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The Haar Wavelets \+ are considered and applied to this problem, and the difficulties analy zed. In an attempt to change the artifacts seen in the Haar decomposit ion, a basis proposed by Richard Bartels is applied and experimented w ith as well. Both bases turn out to introduce too many artifacts into \+ the image to challenge the JPEG encoding, but the COE basis produces b etter results than the Haar basis as expected." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Motivation" }} {PARA 0 "" 0 "" {TEXT -1 383 "In the interests of understanding wavele t mathematics, it is much easier to follow some hands-on examples as o pposed to reading proofs of the mathematics or abstract statements. By directly applying the wavelet transform, the paper hopes to show the \+ reader how some simple wavelets work, by way of introduction to the he avier mathematics that are justified by these initial attempts." }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "The Haar Transform" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 213 "The Haar transform uses square pulses to approximate the original function. The basis functions for Haar wavel ets at some level all look like a unit pulse, shifted along the x-axis . The basis functions are called " }{TEXT 261 7 "scales " }{TEXT -1 61 "in wavelet terminology, and are usually denoted as functions " } {XPPEDIT 19 1 "phi(t):" "-%$phiG6#%\"tG" }{TEXT -1 8 ", where " } {XPPEDIT 19 1 "t:;" "I\"tG6\"" }{TEXT -1 58 " denotes time. The Haar s cales are all of the unit pulses " }}{PARA 279 "> " 0 "" {XPPEDIT 19 1 "phi(t) := PIECEWISE([1, -t <= 0 and t-1 < 0],[0, otherwise]);" ">-% $phiG6#%\"tG-%*PIECEWISEG6$7$\"\"\"31,$F&!\"\"\"\"!2,&F&\"\"\"\"\"\"F/ F07$F0%*otherwiseG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 15 " Th e functions " }{XPPEDIT 19 1 "phi(t - s):" "-%$phiG6#,&%\"tG\"\"\"%\"s G!\"\"" }{TEXT -1 36 " are the shifted pulses, shifted by " }{XPPEDIT 19 1 "s:" "I\"sG6\"" }{TEXT -1 21 " units to the right. " }}{PARA 13 " " 1 "" {INLPLOT "6%-%'CURVESG6$7co7$$!\"#\"\"!F*7$$!1LLL$Q6G\">!#:F*7$ $!1nm;M!\\p$=F.F*7$$!1LLL))Qj^'***!#;F*7$$!1+ +++0\"*H\"*FPF*7$$!1++++83&H)FPF*7$$!1LLL3k(p`(FPF*7$$!1nmmmj^NmFPF*7$ $!1ommm9'=(eFPF*7$$!1,++v#\\N)\\FPF*7$$!1pmmmCC(>%FPF*7$$!1*****\\FRXL $FPF*7$$!1+++D=/8DFPF*7$$!1mmm;a*el\"FPF*7$$!1pmm;Wn(o)!#!#=F*7$$\"11iT&Q.d\"y!#>$\"\"\"F*7$$\"17mT5!*\\ PNFcpFhp7$$\"1-;H#oFMH'FcpFhp7$$\"1$fmTNc$\\!