TOPOGRAPHIC OBJECT RECOGNITION THROUGH SHAPE

Transcription

1 TOOGAHIC OBJECT ECOGNITION THOUGH SHAE Laura Keyes and Adam Wnstanley Techncal eport Submtted to Ordnance Survey, Southampton March 00 Department of Computer Scence Natonal Unversty of Ireland, Maynooth Co. Kldare Ireland - -

2 ABSTACT Automatc structurng feature codng and obect recognton of topographc data, such as that derved from ar survey or raster scannng large-scale paper maps, requres the classfcaton of obects such as buldngs, roads, rvers, felds and ralways. The recognton of obects n computer vson s largely based on the matchng of descrptons of shapes. Fourer descrptors, moment nvarants, boundary chan codng and scalar descrptors are methods that have been wdely used and have been developed to descrbe shape rrespectve of poston, orentaton and scale. The applcablty of the above four methods to topographc shapes s descrbed and ther usefulness evaluated. All methods derve descrptors consstng of a small number of real values from the obect s polygonal boundary. Two large corpora representng data sets from Ordnance Survey maps of urbec and lymouth were avalable. The effectveness of each descrpton technque was evaluated by usng one corpus as a tranng-set to derve dstrbutons for the values for supervsed learnng. Ths was then used to reclassfy the obects n both data sets usng each ndvdual descrptor to evaluate ther effectveness. No ndvdual descrptor or method produced consstent correct classfcaton. Varous models for the fuson of the classfcaton results from ndvdual descrptors were mplemented. These were used to eperment wth dfferent combnatons of descrptors n order to mprove results. Overall results show that Moment Invarants fused wth the mn fuson rule gave the best performance wth the two data sets. Much further wor remans to be done as enumerated n the concludng secton. - -

3 TABLE OF CONTENTS ABSTACT Chapter : INTODUCTION Chapter : SHAE-BASED DESCITION. Fourer Descrptors. Moment Invarants.3 Scalar Descrptors Chapter 3: CLASSIFICATION 3. Supervsed v Unsupervsed Classfcaton 3. Classfcaton usng Bayes Theorem 3.3 Implementng Bayesan Classfcaton Chapter 4: COMBINING CLASSIFIES 4. The Fuson Model 4. Theory 4.. The roduct ule 4.. Sum ule 4.3 Classfer Combnaton 4.3. Maorty Vote ule 4.3. Mn ule Ma ule Medan ule 4.4 Implementng Data Fuson Chapter 5: EXEIMENTAL ESULTS 5. Indvdual descrptors Chapter 6: CONCLUSIONS EFEENCES - 3 -

4 Append : esults from urbec data set Append : esults from lymouth data set Append 3: Classfcaton code Append 4: Data Fuson code Append 5: Summary of classfcatons by descrptor method and feature type - 4 -

5 Chapter : INTODUCTION The Intellgent and Graphcal esearch Group wthn the Department of Computer Scence at Natonal Unversty of Ireland, Maynooth NUIM s researchng nto the automatc recognton of features and obects on topographc maps. The man applcaton of ths wor s the automatc structurng of topographc data for computer cartography and GIS systems. The technques beng evaluated can be dvded nto two broad categores: recognton based on solated shape descrbed here, and recognton based on contet. In shape-based classfcaton, the shape of each obect s descrbed usng a small number of descrptor values typcally 7 to 5 real numbers. ecognton s based on matchng the descrptors of each shape to standard values representng typcal shapes and choosng the closest match. Several types of descrptor values have been developed mostly n the feld of computer vson. esearch at NUIM so far has concentrated on four technques: scalar descrptors area, dmenson, elongaton, number of corners etc., Fourer descrptors, moment nvarants and boundary chan encodng. These technques are well understood when appled to mages and can be normalsed to descrbe shapes rrespectve of poston, scale and orentaton. They can also be easly appled to vector graphcal shapes. Wor carred out to date ncludes the obect recognton and classfcaton of buldngs and parcels from test data provded by the Isle of Man government usng three of the above mentoned technques namely Fourer descrptors, moment nvarants and scalar descrptors. esults ndcate that no one shape technque alone s powerful enough for the tas - n dfferent stuatons one technque wll perform better than the others and produce sgnfcant results e.g. dstngushng buldngs from lnear features n bult-up areas usng the moment nvarants method. In order to test these technques further, they were evaluated on a corpus of topographc data provded by OSGB usng the feature codes obect types used n the large-scale OS GB topographc database. The most sgnfcant ams were to: statstcally analyse the range of descrptor values obtaned by each method both wthn and between each OS feature type; evaluate classfcaton performance of each method on all polygons through comparson wth orgnal data; - 5 -

6 nvestgate possble mprovement n performance by evaluatng strateges of combnng methods; and evaluate performance of methods n detectng msclassfed features n orgnal data. Ths report descrbes the results of ths eercse. It contans the followng sectons:. The man tass and ams of the proect;. A descrpton of the mplementaton and ntegraton of the software modules for ndvdual methods; 3. An evaluaton of each method; 4. A comparson between methods; 5. Combnaton and selecton of methods for optmal results; 6. Conclusons; 7. Suggestons for future research derved from the conclusons

