PAGEPAGE1中英文對(duì)照外文翻譯文獻(xiàn)(文檔含英文原文和中文翻譯)譯文：基于局部二值模式多分辨率的灰度和旋轉(zhuǎn)不變性的紋理分類摘要：本文描述了理論上非常簡(jiǎn)單但非常有效的，基于局部二值模式的、樣圖的非參數(shù)識(shí)別和原型分類的，多分辨率的灰度和旋轉(zhuǎn)不變性的紋理分類方法。此方法是基于結(jié)合某種均衡局部二值模式，是局部圖像紋理的基本特性，并且已經(jīng)證明生成的直方圖是非常有效的紋理特征。我們獲得一個(gè)一般灰度和旋轉(zhuǎn)不變的算子，可表達(dá)檢測(cè)有角空間和空間結(jié)構(gòu)的任意量子化的均衡模式，并提出了結(jié)合多種算子的多分辨率分析方法。根據(jù)定義，該算子在圖像灰度發(fā)生單一變化時(shí)具有不變性，所以所提出的方法在灰度發(fā)生變化時(shí)是非常強(qiáng)健的。另一個(gè)優(yōu)點(diǎn)是計(jì)算簡(jiǎn)單，算子在小鄰域內(nèi)或同一查找表內(nèi)只要幾個(gè)操作就可實(shí)現(xiàn)。在旋轉(zhuǎn)不變性的實(shí)際問(wèn)題中得到了良好的實(shí)驗(yàn)結(jié)果,與來(lái)自其他的旋轉(zhuǎn)角度的樣品一起以一個(gè)特別的旋轉(zhuǎn)角度試驗(yàn)而且測(cè)試得到分類,證明了基于簡(jiǎn)單旋轉(zhuǎn)的發(fā)生統(tǒng)計(jì)學(xué)的不變性二值模式的分辨是可以達(dá)成。這些算子表示局部圖像紋理的空間結(jié)構(gòu)的又一特色是，由結(jié)合所表示的局部圖像紋理的差別的旋轉(zhuǎn)不變量不一致方法，其性能可得到進(jìn)一步的改良。這外文翻譯：基于局部二值模式多分辨率的灰度和旋轉(zhuǎn)不變性的紋理分類（節(jié)選）外文翻譯：基于局部二值模式多分辨率的灰度和旋轉(zhuǎn)不變性的紋理分類（節(jié)選）PAGEPAGE7些直角的措施共同證明了這是旋轉(zhuǎn)不變性紋理分析的非常有力的工具。關(guān)鍵詞：非參數(shù)的，紋理分析，Outex，Brodatz，分類，直方圖，對(duì)比度灰度和旋轉(zhuǎn)不變性的局部二值模式我們通過(guò)定義單色紋理圖像的一個(gè)局部鄰域的紋理T，如P（P>1）個(gè)象素點(diǎn)的灰度級(jí)聯(lián)合分布，來(lái)描述灰度和旋轉(zhuǎn)不變性算子：g )P1Tt(gg )P1c 0

（1）其中，gc為局部鄰域中心像素點(diǎn)的灰度值，gp（p=0，1…P-1）R(R>0)鄰域內(nèi)對(duì)稱的空間象素點(diǎn)集的灰度值。圖1c g的坐標(biāo)是（0,0g的坐標(biāo)為(Rsin(2p/P),Rcos(p/P))1舉例說(shuō)明了圓形對(duì)稱鄰域集內(nèi)各種不同的（P,。不完全落在中心點(diǎn)鄰域內(nèi)的像素點(diǎn)的灰c 灰度不變性的達(dá)成作為灰度不變性的第一步，在不丟失任何圖像信息的前提下，我們從圓形對(duì)稱鄰域集gp（p=0,……P-1）中減去中心點(diǎn)（gc）的灰度值，即令：Tt(g,g g,ggc 0 c 1 c

g) （2）,,g P1然后，我們假設(shè)差分gP

ggc

，這樣我們就可以把式（2）式分解為：Tt(g)t(g g,ggc 0 c 1 c

g) （3）,,g P1實(shí)際上，嚴(yán)格的獨(dú)立性是無(wú)法達(dá)成的，因此，被分解的因式只是聯(lián)合分布的一個(gè)近似值。然而，當(dāng)我們?cè)谛D(zhuǎn)中可以保持灰度不變性的話，我們?cè)敢獬袚?dān)丟失一些圖像小信息的可能。也就是說(shuō)，因式t(gc