*FcpFhp7$$\"1e;zp87c9F\\p Fhp7$$\"1cm;/rI2?F\\pFhp7$$\"1_m\"Hdy'4JF\\pFhp7$$\"1[mmT+07UF\\pFhp7$ $\"1Tm;zHz;kF\\pFhp7$$\"1Mmm;f`@')F\\pFhp7$$\"1JLLL1+Y7FPFhp7$$\"1)*** *\\nZ)H;FPFhp7$$\"1lmm;$y*eCFPFhp7$$\"1*******R^bJ$FPFhp7$$\"1'*****\\ 5a`TFPFhp7$$\"1(****\\7RV'\\FPFhp7$$\"1'*****\\@fkeFPFhp7$$\"1JLLL&4Nn 'FPFhp7$$\"1*******\\,s`(FPFhp7$$\"1lmm\"zM)>$)FPFhp7$$\"1*******pfa<* FPFhp7$$\"1km;zy*zd*FPFhp7$$\"1HLLeg`!)**FPFhp7$$\"1m\"H#3Mo+5F.F*7$$ \"1+]i5KJ.5F.F*7$$\"1L3-8I%f+\"F.F*7$$\"1mmT:Gd35F.F*7$$\"1L$3-UKQ,\"F .F*7$$\"1+++D?4>5F.F*7$$\"1LLeM7hH5F.F*7$$\"1mm;W/8S5F.F*7$$\"1LLLj)o6 1\"F.F*7$$\"1++]#G2A3\"F.F*7$$\"1mm\"H3XL7\"F.F*7$$\"1LLL$)G[k6F.F*7$$ \"1++]7yh]7F.F*7$$\"1nmm')fdL8F.F*7$$\"1nmm,FT=9F.F*7$$\"1LL$e#pa-:F.F *7$$\"1+++Sv&)z:F.F*7$$\"1LLLGUYo;F.F*7$$\"1nmm1^rZF.F*7$$\"\"#F*F*-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%+ AXESLABELSG6$%\"tG%!G-%%VIEWG6$;F(F\\y%(DEFAULTG" 2 233 189 189 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 88 137 0 0 0 0 0 1 }}{PARA 257 "" 0 "" {TEXT 262 52 "Figure 1: Plot of the unshifted Haar scale f unction." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "To move to a higher l evel of detail, we consider the functions " }{XPPEDIT 19 1 "phi(2*t - \+ s):" "-%$phiG6#,&*&\"\"#\"\"\"%\"tGF(F(%\"sG!\"\"" }{TEXT -1 75 ", whi ch are the same bars halved in width. Suppose we have a step function \+ " }{XPPEDIT 19 1 "q:" "I\"qG6\"" }{TEXT -1 129 " whose value is 5 on t he interval [0, 0.5) and whose value is 3 on the interval [0.5, 1), an d 0 otherwise. Then we can represent " }{XPPEDIT 19 1 "q:" "I\"qG6\"" }{TEXT -1 17 " as the function " }{XPPEDIT 19 1 "5*phi(2*t) + 3*phi(2* t - 1):" ",&*&\"\"&\"\"\"-%$phiG6#*&\"\"#F%%\"tGF%F%F%*&\"\"$F%-F'6#,& *&\"\"#F%F+F%F%\"\"\"!\"\"F%F%" }{TEXT -1 62 ". The closest representa tion in the lower resolution would be " }{XPPEDIT 19 1 "4*phi(t):" "*& \"\"%\"\"\"-%$phiG6#%\"tGF$" }{TEXT -1 71 ", which is determined by av eraging the previous coefficients (clearly, " }{TEXT -1 152 "the avera ge of 5 and 3 is 4). The question is how to get from the lower resolut ion space back to the higher resolution space? The answer is to know t he " }{TEXT 263 9 "wavelets," }{TEXT -1 363 " or the functions that sp an the space that contains the information we have discarded in moving from one resolution to another. The wavelet can be expressed as a lin ear combination of the basis vectors for the higher detail space, sinc e it lies inside the higher resolution space which is spanned by those basis functions. For the Haar case, there is one wavelet " }{XPPEDIT 19 1 "psi := t -> -phi(2*t) + phi(2*t - 1):" ">%$psiGR6#%\"tG7\"6$%)op eratorG%&arrowG6\",&-%$phiG6#*&\"\"#\"\"\"F&F2!