7 Chapter : Shape-based Classfcaton Topographc data capture for large-scale maps typcally depcted at :50 and :500 conssts of two parts: the dgtsaton of the geometry and the addton of attrbutes ndcatng the feature and/or obect type beng depcted. Whereas the former can be automated usng mage processng and smlar technques, the latter s often a manual tas. One possble means of automaton s obect recognton through shape. Ths proect uses shape recognton technques borrowed from the feld of computer vson to descrbe a measurement of shape to characterse and classfy features on maps. The man applcaton of ths wor s the automatc structurng of topographc data for computer cartography and Geographcal Informaton Systems GIS. ecognton of obects s largely based on the matchng of descrpton of shapes wth a database of standard shapes. Numerous shape descrpton technques have been developed such as, Fourer descrptors, moment nvarants and scalar features area, number of ponts, etc.. revous wor has evaluated these technques on topographc obects as depcted n large-scale mappng. Unle many applcatons where the shape categores are very eact for eample, dentfyng a partcular type of arcraft n a scene, ths problem requres the classfcaton of a partcular shape nto a general class of smlar obect shapes, for eample, buldng, road or parcel. Each technque proved partally successful n dstngushng classes of obect although no one technque provded a general soluton to the problem. As part of ths report these technques are further evaluated on a real-world problem usng a corpus of topographc data provded by Ordnance Survey n Great Brtan OS GB. The data set conssts of the features codes obect types used on the large-scale OS GB topographc database. Ths report bulds on prevous wor carred out to produce an accurate combned methodology for the classfcaton of general shapes on maps. The followng sectons ntroduce each of the above named shape recognton technques ndvdually and descrbe how they are appled as general classfers to broad classes of topographc shape buldngs, felds and road etc.. The overall mplementaton of the proect and eperment s outlned and sets out the most sgnfcant ams of the report. An - 7 -

8 evaluaton and comparson s made of the effectveness of each technque n recognsng features and obects. A data fuson technque s then proposed and evaluated. Ths allows the combnng of the results of the Fourer descrptor, moment nvarants and scalar descrptor technques respectvely, to gve an overall score for each canddate obect category. The purpose of ths report s to draw from our results the man conclusons and see f they are applcable to OS. The recognton and descrpton of obects plays a central role n automatc shape analyss for computer vson and t s one of the most famlar and fundamental problems n pattern recognton. Common eamples are the readng of alphabetc characters n tet and the automatc dentfcaton of arcraft. Most applcatons usng Fourer descrptors, moment nvarants and scalar descrptors for shape recognton deal wth the classfcaton of such defnte shapes. To dentfy topographc obects each of the technques needs to be etended to deal wth general categores of shapes, for eample houses, parcels and roads. The data used for the eperments descrbed n the followng sectons was etracted from vector data sets representng large-scale :50 plans of the urbec and lymouth areas n Great Brtan Ordnance Survey. The data had been pre-processed to etract mnmal closed polygons and OS feature codes had been appled. An nterpolaton method was appled to sample the shape boundary at a fnte number N of equdstant ponts. These ponts are then stored n the approprate format for processng wth each shape descrpton technque. The shapes can then be descrbed usng a small set of descrptor values typcally 7 to 0 real numbers. The recognton s based on matchng the descrptors of each shape to standard values representng typcal shapes and choosng the closest match.. Fourer Descrptors.. Bacground Fourer transform theory Gonzalez and Wntz 977 has played a maor role n mage processng for many years. It s a commonly used tool n all types of sgnal processng and s defned both for one and two-dmensonal functons. In the scope of ths paper, - 8 -

9 the Fourer transform technque s used for shape descrpton n the form of Fourer descrptors. The Fourer descrptor s a wdely used all-purpose shape descrpton and recognton technque Granlund 97, Wnstanley 998. The shape descrptors generated from the Fourer coeffcents numercally descrbe shapes and are normalsed to mae them ndependent of translaton, scale and rotaton. These Fourer descrptor values produced by the Fourer transformaton of a gven mage represent the shape of the obect n the frequency doman Wallace and Wntz 980. The lower frequency descrptors store the general nformaton of the shape and the hgher frequency the smaller detals. Therefore, the lower frequency components of the Fourer descrptors defne a rough shape of the orgnal obect.. Theory The Fourer transform theory can be appled n dfferent ways for shape descrpton. One method wors on the change n orentaton angle as the shape outlne s traversed Zahn and oses 97, but for the purpose of ths paper the followng procedure was mplemented Wood 986. The boundary of the mage s treated as lyng n the comple plane. So the row and column co-ordnates of each pont on the boundary can be epressed as a comple number, + y where s sqrt -. Tracng once around the boundary n the counter-clocwse drecton at a constant speed yelds a sequence of comple numbers, that s, a one-dmensonal functon over tme. In order to represent traversal at a constant speed t s necessary to nterpolate equ-dstant ponts around the boundary. Traversng the boundary more than once results n a perodc functon. The Fourer transform of a contnuous functon of a varable s gven by the equaton: F u f u e πu d When dealng wth dscrete mages the Dscrete Fourer Transform DFT s used. So equaton transforms to: F N N 0 π N u f u e - 9 -

10 The varable s comple, so by usng the epanson e[- A] cos A. sn A where N s the number of equally spaced samples, equaton becomes: F N N 0 u f + y. cos A.sn A 3 where A πu/. The DFT of the sequence of comple numbers, obtaned by the traversal of the obect contour, gves the Fourer descrptor values of that shape. The Fourer descrptor values can be normalsed to mae them ndependent of translaton, scale and rotaton of the orgnal shape. Smply, translaton of the shape by a comple quantty havng and y components, corresponds to addng a constant + y to each pont representng the boundary. Scalng a shape s acheved by multplyng all co-ordnate values by a constant factor. The DFT results n all members of the correspondng Fourer seres beng multpled by the same factor. So by dvdng each coeffcent by the same member, normalsaton for sze s acheved. otaton normalsaton s acheved by fndng the two coeffcents wth largest magntude and settng ther phase angle equal to zero Keyes and Wnstanley Fourer Descrptors of cartographc shapes To apply the Fourer descrptor technque to cartographc data, the ponts are stored as a seres of comple numbers and then processed usng the Fourer transform resultng n another comple seres of the same length N. If the formula for the dscrete Fourer transform were drectly appled each term would requre N teratons to sum. As there are N terms to be calculated, the computaton tme would be proportonal to N. So the algorthm chosen to compute the Fourer descrptors was the Fast Fourer Transform FFT for whch the computaton tme s proportonal to NlogN. The FFT algorthm requres the number of ponts N defnng the shape to be a power of two. In the case of ths proect t was decded to use 5 sample ponts