)在（3）中描述了圖像的全局亮度，但并不為紋理分析提供有用信息。因此，原始的聯(lián)合灰度級(jí)因式（1）的許多紋理特征信息可由聯(lián)合差分因式表達(dá)[1]：

,g P1Tt(g g,,g P10 c 1 c

g) （4）c這是一個(gè)有高度識(shí)別能力的紋理算子，可以算出P空間中各種模式下每個(gè)像素點(diǎn)鄰域的直方圖。對(duì)于固定的區(qū)域，在各個(gè)方向的差別為零。在一個(gè)慢慢傾斜的邊緣，該算子可算出沿傾斜方向差分最大的點(diǎn)和差分為零的點(diǎn)，對(duì)于斑點(diǎn)而言，各個(gè)方向的差分都是很大的。有正負(fù)之分的差分gP

gc化具有不變性。我們所得到的關(guān)于灰度計(jì)數(shù)不變性只考慮差分符號(hào)而非它們的精確值：Tt(s(g g),s(gg0 c 1 c

s(g g)) (5)p1 c其中，

1,x0s(x)0,x0

(6)通過(guò)為每一個(gè)s(gP

g2p，我們把式（5）轉(zhuǎn)換為一個(gè)獨(dú)c特的LBPP,R

碼來(lái)刻畫局部圖像紋理的空間結(jié)構(gòu)的特性：p1LBPP,R

p0

s(gp

g)2p (7)cLocalBinaryPatterns這個(gè)名字反映了LBP算子的泛函性，即第一個(gè)局部鄰域點(diǎn)的灰度值是中心像素點(diǎn)進(jìn)入二值模式的開(kāi)始。LBPP,R

算子是通過(guò)對(duì)灰度的任何單調(diào)變化定義不變量，也就是，只要保持圖像灰度值的順序不變，LBPP,R

算子所產(chǎn)生的LBP碼就不變。P=8,R=1LBP8,1

這與我們?cè)谖墨I(xiàn)[2]中提到的LBP是類似的。LBP8,1

和LBP之間有兩個(gè)不同點(diǎn)：1)鄰域集內(nèi)的像素點(diǎn)被編入索引以形成一個(gè)循環(huán)鏈，2)對(duì)角線上像素點(diǎn)的灰度值由插值法確定。兩者的修改都必需獲得圓形對(duì)稱鄰域集，這考慮到源自LBPP,R

的旋轉(zhuǎn)不變式之一。旋轉(zhuǎn)不變性的達(dá)成由鄰域集中P個(gè)像素點(diǎn)對(duì)應(yīng)2P個(gè)不同的二值模式，LBPP,R

算子會(huì)生成2P個(gè)不同的輸出值。當(dāng)圖像被旋轉(zhuǎn)時(shí)，g 的灰度值會(huì)對(duì)應(yīng)地繞著g的四周沿著圓周的邊界移動(dòng)。g始終P 0 0被指定為元素(0,R)的灰度值，而恰恰gc

旋轉(zhuǎn)一個(gè)特定的二值模式后自然生成一個(gè)不同的LBPP,R

值。這不適用于只由0s（或1s）組成的旋轉(zhuǎn)任何角度始終保持不變的模式。為了要消除旋轉(zhuǎn)的影響，也就是，要分配一個(gè)獨(dú)特的標(biāo)識(shí)符給每個(gè)旋轉(zhuǎn)不變性的局部二值模式，我們定義：LBPriP,R

min{ROR(LBPP,R

,i)i0,1,

(8),P其中ROR(x,i)P-位元,P只簡(jiǎn)單對(duì)應(yīng)于被多次順時(shí)針?lè)较蛐D(zhuǎn)的鄰域集因而最有效位元的一個(gè)最大碼從g 啟動(dòng),P1為0。LBPriP,R

