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Time Frequency Analysis and Wavelet Transform Tutorial
Haar Transform and Its Applications
Pei-Yu Chao 趙珮妤 D00945005
Abstract The The Haar Haar tran transfo sform rm is one one of the the simpl simplest est and and basic basic trans transfo form rmat atio ion n from from the the space/time domain to a local frequenc domain! "hich re#eals the space/time-#ariant spec spectr trum um$$ The The attr attrac acti tin% n% feat featur ures es of the the Haar Haar tran transf sfor orm! m! incl includ udin in% % fast fast for for implementation and able to analse the local feature! ma&e it a potential candidate in modern electrical and computer en%ineerin% applications! such as si%nal and ima%e compression$ 'n this tutorial! the mathematics and applications of Haar transform "ill be e(plored$
1
Chapt Chapter er 1 Intro Introdu duct ctio ion n The Haar transform "as proposed in )9)0 b a Hun%arian mathematician *lfred Haar +),$ +),$ The Haar transform is one of the earliest transform functions proposed$
Con#entionall! ourier transform has been used e(tensi#el to analse the spectral conten contentt of a si%nal si%nal$$ Ho"e#e Ho"e#er! r! ourier ourier transfor transform m is not able to represe represent nt a nonnonstationar stationar si%nal si%nal adequatel. adequatel. "hereas time-frequenc time-frequenc analsis function! e$%$! Haar transform! transform! is found found effecti#e as it pro#ides a simple approach approach for analsin% analsin% the local aspects of a si%nal$
The The Haar Haar trans transfo form rm uses uses Haar Haar func functi tion on for for its its basi basis$ s$ The The Haar Haar func functi tion on is an orthon orthonorm ormal! al! rectan% rectan%ula ularr pair$ pair$ Compare Compared d to the ourie ourierr transf transform orm basis basis functio function n "hich onl differs in frequenc! the Haar function #aries in both scale and position$
The Haar transform transform is compact! compact! dadic and orthonorma orthonormal$ l$ The Haar transform ser#es as a prototpe for the "a#elet transform! and is closel related to the discrete Haar "a#elet transform +,$ +,$
Chapter Chapter 2 The Haar Haar Transform ansform 2.1 2.1 Th Thee Haar Haar Func Functi tion on The famil of N of N Haar Haar functions functions
are defined on the inter#al inter#al
+,$ +,$ The
shape of the Haar function! of an inde( k ! is determined determined b t"o parameters1 p parameters1 p and and q! "here 2
and & is in a ran%e of
2hen
$
! the Haar function is defined as a constant
. "hen
!
the Haar function is defined as
rom the abo#e equation! one can see that p determines the amplitude and "idth of the non-3ero part of the function! "hile q determines the position of the non-3ero part of the Haar function +,$
2.2 The Haar Matrix The discrete Haar functions formed the basis of the Haar matri( H
"here
and
is the ronec&er product$
The ronec&er product of
! "here
is an
matri( and
is a
3
matri(! is e(pressed as
2hen
"here
is a
matri(! and
is a Haar function$
The Haar matri( is real and ortho%onal! i$e$!
! i$e$!
*n un-normali3ed -point Haar matri(
is sho"n belo" +,
4
rom the definition of the Haar matri( H! one can obser#e that! unli&e the ourier transform! H matri( has onl real element 6i$e$! )! -) or 07 and is non-smmetric$
The first ro" of H matri( measures the a#era%e #alue! and the second ro" H matri( measures a lo" frequenc component of the input #ector$ The ne(t t"o ro"s are sensiti#e to the first and second half of the input #ector respecti#el! "hich corresponds to moderate frequenc components$ The remainin% four ro"s are sensiti#e to the four section of the input #ector! "hich corresponds to hi%h frequenc components$ i%$ ) sho"s the Haar function at each ro" of H matri($ 8otice the "idth and location of the Haar function is chan%ed$ The Haar function "ith narro"er "idth is responsible for analsin% the hi%her frequenc content of the input si%nal$
i%$ ) Haar functions for composin% -point Haar transform matri( +,$
5
The in#erse & -point Haar matri( is described as
or
+,
! un-normalised in#erse -points Haar transform$
2.3 The Haar Transform The Haar transform
of an N -input function
is the
element #ector
The Haar transform cross multiplies a function "ith Haar matri( that contains Haar functions "ith different "idth at different location$
or e(ample1
6
The Haar transform is performed in le#els$ *t each le#el! the Haar transform decomposes a discrete si%nal into t"o components "ith half of its len%th1 an appro(imation 6or trend7 and a detail 6or fluctuation7 component$ The first le#el of
appro(imation
for
is defined as
! "here
is the input si%nal$ The multiplication of
that the Haar transform preser#es the ener% of the si%nal$ The #alues of
the a#era%e of successi#e pairs of
represents
#alue$
The first le#el detail
for
ensures
is defined as
$ The #alues of
represents the difference of successi#e pairs
7
of
#alue$
The first le#el Haar transform is denoted as
$ The in#erse of this transformation can
be achie#ed b
The successi#e le#el of Haar transform! the appro(imation and detail component are calculate in the same "a! e(cept that these t"o components are calculated from the pre#ious appro(imation component onl$
*n e(ample1
! the first le#el appro(imation and detail
components are
Chapter 3 Application 3.1 Sinal Compression et:s define the ener% E of a si%nal ; as the sum of the square of its #alue! i$e$!
or
!