11 The FFT algorthm s appled to these 5 coeffcents. The lst s normalsed for translaton, rotaton and scale. Ths results n the frst two terms always havng the values 0 and.0 respectvely whch maes them redundant for classfcaton. Calculaton of the Fourer Spectrum bulds a new lst and dsposes of the Fourer transform lst. The result s 50 Fourer descrptor terms. The nature of the Fourer transform means that general shape nformaton s modelled n the frst few terms whle the later terms reflect small detal. Therefore n shape classfcaton, a lmted number of terms are used. In ths proect, the frst 6 terms are used n the evaluaton.. Moment Invarants.. Bacground Moment Invarants have been frequently used as features for mage processng, remote sensng, shape recognton and classfcaton. Moments can provde characterstcs of an obect that unquely represent ts shape. Invarant shape recognton s performed by classfcaton n the multdmensonal moment nvarant feature space. Several technques have been developed that derve nvarant features from moments for obect recognton and representaton. These technques are dstngushed by ther moment defnton, such as the type of data eploted and the method for dervng nvarant values from the mage moments. It was Hu Hu, 96, that frst set out the mathematcal foundaton for twodmensonal moment nvarants and demonstrated ther applcatons to shape recognton. They were frst appled to arcraft shapes and were shown to be quc and relable Dudan, Breedng and McGhee, 977. These moment nvarant values are nvarant wth respect to translaton, scale and rotaton of the shape. Hu defnes seven of these shape descrptor values computed from central moments through order three that are ndependent to obect translaton, scale and orentaton. - -

12 Translaton nvarance s acheved by computng moments that are normalsed wth respect to the centre of gravty so that the centre of mass of the dstrbuton s at the orgn central moments. Sze nvarant moments are derved from algebrac nvarants but these can be shown to be the result of a smple sze normalsaton. From the second and thrd order values of the normalsed central moments a set of seven nvarant moments can be computed whch are ndependent of rotaton... Theory Tradtonally, moment nvarants are computed based on the nformaton provded by both the shape boundary and ts nteror regon Hu 96. The moments used to construct the moment nvarants are defned n the contnuous but for practcal mplementaton they are computed n the dscrete form. Gven a functon f,y, these regular moments are defned by: p q Μ pq y f, y ddy 4 M pq s the two-dmensonal moment of the functon f,y. The order of the moment s p + q where p and q are both natural numbers. For mplementaton n dgtal form ths becomes: Μ pq Χ Υ p y q f, y 5 To normalse for translaton n the mage plane, the mage centrods are used to defne the central moments. The co-ordnates of the centre of gravty of the mage are calculated usng equaton 5 and are gven by: Μ Μ 0 00 Μ y Μ The central moments can then be defned n ther dscrete representaton as: µ pq Χ Υ p y y q - -

13 7 The moments are further normalsed for the effects of change of scale usng the followng formula: η pq µ 8 Where the normalsaton factor: γ p + q / +. From the normalsed central moments a set of seven values can be calculated and are defned by: pq µ γ 00 φ η 0 + η 0 φ η 0 - η 0 + 4η φ 3 η 30-3η + η 03-3η φ 4 η 30 + η + η 03 + η φ 5 3η 30-3η η 30 + η [η 30 + η 3η + η 03 ] + 3η - η 03 η + η 03 [3η 30 + η η + η 03 ] φ 6 η 0 - η 0 [η 30 + η η + η 03 ] + 4η η 30 + η η + η 03 φ 7 3η - η 03 η 30 + η [η 30 + η - 3η + η 03 ] + 3η - η 30 η + η 03 [3η 30 + η η + η 30 ] 9 These seven nvarant moments, φ I, I 7, set out by Hu, were addtonally shown to be ndependent of rotaton. However they are computed over the shape boundary and ts nteror regon and so are not easly derved from vector graphcs

14 ..3 New moments For the purpose of ths proect, an algorthm was mplemented that calculates the moment nvarants usng the shape boundary alone. These can be proven to be nvarant under obect translaton, scale and rotaton Chaur-Chn Chen 993. Then, usng the same notaton for convenence, the moment defnton n equaton 4 can be epressed as: Μ pq C p y q ds For p, q 0,,,3, where c s the lne ntegral along the curve C and ds d + dy. The central moments can be smlarly defned as: 0 µ pq C p q y y ds Gven that the centrods are as n the orgnal method: Μ Μ for a dgtal mage, then equaton becomes 0 00 Μ y Μ 0 00 µ pq Χ, Υ C p y y q 3 Thus the central moments are nvarant to translaton. These new central moments can also be normalsed such that they are scalng nvarant. η pq µ 4 where the normalsaton factor s: γ p + q +. The seven moment nvarant values can then be calculated as before usng the results obtaned from the computaton of equaton s 0 to 4 above. pq µ γ

15 Usng the same data sets as n the Fourer descrptor method descrbed earler, the moments technque s appled. However, for moments the ponts etracted from the map are stored not as comple numbers but represent the and y co-ordnates of the polygonal shape. These ponts are processed by a moment transformaton on the outlne of the shape, whch produces seven moment nvarant values that are normalsed wth respect to translaton, scale and rotaton usng the formulae above. The resultng set of values can be used to dscrmnate between the shapes..3 Scalar Descrptors Scalar descrptors are based on scalar features derved from the boundary of an obect. They use numerous metrcs of the obect as shape descrptors. Smple eamples of such features nclude: the permeter length; the area of the shape; the elongaton.e. rato of the area of a shape to the square of the length of ts permeter A/ ; the number of nodes unctons n the boundary; the number of sharp corners. Many other scalar descrptors can be devsed