量化了對(duì)特定的微特征的個(gè)別旋轉(zhuǎn)不變性模式的發(fā)生統(tǒng)計(jì)學(xué)；因此該模式可作2舉例說(shuō)明了當(dāng)P=8時(shí)的368,1#0檢測(cè)到明亮的斑點(diǎn)，#8有暗點(diǎn)和平坦的區(qū)域，#4有邊緣。如果我們?cè)O(shè)定R=1LBPri[3]中指定了的8,1LBPROT。圖2基于均衡模式改進(jìn)的旋轉(zhuǎn)不變性和有角空間的更佳量化然而，我們的實(shí)際經(jīng)驗(yàn)已經(jīng)顯示LBPROT同樣不能提供非常好的識(shí)別，這點(diǎn)我們也總結(jié)在[3]。這有兩個(gè)原因：1)LBPROT36常大，2)45°間隔的粗糙量化。我們已經(jīng)觀測(cè)得知，特定的LBP可描述絕大多數(shù)的基本紋理特征，有時(shí)可描述超過(guò)90%3×33節(jié)中加以2(0)U值“模式“模式”下的空間變換碼的跳躍）000000002和111111112U02U220/1的跳躍。類似的，其它27種模式的U值至少為4。我們指定U值不大于2的為“均衡”模式，并提出了替代LBPriP,R

的基于灰度和旋轉(zhuǎn)不變紋理的算子如下： P1s(g g)當(dāng)U(LBP

)2LBPriu2

P0 p c其它

P,R

(9)P,R

P1其中U(LBPP,R

)s(g

g)s(gc

g)|P1|s(gc p1

g)s(gc

g)| (10)criu2U2P+1P(9)指定了一個(gè)獨(dú)特的標(biāo)識(shí)給這些像素點(diǎn)對(duì)應(yīng)模式（0P）12通過(guò)圖案把“均衡”模式表示出來(lái)了。在實(shí)踐中，從LBPP,R

LBPriu2P+2個(gè)不同的輸出值，是基于2p個(gè)元P,R素的查找表的最佳實(shí)現(xiàn)。紋理分析中最終使用的紋理特征是算子作用在紋理樣本之上所得值（即模式標(biāo)識(shí)）的累計(jì)直方圖。相對(duì)于全獨(dú)立模式的直方圖，“均衡”模式的直方圖之所以能提供更好的識(shí)別力，歸結(jié)為它們的統(tǒng)計(jì)特性的差別。全模式累計(jì)直方圖中的“非均衡”模式的相關(guān)比例很小，因而它們的概率得不到可靠的估計(jì)。對(duì)樣本和模型直方圖的相異點(diǎn)分析中的有噪估計(jì)會(huì)使效果變差。LBPri）8個(gè)像素點(diǎn)所提供的8,1有角空間45°角粗量化的制約。因?yàn)橛薪强臻g的量化被定義為(360°/P)，所以要使用一個(gè)更大的P和RR（9對(duì)于R=2P個(gè)元素的查找表的有效執(zhí)行，要求為P設(shè)定一P值最大為2416MB的查詢表。局部圖像紋理對(duì)比度的旋轉(zhuǎn)不變量方差的量度LBPriu2算子是一個(gè)灰度不變性方法，也就是，它的輸出值不受任何灰度轉(zhuǎn)化的影響。P,R它是空間模式的優(yōu)良方法，但根據(jù)定義，丟失了對(duì)比度。如果灰度不變性不是必需的，而我們又想要合并局部圖像紋理的對(duì)比度，則可用旋轉(zhuǎn)不變量來(lái)衡量局部方差：VARP,R

1P1(gP p0

)2,

1P

P1gpp0

(11)VAR

P,R

LBPriu2和P,R

P,R

是互相補(bǔ)充的，它們的聯(lián)合LBPriu2/VARP,R

P,R

的數(shù)學(xué)期望是局部圖像紋理旋轉(zhuǎn)不變量強(qiáng)有力的衡量。鑒于此，即使我們?cè)诒狙芯恐邢拗莆覀冏约河玫骄哂邢嗤?P,R)值的LBPriu2和VARP,R

P,R

算子，也不會(huì)影響我們使用作用于不同鄰域的算子的聯(lián)合分布。非參數(shù)的分類法則在分類階段，我們求出樣本和模型直方圖的相異值作為擬合度測(cè)試，這個(gè)值由非參數(shù)的統(tǒng)計(jì)檢驗(yàn)來(lái)衡量。通過(guò)非參數(shù)檢驗(yàn)，關(guān)于紋理分類的假設(shè)，我們可以避免任何可能的錯(cuò)誤。有許多眾所周知的擬合度統(tǒng)計(jì)量，諸如2統(tǒng)計(jì)量和G（對(duì)數(shù)似然比）統(tǒng)計(jì)量[4]。本研究中，測(cè)試樣本S被指派給M模型類，它的極大對(duì)數(shù)似然統(tǒng)計(jì)量為：L(S,M)

