計椚醐卸学会刺ほ糟17三相相集会（199臥11．20〕

帥粥17針1マノ＼ラ ノ ヒご1スの圧巨離を月∃t、た・米 ドル泉［∈幣の言散男り
1N S P E C T T ON O F U■S ＄ 臼lI．IJSU SING MAITA L AN O f3IS T〕IS T A N C E
垣谷 学∃＝

坂野 進可＝Manab11Kakitani， Susu皿u Sakano＊日本犬苧工学部＊Cロ11ege of Engineering，Nih（〕n University

キ・−−ワーー ド

マハラjビス鯛蔚（皿ahalaIlObis distance〕，醐錮〔pattern recognitj．on）
米r］L紙幣（US dollar bills〕．品質工学（Quality Engineering〕

連絡先‡963一呂642剛軸帥配芋相田］拙守工鞘鮎工期拍トロニク絹究室Tel∴ D24−956一日了74．Fax，：D24−956−呂日6ne山aii二 Sak那0如ech．ce．nihon−u．aC．jp

A Nl：W APPROACH FOR INSPl王CT T ON OF

PRINTED MATTERS ANl〕 BILLSCqll曙eOfl王l喝in托ri喝，Nihon Univer由ty

ABSTRACT

Muchattentionhasbeenfocusedon Mahalan口bisdistarlCざforamethodoftreatingmulti
dirDenSionalinfornLation system．such as，pattern reC鴨nition．The use of Mahalanobi6

distanceis proposed by Dr．G．Taguchiand some researches have been川ade onit．
In thisl迫匹r，We prOPOSe a printed Tna■亡ter recogniti（n rnethod usiT唱 Mahalanobis

distance．The billmoney of L［S dollar js adopted as one of the printed皿atterS8．nd
itis shovn that good results areobtained for the recロgTlitionof US bi11s．Adoptingthe protx）Sed met†10d to recognition pTObleq）S，the e王fectiveness of theTnethodis sho町l．

INTRODUCT10N
The use of Mahalanobis distanceIneth口d of treating d）ultiTdimensionalinf〔Jrnation
SyStem for health exaninations vas proposed byI〕r．G．Taguchi．Suchapplications

fmr medicalfurecasting and treatnent havebeen developingin the past severalyears．
Many researches have been阻deon neuralnetworks for pattern rec昭njtion wi［h proEni−si■ng reSults，Thesemethodus占slabvaluesand theslabvalues are thesuTmfinput
Pixels generated EIS Characteristics of theinputs byq］at；ks．l−Towever，naSkopti皿izati口n
has not been elEIbrated sufficiently and the methods have tロbeiLnprOYed stillmore．
We propse the recognitionEnethcd using Mahalanobis distance，Theda亡awhi亡h arecullected frロ皿a n13rmalgl−OuPOf the ret：Ognized objects are process巳d by constructir唱aMahalan口bis Spacein theanalysis． The6e dataiIIClude vari．ousitenswithdifferent

dimensions and many of theseitems ar，e Emtua71y corrclated． The quadratic value from
the Mahalanobis Space，Called Mahalanobis distance．is used for data analysis．
The standard Mahalanobis Spaceis calculated fro咄theinverse Tnatri貰 Of the correq
lation coefficients．ln the fir6t Part Of this paper，theanalysis of Mahalanobis
distanceare e叩1ained and the applicatiol10f thepr口PSed methcd to the匹ttem reCOg¶
rlitionof the US d〔】1lar billmoneyis shwnin theIast part of this paper．

MA‖ALANOBIS D r STANCEStaTdard Mahalux）bis SI迫Ce

As an exaTuPle，We COnSider the caseof two variables Xlar］d Xコ． Tf thereis noc口rreLation betveen X．aTld‡J，then，the Mahalan□bis distance b巳COIneS a Circle shownin F．igllrel（a）．If thereisanycorrelation between X．and R」．the Mahalar）Obis

distanceis shownin Figurel（b〕．We can calcl】1ate the distance between O and A
and the distance between O and B using the invefse maLrix of the standardized corre‑
1ELtiort coefficients．This distaTICeis called a Mahalanobis distance．

X■Ⅹl（b） dny COrrelation（a） rlU Cロrre】atlon

Fig．1 Mahalanobis SI］a⊂巳and Mdha）anobis dj．stance
CaluculatiorL Of Mahala7t）bis dist且nCe

The data are starldardized using the nexLt equation．
Ⅹij ￣mi〔i＝1．2 j＝1，2．■・n〕

Ⅹ上1

（1〕

亡ri

Where・Ln⊥is theavera芦eOfthedata嵐JJand ロ■iis thestandardized deviation．

■Thecorrelationcoefficientsarecaiculatcdfromthenextequationandthecorrelation
matrixis expressed as equatjon（3），tご、）