16 Chapter 3: Classfcaton 3. Supervsed v Unsupervsed Classfcaton Shape descrpton technques, such as those descrbed n chapter two, generally characterse an obect s shape as a set of real numbers. Classfcaton of obects based on shape therefore conssts of comparng these descrptors. Two general forms of classfcaton are possble: unsupervsed and supervsed. Unsupervsed learnng occurs where the dstrbuton of descrptor values of obects n a data-set s analysed. Clusters of obects of smlar shape are dentfed. These are assumed to represent a class of smlar obects. In ths scheme, the classes dentfed emerge from the analyss of the data-set and can depend both on that analyss and the data-set n use. Supervsed learnng occurs when the classes to whch obects are to be assgned are decded beforehand. Values of descrptors that characterse each obect class are determned n some way and obects are classfed through the smlarty of ther descrptors to these characterstc values. Supervsed learnng therefore requres a way to determne some norms for the values of a partcular class and a way to measure whether the descrptor values of an unclassfed obect belong to the group defned by those norms. A common method to determne the norms for a class s to tae a sample of shapes we now to belong to that class and calculate the mean or medan values for each descrptor. In addton, a measure of the dstrbuton of values wthn the sample can be made. Classfcaton then conssts to comparng the values of ts descrptors wth that of the mean, possbly tang nto account the dstrbuton for the class. Gven two sets of descrptors, how do we measure ther degree of smlarty? If two shapes, A and B, produce a set of values represented by a and b then the dstance between them can be gven as c a b. If a and b are dentcal then c - 6 -

17 wll be zero. If they are dfferent then the magntudes of the coeffcents n c wll gve a reasonable measure of the dfference. It proves more convenent to have one value to represent ths rather than the set of values that mae up c. The easest way s to treat c as a vector n a mult-dmensonal space, n whch case ts length, whch represents the dstance between the planes, s gven by the square root of the sum of the squares of the elements of c. In ths way classfcaton can be performed by choosng the class mean that s closest to the shape to be classfed. Earler wor on ths proect used ths dstance measure n classfcaton wth some lmted success Keyes and Wnstanley 999, 000. However, ths method taes no account of the dstrbuton of descrptor values for each class. Therefore t was decded to ncorporate the nformaton gven by the dstrbuton usng Bayesan statstcs. 3. Classfcaton usng Bayes Theorem Bayesan statstcs allows us to use the dstrbuton of the values for each descrptor for each class of obect n determnng the probablty that a partcular obect belongs to that class. Gven a partcular value for a descrptor, we can calculate the lelyhood of that value occurrng n the dstrbuton of values for a partcular class. Applyng Bayes theorem, we can calculate from ths the probablty of the obect belongs to that class. We can calculate such a probablty for each class. We then decde that the obect belongs to the class for whch t that descrptor gves the hghest probablty. The obectve s to desgn classfers that wll classfy an obect n the most probable of the classes gven. For eample, n the eperment descrbed later n ths report, our classfcaton tas has s classes, Buldngs, Defned Natural Land Cover, Multple Surface Land, General Unmade Land, Made oad and oad Sde,,..., 6 respectvely, and an unnown feature type taen from the data-set for eample a buldng represented by the feature vector. From ths the condtonal or posteror probabltes,,,...,6 can be formed whch represent the probablty that the unnown feature type belongs to the respectve class gven that the - 7 -

18 correspondng feature vector taes on the value. To calculate the posteror probabltes, Bayes decson theory prncples are appled. The frst step nvolves the calculaton of the pror probabltes for each class. Tae for eample the Buldng class and Defned Natural Land Cover Defned Land class. Then, and denote the probabltes of a feature type belongng to ether class or respectvely before we have consdered any descrptor values. As we have a prevously classfed data-set, we can estmate ths pror probabltes as: NumberOfBuldngs TotalNumberOFeatures NumberOfDefndLAnd TotalNumberOFeatures Gven these probabltes and the frst crteron for decdng whether an observed feature type s of type Buldng or Defned Land would smply be to tae the class wth the larger probablty, whch can be wrtten as: f then f < then Better probablty results can generally be obtaned by consderng addtonal nformaton about the features such as the mean and standard devaton of each class. Let ths addtonal nformaton be dentfed by the descrptor vector usng feature vector to represent more than one sngle measured feature. Usng ths nformaton the condtonal probabltes dscussed earler can be formed. The classfcaton crteron can now be descrbed as: f > then decde and f > then decde - 8 -

19 - 9 - Bayes laws can be appled to these condtonal probabltes to redefne them n terms of ther densty functons, whch are denoted by f and f. The dervaton of the new classfcaton crteron, now n terms of the condtonal densty functons f and f states that f f So equaton above can be rewrtten as: then f f f f f then < f f f f f Bayes decson rule s obtaned by elmnatng the denomnator and s as follows: then f f f then f f f <? f f L T From the condtonal densty functons a lelhood rato h L and threshold T can be obtaned. Usng these functons the above crteron now be epressed as:

20 T T < L then L then whch reads f T L then decde else decde. Ths crteron can be generalsed qute easly to stuatons nvolvng more than two classes and multple dmensonal feature spaces. So, let be the number of classes nvolved n ths proect whch equals s and usng the respectve condtonal densty functons f the Bayesan classfcaton can now be wrtten as follows: f f Ma{ f } then select, 3.3 Implementng Bayesan Classfcaton Applyng Bayes Theorem to classfcaton therefore requres: the calculaton of pror probabltes of each class occurrng the modellng of a dstrbuton functon of the lelhoods of values occurrng for each class Both of these were estmated through an analyss of the classfcaton of a data-set provded by Ordnance Survey. The dstrbuton functon for each descrptor was appromated as a normal curve, modelled from the means and standard devatons calculated from the data-set. Usng Bayesan classfcaton a class can be assgned to each obect based on the value of one descrptor. Ths s accompaned by a probablty estmate that the classfcaton s correct. We are evaluatng three shape descrpton methods, each contanng several descrptors 5 descrptors n all. If, as s lely, these dsagree as to the classfcaton, we requre a method of combnng them to produce an overall consensus as to the correct classfcaton

21 Chapter 4: Combnng Classfers When settng out to desgn a shape recognton system the ultmate goal s to acheve the best possble classfcaton performance. Attanng ths goal nvolves the applcaton of sutable classfcaton schemes/technques to the problem. Tradtonally an analyss of the results produced by each technque became the bass for choosng one of the classfers as a fnal soluton. However, t has been observed n many studes that although one technque would yeld the best performance, the set of shapes mss-classfed by the dfferent classfers would not necessarly overlap. Ths suggests that dfferent classfer technques can offer complementary descrptons of the shapes to be classfed, whch leads to the combnng of the classfers for mproved performance. 4. The Fuson Model Usng and combnng multple learned classfcaton models for ncreasng accuracy and effcency s an area attractng much nterest recently. The central problem nvolved s how to ntegrate several classfers or eperts to produce a sngle fnal classfcaton. Fgure, llustrates the decson combnaton topology used n ths report. map feature data set Fourer descrptors Moment nvarants Scalar descrptors Σ data fuson algorthm classfed feature Fgure, Decson combnaton topology used for fusng the results of three shape recognton methods. - -

22 The approach taen here to the fuson of the recognton technques used follows a classfer combnaton scheme developed by Kttler et al[998]. 4. Theory The fuson of ndvdual classfers s based on a theoretcal framewor set out n Bayes theorem. Before fuson can tae place probabltes must be assgned or calculated ndcatng the lelhood that a partcular obect belongs to each avalable class. Consderng the classfcaton problem, where Z s to be assgned to one of m possble classes,...,,assume there are classfers each representng the gven pattern m by a dstnct measurement vector, the measurement vector used by the th classfer beng denoted by. In the measurement space each class ϖ s modelled by the probablty densty functon p and ts pror probablty of occurrence s denoted. The models are consdered to be mutually eclusve whch means that only one class can be assocated wth each obect. Accordng to Bayes theorem, gven measurements,,..., the pattern, Z, should be assgned to the class provded the a posteror probablty of the nterpretaton s mamum,.e. assgn Z f,..., ma,..., The Bayesan decson rule 5 states that n order to use all the avalable nformaton correctly to reach a decson, t s essental to compute the probabltes of the varous hypotheses by consderng all the measurements smultaneously. Ths s however a very epensve computaton therefore rule 5 s smplfed and epressed n terms of decson support computatons performed by the ndvdual classfers, each eplotng only the nformaton conveyed by the vector. ewrtng the a posteror probablty,..., usng Bayes theorem we have: 5 - -

23 - 3 -,...,,...,,..., p p 6 where,..., p s the uncondtonal measurement ont probablty densty. Ths can be epressed n terms of the condtonal measurement dstrbutons as follows: m p p,...,..., 7 Therefore, n the followng placng the concentraton only on the numerator terms of The roduct ule The measurement,... p represents the ont probablty dstrbuton of the descrptor values etracted by the classfers. Treatng the representatons used as condtonally statstcally ndependent as outlned by Kttler et al, the followng can be obtaned, p p,..., 8 where p s the measurement model of the th representaton. Substtutng from 8 and 7 nto 6 gves: m p p,..., 9 and usng 9 n 5 gves the followng decson rule. assgn Z f ma m p p

24 uttng ths n terms of the a posteror probabltes yelded by the respectve classfers: 0 assgn Z f p m ma p The decson rule n quantfes the lelhood of a hypothess by combnng the a posteror probabltes produced by the ndvdual classfers by means of a product rule. It can be a severe rule of combnng the classfer outputs as a sngle descrptor to nhbt a partcular nterpretaton by outputtng a close to zero probablty for t. 4.. Sum ule Kttler et el [998] developed a scheme for the fuson of ndvdual classfers called the sum rule whch he based on the above theoretcal framewor. Consderng the decson rule n and based on the assumpton that the a posteror probabltes computed by the respectve classfers wll not devate dramatcally from the pror probabltes, the posteror probabltes can then be epressed as: + δ where δ satsfes δ <<. Substtutng for the posteror probabltes n gves: + δ Epandng the product and neglectng any terms of second and hgher order, we can appromate the rght hand sde of 3 as: 3 + δ + δ 4-4 -

25 By substtutng 4 and nto the sum decson rule s obtaned. assgn Z f + m ma[ + ] 5 The sum of the classfers s obtaned for each class and the lelhood class computed by tang the mamum a posteror probabltes produced by the sum combnaton scheme. 4.3 Classfer Combnaton The product and sum decson rules n and 5 form the basc schemes for classfer combnaton. Many combnaton strateges can be developed from these rules by notng that: mn ma 6 Ths shows that the product and sum rules can be appromated by the upper or lower bounds suggested by 6, as approprate. Also the hardenng of the a posteror probabltes to produce bnary valued functons as f ma 0 otherwse 7 result n the combnng of a decson outcome rather than ust the combnng of posteror probabltes. From these appromatons the followng rules can be constructed. All the combnaton schemes and ther relatonshp are represented n Fgure