SlogM (12)b b其中，B為bin的數(shù)量，S和M 分別對(duì)應(yīng)樣本和模型的直方圖維值（bin）為b的概b b率。方程式（12）是G（對(duì)數(shù)似然比）統(tǒng)計(jì)量的直接簡(jiǎn)化：G(S,M)2BSlogSb

2B[SlogS Slog

] (13)b Mb

b b bb1其中，表達(dá)式右邊的第一項(xiàng)可以忽略地看作是給定的常數(shù)S。L是一個(gè)非參數(shù)假設(shè)，用于衡量樣本S的似然度，是來(lái)自紋理類別還是基于預(yù)分類紋MLBPriu2/VARP,R

P,R

(12)的情況下，可以直接方式徹底掃描二維直方圖。LBPriu2有一個(gè)離散輸出值(0→P+1)P,R化，但算子的輸出值直接被累計(jì)成P+2維的直方圖。每維都能有效提供一個(gè)在紋理樣本或原型中遇到的對(duì)應(yīng)模式的概率的估計(jì)量。因?yàn)橹挥幸粋€(gè)模式小子集可以幾乎包含一個(gè)給定的模式，所以毗連的鄰域之間的空間依存關(guān)系是固有地存在于直方圖中的。方差量度VAR 有一個(gè)連續(xù)值的輸出，因此，需要特征空間的量化。這可通過(guò)在總分P,RB維。因此，直方圖的維數(shù)的刪除數(shù)值對(duì)應(yīng)組合數(shù)據(jù)的百分位100/大可能導(dǎo)致稀疏且不穩(wěn)定的直方圖。根據(jù)經(jīng)驗(yàn)方法，統(tǒng)計(jì)學(xué)文獻(xiàn)時(shí)常建議平均每維10個(gè)條目應(yīng)該是足夠的。在實(shí)驗(yàn)方面，我們?cè)O(shè)定B的數(shù)值，以便這一個(gè)條件得到滿足。多分辨率分析我們已經(jīng)描述了一般旋轉(zhuǎn)不變算子作用于P像素點(diǎn)以RPR重算子所提供的聯(lián)合信息來(lái)完成。本研究中，我們通過(guò)定義來(lái)直接實(shí)現(xiàn)多分辨率分析，聚合相異度相當(dāng)于對(duì)應(yīng)LN的對(duì)數(shù)似然和。算子定義如下：L Nn1

