A New Clustering Method for Landsat Images Using Local Maximums of a Multidimensional Histogram

Transcription

1 Purdue University Purdue e-pubs LARS Sympsia Labratry fr Applicatins f Remte Sensing A New Clustering Methd fr Landsat Images Using Lcal Maximums f a Multidimensinal Histgram K. Matsumt M. Naka H. Yamamt Fllw this and additinal wrks at: Matsumt, K.; Naka, M.; and Yamamt, H., "A New Clustering Methd fr Landsat Images Using Lcal Maximums f a Multidimensinal Histgram" (1981). LARS Sympsia. Paper This dcument has been made available thrugh Purdue e-pubs, a service f the Purdue University Libraries. Please cntact epubs@purdue.edu fr additinal infrmatin.

2 Reprinted frm Seventh Internatinal Sympsium Machine Prcessing f Remtely Sensed Data with special emphasis n Range, Frest and Wetlands Assessment June 23-26, 1981 Prceedings Purdue University The Labratry fr Applicatins f Remte Sensing West Lafayette, Indiana USA Cpyright 1981 by Purdue Research Fundatin, West Lafayette, Indiana All Rights Reserved. This paper is prvided fr persnal educatinal use nly, under permissin frm Purdue Research Fundatin. Purdue Research Fundatin

3 A NEW CLUSTERING METHOD FOR LANDSAT IMAGES USING LOCAL MAXIMUMS OF A MULTI-]IMENSIONAL HISTOGRAM K. MATSUMOTO, M. NAKA, H. YAMAMOTO Natinal Aerspace Labratry Tky, Japan I. ABSTRACf This paper describes a new clustering methd fr a huge satellite image. This methd is cmpsed f tw majr schemes, a multi-layered hashing scheme fr multi-dimensinal histgram (MDH) and a histgram clustering scheme using MDH. The MDH cnstructin scheme wrks in 3 stages. In the 1st stage it cnstructs a few histgram layers fr each part f an image, and in the 2nd and 3rd stages histgram layers are cmbined int ne histgram. The clustering scheme searches fr lcal maximums f MDH, and decides clusters arund lcal maximums as sets f hyperrectangles. The majr parameters f the clustering scheme are briefly discussed, and sme results are als shwn. II. INTRODUCflON Clustering technique is imprtant as an unsupervised classifier in autmated satellite image analysis. Because f the huge amunt f image data, hwever, it is very difficult fr current clustering methds t analyze mre than a small sub-frame f a satellite image. There are tw majr appraches t clustering. One is an ISODATA-like methd that requests sme reasnable initial partitins and then cnverges t ptimal partitins using an iterative prcedure. Anther apprach is based n bserving a distributin pattern by a histgram. 3 This methd segments the data in accrdance with a frm f the distributin pattern using sme pattern recgnitin techniques. In the frmer apprach, the cnvergence t a subptimal segmentatin is pssible under reasnable initial partitins. But in this methd there are many unslved prblems; fr example, finding a reasnable methd t estimate the initial partitins, and aviding the pssibility f multiple slutins. l In the latter apprach, thugh there are sme difficult prblems in getting enugh statistics t bserve the distributin pattern and in recgnizing the pattern, it has the advantage ver the frmer apprach f nt needing any initial partitins. In this paper a new methd using the latter apprach is described. This methd derives cluster frms frm an MDH. The basic idea is as fllws: Since a histgram is a sample f the mixed cumulative distributin functin, if we assume the prbability density functin f each cluster t be unimdal, the frequency value crrespnding t the center f a cluster shuld be a lcal maximum. Als, the bundary between tw adjacent clusters shuld be a valley r a ridge f the histgram. Therefre, basically, it wuld seem t be very natural t recgnize clusters by searching fr lcal maximums and then setting the bundaries in the valleys f the histgram. Hwever, there are the fllwing prblems in putting this basic idea int practice: (1) Efficient cnstructin scheme f an MDH. (2) Lcal maximum searching algrithm using an MDH. (3) Expressin f a cluster bundary in an MDH. (4) Prcessing f the sampling errr f an MDH. In sectin III f this paper prblem (1) will be discussed, and prblems (2)-(4) will be discussed in sectin IV. This new methd has been implemented n the large scale vectr prcessr F ACOM 230{75 APU (called the APU). Sme f the results are shwn in this paper. Ill. CONSTRUCflON OF MULTI-DIMENSIONAL HISTOGRAM There are tw pssible methds in cnstructing an MDH, (a) the usual methd using a multi-dimensinal array, and (b) a methd using a histgram table. (a) is the simplest methd, but sme resampling prcess and sme degradatin 1981 Machine Prcessing f Remtely Sensed Data Sympsium 321