\"\"-F.6#,&*&\"\"#F2F&F 2F2\"\"\"F3F2F+F+F+" }{TEXT -1 42 ", which looks like a zig-zag square wave: " }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7^p7$$!\"#\"\"!F*7 $$!1LLL$Q6G\">!#:F*7$$!1nm;M!\\p$=F.F*7$$!1LLL))Qj^'***!#;F*7$$!1++++0\"*H\"*FPF*7$$!1++++83&H)FPF*7$$!1LLL3k(p`(FPF *7$$!1nmmmj^NmFPF*7$$!1ommm9'=(eFPF*7$$!1,++v#\\N)\\FPF*7$$!1pmmmCC(>% FPF*7$$!1*****\\FRXL$FPF*7$$!1+++D=/8DFPF*7$$!1mmm;a*el\"FPF*7$$!1pmm; Wn(o)!#!#=F*7$$\"11iT&Q.d\"y!#>$! \"\"F*7$$\"17mT5!*\\PNFcpFhp7$$\"1-;H#oFMH'FcpFhp7$$\"1$fmTNc$\\!*FcpF hp7$$\"1e;zp87c9F\\pFhp7$$\"1cm;/rI2?F\\pFhp7$$\"1_m\"Hdy'4JF\\pFhp7$$ \"1[mmT+07UF\\pFhp7$$\"1Tm;zHz;kF\\pFhp7$$\"1Mmm;f`@')F\\pFhp7$$\"1JLL L1+Y7FPFhp7$$\"1)****\\nZ)H;FPFhp7$$\"1lmm;$y*eCFPFhp7$$\"1*******R^bJ $FPFhp7$$\"1'*****\\5a`TFPFhp7$$\"1'***\\(3S*eXFPFhp7$$\"1(****\\7RV' \\FPFhp7$$\"1Z7.K?Z#*\\FPFhp7$$\"1'\\i!R\\g?]FP$\"\"\"F*7$$\"1ZP4Yyt[] FPF`t7$$\"1(*\\7`2(o2&FPF`t7$$\"1(\\(=nl8L^FPF`t7$$\"1(**\\7Q-%*=&FPF` t7$$\"1(*\\P4S$>I&FPF`t7$$\"1(***\\PcY9aFPF`t7$$\"1(**\\P*)G&RcFPF`t7$ $\"1'*****\\@fkeFPF`t7$$\"1jmmT30piFPF`t7$$\"1JLLL&4Nn'FPF`t7$$\"1**** ***\\,s`(FPF`t7$$\"1lmm\"zM)>$)FPF`t7$$\"1*******pfa<*FPF`t7$$\"1km;zy *zd*FPF`t7$$\"1HLLeg`!)**FPF`t7$$\"1m\"H#3Mo+5F.F*7$$\"1+]i5KJ.5F.F*7$ $\"1L3-8I%f+\"F.F*7$$\"1mmT:Gd35F.F*7$$\"1L$3-UKQ,\"F.F*7$$\"1+++D?4>5 F.F*7$$\"1LLeM7hH5F.F*7$$\"1mm;W/8S5F.F*7$$\"1LLLj)o61\"F.F*7$$\"1++]# G2A3\"F.F*7$$\"1mm\"H3XL7\"F.F*7$$\"1LLL$)G[k6F.F*7$$\"1++]7yh]7F.F*7$ $\"1nmm')fdL8F.F*7$$\"1nmm,FT=9F.F*7$$\"1LL$e#pa-:F.F*7$$\"1+++Sv&)z:F .F*7$$\"1LLLGUYo;F.F*7$$\"1nmm1^rZF.F*7$$\"\"#F*F*-%'COLOURG6&%$RGBG$\"#5FipF*F*-%+AXESLABELSG6$%\"tG% !G-%%VIEWG6$;F(F_[l%(DEFAULTG" 2 217 177 177 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 }}{PARA 258 "" 0 "" {TEXT 264 26 "Figure 2: The Haar wavelet" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "To reconstruct the original functi on, we can easily see that " }{XPPEDIT 19 1 "q := t -> 4*phi(t) - psi( t):" ">%\"qGR6#%\"tG7\"6$%)operatorG%&arrowG6\",&*&\"\"%\"\"\"-%$phiG6 #F&F/F/-%$psiG6#F&!\"\"F+F+F+" }{TEXT -1 264 ". In general, we can tak e the two coefficients of the high resolution basis and re-express the m as coefficients of the lower resolution basis plus the wavelets - al l we are really doing is a change of basis, which we might usually wri te as a matrix multiplication!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "To write this out with matrices, we would construct a matrix " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'MATRIXG6#7$7$#\"\"\"\"\"#F*7$ #!