1rl．；（、31r∴ l

TheinverseⅢatrix Ais the nex亡equation．

A ＝ R・′1［a⊥・i］

州

Therl，the MahalarK）bis distamceis calculated froE）the TleXt equEttion，
Ⅹi−mi1

Dコ ＝冊¶−∑al」≠二X．巨 −−m，X（一5こ〕

2L「．

打i

The standard Mahalanubj，S Spaceis c（〕TtStruCted fr口n］the datain normalcDndition andthe Mahalan（］bi与distances arelocated nearl．0． The datain abnorTnalcondition shDVS
the value far froThl，0．We can classify the cases of the nor血at COndition and the ab−
normalc□nditi（〕n uSing this standardized Mahalanobis Space and Mahalanobis dis−tance．

FloY Of callR：ulatil唱

The flov of calculating the satndard Maharanobis Spaceis shovnin Figure2．

＾PPrJICAT T ON TO INSPECT10N OF US BILLS
We apply the proTX］StX］皿ethod to theins匹Ctionof US d〔）11ar bj11s．
Rec曙Ilitjon usinE a huJnan pjcture
The htLnan Picturei6drawn at thecenter part of US dollar bi）1（Figut・e3〕．We

ad（】pt the human pictl］reS（Jf ea亡h dロ1lar bi11s as the data and transfor皿the do11ar data
into frequency domaj．n by FFT and use amplitudes□f Fuurier coefficient古aS theinput data of亡he caluculating standard Maharanobis Space．Weget ti皿e Seriesdata

using the uutput扇卯als ofl．Dand PD sensors（Figure A）．

human plcture

Fig．4 Senso1・OUtput Signals

Fig．3Ⅰ・1uman picture of US bill
Fourier trant；for皿The tj皿e Serie6data of＄lis obtajned by the output signalt；Of sensors and thedatais

ShuYn jnl■11igure5．We transform this dataintLO the a叩1itude of fourier c（〕efficients
Vjth treatinE Of fouri．er transfoIⅦ．Fourier coefficient6＾（n）and B（Tl）are calcu］atedby the next equations．

A（n〕＝一〔k−1）n）

1革＿】f（k〕cus‡（臣N

2打f3〔n〕＝ ，Ni：．f（k〕sin‡L¶（K−1〕nI（了．）M

Theamplitude of Fourier coefficientsisobtained using thenext equation（Figure6），
血p（11〕

Amp〔巾 ＝A（n）コ十B（n〕：ま ‥・（8〕3U

Pixel訓0value】00

00 20 40 6口 8DlO口120】一40（time〕
Fig．5′Iljme SeriefS data of＄1

り 20 40 60 日O（打equency〕

Fig．6 ＾mpl．jtuLleof c□efficientsConstn）Ctjonof standard Maharanobis S脚CCAs shownin Fi即re6，amplitudeof fourier coefficientsof＄1has nineprojections
and thesepTDjections of frequency arecharacteristics of＄1bi11．We treat theheightsロf these nineprojections as theinput data and we colIe亡l：the dataロf reat＄1bills，The standard Mahalanobis SEnCeiscalculated using3O sheetsof＄1bill．
Each bi11s have different heights of frequency projections becauseof thevarietj巳S．Of

their used years， StaTldardized datais e耳preSSed as follows：Ⅹji ￣ml〔i＝1〜9．j＝1〜30）

Ⅹii（臣ロ1油ere．x＝isinput data and mli＄aVerage ar）d 亡rlis standardized devjation．
The correlation t迫atrixis亡alculated froⅧthe next equation．
1 r■コrll・・・rtH rl，rコ11 r急1・・・rコ］．r」りr∃lrlコ1・・・「川 rうりi−10）● … ＝ ■ ■ ■ ● ■■●●■■■■●●●●r。Irリ∠rt寸1・・・r．．HlEach ri）is calculated from the next equations，

13ハリr】」＝r上．＝−（Ⅹ．．エコー＋Ⅰ．ユ‡上コ＋・・・・・＋Ⅹ■上り X上上．．＋X15t．X上川）1

r■．1＝「弓，＝−（Xll‡。，十Xl」Ⅹヨ。十…‥＋X．上り Xミ1‖ 一トⅩl引，よ1ミt、〕30＝ ＝ ＝ ＝ ＝ ■ ● ● ■ ■ ■ ＝ ＝ ■ ■ ● ＝ ＝ ■ ■ ● ■… ＝ ＝ ＝ … ＝ ● ■ ＝ … ＝ ＝ ■ ■ ■ ■ ■ ■t：ll）

Theinv占TSen）atrixof the correlationEnatrixis calculated using eq（4）and Mahala−nobis distanceD⊥ 上is expressed as thelleXt eqtJation．
1Dk∑∑a⊥」×Ⅹ上kXX．ikrけ】