L(Sn,Mn) (14)Sn和Mn提取的對(duì)應(yīng)樣本和模型直方圖。這個(gè)表達(dá)式是基于G統(tǒng)計(jì)量(13)的特性的疊加，即，幾個(gè)G檢驗(yàn)結(jié)果可以歸納出個(gè)有深遠(yuǎn)意義的結(jié)果。如果X和Y是獨(dú)立隨機(jī)事件，且S ，S ，M 和M 分別為SX Y X YM的邊緣分布，則G(SXY,MXY)G(SX,MX)G(SY,MY)[5]通常，不同紋理特征之間的獨(dú)立性假設(shè)是站不住腳的。然而，由于統(tǒng)計(jì)學(xué)的偏差以及高維直方圖的計(jì)算復(fù)雜度，精確的聯(lián)合概率估計(jì)是不可行的。例如，LBPriu2，LBPriu2和8,R 16,RLBPriu2的疊加直方圖包含4680（10×18×26）個(gè)單元。為了滿足統(tǒng)計(jì)可靠性的第一法則，24,R即，平均每單元至少要有10個(gè)條目，圖像大小至少為（216+2R）(216+2R)個(gè)像素。因此，高維直方圖只有當(dāng)真實(shí)圖像大的時(shí)候才可靠，這使之變的不切實(shí)際。大的多維直方圖的計(jì)算在計(jì)算速度和內(nèi)存消耗上也是很可觀的。最近，我們?cè)诩y理分割中也成功使用了這種方法，為多分辨率分析中獨(dú)立直方圖的合并做了大量不同選項(xiàng)的比較[6]。本研究中，我們限制至多三個(gè)算子的合并。PAGEPAGE16MultiresolutionGray-ScaleandRotationInvariantTextureClassificationwithLocalBinaryPatternsAbstract：Thispaperpresentsatheoreticallyverysimple,yetefficient,multiresolutionapproachtogray-scaleandrotationinvarianttextureclassificationbasedonlocalbinarypatternsandnonparametricdiscriminationofsampleandprototypedistributions.Themethodisbasedonrecognizingthatcertainlocalbinarypatterns,termed“uniform” ,arefundamentalpropertiesoflocalimagetextureandtheiroccurrencehistogramisproventobeaverypowerfultexturefeature.Wederiveageneralizedgray-scaleandrotationinvariantoperatorpresentationallowsfordetectingthe“uniform”patternsforanyquantizationoftheangularspaceandforanyspatialresolutionandpresentsamethodforcombiningmultipleoperatorsformultiresolutionanalysis.Theproposedapproachisveryrobustintermsofgray-scalevariationssincetheoperatoris,bydefinition,invariantagainstanymonotonictransformationofthegrayscale.Anotheradvantageiscomputationalsimplicityastheoperatorcanberealizedwithafewoperationsinasmallneighborhoodandalookuptable.Excellentexperimentalresultsobtainedintrueproblemsofrotationinvariance,wheretheclassifieristrainedatoneparticularrotationangleandtestedwithsamplesfromotherrotationangles,demonstratethatgooddiscriminationcanbeachievedwiththeoccurrencestatisticsofsimplerotationinvariantlocalbinarypatterns.Theseoperatorscharacterizethespatialconfigurationoflocalimagetextureandtheperformancecanbefurtherimprovedbycombiningthemwithrotationinvariantvariancemeasuresthatcharacterizethecontrastoflocalimagetexture.Thejointdistributionsoftheseorthogonalmeasuresareshowntobeverypowerfultoolsforrotationinvarianttextureanalysis.IndexTerms：Nonparametric,textureanalysis,Outex,Brodatz,distribution,histogram,contrast.GRAYSCALEANDROTATIONINVARIANTLOCALBINARYPATTERNSWestartthederivationofourgrayscaleandrotationinvarianttextureoperatorbydefiningtextureTinalocalneighborhoodofaonochrometextureimageasthejointdistributionoftheg )g )P1Tt(gc

,g0

（1）where grayvalue gccorrespondsto the grayvalueof the center pixel of thelocalneighborhoodand

correspondtothegrayvaluesofPequallyspacedpixelsponacircleofradiusR(R>0)thatformacircularlysymmetricneighborset.If the coordinates of

are (0,0), then the coordinates of g

are given by(R

sin(2p/P),

R

cp/P))

pFig.1illustratescircularlysymmetricneighborsetsforvarious(P,R).Thegrayvaluesofneighborswhichdonotfallexactlyinthecenterofpixelsareestimatedbyinterpolation.AchievingGray-ScaleInvarianceAsthefirststeptowardgray-scaleinvariance,wesubtract,withoutlosinginformation,thegrayvalueofthecenterpixel(gc

)fromthegrayvaluesofthecircularlysymmetricneighborhoodg （p=0,……P-1）,giving:P

Tt(g,g0

g,gc

g,,,g P1

g) （2）cNext,weassumethatdifferences gPfactorize(2):

g areindependentof gc

,whichallowsustoTt(gc

)t(g0

g,gc

gc

g) （3）,,g P1Inpractice,anexactindependenceisnotwarranted;hence,thefactorizeddistributionisonlyanapproximationofthejointdistribution.However,wearewillingtoacceptthepossiblesmalllossininformationasitallowsustoachieveinvariancewithrespecttoshiftsingrayscale.Namely,thedistribution t(gc