\"\"F,F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 382 "The observant read er will notice that we seem to have neglected some scale factors in th e non-matrix version. It is convenient to do so when we are looking at things loosely, but of course matrices won't permit such sloppy const ructions. The precision of matrices is strong encouragement for gettin g to matrix form as soon as possible. Now we can consider our little v ector of data " }{XPPEDIT 19 1 "v := [5, 3]:" ">%\"vG7$\"\"&\"\"$" } {TEXT -1 25 " and compute the product " }{XPPEDIT 19 1 "M*v:" "*&%\"MG \"\"\"%\"vGF$" }{TEXT -1 29 ". Doing so yields the result " }{XPPEDIT 19 1 "[4, -1]:" "7$\"\"%,$\"\"\"!\"\"" }{TEXT -1 39 " which correspond s to the equation for " }{XPPEDIT 19 1 "q" "I\"qG6\"" }{TEXT -1 24 ", \+ expressed in terms of " }{XPPEDIT 19 1 "phi(t):" "-%$phiG6#%\"tG" } {TEXT -1 5 " and " }{XPPEDIT 19 1 "psi(t):" "-%$psiG6#%\"tG" }{TEXT -1 43 " instead of in terms of the original basis " }{XPPEDIT 19 1 "ph i(2*t);" "-%$phiG6#*&\"\"#\"\"\"%\"tGF'" }{TEXT -1 5 " and " } {XPPEDIT 19 1 "phi(2*t - 1):" "-%$phiG6#,&*&\"\"#\"\"\"%\"tGF(F(\"\"\" !\"\"" }{TEXT -1 96 ". Asking how to get back to the original basis co uldn't be clearer now: we simply need multiply " }{XPPEDIT 19 1 "[4, - 1];" "7$\"\"%,$\"\"\"!\"\"" }{TEXT -1 19 " by the inverse of " } {XPPEDIT 19 1 "M" "I\"MG6\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 15 "The inverse is " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'M ATRIXG6#7$7$\"\"\"!\"\"7$F(F(" }}{PARA 0 "" 0 "" {TEXT -1 52 "and the \+ result of multiplying is [5, 3] as expected." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 43 "Compressing Images with the Haar Transform" }} {PARA 0 "" 0 "" {TEXT -1 257 "The Haar basis is known to be poor for i mage compression because it introduces rectangular artifacts into the \+ image. To understand why this is, we invest a bit of time developing a nd applying the Haar transform to images to see exactly what the resul ts are." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 268 "In order to apply the \+ transform to two dimensional data, we can apply the Haar transform fir st in one dimension, then in the other. Doing so and working out the n umbers provides us with a matrix for changing from the high resolution space to the low resolution space: " }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%'MATRIXG6#7&7&#\"\"\"\"\"%F(F(F(7&F(F(#!