Analysis of Maximum Likelihood Classification. on Multispectral Data

Transcription

1 Appled Mathematcal Scences, Vol. 6, 0, no. 9, Analyss of Maxmum Lkelhood Classfcaton on Multspectral Data Asmala Ahmad Department of Industral Computng Faculty of Informaton and Communcaton Technology Unverst Teknkal Malaysa Melaka Hang Tuah Jaya, 7600 Duran Tunggal, Melaka, Malaysa Shaun Quegan School of Mathematcs and Statstcs Unversty of Sheffeld Sheffeld, Unted Kngdom Abstract The am of ths paper s to carry out analyss of Maxmum Lkelhood (ML) classfcaton on multspectral data by means of qualtatve and quanttatve approaches. ML s a supervsed classfcaton method whch s based on the Bayes theorem. It makes use of a dscrmnant functon to assgn pxel to the class wth the hghest lkelhood. Class mean vector and covarance matrx are the key nputs to the functon and can be estmated from the tranng pxels of a partcular class. In ths study, we used ML to classfy a dverse tropcal land covers recorded from Landsat 5 TM satellte. The classfcaton s carefully examned usng vsual analyss, classfcaton accuracy, band correlaton and decson boundary. The results show that the separaton between mean of the classes n the decson space s to be the man factor that leads to the hgh classfcaton accuracy of ML. Keywords: ML, Classfcaton, Decson Boundary

2 646 Asmala Ahmad and Shaun Quegan Introducton Maxmum Lkelhood (ML) s a supervsed classfcaton method derved from the Bayes theorem, whch states that the a posteror dstrbuton P( ω),.e., the probablty that a pxel wth feature vector ω belongs to class, s gven by: ( ω) ( ω ) P( ) P( ω) P P = () where P(ω ) s the lkelhood functon, P() s the a pror nformaton,.e., the probablty that class occurs n the study area and P ( ω) s the probablty that ω s observed, whch can be wrtten as: M ( ) = P( ω ) P( ) P ω () = where M s the number of classes. P ( ω) s often treated as a normalsaton M constant to ensure P( ω ) sums to. Pxel x s assgned to class by the rule: = x f P( ω) > P( ω) for all (3) ML often assumes that the dstrbuton of the data wthn a gven class obeys a multvarate Gaussan dstrbuton. It s then convenent to defne the log lkelhood (or dscrmnant functon): t N g ( ω ) = ln P( ω ) = ( ω μ ) C ( ω μ ) ln( π) ln( C ) (4) Snce log s a monotonc functon, Equaton (3) s equvalent to: x f g (ω) > g (ω) for all. (5) Each pxel s assgned to the class wth the hghest lkelhood or labelled as unclassfed f the probablty values are all below a threshold set by the user [9]. The general procedures n ML are as follows:. The number of land cover types wthn the study area s determned.. The tranng pxels for each of the desred classes are chosen usng land cover nformaton for the study area. For ths purpose, the Jeffres-Matusta (JM) dstance can be used to measure class separablty of the chosen tranng pxels.

3 Analyss of maxmum lkelhood classfcaton 647 For normally dstrbuted classes, the JM separablty measure for two classes, J, s defned as follows [4]: J α ( e ) = (6) where α s the Bhattacharyya dstance and s gven by [4]: C + C t ( C + C) α = ( μ ) ( ) ln μ μ μ + (7) 8 C C J ranges from 0 to.0, where J >.9 ndcates good separablty of classes, moderate separablty for.0 J.9 and poor separablty for J <.0 []. 3. The tranng pxels are then used to estmate the mean vector and covarance matrx of each class. 4. Fnally, every pxel n the mage s classfed nto one of the desred land cover types or labelled as unknown. In ML classfcaton, each class s enclosed n a regon n multspectral space where ts dscrmnant functon s larger than that of all other classes. These class regons are separated by decson boundares, where, the decson boundary between class and occurs when: g (ω) = g (ω) (8) For multvarate normal dstrbutons, ths becomes: N t ( ω μ ) C ( ω μ ) ln( π) ln( C ) t N ( ω μ ) C ( ω μ ) ln( π) ln( C ) = 0 (9) whch can be wrtten as: t t ( μ ) C ( ω μ ) ln ( C ) + ( ω μ ) C ( ω μ ) + ln ( C ) = 0 ω (0) Ths s a quadratc functon n N dmensons. Hence, f we consder only two classes, the decson boundares are conc sectons (.e. parabolas, crcles, ellpses or hyperbolas).