)in(3)describestheoverallluminanceoftheimage,whichisunrelatedtolocalimagetextureand,consequently,doesnotprovideusefulinformationfor,g ,g P1Tt(g0

g,gc

gc

g) （4）cThisisahighlydiscriminativetextureoperator.ItrecordstheoccurrencesofvariouspatternsintheneighborhoodofeachpixelinaP-dimensionalhistogram.Forconstantregions,thedifferencesarezeroinalldirections.Onaslowlyslopededge,theoperatorrecordsthehighestdifferenceinthegradientdirectionandzerovaluesalongtheedgeand,foraspot,thedifferencesarehighinalldirections.Signeddifferencesgp-gcarenotaffectedbychangesinmeanluminance;hence,thedifferencedistributionisinvariantagainstgray-scaleshifts.Weachieveinvariancewithrespecttothescalingofthegrayscalebyconsideringjustthesigns ofthedifferencesinsteadoftheirexactvalues:Tt(s(g0

g),s(gc 1

g), s(gc

p1

g)) (5)cwhere

1,x0s(x)0,x0

(6)Byassigningabinomialfactor 2p foreachsigns(gP

g),wetransform(5)intoauniquecLBPP,R

numberthatcharacterizesthespatialstructureofthelocalimagetexture:p1LBPP,R

p0

s(gp

g)2p (7)cThenameaLocalBinaryPatternoreflectsthefunctionalityoftheoperator,i.e.,alocalneighborhoodisthresholdedatthegrayvalueofthecenterpixelintoabinarypattern. LBPP,Roperatorisbydefinitioninvariantagainstanymonotonictransformationofthegrayscale,i.e.,aslongastheorderofthegrayvaluesintheimagestaysthesame,theoutputofthe LBPP,Roperatorremainsconstant.Ifweset(P=8;R=1),weobtain LBP8,1[2].Thetwodifferencesbetween LBP8,1

,whichissimilartotheLBPoperatorweproposedand LBPare: 1)Thepixelsintheneighborsetareindexedsothattheyformacircularchainand 2)thegrayvaluesofthediagonalpixelsdeterminedbyinterpolation.Bothmodificationsarenecessarytoobtainthecircularlysymmetricneighborset,whichallowsforderivingarotationinvariantversionof LBP .P,RAchievingRotationInvarianceThe LBPP,R

operator produces 2P different output values, corresponding to the 2PdifferentbinarypatternsthatcanbeformedbythePpixelsintheneighborset.Whentheimageisrotated,thegrayvalues gP

willcorrespondinglymovealongtheperimeterofthecirclearound g.Since g isalwaysassignedtobethegrayvalueofelement(0;R)totherightof0 0g rotatingaparticularbinarypatternnaturallyresultsinadifferent LBPc P,R

value.Thisdoesnotapplytopatternscomprisingofonly0s(or1s)whichremainconstantatallrotationangles.Toremovetheeffectofrotation,i.e.,toassignauniqueidentifiertoeachrotationinvariantlocal,P,PLBPriP,R

min{ROR(LBPP,R

,i)i0,1,

(8)whereROR（x;i）performsacircularbit-wiserightshiftontheP-bitnumberxitimes.Intermsofimagepixels,（8）simplycorrespondstorotatingtheneighborsetclockwisesotimesthatamaximalnumberofthemostsignificantbits,startingfrom gP1,is0.LBPriP,R

quantifies the occurrence statistics of individual rotation invariant patternscorrespondingtocertainmicrofeaturesintheimage;hence,thepatternscanbeconsideredasfeaturedetectors.Fig.2illustratesthe36uniquerotationinvariantlocalbinarypatternsthatcanoccurinthecaseofP=8,i.e., LBPriP,R

canhave36differentvalues.Forexample,pattern#0detects bright spots, #8 dark spots and flat areas, and #4 edges. If we set R=1,LBPriP,R

correspondstothegray-scaleandrotationinvariantoperatorthatwedesignatedasLBPROTin[3].ImprovedRotationInvariancewith“Uniform” PatternsandFinerQuantizationoftheAngularSpaceOurpracticalexperience,however,hasshownthatLBPROTassuchdoesnotprovideverygooddiscrimination,aswealsoconcludedin[3].Therearetworeasons:Theoccurrencefrequenciesofthe36individualpatternsincorporatedinLBPROTvarygreatlyandthecrudequantizationoftheangularspaceat45°intervals.Wehaveobservedthatcertainlocalbinarypatternsarefundamentalpropertiesoftexture,providingthevastmajority,sometimesover90percent,ofall3×3patternspresentinobservedtextures.ThisisdemonstratedinmoredetailinSection3withstatisticsoftheimagedatausedintheexperiments.Wecallthesefundamentalpatterns“uniform”astheyhaveonethingincommon,namely,uniform circularstructurethatcontainsveryfewspatialtransitions.“Uniform”patternsareillustratedonthefirstrowofFig.2.Theyfunctionastemplatesformicrostructuressuchasbrightspot(0),flatareaordarkspot(8),andedgesofvaryingpositiveandnegativecurvature(1-7).Toformallydefinethe“uniform”patterns,weintroduceauniformitymeasureU(“pattern”),whichcorrespondstothenumberofspatialtransitions(bitwise0/1changes)inthe“pattern”.Forexample,patterns 000000002 and111111112 haveUvalueof0,whiletheothersevenpatternsinthefirstrowofFig.2haveUvalueof2asthereareexactlytwo0/1transitionsinthepattern.Similarly,theother27patternshaveUvalueofatleast4.WedesignatepatternsthathaveUvalueofatmost2as“uniform”andproposetheoperatorforgray-scaleandrotationinvarianttexturedescriptioninsteadof LBPri :P,R P1s(g g