4 f data accuracy are inevitable, because the data length f an MSS pixel is 27 bits and it requires a huge memry capacity f 1.3 x 10 8 wrds even fr the current LANDSAT MSS image. In methd (b), the table length and the table search methd may becme prblems. But Shlien shwed that it is pssible t gather mre than 75%-90% pixels frm a full frame LANDSAT MSS image using the duble-hashing scheme and abut 10,000 histgram cells. 2 In the fllwing discussin the MDH is cnstructed by the latter histgram table methd using the duble-hashing scheme. The reasn is that it easily deals with multi-band data, des nt require a huge memry, and maintains data accuracy. Even in this methd, hwever, the fllwing prblems remain, sme f which were pinted ut by Shlien: (1) Limitatin f table length; especially fr a full frame image. (2) Difficulty in cmputing the hashing scheme fr a large number f bands. (3) Efficiency f the hashing scheme. A. THE VECTOR FORM HASHING SCHEME FOR A LARGE NUMBER OF BANDS On the vectr prcessr APU, each scan line pixel is prcessed simultaneusly. Frm eq. (1)--(12) hash prbes' vectr f each scan line pixel are calculated and shifted by eq. (12) simultaneusly. Using the table address determined by the hash prbes' vectr, each pixel f a scan is tried sequentially, and the cnflicted pixels are gathered and shifted by eq. (12) fr the next trial. v!< = ~,I Q=l 4. (Q-l) sj Q+ 4 (k_l)' A b ~ {bl, bz,.. 'b n } (j = I,...,n) (k=i,...,m) {3~ = 1, (3~ = md (a, Ph) (h=1. 2) {3: = md ({3k-l (3~, Ph) 'Yk h,= md (V~, Ph) '1 I a~ = md {~ 'Y~: {3~, PI} + 1 I k=l'l b l = md {~ 'Y~: {3~, Pz} + 1 k=l 'I a i + 1 = md { a i + b, PI } + 1 (1) (2) (3) (4) (5) (6),(7) (8) (9) (10) (11) (12) where a = 2 28 A = 27 Ph = the largest prime number pair which des nt exceed table length. S~ the spectral value f j th pixel. n : the pixels' width f a scan line. Fr a large number f bands, during the calculatin f a! and b. the m wrds divisin algrithm is necessary in the u1sual m~thd, and it requires annying calculatins. Hwever, if eq. (6)-(11) are used t calculate a! and b., even fr 1 J a large number f bands, nly a slight additinal cmputing time is needed. Since,B~ can be determined previusly, nly three (+,X,7) additinal single precisin peratins are needed fr each additinal wrd expressing a hash vectr V. By eq. (6)-(11), the hash prbes can be calculated efficiently even fr a large number f bands and even with a small wrd-length cmputer. B. MULTI-LAYERED HASHING SCHEME The degradatin f the efficiency f a hashing scheme becmes a serius prblem when cnstructing the MDH f a huge satellite image using a limited histgram table length. The degradatin f the efficiency and the difficulty f a new histgram cell's registratin becme mre serius as the number f prcessed scan lines increases, and as the number f blank entries in a histgram table decreases (Fig. 1). The usual way t maintain hashing scheme efficiency is t remve cells with less frequency frm the table and t register all ther cells again by the same hashing scheme, which hereafter will be called "re-hashing". But this simplest way des nt wrk well, because during the re-hashing prcess sme cells can nt be registered again, and because nce the re-hashing is activated, it is activated fr almst all scan lines. Especially, re-hashing activatin in almst every scan indicates the pssibility f missing a relatively large cluster which appears in the latter part f an image. T deal with this prblem, the fllwing multi-layered hashing scheme was develped. stf'p 1: Cnstruct the MDH alng scan lines by the usual hashing scheme using finite prbes. When a certain cnditin is satisfied, cmplete a current histgram layer, and stre its table in file A. Then, clear the table t begin the next histgram layer cnstructin. While repeating this prcess, als stre the rejected pixels in file B. step 2: After prcessing all pixels f an image, get the histgram tables f each layer t add ne by ne sequentially by the duble-hashing scheme using the same Machine PrceSSing f Remtely Sensed Data Sympsium