26 4.3. Maorty Vote ule Usng the sum rule from 5 and the above assumpton of equal prors and by hardenng the probabltes accordng to 7 gves: assgn Z f m ma 8 When calculatng the maorty vote rule for each class, the sum on the rght hand sde counts the votes receved for the ndvdual classfers. The class, whch receves the largest number of votes, s selected as the maorty decson and the fnal sngle classfcaton Mn ule Startng wth the product rule n and boundng the product of posteror probabltes from the above we obtan assgn Z f mn m ma mn 9 whch under the assumpton of equal prors smplfes to assgn Z f mn m ma mn 30 The mn rule combnaton scheme quantfes the lelhood of a gven shape belongng to a partcular class by determnng the mnmum a posteror probablty for each class. The fnal decson s then based on the mamum of the obtaned mnmum probabltes for each ndvdual classfer Ma ule Startng from the sum rule n 5 and appromatng the sum by the mamum of the posteror probabltes gves assgn Z f - 6 -

27 - 7 - ] ma ma[ ma m whch, under the assumpton of equal prors, smplfes to assgn Z f ma ma ma m 3 Ths strategy obtans a decson by computng the mamum posteror probablty for each class and then tang the mamum of these values Medan ule Under the assumpton of equal prors the sum rule n 5 can be vewed to be computng the average a posteror probablty for each class over all the classfer outputs, assgn Z f ma m 33 That s, the rule assgns a shape to the class n whch the average posteror probablty s mamum. However t s possble that a classfer mght output an a posteror probablty for some class whch s a outler. Such an output would affect the average, whch could lead to an ncorrect decson. Another robust method for fndng the mean s the medan. The followng rule bases the combned decson on the medan of the posteror probabltes. assgn Z f ma m med med 34

28 4.4 Implementng Data Fuson All the methods of data fuson descrbed above were mplemented. Each shape was classfed by ndvdual descrptors wth accompanyng measure of certanty or confdence. These were then fused n each of 8 methods and then the resultng classfcatons measured aganst the nown classes the obect belonged to

29 Chapter 5: Epermental esults and Conclusons Two data sets were provded by Ordnance Survey to evaluate shape classfcaton on topographc data: urbec and lymouth. Because supervsed learnng was beng used, t was necessary to have an eample data set to derve the statstcs to provde the lelhood dstrbutons for each descrptor for each class of obect. It was decded to use the urbec data set for ths purpose. The urbec data for all the polygons representng s of the most common feature types Table were etracted. The four scalar descrptors for each polygon were calculated drectly from these boundares. Each boundary was sampled at 8 ponts and ths sampled boundary used to calculate the Fourer descrptors FDs and moment nvarants MI. The samplng results n 8 FDs but t s nown that most of the shape nformaton s contaned n the frst few. Therefore the frst 6 were used wth FD0 and FD bengs redundant due to normalsaton. Label used here OS Type OS Descrpton buldng 03 Buldng Type A defned land Defned Natural Land Cover 3 multple surface land Multple Surface Land 4 unmade land General Unmade Land 5 road Made oad 6 roadsde 6033 oadsde Unnown Land Table : Obect types used n classfcaton eperment. Therefore for each polygon we have 5 descrptor values four scalars, seven moment nvarants and fourteen Fourer Descrptors FD to FD6. The values for each descrptor obtaned by all polygons for each of the s chosen feature codes n the urbec data were statstcally analysed to obtan measures of the mean and standard devaton. A normal dstrbuton was assumed. These dstrbutons were then used to - 9 -

30 classfy each polygon n both data sets usng each ndvdual descrptor.e. 5 results per polygon. The ndvdual results from ndvdual descrptors for each polygon were then fused usng each method descrbed n chapter 4 producng an overall result treatng all 5 descrptors equally. Ths was also done for each of the three descrptor types. Fnally, the results for each descrptor type were then fused to produce an overall result from all descrptors. Detaled results are tabulated n append urbec and lymouth. Table shows the performance of the ndvdual classfers on the lymouth data set. It can be seen that as epected performance was varable dependng on descrptors and fuson method used. Best performer was Moment Invarants fused usng the Mn rule 8% correctly classfed. oorest were most of the descrptors usng the Sum rule adusted.e. normalsed % whch s symptomatc of the theoretcally wea bass for ths method [Kttler 998]. It s also note-worthy that fusng all methods usng the technques descrbed here produces poorer performance than Moment Invarants alone. 7.3 erformance of fused descrptors over all selected features Number of polygons processed: All 5 Descrptors ALL maorty ma mn medan sum sum ad product product ad number percent Scalar Descrptors SCALAS maorty ma mn medan sum sum ad product product ad number percent Fourer Descrptors FOUIES maorty ma mn medan sum sum ad product product ad number percent Moment Invarants MOMENTS maorty ma mn medan sum sum ad product product ad number percent Maorty of 3 methods MAJOITY maorty ma mn medan sum sum ad product product ad number percent Table : Summary of performance of fuson of descrptors on all features n lymouth data set showng number and percentage correctly classfed

31 The remander of ths chapter conssts of the followng sectons:. An evaluaton of each ndvdual descrpton methods.e. FDs, MIs and scalars;. A comparson between methods; 3. An evaluaton of Fuson methods; 4. Conclusons; 5. Suggestons for future research derved from the conclusons. 5. erformance of ndvdual descrptors In ths secton a sample of the results produced by the applcaton of the Fourer descrptor, moment nvarants and scalar descrptor technques are presented to evaluate and compare ther usefulness n shape dscrmnaton of general topographc features. Fgure 3 plots the average values, obtaned for fve categores of obects from a sample data set usng the Moment Invarant method n ths eample. Ths shows some separaton between classes n the n-descrptor space. However, n order to classfy shapes wth any degree of certanty, the varaton wthn classes must n general be less than that between classes IM IM IM Buldng arcel oad alway Stream Fgure 3. Average descrptor values of fve sample shapes Moment Invarants usng Isle of Man data

