

a b

c d

Figure Data compression example using fourlevel pyramids The

pyramid data was compressed to a total of bits ie total rstorder

entropywas bitpixel a Original Lena image at  pixels

b Compressed using tap separable QMF bank c Compressed using

tap asymmetrical lter bank d Compressed using ring hexagonal QMF bank

SUBBAND IMAGE CODING

a b

c d

Figure Data compression example using fourlevel pyramids The

pyramid data was compressed to a total of bits ie total rstorder

entropywas bitpixel a Original Lena image at  pixels

b Compressed using tap separable QMF bank c Compressed using

tap asymmetrical lter bank d Compressed using ring hexagonal QMF bank

Chapter Subband Transforms

Several examples servetoillustrate these prop erties The Gab or basis

functions are welllo calized but the severe nonorthogonalityofthe transform

leads to sampling functions which are very p o orly lo calized The blo ck DCT

is an equalwidth subband transform with p o or frequency lo calization The

LOT enhancementprovides improved frequency lo calization The Laplacian

pyramid is an example of an o ctavewidth subband transform that is non

orthogonal nonoriented and overcompleteits prop erties are nonoptimal for

co ding still images but maybeadvantageous for co ding moving images

Subband transforms based on banks of QMFs are welllo calized orthogo

nal and can b e applied recursively to form o ctavewidth subbands Separable

application of these transforms oers orientation sp ecicityin some but not

all of the subbands Nonseparable orthogonal subband transforms based on

hexagonal sampling oer orientation sp ecicityinall of the subbands although

they are more dicult to implement These orthogonal subband transforms

are highly eectivein image co ding applications and may also b e appropriate

for applications in image enhancementandmachine vision tasks

App endix Filters

In this app endix we discuss the design of QMFs presenta set of example

lter kernels and compare their theoretical energy compaction prop erties to

those of the DCT and LOT transforms

Filter Design

A go o d QMF is one that satises the constraintgiven in equation

In addition one would likethesubband images to havea minimal amountof

aliasing The ob jective then is to design lters with small regions of supp ort

that satisfy b oth of these constraints Assuming symmetric linear phase lter

designs a lter of size N is determined bya setofdNe free parameters

where de indicates the ceiling function Therefore lters maybedesigned by

minimizing an error function dened on the space of these free parameters

For a xed lter size we dene a frequencydomain lter bank error func

tion as the maximal deviation of the overall lter bank resp onse given in

equation from its ideal value

n o

 

E max f jF j jF j

 

where ranges over the samples in the frequency sp ectrum The function

f isa frequency weighting function roughly matched to the sensitivityof 

SUBBAND IMAGE CODING

n QMF QMF QMF

Table Oddlength QMF kernels Half of the impulse resp onse sample

values are shown for eachofthe normalized lowpass QMF lters All lters

are symmetric ab out n The appropriate highpass lters are obtained

n

bydelaying byonesample and multiplying with the sequence 

the human visual system and the statistics of images

f j j



Wealso dene an intraband aliasing error function

E maxff jF F jg

 



where the function f isdened as





f j j



The frequency vector ranges over all of the samples in the frequency sp ec

trum except for the p ointat Aliasing within subbands cannot b e elimi

nated at this p ointbecause the overall lter bank resp onse at this p ointwould

then b e forced to zero violating the constraintinequation

Finallywecombine the two error functions as a weighted sum

E E E

 

Given a set of values for the free parameters wecan construct a kernel and

compute the value of the error function E Todesign lters we used a downhill

simplex metho d to searchthe space of free parameters for minima in E The

weighting factor was adjusted to givea lter bank resp onse error E less



than a xed threshold A set of example o ddlength lter kernels are given in

table

The same design technique was used for multidimensional nonseparable

lters For the hexagonally symmetric lters the free parameters comprise a

wedgeshap ed region covering approximately one twelfth of the kernel The

Chapter Subband Transforms

twoerror functions are dened in the same manner as for the onedimensional

lters The frequency vector now ranges over all of the samples in the two

dimensional frequency sp ectrum except for those in a hexagonal b oundary

containing the p oint A set of kernel values is given in table



In table wegiveseveral inverse lters for the tap asymmetrical system

describ ed in section These kernels were designed byminimizing the

maximal absolutevalue reconstruction error for a step edge input signal

Filter Compaction Prop erties

An optimal transform for data compression should minimize the bit rate for a

given allowable error in the reconstructed image If the basis functions of the

transform are orthonormal and if exp ected mean square dierence is used as

an error measure this is equivalenttomaximizing the following expression for

the gain in co ding over PCM

N 

X



j

N

j 

G

N

N 

Y



j

j 



where is the variance of the j th transform co ecient

j

This measure was computed for some of the onedimensional QMF lters

given in this App endix as displayed in table Values were computed as

suming Markovsecond order signal statistics where the auto correlation matrix

R is a symmetricToeplitz matrix of the form

xx

 N

N 

 N 

R

xx

N N  N 

and where is the intersample correlation co ecient A value of

was used to compute the numbers given in table The compaction values

are given for a onedimensional image of size N with the QMF lter

kernels reected at the edges in a manner that preserves the orthogonalityof

the basis set Comparable values for the p ointLOTwithkernel sizes

L and a p ointblock DCT and a p ointblockDCT are also given

SUBBAND IMAGE CODING

j k l l k j

k g h i h g k

l h e f f e h l

l i f c d c f i l

k h f d b b d f h k

j g e c b a b c e g j

k h f d b b d f h k

l i f c d c f i l

l h e f f e h l

k g h i h g k

j k l l k j

Table Ahexagonal lter The letters refer to the free parameters

see text Only the lowpass lter is shown The three highpass lters are

formed bymodulating and shifting the lowpass

Parameter ring ring ring

a

b

c

d

e

f

g

h

i

j

k

l

Table Some example hexagonal lter co ecientvalues The parame

ter letters corresp ond to the diagram shown in table

Chapter Subband Transforms

n

Table Filter impulse resp onse values for and tap inverses

for the tap system describ ed in section Half of the impulse resp onse

sample values are shown for eachofthe normalized lowpass lters All lters

are symmetric ab out n The appropriate highpass lters are obtained

n

bymultiplying with the sequence  and shifting by one pixel

The tap subband lter gives slightly b etter value than the p oint DCT

and the tap subband lter is substantially b etter These comparisons do not

necessarily corresp ond to measurementsofsub jectivequalityhowever since

they are based on a crude Markov statistical mo del of images and since they

assume an MSE error measure Wehave found that images compressed with

atap subband transform are p erceptually sup erior to the p oint DCT

primarily b ecause of the absence of the blo ckartifacts Wealsond that the

tap QMF is preferable to the tap QMF the tap lter pro duces more

aliasing but the Gibbs ringing is more noticeable with the tap lter We

have not p erformed anycoding exp eriments using the LOT and so cannot

commentonits p erformance

lter G

PCM

QMF

QMF

QMF

fastLOT

DCT

DCT

Table Theoretical co ding gains over PCM for fourlevel QMF pyra

mids the fast LOT with N andL andthe blo ckDCTValues

were computed assuming rstorder GaussMarkov signal statistics with

on a a onedimensional image of size

SUBBAND IMAGE CODING

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