9For e粥川Ip】e，Ⅹ1l＝0．0．‡Jl＝−0．62，‡ぅ、＝0．了乱‡．1ユニ1．32．Ir■＝−0．48．X．．．＝−0．51．

X7■＝−0．56．XHl＝0．0．X。t＝□．30，t】1enI〕1コ＝0．d4．

The data tnse of Mahalanobis S脚Cei6COnStruCted frロm the Maf］alan血is dintances
Of30sheets of＄l． When the Mahalanobis distances of s亡n］er巳mOdeled‡1bilIsareCalculated．their numbers are far from】．0，

InsFX：tjon e畔riDentSThe standard Maha］anobis Spaceis constructedofcalculating the Mahalanobis disT

tanceofreal＄1bi11sandusingthe30sheetsofther占nde】ed＄1bi11stheinspection
experiments are practised，The result of the ex匹riments are shovniTlneXt Figlre，nu山berロf times

real＄1bills／re叩deled＄1bills

丘／／ーl
一■

一ー■I■■■l2■r■■■l■■■−■■■

一qI■■■■−■

亡じ■

〆／

ll
l1
11nU

0 2 4 6 旨 10121d1618
Mahalanobis distanceFig，3 Result ofins匹Ctiロn eXperi皿entS
As shovnin the above ex匹riments result，WeCarL discern between the realg00d bills

and the remodeled bad bills using tLle prDPSed Mallalanobis S脚Ce and Mahalanobis
distance．

SELECTION OF PAR＾METER ‡T打MS
Experi皿ent uSingor，tho即Ⅰ岨1array
As the stx〕Ve mentioned，Ve uSe the nineprojectionsof frequeTICyaC）Plitudeas theinput
dataamd甘e get theadequate result of theinspection e叩eri山entS．口utifve selecttheinputdata，WeWillbe able toirlS匹Ct the billsⅧOre eaSily，W巳praCtise theex匹riment of parameters selc仁tion．The experimentis practised usingl−1コOrthogonal

ELrray．The case of taking use of the paraJneterSitenlCOrreS脚nds to the firstlevelin
the exFerimellt aTLd the case口f taking TX）uSeOftheiteTn印rreS匹nds to the secnrldlevel

The effect of the experiment js shovnin the next SN ratjo，
1

1

1

1

町 ＝ −王og（−（−−−−−＋一冊＋ ‥・十伽−mnlコ D」■j〕1

（12）

D．」、コ

ThelayoLlt Of paran）eterS and the SN rati〔Jare ShowrLin Tablel．SN ratiois calcuY
lated using the experiment data．Table2shovs the effectof the parameteriteE）S．
As shovnir）Table2，itemlanditem2 become the mainitedlS for th巳input血ta．
Ea亡hitems correspond to the first projectionand the second projectionof the frequ−ency，Theamplitudesoflowfrequencyinfluen仁e theinspection ofbills．

Table．1 Layout and SN ratio

Table．2 Effect ofitems

17．3672
17．0434

31ユ．1708

4】6．T796﹁nU

11．D558
11．4537
11．1568
11．4167
11．3695】．111】11

15．1176111112211222111212212122122112221222122l．1221212222112221222111122121212211212

1G．呂399】．乱3582■巳U］．1．66】71】．8d64712」4911

3．4別口

12

13．2495 −2．0795
13．363丁 −2．3079

7．501呂10．】＿98日

13．2626 −2．1ロ5日

13．0028 −1．5861
1．3．0500 轟1．6帥5

11．印13 0．8170■‖コ3，26061− 2
9．41594．0219leve1212．9658 −1．5121

1云．6128n爪Y12leVel 1
16．9177
14．2207

SN ratio（db〕

l2ワ﹈

NO 12345fj7

ハU匹

2．2671NThe result of the ex匹rj．n）ent uSing theatx］Ve t佃items shovsirL Figure4，We can

COnStruCt the billsinspection町SteZd u6ing thelow frequencyitem60f ti皿eSeriesdataof US doJlar bi11s．
number of ti皿e呂／rea−＄1／

reⅢ】Odeled＄1／ll
I II Ill

ル： ‖r‖llllll＿＿】0‖II1】lllllll

2u3□
Mahalanobjs DistancelllldO

Fig．4 Ex匹riment resu］t after selecting問．rametCrit印SCONCLUS T ON

Theuseof Mahalanobis distance nethodof trea■亡ingn］ulti−dinensiロnalinforTnatiDn

SYSteⅧfor US billsinspectionwasproposed．ThestandardMahalanDbis Space vas
used for the recognitionofpictures．The Mhalanobis distance was calculated from theinverseTnatrixofthecorrelationcoefficients．ThepropsedLuethdcanbeappliednot
Only to bi11sir）SI光Ction but also to parinted matters recogTlition．

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