8
rom this e(ample! "e can see that Haar transform preser#e ener%! i$e$!
and
$ urthermore! "e can see
that the ener% of the appro(imation component is much hi%her than the ener% of
detail component$ 'n the first le#el of transformation! the ener% of the appro(imation
is about
of the ener% of the si%nal ! and in the second le#el of
transformation! the ener% of the appro(imation
is about
2hich means that! after first le#el of Haar transform!
into a si%nal
$
of ener% is concentrated
that is half of the len%th of ! and after the second Haar transform!
of the ener% is concentrated into a si%nal
that is quarter of the len%th of $
This is called the compaction of ener% +4,! and it "ill occur "hene#er the ma%nitude of the detail component is si%nificantl smaller than the appro(imation component$ Thus! compression "ithout seriousl affectin% the information of the ori%inal si%nal can be achie#ed$
9
There are t"o basic cate%ories of compression techniques +4,$ The first cate%or is the lossless compression$ *s the name stand! the de-compressed si%nal is error-free$ Tpical lossless methods are Huffman compression! <2 compression! arithmetic compression and run-len%th compression$
The other tpe is the loss compression$ =#en thou%h this tpe of compression method produces error in the de-compressed si%nal! the error should onl b mar%inal$ The ad#anta%e of loss techniques is that hi%her compression ratio can be achie#ed! "hen compared to the lossless compression technique$ The Haar transform is a tpe of loss compression$
The steps in#ol#ed in a simple si%nal compression are described in i%$ $
This al%orithm is applied to a si%nal sho"n in i%$ 6a7! and the outcome of a )0-le#el Haar transform is sho"n in i%$ 6b7$ * threshold of 0$5? is chosen based on the cumulati#e ener% distribution of the Haar transformed si%nal$ Thereafter! compressed si%nal is obtained #ia in#erse Haar transform! "hich is sho"n in i%$ 6c7$ The compressed si%nal is almost identical to the ori%inal si%nal$ The ma(imum error 10
calculated o#er all #alues of appro(imated si%nal is no more than
+4,$
Hence! a compression factor of 0 "ith minimal error is achie#ed$
i%$ @i%nals durin% the steps of compression$ 6a7 The ori%inal si%nal! 6b7 )0-le#el Haar transform of the ori%inal si%nal! and 6d7 is the compressed si%nal 6in#erse Haar transform7
2hen the same al%orithm is applied to the si%nal sho"n in i%$ 46a7! performance of the si%nal compression is poorer$ The compressed si%nal! as sho"n in i%$ 46c7! has hi%her error "ith lo"er compression ratio 6)01)7$ 11
i%$ 4 @i%nals durin% the steps of compression$ 6a7 The ori%inal si%nal! 6b7 )-le#el Haar transform of the ori%inal si%nal! and 6d7 is the compressed si%nal 6in#erse Haar transform7
3.2 !e"noisin 2hen si%nal is recei#ed after transmission o#er some distance! it is often distorted b noise$ De-noisin% is a process "hich is used to reco#er the noise-buried speech! "hich enhances the reco%nisabilit of the speech si%nal$
The steps in#ol#ed in a simple de-noisin% process are described in i%$ 5$ *fter Haar transform is performed! a thresholdin% is used! i$e$! an #alues of the transformed si%nal lie belo" the noise threshold is set to 0$ Thereafter! in#erse Haar transform is 12
performed to re#eal the appro(imated si%nal$
i%$ 5 >loc& dia%ram illustrate de-noisin% T"o si%nal from i%$ 46a7 and i%$ 56a7 are distorted "ith additi#e noise! "hich are sho"n in i%$ ?6a7 and i%$ A6a7$ The de-noisin% process is applied to the noisedistorted si%nals$ 't can be clearl obser#ed that there are lar%e numbers of fluctuation in the Haar transformed si%nal "hich is contributed b the random noise$ *fter thresholdin% and in#erse Haar transform! the de-noised si%nals are re#ealed! "hich are sho"n in i%$ ?6d7 and i%$ A6d7$
i%$ ? @i%nals durin% the steps of de-noisin%$ 6a7 The ori%inal si%nal 6noise-distorted7!