ifU(LBP

)2LBPriu2 P0 p

P,R

(9)P,R P

otherwisewhere

U(LBPP,R

)s(g

g)s(gc

g)|P1|s(gc p1

g)s(gc

g)| (10)cSuperscriptriu2reflectstheuseofrotationinvariant"uniform"patternsthathaveUvalueofmost2.Bydefinition,exactlyP+1"uniform"binarypatternscanoccurinacircularlysymmetricneighborsetofP pixels.Equation(9)assignsauniquelabeltoeachofthemcorrespondingtothenumberofa1obitsinthepattern(0P),whilethe"nonuniform"patternsaregroupedunderthe"miscellaneous"label(P+1).InFig.2,thelabelsofthe"uniform"patternsaredenotedinsidethepatterns.Inpractice,themappingfrom LBPP,R

to LBPriu2,whichhasP+2distinctoutputP,Rvalues,isbestimplementedwithalookuptableof 2P elements.Thefinaltexturefeatureemployedintextureanalysisisthehistogramoftheoperatoroutputs(i.e.,patternlabels)accumulatedoveratexturesample.Thereasonwhythehistogramof"uniform" patterns provides better discrimination in comparison to the histogram of individualpatternscomesdowntodifferencesintheirstatisticalproperties.Therelativeproportionof"nonuniform"patternsofallpatternsaccumulatedintoahistogramissosmallthattheirprobabilitiescannotbeestimatedreliably.Inclusionoftheirnoisyestimatesinthedissimilarityanalysisofsampleandmodelhistogramswoulddeteriorateperformance.WenotedearlierthattherotationinvarianceofLBPROT(LBPri)ishamperedbythecrude458,1quantizationoftheangularspaceprovidedbytheneighborsetofeightpixels.AstraightforwardfixistousealargerPsincethequantizationoftheangularspaceisdefinedby(360°/P).However,certainconsiderationshavetobetakenintoaccountintheselectionofP.First,PandRarerelatedinthesensethatthecircularneighborhoodcorrespondingtoagivenRcontainsalimitednumberofpixels(e.g.,nineforR=1),whichintroducesanupperlimittothenumberofnonredundantsamplingpointsintheneighborhood.Second,anefficientimplementationwithalookuptableof 2P elementssetsapracticalupperlimitforP.Inthisstudy,weexplorevaluesupto24,whichrequiresalookuptableof16MBthatcanbeeasilymanagedbyamoderncomputer.RotationInvariantVarianceMeasuresoftheContrastofLocalImageTextureThe LBPriuP,R

operatorisagray-scaleinvariantmeasure,i.e.,itsoutputisnotaffectedbyanymonotonicransformationofthegrayscale.Itisanexcellentmeasureofthespatialpattern,butit,bydefinition,discardscontrast.Ifgray-scaleinvarianceisnotrequiredandwewantedtoincorporatethecontrastoflocalimagetextureaswell,wecanmeasureitwitharotationinvariantmeasureoflocalvariance:VARP,R