5 ~. length table. In the case f cllisins, hwever, fr the current cell take away the table entry f a cell which has the least frequency value alng the cllisin chain. Als during this prcess, stre the rejected r replaced cells in fileb. step 3: After adding the histgrams f all layers, try again t register all pixels and cells in file B int the current histgram table by the usual duble-hashing scheme. The basic idea f this methd is the cmbinatin f high speed prcessing f the huge, raw-image data (step I), and the elabrate and time cnsuming prcess f selecting the larger frequency cells fr the cmpressed data (step 2). This cmbinatin f step I and step 2 nt nly saves cmputing time, but als saves relatively large clusters r cells that appear in the latter part f an image. As the pssible cnditins in step I, (a) the number f used entries in a table, and (b) the number f rejected pixels, are tested. A suitable cnditin is t have the ability t maintain high efficiency while generating a small number f histgram layers. Hwever, these tw cnditins are nt cmpatible. A few results in Fig. I-c,d shw that (b) has a linear relatinship t the efficiency, but (a) des nt have, and als that (a) incurs the pssibility f cmpleting a layer t early, befre the degradatin f efficiency ccurs. Therefre (b) is mre suitable. Theretically, withut step 3 the resulting histgram table is nt a true sample f the MDH because, after cmpletin f step 2, the pssibility remains that sme pixels r cells in file B have crrespnding cells in the resulting table. Hwever, the amunt f such data is very small, as shwn in Table I, and cst perfrmance f step 3 is rather lw. Therefre, it is pssible t take away step 3, when higher speed prcessing is required. Table I is an example f cnstructing an MDH fr a full frame image f a LANDSAT scene ID.II The effect f this methd is remarkable. In nly 57.3% f the cmputing time, it gathers 25.2% mre data than the basic methd using the same table length. IV. MULTI-DIMENSIONAL HISTOGRAM CLUSTERING Fr an MDH and fr clusters cntained in a raw image data, the fllwing are assumed: (AI) An MDH is gd enugh t bserve the distributin pattern f a raw image. (A2) Each cluster has a unimdal distributin. (A3) The frequency difference, between a peak f a cluster and valleys crrespnding t the cluster's bundaries, is larger than the sampling errr. The basic idea described here is t search fr a lcal maximum f an MDH as a peak f a cluster, t gradually extend the cluster's area frm the peak, and t determine its bundaries at the pints where it cnnects t the area f anther cluster. In this idea, the recgnitin f a true peak and extentin f an area are the mst imprtant prcesses. The majr reasn fr nt recgnizing a true peak is the appearance f false peaks due t sampling errr. The extentin prcess is deeply related t the expressin f a cluster's area. The algrithm prpsed here simultaneusly carries ut the three functins: searching fr lcal maximums, determining the bundaries f clusters, and rejecting false peaks due t sampling errr. Basic terminlgies are defined as fllws: cluster: A candidate whse smallest depth f valleys surrunding it is mre than N. candidate: A set f adjining but nnintersecting areas which is develped arund an islated cell. islated cell: A histgram cell which des nt cnnect t any area f clusters r candidates. The algrithm is as fllws (Fig. 2): step 1: Srt the MDH in descending rder f frequency. step 2: Set prcessing level LI at the mst frequent value f the MDH, and make a candidate area f the "recrd" LI arund the mst frequent cell. step 3: Update the current prcessing level, ~+l = Lj-L'>Lj. step 4: Establish sme candidates as clusters if their "recrds" are lder than L j N. This cnditin means that the smallest depth f valleys exceeds threshld N. step 5: Carry ut step 6 and step 7 sequentially fr the cells whse frequency value f is L j > f 2 ~+ 1. step 6: Lk ver fr the spatial cnnectedness between the current cell and the areas f clusters and candidates. step 7: (case: islated) Make a new candidate area f a "recrd" Li+ 1 arund the cell. (case: cnnected t nly ne area) Jin the cell t the cnnected area and redefine that area. (case: cnnected t mre than ne area) Jin the cell t a suitable area and merge ther areas t that area. step 8: If there are any unprcessed cells, g t step 3. The definitins f an area, cnnectedness, redefinitin f an area, and merging are as fllws. These definitins are 1981 Machine Prcessing f Remtely Sensed Data Sympsium 323