\"\"F*F,7&F(F,F(F,7&F(F,F, F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "The inverse transform is: \+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7&\"\"\"F(F(F(7&F(F( !\"\"F*7&F(F*F(F*7&F(F*F*F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 408 "W e can apply this transform once to the image, and create four smaller \+ \"images\", one which is the averages of the pixels, and is the coeffi cients of the low resolution space, and the other three which are the \+ wavelet coefficients that allow us to reconstruct the image. Looking a t an image that has been decomposed in this way gives us an immediate \+ intuitive idea of what we know is happening mathematically." }}{PARA 259 "" 0 "" {BITMAP 136 195 195 1 "?TMwiG=:;JUJNJ>k[k:>=>::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::>fq?f:wc:?b I;:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::B:< Jjreywe;uW;ERi=::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::JZZ:>FxDBR::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::>BB:;]i<<@yyyYyyyyu::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::;uW<@;<F<@;<F<@;<F<@;<:<:yyyxI:B:>:xIyA:::::::::::::::::::::::::::::::::::::: :::::::::>F<@;<:<:yay=:<:;Zymy;::::::::::::::::: ::::::::::::::::::::::::::::::;]ZKZZ:>FxDBRvYxI:::::B:>:xIyAZ:J:ryZ::: xIya:J:::yayM:B:vYxI;Z::jysyZ:J:ryvY::<:;:vYxIB:>:xIya:J:::yA>:xIyA>:< :yayM:B::vYxIB:>::jy;;Zymy;:::::::::::::::;]ZKZZ:>FxDBRvYxI:::::B:>:xI 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=>:;:;]Z;J:>:>F<@;<F<@;<F:::::JZ Z:vOJjB:F:BB:y[yJjBR>BB:;]i<< @y[yjywYvIxAyay=::xAryvIjYrI:vWxH::rIvAjqsuvAx=YaimYrIvWxHy[yjqsuvAx=Y aimYrI::::::::::::vOJj::::::;<F<@;<BB:y[yJjBR>BB:;]i<<@y[yjywYvIjysYvI::xI:yaykYrI:ZimqsIvAxHYayjY:Y aimYrIvWxHy[yjqsuvAx=YaimYrIvWxHy[y::::::::::::jQ;;=:::::>BB:Y?>FF <@::B:BB:y[yjQ;;=:Z:B:BR>F:JZZ:vAx=;]ZKZZ:>FxDBRvAx=yqykyryZ:>Zy myK:vYxIxAryZykYrI:ZimqsIZimYrIZimq;rIZimY:YAZy:xHy[yjqsuvAx=::::::::: :::Y?>F:::::JZZ:vOJjB:F:J:>::>F<@:;<F<@ ;<F<@;<FJZZ:vAx=;]ZKZZ:>FxDBRvAx=yqy;:J:ZyA:ry;:xI:jYrI::vWxH:y[y:::::: :::jqsuvAx=::::::::::::Y?>F:::::JZZ:vOJjB:<::::JjBR>BB:y[yjQ;;=>:;:;]Z ;J:>:>F<@;<F<@;<F:::::J ZZ:vAx=;]ZKZZ:>FxDBRvAx=yqy;J:Zy]::yI:jy=:vY:vAx=::ruvWx=y;:::::::::vW xHy[y::::::::::::jQ;;=:::::>BB:Y?>F:::::JZZ:vAx=Y?