32 To evaluate each of the methods as shape recognton technques, several shapes from the map buldngs, parcels and roads were used as test shapes. For the Fourer descrptor and moment nvarants methods, the descrptor values used to descrbe the obects are computed from the equally spaced,y ponts along the boundary of each of the test shapes usng the formulae derved n chapter. The scalar descrptors are calculated from the boundary of the obects also, usng the scalar shape recognton aspects descrbed n chapter. The aspects used are: area; permeter length; elongaton; and number of ponts. Table 3 s an eample of the frst 6 low-order Fourer descrptors obtaned for a house shape, FD0 represents the frst descrptor value Table 3: Fourer descrptor values calculated for a house shape. From nspecton of the values produced for each polygon, most of the shape nformaton s descrbed by the frst few descrptors and so only the frst 6 terms were used for comparson, rememberng that due to the normalzaton procedures, FD0 and FD are redundant. Table 4 s an eample of a set of seven nvarant moments IM obtaned for a house, road and parcel shape startng a nde IM0. Buldngs oads arcels IM IM.47573e IM e IM e e-05 IM4.4389e e-05 IM e e e-07 IM6.9403e e e- Table 4: Moment nvarant values calculated for house, road and parcel shapes. In ths paper each of the shape descrpton technques, Fourer descrptors, moment nvarants and scalar descrptors, were computed for three types of feature, namely buldngs, parcels and roads n s dfferent sub-categores used n Ordnance Survey large-scale data-sets Table

33 Fgure 4, shows a plot of the mean values for each of these categores n threedmensonal space usng the moments nvarants method n ths eample. 0 0 IM buldn g * defnedland < surface land o unm ade-lan d road roadsde IM IM 0 Fgure 4: Average moment nvarants IM of s shape categores urbec data A sample of the results produced by the applcaton of the Fourer descrptors s presented to evaluate ther usefulness n the shape dscrmnaton. These results obtaned for each data set were plotted usng the Fourer descrptor s FD, FD3, FD4 to observe how well the formed separate groups. Fgure 5 a and b and Fgure 6 a and b below show the degree to whch these data set cluster n FD, FD3, FD4 space. Note, that due to normalsaton the frst two terms obtaned n the Fourer descrptors set, FD0 0 and FD are redundant FD 4 0. Defnedland and buldngs FD Buldng And oad FD FD FD FD Fgure 5 a: Clusterng of the polygon shapes, buldngs and defned natural land cover n three-dmensonal space of the features FD, FD3 and FD4, b:

34 Cluster of the polygon shapes, buldngs and made-road n three-dmensonal space FD, FD3 and FD Made-road oad sde FD Surface Land Unmade Land Buldng FD Fgure 6 a: Clusterng of the polygon shapes, made-road and roadsde n threedmensonal space of the features FD, FD3 and FD4, b: Clusterng of the polygon shapes, surface land, unmade-land and buldngs n three-dmensonal space FD, FD3 and FD FD FD.5 As thses plots show, often no two feature classes are completely dstnct from each other. Ths evdence therefore ndcates that Fourer descrptors are not very good for use n shape descrpton where the data sets are of a very general shape. To show ths mathematcally the repeatablty functon was computed for each of the s map categores. Table 5 shows these measurements n FD as t s the most sgnfcant descrptor value. The repeatablty of the measurements of each class s represented as three tmes the standard devaton and can be seen n the shaded dagonal column of the table. The repeatablty of each class s szeably larger than the dstance between the mean values for all the s classes whch shows that the classes are not dstnce enough to conclude any sgnfcant postve results. Buldngs Defnedland Surfaceland Unmade-land Madeoad oadsde No. polygons Buldngs Defnedland Surfaceland Unmade-land Madeoad oadsde 0.7 Table 5: Comparson of repeatablty wthn feature classes and dstance between classes for the Fourer descrptor technque n FD. A sample of the results produced by the applcaton of the moment nvarants technque was also evaluated. The Fgure 7 shows plots obtaned for the moment

35 nvarants technque for a sample of each feature type, each plot showng the degree to whch each set of obects cluster n ther three-dmensonal space defnedland unmadeland 0-0 IM IM 0-0 surfaceland 0-30 buldng IM IM IM 0-30 buldngs 0-0 IM Madeoad defnedland and unmadeland IM IM buldngs IM Fgure 7. Clusterng of the polygon shapes, buldngs and made-roads, n threedmensonal space of the features IM0,IM and IM. Fgure 7 shows the degree to whch the data sets, buldng and defned land cover cluster and also n a cluster plot of the data sets, defned land cover and unmade-land. In contrast, t can be seen how the features buldngs and roads separate when plotted. To measure the clusterng obtaned, the repeatablty functon and mean value measurements were computed for each set or the sample shapes. The results can be seen n table 6. Only the frst moment nvarants measure, IM0 s used here to mae t easer to read the table as t s the most sgnfcant moment result. Buldngs Defnedland Surfaceland Unmade-land Madeoad oadsde No. polygons Buldngs 5.005e e e e-004 Defnedland e e e-004 Surfaceland e e-004 Unmade-land Madeoad e-004 oadsde