4 648 Asmala Ahmad and Shaun Quegan Methodology The study area was located n Selangor, Malaysa, coverng approxmately 840 km wthn longtude 0 0 E to 0 30 E and lattude 99 N to 3 5 N (Fgure ). The satellte data come from bands ( µm), ( µm), 3 ( µm), 4 ( µm), 5 ( µm) and 7 ( µm) of Landsat-5 TM dated th February 999. The satellte records surface reflectance wth 30 m spatal resoluton from a heght of 705 km. Pror to any data processng, maskng of cloud and ts shadow were carred out based on threshold approach [8], []. Vsual nterpretaton of the Landsat data (Fgure (b)) was carred out to dentfy man land covers wthn the study area. The task was aded by a reference map (Fgure (a)), produced n October 99 by the Malaysan Surveyng Department and Malaysan Remote Sensng Agency usng ground surveyng and SPOT satellte data. man classes were dentfed,.e. water, coastal swamp forest, dryland forest, ol palm, rubber, ndustry, cleared land, urban, sedment plumes, coconut and bare land. Fg.. The study area from (a) the land cover map and (b) the Landsat-5 TM wth bands 5 4 and 3 assgned to the red, green and blue channels. Cloud and ts shadow are masked n black. Tranng areas were establshed by choosng one or more polygons for each class. Pxels fall wthn the tranng area were taken to be the tranng pxels for a partcular class. In order to select a good tranng area for a class, the mportant propertes taken nto consderaton are ts unformty and how well they represent the same class throughout the whole mage [5]. Class separablty of the chosen tranng pxels were determned by means of the JM dstance. Ffty pars have JM dstance between.9 and.0 ndcatng good separablty, four from.0 to.9 ndcatng moderate separablty and one less than.0 ndcatng poor separablty. The worst separablty, possessed by the urban ndustry par (0.947), was expected snce both have qute smlar spectral characterstcs. For each class, these tranng pxels provde values from whch to estmate the mean and covarances of the spectral bands used. These nformaton are to be used by the ML classfer to assgn pxels to a partcular class.

5 Analyss of maxmum lkelhood classfcaton Analyss of ML classfcaton 3. Vsual Analyss The outcome of ML classfcaton after assgnng the classes wth sutable colours, s shown n Fgure : coastal swamp forest (green), dryland forest (blue), ol palm (yellow), rubber (cyan), cleared land (purple), coconut (maroon), bare land (orange), urban (red), ndustry (grey), sedment plumes (sea green) and water (whte). Clouds and ther shadows are masked black. The areas n terms of percentage and square klometres were also computed; the classes wth the largest area are ol palm, cleared land and ndustry. Although beng smlar, coastal swamp forest and dryland forest can be clearly seen n the south-west and northeast of the classfed mage, as ndcated by the reference map. Coastal swamp forest covers most of the Island and coastal regons n the south-west of the scene. Most of the dryland forest can be recognsed as a large straght-edged regon n the north-east. Ol palm and urban domnate the northern and southern parts respectvely. Rubber appears as scattered patches that mostly are surrounded by ol palms. Industry can be recognsed as patches near the urban areas, especally n the south-west and north-east. Coconut can be seen n the coastal area n the north-west of the mage. A qute large area of bare land can be seen n the east, whle cleared land can be seen mostly n the north, south and south-east of the mage. Class Colour Area (km ) Area (%) Urban Ol palm Coastal swamp forest Industry Dryland forest Rubber 0..4 Coconut Cleared land Bare land Sedment plumes Water Fg.. ML classfcaton usng band,, 3, 4, 5 and 7 of Landsat TM and the class areas n terms of square klometre and percentage. 3. Accuracy Analyss Accuracy assessment of the ML classfcaton was determned by means of a confuson matrx (sometmes called error matrx), whch compares, on a class-byclass bass, the relatonshp between reference data (ground truth) and the correspondng results of a classfcaton [9]. Such matrces are square, wth the number of rows and columns equal to the number of classes,.e.. For all