>F:::::JZZ:vAx=;]ZKZ Z:>FxDBRvAx=yqy;:<:vY:Z::yA:y[y::::::::::::::::::::::::::::jQsey?::::: :F<@;<F<@y[yjywY::>:yayA:<:yqy K:Zy=vAx=::::::::::::::::::::::::::::::::::::::::::::::::::::::;]ZKZZ: >FxDBRvAx=:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::;]ZKZZ:>FxDBR:::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::>BB:;]i<<@::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::;<F<@:::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::B:<5:" }}{PARA 260 "" 0 "" {TEXT 265 71 "Figur e 3: A small picture of Lena after one pass of the Haar transform." }} {PARA 0 "" 0 "" {TEXT -1 95 "We can see clearly from the image that on e quarter of it is a smaller version of the image - a " }{TEXT 266 17 "lower resolution " }{TEXT -1 209 "version. The other three are the de tails - they looked like the edges of the image have been embossed int o grey sheets of no detail at all. What have we gained? We can go back to the original at any time, but " }{TEXT 267 55 "we have extracted t he structure of the underlying image" }{TEXT -1 207 ". A simple compre ssor, such as the standard UNIX compress utility, can now compress thi s version of the image to one half of its original size - whereas virt ually no compression was possible on the original." }}{PARA 0 "" 0 "" {TEXT -1 233 "However, we would like much more than 50% compression, w ith minor loss in image quality. To attempt to get even better results out of the wavelet transform for compression, we would usually quanti ze the coefficients of the wavelets. " }}}{EXCHG {PARA 13 "" 1 "" {INLPLOT "6#-%'CURVESG6$7[[l7$\"\"!F(7$$\"\"\"F(F(7$$\"\"#F(F(7$$\"\"$ F(F(7$$\"\"%F(F(7$$\"\"&F(F(7$$\"\"'F(F(7$$\"\"(F(F(7$$\"\")F(F(7$$\" \"*F(F(7$$\"#5F(F(7$$\"#6F(F(7$$\"#7F(F(7$$\"#8F(F(7$$\"#9F(F(7$$\"#:F (F(7$$\"#;F(F(7$$\"#F(F(7$$\"#?F(F(7$$\"#@F(F(7 $$\"#AF(F(7$$\"#BF(F(7$$\"#CF(F(7$$\"#DF(F(7$$\"#EF(F(7$$\"#FF(F(7$$\" #GF(F(7$$\"#HF(F(7$$\"#IF(F(7$$\"#JF(F(7$$\"#KF(F(7$$\"#LF(F(7$$\"#MF( F(7$$\"#NF(F(7$$\"#OF(F(7$$\"#PF(F(7$$\"#QF(F(7$$\"#RF(F(7$$\"#SF(F(7$ $\"#TF(F(7$$\"#UF(F(7$$\"#VF(F(7$$\"#WF(F(7$$\"#XF(F(7$$\"#YF(F(7$$\"# ZF(F(7$$\"#[F(F(7$$\"#\\F(F(7$$\"#]F(F(7$$\"#^F(F(7$$\"#_F(F(7$$\"#`F( F(7$$\"#aF(F(7$$\"#bF(F(7$$\"#cF(F(7$$\"#dF(F(7$$\"#eF(F(7$$\"#fF(F(7$ $\"#gF(F(7$$\"#hF(F(7$$\"#iF(F(7$$\"#jF(F(7$$\"#kF(F(7$$\"#lF(F(7$$\"# mF(F(7$$\"#nF(F(7$$\"#oF(F(7$$\"#pF(F(7$$\"#qF(F(7$$\"#rF(F(7$$\"#sF(F (7$$\"#tF(F(7$$\"#uF(F(7$$\"#vF(F(7$$\"#wF(F(7$$\"#xF(F(7$$\"#yF(F(7$$ \"#zF(F(7$$\"#!)F(F(7$$\"#\")F(F(7$$\"##)F(F(7$$\"#$)F(F(7$$\"#%)F(F(7 $$\"#&)F(F(7$$\"#')F(F(7$$\"#()F(F(7$$\"#))F(F(7$$\"#*)F(F(7$$\"#!*F(F (7$$\"#\"*F(F(7$$\"##*F(F(7$$\"#$*F(F-7$$\"#%*F(F(7$$\"#&*F(F-7$$\"#'* F(F07$$\"#(*F(F*7$$\"#)*F(F37$$\"#**F(F07$$\"$+\"F(F-7$$\"$,\"F(F67$$ \"$-\"F(F97$$\"$.\"F(F97$$\"$/\"F(FB7$$\"$0\"F(FB7$$\"$1\"F(F`o7$$\"$2 \"F(Fjn7$$\"$3\"F(FZ7$$\"$4\"F(Fep7$$\"$5\"F(Fgq7$$\"$6\"F(Fbs7$$\"$7 \"F(Fat7$$\"$8\"F(Fcx7$$\"$9\"F(Fh\\l7$$\"$:\"F(Fi[l7$$\"$;\"F(F]al7$$ \"$<\"F($\"$X\"F(7$$\"$=\"F($\"$.