36 Table 6: Comparson of repeatablty wthn feature classes and dstance between classes for the moment nvarants technque n IM0. Each output for the moment nvarants method n the shape recognton of general shapes on maps, show that there s a sgnfcant separaton occurrng between most of the classes. Although overlap does est also seen by the human eye good classfcaton occurs. On eamnng Table 6 more closely t can be seen that the repeatablty for the buldngs s smaller than the dstance between the mean values for all categores ecept for the surface land data set though these values are close. Ths s also true for the repeatablty measure for the surface land class where the dstance between the means values s larger ecept for buldngs. Comparng the fgures obtaned for the other data sets we see that for many the repeatablty measure s larger but stll close to the mean dstance for most cases. As presented above for the Fourer descrptor and moment nvarants methods, a sample of the results produced by applyng the scalar descrptor technque to the data set s evaluated also. Fgures 8 to show the resultng cluster graphs and the degree to whch the features separate n the three-dmensonal space of area, permeter and number of ponts. 0 3 NO OF OINTS 0 0 Defned land EIMETE AEA Buldng

37 Fgure 8 Clusterng of the polygons, buldngs and defned land cover, n the three-dmensonal Defned land space area, permeter and number of ponts and Unmade land NO OF OINTS EIMETE AEA Fgure 9 Clusterng of the polygons, defned land cover and unmade-land, n the three-dmensonal space area, permeter and number of ponts 0 NO OF OINTS Buldng EIMETE 0 0 AEA 0 4 Made-road Fgure 0 Clusterng of the polygons, buldngs and made-road, n the threedmensonal space area, permeter and number of ponts

38 0 4 NO OF OINTS Surface land Unmade land EIMETE 0 0 Buldngs AEA Fgure Clusterng of the polygons, buldngs, surface land and unmade land, n the three-dmensonal space area, permeter and number of ponts Fgure 8 shows the cluster plot of the data sets defned natural land cover and buldngs. In Fgure 9 a cluster plot of the features defned natural land cover and unmade land. Fgure 0 and Fgure show the degree to whch the data sets buldngs and made-roads cluster and the degree to whch the data sets buldngs, surface land and unmade land cluster. To analyss the results further the results are agan represented mathematcally, n ths case by computng the repeatablty functon and mean value measurements for the area, whch s consdered the most sgnfcant feature descrptor for the scalars. The table for the repeatablty and mean values s as follows: Buldngs Defnedland Surfaceland Unmade-land Madeoad oadsde No. polygons Buldngs e e e Defnedland.0665e e e e e+04 Surfaceland.7478e e Unmade-land.575e e e+04 Madeoad e oadsde.758e+03 Table 7: Comparson of repeatablty wthn feature classes and dstance between classes for the scalar descrptor technque n area

39 The outputs obtaned for the scalar descrptor method of general shapes on maps show that there s a sgnfcant dstncton between the maorty of the classes. Some overlap ets but overall classfcaton s good. On eamnaton, table 7 shows, especally for the buldng features, that the repeatablty s smaller than than the dstance between the mean values whch ndcates good classfcaton performance for the scalar method. As shape descrptor technques the evdence publshed to date s that all three technques evaluated, Fourer descrptors, moment nvarants and scalar descrptors, are very good features to use when dealng wth very specfc shapes such as a partcular arcraft or alphanumerc character. On nvestgaton of ther usefulness for the shape descrpton of general shapes on maps, for eample houses, roads, parcels etc. the Fourer descrptors do not appear to be very successful. However, the moment nvarants technque proved to be sgnfcantly more successful n ts tas and specfc scalar measures are also very dscrmnatory. Ths s llustrated by the pe charts n Fgure derved from the results summary n Append 5. Each chart shows the classfcaton results on obects belongng to each of the s feature types consdered. For eample, scalar descrptors correctly classfed almost 00% of buldngs. Scalar descrptor method < % % 3% < % 5% % 36% 59% 00% Buldng < % Defned Natural land 8% 84% Surface Land % Buldng DefnedLan dsurfacelan dunmadeland Madeoad oadsde 9% < % 7% 00% 98% General Unmade Land Made oad oad Sde

40 Fourer Descrptor method < % % 7% 3% 4% 99% 9% 93% Buldng Defned Natural Land Surface Land < % % % < % Buldng DefnedLand SurfaceLand UnmadeLand Madeoad oadsde 00% 98% 98% Unmade Land Made oad oad Sde Moment Invarants method < % < % % 3% < % 6% 6% 34% 53% 00% Buldng Defned Natural Land Surface Land < % 3% 6% 3% 8% Buldng DefnedLand SurfaceLand UnmadeLand Madeoad oadsde < % 3% 48% 63% < % 6% 00% 8% Unmade Land Made oad oad Sde Fgure ecognton performance of descrptor methods by feature type

41 5. Fuson methods S methods of data fuson were mplemented: maoty vote, ma rule, mn rule, medan rule, sum rule and product rule. Two of these sum and product had two versons whether they ncluded or ecluded the adustment for normalsaton. They were appled to fuse the classfcaton results gven by the descrptors obtaned from each polygon n three ways: Each descrptor 5 n all treated equally to obtan a global result Table 8, secton 7.3. Each descrptor fused nto ts group 3 groups.e. scalar, FD and MI to obtan a result for each group Table 8, secton Each group result fused to obtan an overall result Table 8, secton Table 8 shows that, wth notable eceptons, the classfcaton accuracy obtaned was farly consstent no matter whch was used. Best performer was the mn rule followed by the product rule. Worst performer by far was the normalsed sum rule. Ths confrms the arguments n [Kttler 998] whch questons the theoretcal bass of the sum rule. 7.3 erformance of fused descrptors over all selected features Number of polygons processed: All 5 Descrptors ALL maorty ma mn medan sum sum ad product product ad number percent Scalar Descrptors SCALAS maorty ma mn medan sum sum ad product product ad number percent Fourer Descrptors FOUIES maorty ma mn medan sum sum ad product product ad number percent Moment Invarants MOMENTS maorty ma mn medan sum sum ad product product ad number percent Maorty of 3 methods MAJOITY maorty ma mn medan sum sum ad product product ad number percent Table 8: Summary of performance of fuson of descrptors on all features n lymouth data set showng number and percentage correctly classfed