13
6b7 )0-le#el Haar transform of the ori%inal si%nal$ The t"o hori3ontal lines represents the noise threshold B 0$5$ 6c7 The si%nal after thresholdin%! and 6d7 is the de-noised si%nal 6in#erse Haar transform7
i%$ A @i%nals durin% the steps of de-noisin%$ 6a7 The ori%inal si%nal 6noise-distorted7! 6b7 )-le#el Haar transform of the ori%inal si%nal$ The t"o hori3ontal lines represents the noise threshold B 0$$ 6c7 The si%nal after thresholdin%! and 6d7 is the de-noised si%nal 6in#erse Haar transform7
't "as found that i%$ ?6d7 is an closer appro(imate of the ori%inal! non-noisedistorted si%nal than i%$ A6d7$ 't is due to that the ener% of the si%nal i%$ ?6a7 is concentrated into a fe" hi%h-ener% #alues and additi#e noise is concentrated in lo"ener% #alue after Haar transformation$ Therefore! it is possible to se%re%ate the si%nal component from the noise component. "hereas in the case sho"n in i%$ A! the ener% of the si%nal is not concentrated into a fe" hi%h-ener% #alue! e$%$! it is spread across se#eral #alue! and the noise contaminated those transformed si%nal #alue "hich 14
ma&e the thresholdin% technique less effecti#e$
3.3 Imae Compression Haar transform can be used in compressin% an ima%e of
! "here both
and
are multiple of t"o$ 'ma%e compression is an e(pansion of one-dimensional si%nal compression$ To illustrate the process! a simple e(ample is sho"n belo" +A,$ * t"o-dimensional input si%nal matri( S is set to be1
irstl! the first-le#el Haar transform is applied to the ro"s of the input si%nal @$ The first appro(imation and detailed matri( of the ro"s are obtained$
@econdl! first-le#el Haar transform is applied to the columns of the resultant matri(
$ The first appro(imation and detailed matri( of the columns are obtained$
ollo"in% denotation is used1
15
A1 appro(imation area that includes information of the a#era%e of the ima%e$
H1 hori3ontal area that includes information about the #ertical ed%es/details in
the ima%e$
#1 #ertical area that includes information about the hori3ontal ed%es/details in
the ima%e$
!1 dia%onal area that includes information about the dia%onal details! e$%$!
corners! in the ima%e$
rom @ection $) "e &no"n that! after Haar transform! the appro(imation component contains most of the ener%$ Hence! it is clear that e(clude information from appro(imation area "ill result in bi%%est distortion to the compressed ima%e. and e(clude information from dia%onal area "ill result in least distortion to the compressed ima%e$
The error 6E@=7 of the reconstructed ima%es are summari3ed in the Table belo" +A,
E@=
i%$ )?A$4?9
i%$ 9 )5$AAA
i%$ )0 ?4$AA
't is clear that as the comple(it of the ima%e increased! i$e$! ena ima%e is the most comple( ima%e of these three! the error of the reconstructed ima%e become %reater$ Hence! the performance of the Haar transform is limited$
Chapter $ Conclusion The bac&%round and deri#ation of the Haar transform is presented in the first half of this tutorial$ The simplicit and localised propert of the Haar transform can be obser#ed$ The applications of the Haar transform are presented in the second half of this tutorial! "here the "or& process! e(amples and the performance of the Haar 18
transform in each of these applications "ere demonstrated$ rom the results sho"n in the application section! it is clear that Haar transform has its limitation! that it ma not be suitable for processin% certain tpes of si%nal$ 8e#ertheless! Haar transform is a %ood time-#ariant spectral tool "hich can be used for applications that requires hi%h memor efficienc$
19
%eferences +), $@$ @tan&o#iF and >$G$ al&o"s&i$ The Haar "a#elet transform1 its status and achie#ementsI$ Computers and Electrical Engineering ! Jol$9! 8o$)! pp$5-44! Ganuar 00$ +, $
2an%$
Haar
transformI$
'nternet
2eb
*ddress1
http1//fourier$en%$hmc$edu/e)?)/lectures/Haar/inde($html! December 04! 00$ +, G$G$ Din%$ Time-requenc *nalsis and 2a#elet TransformI$ ecture 8otes! 8ational Tai"an Kni#ersit$ +4, G$@$ 2al&er$ A Primer on Wavelets and their Scientific Application$ CC Press C! )999$ +5, E$ *l"a&eel and <$ @haaban$ ace eco%nition >ased on Haar 2a#elet Transform and Principal Component *nalsis #ia e#enber%-Earquardt >ac&propa%ation 8eural 8et"or&I$ European ournal of Scientific !esearch! Jol$4! 8o$)! pp$5-)! 0)0$ +?, P$ Por"i& and *$ iso"s&a$ The Haar-2a#elet Transform in Di%ital 'ma%e Processin%1 'ts @tatus and *chie#ementsI$ "achine graphics # vision! Jol$)! 8o$)-! pp$ A9-9! 004$ +A, *$ >hard"aL and $ *li$ 'ma%e Compression Ksin% Eodified ast Haar 2a#elet TransformI$ World Applied Science ournal ! Jol$A! 8o$5! pp$?4A-?5! 009$