#F(7$$\"$>\"F($\"$@#F(7$$\"$?\"F($\"$ t#F(7$$\"$@\"F($\"$e$F(7$$\"$A\"F($\"$m&F(7$$\"$B\"F(F(7$$\"$C\"F(F(7$ $\"$D\"F(F(7$$\"$E\"F(F(7$$\"$F\"F(F(7$$\"$G\"F($\"&#zgF(7$$\"$H\"F(F( 7$$\"$I\"F(F(7$$\"$J\"F(F(7$$\"$K\"F(F(7$$\"$L\"F(F(7$$\"$M\"F($\"$j%F (7$$\"$N\"F($\"$p$F(7$$\"$O\"F($\"$1$F(7$$\"$P\"F($\"$>#F(7$$\"$Q\"F($ \"$%=F(7$$\"$R\"F(F]dl7$$\"$S\"F(Fe_l7$$\"$T\"F(F[]l7$$\"$U\"F(F]x7$$ \"$V\"F(F_v7$$\"$W\"F(Fiu7$FbalF^t7$$\"$Y\"F(Fjq7$$\"$Z\"F(Faq7$$\"$[ \"F(Fgq7$$\"$\\\"F(F]o7$$\"$]\"F(Fjn7$$\"$^\"F(FQ7$$\"$_\"F(Fgn7$$\"$` \"F(FQ7$$\"$a\"F(F97$$\"$b\"F(FH7$$\"$c\"F(F?7$$\"$d\"F(F67$$\"$e\"F(F B7$$\"$f\"F(F-7$$\"$g\"F(F<7$$\"$h\"F(F37$$\"$i\"F(F07$$\"$j\"F(F*7$$ \"$k\"F(F(7$$\"$l\"F(F*7$$\"$m\"F(F*7$$\"$n\"F(F*7$$\"$o\"F(F-7$$\"$p \"F(F(7$$\"$q\"F(F(7$$\"$r\"F(F(7$$\"$s\"F(F*7$$\"$t\"F(F*7$$\"$u\"F(F (7$$\"$v\"F(F(7$$\"$w\"F(F(7$$\"$x\"F(F(7$$\"$y\"F(F(7$$\"$z\"F(F(7$$ \"$!=F(F(7$$\"$\"=F(F(7$$\"$#=F(F(7$$\"$$=F(F(7$FgflF(7$$\"$&=F(F(7$$ \"$'=F(F(7$$\"$(=F(F(7$$\"$)=F(F(7$$\"$*=F(F(7$$\"$!>F(F(7$$\"$\">F(F( 7$$\"$#>F(F(7$$\"$$>F(F(7$$\"$%>F(F(7$$\"$&>F(F(7$$\"$'>F(F(7$$\"$(>F( F(7$$\"$)>F(F(7$$\"$*>F(F(7$$\"$+#F(F(7$$\"$,#F(F(7$$\"$-#F(F(7$FgalF( 7$$\"$/#F(F(7$$\"$0#F(F(7$$\"$1#F(F(7$$\"$2#F(F(7$$\"$3#F(F(7$$\"$4#F( F(7$$\"$5#F(F(7$$\"$6#F(F(7$$\"$7#F(F(7$$\"$8#F(F(7$$\"$9#F(F(7$$\"$:# F(F(7$$\"$;#F(F(7$$\"$<#F(F(7$$\"$=#F(F(7$FbflF(7$$\"$?#F(F(7$F\\blF(7 $$\"$A#F(F(7$$\"$B#F(F(7$$\"$C#F(F(7$$\"$D#F(F(7$$\"$E#F(F(7$$\"$F#F(F (7$$\"$G#F(F(7$$\"$H#F(F(7$$\"$I#F(F(7$$\"$J#F(F(7$$\"$K#F(F(7$$\"$L#F (F(7$$\"$M#F(F(7$$\"$N#F(F(7$$\"$O#F(F(7$$\"$P#F(F(7$$\"$Q#F(F(7$$\"$R #F(F(7$$\"$S#F(F(7$$\"$T#F(F(7$$\"$U#F(F(7$$\"$V#F(F(7$$\"$W#F(F(7$$\" $X#F(F(7$$\"$Y#F(F(7$$\"$Z#F(F(7$$\"$[#F(F(7$$\"$\\#F(F(7$$\"$]#F(F(7$ $\"$^#F(F(7$$\"$_#F(F(7$$\"$`#F(F(7$$\"$a#F(F(-%'COLOURG6&%$RGBG$FF!\" \"F(F(" 2 443 153 153 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 188 -12128 0 0 0 0 0 1 }}{PARA 273 "" 0 "" {TEXT 275 82 "Figure 4a: A \+ plot of the histogram of the wavelet coefficients for the Lena image. 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^`YwgTOeTOc^?j:xgn_l>xijOgTOi^`nZ`pnDg::[we>pj\\ixoixOJRCfbhoOWkY?R;@V :;B:;JbyvBFN@h:>Z:::nAAB:wq>n:;B:;:j;:;:<;?B:;B:=:wqyyY:j :>Z:Fv:>:yyyyyy:::::::::::::::::::::::::::::::::::::::::::N:YD:Sml>d:N HEms>`:>@>Z::::::::kB:;jysy:>:<::::::wAB:::::::::::::yay=J:B:::::::::: :::::::::jysy:>:<:::::::::::::::::::vYxI:;Z:::::::::nwx3=" }}{PARA 270 "" 0 "" {TEXT 273 75 "Figure 10: After reconstruction, triangular \+ artifacts are clearly apparant." }}{PARA 0 "" 0 "" {TEXT -1 267 "Sever al other images were also tested using the COE transform, and similar \+ triangular artifacts appear. Although the triangular artifacts are har der to see than boxy artifacts, especially for small image sizes, the \+ end result cannot compete with other wavelet basis. " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 "Conclusions a nd Future Work" }}{PARA 0 "" 0 "" {TEXT -1 608 "Although the COE did p roduce less distortion in the images than the Haar method, the results do not compare well to other wavelet bases or to JPEG coding. Perhaps there would be some value to introducing random noise during reconstr uction to hide the artifacts, or to researching the anti-aliasing lite rature for techniques that might be successfully applied as a post-pro cess. Some additional measurement could be done to support the intuiti on that the COE basis produces \"better\" images than the Haar transfo rm. A standard error metric could be applied to the image, or a subjec tive experiment carried out." }}{PARA 0 "" 0 "" {TEXT -1 592 "An exper iment with the Haar scales using wavelet vectors that are not orthogon al to one another showed worse artifacts than in the Haar basis used i n this paper's examples. This seems justifiable since two wavelet vect ors that are not orthogonal that are both quantized introduce more err or in the direction that they are not orthogonal, possibly exceeding t he threshold for error that we are trying to control during the quanti zation step. Possibly choosing new COE wavelet vectors that are orthog onal to each other as well as to the scales would improve the COE tran sform's quality as well." }}{PARA 0 "" 0 "" {TEXT -1 316 "The existing literature suggests that bases that overlap more and are smoother pro duce better results with image processing, and some examples of these \+ functions are discussed in [STR96]. In particular the wavelets that ar e used for the FBI fingerprint standard could be examined to see if th ey can be improved upon." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Refe rences" }}{PARA 0 "" 0 "" {TEXT -1 66 "[BAR94] Private communication f rom Richard Bartels to Rob Kroeger." }}{PARA 0 "" 0 "" {TEXT -1 61 "[K RO96] Private communication from Rob Kroeger to the author." }}{PARA 0 "" 0 "" {TEXT -1 42 "[STR96] Gilbert Strang and Truong Nguyen, " } {TEXT 278 25 "Wavelets and Filter Banks" }{TEXT -1 34 ", Wellesley-Cam bridge Press, 1996." }}}}{MARK "1 0" 13 }{VIEWOPTS 1 1 0 1 1 1803 }