1P1(gP p0

)2,

1P

P1gpp0

(11)VAR

P,R

isbydefinitioninvariantagainstshiftsin grayscale.Since LBPriu2P,R

VAR

P,R

arecomplementary,theirjointdistribution LBPriu2/P,R

P,R

isexpectedtobeaverypowerfulrotationinvariantmeasureoflocalimagetexture.Notethat,eventhoughweinthisstudyrestrictourselvestousingonlyjointdistributionsof LBPriu2P,R

and

VAR

P,R

operatorsthathavethesame(P;R)values,nothingwouldpreventusfromusingjointdistributionsofoperatorscomputedatdifferentneighborhoods.NonparametricClassificationPrincipleIntheclassificationphase,weevaluatethedissimilarityofsampleandmodelhistogramsasatestofgoodness-of-fit,whichismeasuredwithanonparametricstatisticaltest.Byusinganonparametrictest,weavoidmakingany,possiblyerroneous,assumptionsaboutthefeaturedistributions.Therearemanywell-knowngoodness-of-fitstatisticssuchasthechi-squarestatisticandtheG(log-likelihoodratio)statistic[4].Inthisstudy,atestsampleSwasassignedtotheclassofthemodelMthatmaximizedthelog-likelihoodstatistic:L(S,M)b1

SlogMb b

(12)whereBisthenumberofbinsand Sb

and Mb

correspondtothesampleandmodelprobabilitiesatbinb,respectively.Equation(12)isastraightforwardsimplificationoftheG(log-likelihoodratio)statistic:G(S,M)2BSlogSb

2B[SlogS Slog

] (13)b Mb

b b b bb1wherethefirsttermoftherighthandexpressioncanbeignoredasaconstantforagivenS.LisanonparametricpseudometricthatmeasureslikelihoodsthatsampleSisfromalternativetextureclasses,basedonexactprobabilitiesoffeaturevaluesofpreclassifiedtexturemodelsM.InthecaseofthejointdistributionLBPriu2/VARP,R

P,R

(12)wasextendedinastraightforwardmannertoscanthroughthetwo-dimensionalhistograms.SampleandmodeldistributionswereobtainedbyscanningthetexturesamplesandprototypeswiththechosenoperatoranddividingthedistributionsofoperatoroutputsintohistogramshavingafixednumberofBbins.Since

riu2hasafixedsetofdiscreteoutputvalues(0 →P,RP+1),noquantizationisrequired,buttheoperatoroutputsaredirectlyaccumulatedintoahistogramofP+2bins.Eachbineffectivelyprovidesanestimateoftheprobabilityofencounteringthecorrespondingpatterninthetexturesampleorprototype.Spatialdependenciesbetweenadjacentneighborhoodsareinherentlyincorporatedinthehistogrambecauseonlyasmallsubsetofpatternscanresidenexttoagivenpattern.VariancemeasureVAR hasacontinuous-valuedoutput;hence,quantizationofitsfeatureP,Rspaceisneeded.Thiswasdonebyaddingtogetherfeaturedistributionsforeverysinglemodelimageinatotaldistribution,whichwasdividedintoBbinshavinganequalnumberofentries.Hence,thecutvaluesofthebinsofthehistogramscorrespondedtothe(100=B)percentileofthecombineddata.Derivingthecutvaluesfromthetotaldistributionandallocatingeverybinthesameamountofthecombineddataguaranteesthatthehighestresolutionofquantizationisusedwherethenumberofentriesislargestandviceversa.Thenumberofbinsusedinthequantizationofthefeaturespaceisofsomeimportanceashistogramswithatoosmallnumberofbinsfailtoprovideenoughdiscriminativeinformationaboutthedistributions.Ontheotherhand,sincethedistributionshaveafinitenumberofentries,atoolargenumberofbinsmayleadtosparseandunstablehistograms. Asaruleofthumb,statisticsliterature oftenproposesthataveragenumberof10entriesperbinshouldbesufficient.Intheexperiments,wesetthevalueofBsothatthisconditionis satisfied.MultiresolutionAnalysisWehavepresentedgeneralrotation-invariantoperatorsforcharacterizingthespatialpatternandthecontrastoflocalimagetextureusingacircularlysymmetricneighborsetofPpixelsplacedonacircleofradiusR.ByalteringPandR,wecanrealizeoperatorsforanyquantizationoftheangularspaceandforanyspatialresolution.Multiresolutionanalysiscanbeaccomplishedbycombiningtheinformationprovidedbymultipleoperatorsofvarying(P;R).Inthisstudy,weperformstraightforwardmultiresolutionanalysisbydefiningtheaggregatedissimilarityasthesumofindividuallog-likelihoodscomputedfromtheresponsesofindividualoperators.L Nn1

L(Sn,Mn) (14)wher