6 6430 Asmala Ahmad and Shaun Quegan classes, the numbers of reference pxels are: rubber (03), water (99), coastal swamp forest (4840), dryland forest (66), ol palm (049), ndustry (350), cleared land (50), urban (309), coconut (59), bare land (33) and sedment plumes (88). The dagonal elements n Table (a) represent the pxels of correctly assgned pxels and are also known as the producer accuracy. Producer accuracy s a measure of the accuracy of a partcular classfcaton scheme and shows the percentage of a partcular ground class that s correctly classfed. It s calculated by dvdng each of the dagonal elements n Table (a) by the total of each column respectvely: aa Producer accuracy 00% c a c = () where, th th caa = element at poston a row and a column c = column sums a The mnmum acceptable accuracy for a class s 90% [7]. Table (b) shows the producer for all the classes. It s obvous that all classes possess producer accuracy hgher than 90%: bare land gves the hghest (00%) and ol palm the lowest (9.4%). The relatvely low accuracy of ol palm s manly because 6% and % of ts pxels were classfed as coconut and cleared land. The msclassfcaton of ol palm pxels to the coconut class s due to the fact that ol palm and coconut have a smlar physcal structure, so tend to have smlar spectral behavour and therefore can easly be msclassfed as each other. User Accuracy s a measure of how well the classfcaton s performed. It ndcates the percentage of probablty that the class whch a pxel s classfed to on an mage actually represents that class on the ground [7]. It s calculated by dvdng each of the dagonal elements n a confuson matrx by the total of the row n whch t occurs: User accuracy 00% c c = () where, c = row sum. Coastal swamp forest, dryland forest, ol palm, sedment plumes, water, bare land and urban show a user accuracy of more than 90%. Rubber, cleared land and ndustry possess accuracy between 70% and 90%, whle the worst accuracy s possessed by coconut (6%). The low accuracy of coconut s because the ol palm pxels tend to be classfed as coconut because they havng smlar spectral propertes to ol palm. A measure of overall behavour of the ML classfcaton can be determned by the overall accuracy, whch s the total percentage of pxels correctly classfed:

7 Analyss of maxmum lkelhood classfcaton 643 aa a= Overall accuracy 00% U c = Q (3) where, Q and U s the total number of pxels and classes respectvely. The mnmum acceptable overall accuracy s 85% [3]. The Kappa coeffcent, κ s a second measure of classfcaton accuracy whch ncorporates the off-dagonal elements as well as the dagonal terms to gve a more robust assessment of accuracy than overall accuracy. It s computed as [6]: κ= c c c U U aa a a a= Q a= Q U ca c a a= Q (4) where ca row sums. The ML classfcaton yelded an overall accuracy of 97.4% and kappa coeffcent 0.97, ndcatng very hgh agreement wth the ground truth. = Table : Confuson Matrx for ML Classfcaton. Overall Accuracy = 97.4% Kappa Coeffcent = 0.97 Ground Truth (Pxels) Class Coastal swamp forest Dryland forest Ol palm Rubber Cleared land Sedment plumes Water Coconut Bare land Urban Industry Total classfed pxels Coastal swamp forest Dryland forest Ol palm ML Classfcaton (pxels) Rubber Cleared land Sedment plumes Water Coconut Bare land Urban Industry Total ground truth pxels (a)

8 643 Asmala Ahmad and Shaun Quegan Class Producer Accuracy User Accuracy (Pxels) (%) (Pxels) (%) Coastal swamp forest 480/ / Dryland forest 66/ / Ol palm 9690/ / Rubber 0/ / Cleared land 73/ / Sedment plumes 804/ / Water 99/ / Coconut 47/ / Bare land 33/ / Urban 54/ / Industry 349/ / (b) 3.3 Correlaton Matrx Analyss Classfcaton uses the covarance of the bands; nonetheless, covarance s not ntutve; more ntutve s correlaton, ρ k,l,.e. covarance normalsed by the product of the standard devatons of bands, k and l : (( k k)( l l) ) C E I μ I μ k,l ρ k,l = = k l k l σσ σσ (5) where C k,l s the covarance between bands k and l, σ k and σ l are the standard devatons of the measurements n bands k and l respectvely, E s the expected value operator, and I k and I l and μ k and μ l are the ntenstes and means of bands k and l respectvely. When usng more than two bands, t s convenent to use a correlaton matrx, where the element n row m and column n that correspond to band k and l s gven by ρ k,l. If m= n, then ρk,l =, so ths wll be the value of the dagonal elements of the matrx. Otherwse, f m n, ρ k,l les between - and. In order to analyse the correlaton matrces, plots of correlaton versus band par for all classes are plotted. Fgure 3 shows correlaton between band pars from selected classes,.e. (a) water, (b) coastal swamp forest, (c) dryland forest, (d) ol palm, (e) urban, (f) cleared land, (g) ndustry and (h) sedment plumes. Each coloured curve represents correlaton between a specfc band (gven by a specfc colour) wth all bands (on the x-axs). Landsat bands, and 3 are located wthn a very close wavelength range of the vsble spectrum, wth ther centre wavelengths dfferng only by about 0. μm. Measurements made from these bands normally exhbt smlar responses and therefore are hghly correlated. Poor correlatons may result from mxed pxel problem

9 Analyss of maxmum lkelhood classfcaton 6433 (exstence of more than one class n a pxel). Correlatons between lowernumbered bands (.e. bands, and 3) and hgher-numbered bands (.e. bands 4, 5, and 6) are much lower because nvolvng non-adacency wavelengths. From Fgure 3, for cleared land and sedment plumes, correlaton n most band pars s qute hgh n ML, especally for bands, and 3, whch corresponds to the hgher accuracy n these classes n ML. For certan classes, such as water (wth very low reflectances), the superorty of ML s even clearer, as shown not only by the correlatons from bands, and 3, but also 4, 5 and 7 n ML that have hgh correlatons. A hgh correlaton s shown by ndustry (wth very hgh reflectances) due to the strong relatonshps of varaton between the brghtness of pxels and mean brghtness n all bands (,, 3, 4, 5 and 7). Fg. 3. Correlatons between band pars for (a) water, (b) coastal swamp forest, (c) dryland forest, (d) ol palm, (e) urban, (f) cleared land, (g) ndustry and (h) sedment plumes. 3.4 Mean, Standard Devaton and Decson Boundary Analyss Despte of beng very smlar, both forests can stll be separated qute effectvely from each other usng ML. Here, we nvestgate further the forests n terms of

10 6434 Asmala Ahmad and Shaun Quegan mean, standard devaton and decson boundary. Fgure 4(a) shows the means and (b) standard devaton of coastal swamp forest and dryland forest classes n ML. The means are almost the same partcularly n bands, and 3. The standard devaton of coastal swamp forest s bgger than dryland forest n most of the bands, except band 5. Fg. 4. (a) Means of coastal swamp forest and dryland forest classes n ML classfcaton. DLF and CSF are dryland forest and coastal swamp forest respectvely. (b) Standard devatons of the coastal swamp forest and dryland forest classes n ML classfcaton We subsequently generated the decson boundares usng Equaton (0) between coastal swamp forest and dryland forest. Fgure 5 shows 5 sets of decson boundares; M and M are the means for dryland forest and coastal swamp forest respectvely, Band k Vs. Band l denotes that the vertcal axs s band k whle horzontal axs s band l and CSF and DLF ndcate coastal swamp forest and dryland forest respectvely. The decson boundares formed by the ML have the form of conc sectons,.e. pars :, 3:, 7:, 3: and 7: form an ellptc curve, pars 5:, 5:, 5:3, 7:3 and 7:5 form a parabolc curve and pars 4:, 4:, 4:3, 5:4 and 7:4 form a hyperbolc curve. Most of the boundares are owned by dryland forest swamp forest due to the smaller standard devaton of dryland forest than coastal swamp forest n most of the bands. In most bands (except band 4), the dfference between the means s bg enough that M and M are located n the dfferent sde of the boundary. Hence, ML can effectvely separate between the forests due to ts ablty n postonng the means n the dfferent sde of the decson boundary.

11 Analyss of maxmum lkelhood classfcaton 6435 Fg. 5. Decson boundares between coastal swamp forest and dryland forest for ML classfcaton. 4 Conclusons In ths study, detal analyses of ML classfcaton for tropcal land covers n Malaysa have been carred out, n whch lead to a number of conclusons. ML classfes the classes that exst n the study area wth a good agreement wth the reference map. ML classfed the study area nto classes, wth accuracy 97% (κ = 0.97). ML classfes pxels based on known propertes of each cover type, but the generated classes may not be statstcally separable. The band correlaton of classes wth hgh reflectance, e.g. ndustry, s hgh for all band pars n ML because of the strong relatonshps of varaton between the brghtness of pxels

12 6436 Asmala Ahmad and Shaun Quegan and mean brghtness n all bands. The separaton between mean of the classes n the decson space s beleved to be one of the man factors that leads to the hgh classfcaton accuracy of ML. References [] A. Ahmad, and S. Quegan, Cloud maskng for remotely sensed data usng spectral and prncpal components analyss, Engneerng, Technology & Appled Scence Research (ETASR), (0), 5. [] ENVI, User s gude, Research Systems Inc., USA, 006. [3] J. Scepan, Thematc valdaton of hgh-resoluton global land-cover data sets, Photogrammetrc Engneerng and Remote Sensng, 65 (999), [4] J.A. Rchards, Remote sensng dgtal mage analyss: An ntroducton. Sprnger-Verlag, Berln, Germany, 999. [5] J.B. Campbell, Introducton to remote sensng, Taylor & Francs, London, 00. [6] J.R. Jensen, Introductory Dgtal Image Processng: A Remote Sensng Perspectve, Pearson Prentce Hall, New Jersey, USA, 996. [7] M. Story and R. Congalton, Accuracy assessment: a user's perspectve, Photogrammetrc Engneerng and Remote Sensng, 5 (986), [8] S.A. Ackerman, K.I. Strabala, W.P. Menzel, R.A. Frey, C.C. Moeller and L.E. Gumley, Dscrmnatng clear-sky from clouds wth MODIS, Journal of Geophyscal Research, 03 (998), [9] T.M. Lllesand, R.W. Kefer and J.W. Chpman, Remote Sensng and Image Interpretaton, John Wley & Sons, Hoboken, NJ, USA, 004. Receved: August, 0