6 cnvinient fr cmputatin and require less memry space. area: A hyperrectangle. Its attributes are a "recrd" and a status f either a cluster r a candidate. Each area is registered in an area table. cnnectedness,' 8-neighburs r 4-neighburs. redefinitin f an area: (a) Extend an area if and nly if the area des nt intersect with any ther areas after extentin f the area, and the rati f the number f cells included in the area t the area's vlume is greater than the threshld R. (b) Otherwise, make a new adjining area whse attributes are the same as thse f that area. merging: (a) Between candidates r between a candidate and a cluster, make them equal in their attributes. (b) Between clusters, inhibit merging. The threshld N f valley depths and the threshld R f the rati f cells t vlume are the majr parameters in this methd. Since R cntrls the fidelity in the extentin f a cluster's area as a set f hyperrectangles, the cluster's frm becmes mre cnscientius as R is clser t 1. In such a case, hwever, either a very large memry space is t be prvided, r this algrithm is t stp at an earlier stage because f the limited length f the area table. As fr the threshld N, N indicates the amunt f sampling errr. An unsuitably small N leads t t many clusters, and, vice ver'la, a very large N leads t t few clusters. The apprpriate values f Rand N have t be determined in accrdance with the applicatin field. Table 2 illustrates the gd perfrmance f this clustering methd fr the synthetic data, which have 20,000 samples frm a 10 class mixture in 4 dimensinal space using a Gaussian randm number generatr and a randm cluster generatr. Fig. 3 is an example applying this clustering methd t a LANDSAT scene ID In bth examples, the MDH is cnstructed by the multilayered hashing scheme described in sectin III VI. REFERENCES R.O. Duda and P.E. Hart: Pattern Classificatin and Scene Analysis. Jhn Wiley & Sns, S. Shlien: Practical Aspects Related t Autmated Classificatin f LANDSAT Imagery Using Lkup Tables. CCRS Reprt 75-2, M. Gldberg and S. Shlien: A Clustering Scheme fr Multispectral Images. IEEE SMC-8, N.2, pp86-92, l 10.0.t..0 '" E 5.0 ::l Z c: '" ~ a: "" '" g 10.0 :e 'iii' a:..0 '".t : Rejectin Rati= 1.0% Number f Prcessed Scan lines Number f Prcessed Scan Lines 3.0 ' E ~ c: 1.0 '" ~ : Rejectin Rati= 1.0% Sub Frame B (a) (b) (c) V. CONCLUSION Lad Factr (%) 80 Fr a huge satellite image, this paper has described a new clustering methd ushg lcal maximums f an MDH and an efficient histgram cnstructin methd using multi-layered histgram tables. Results are presented which shw the utility and efficiency f these methds. The ptimal values fr Rand N fr each applicatin field are items fr future study. A quantitative evaluatin f the clustering results is currently under way. These methds can be applied t classical clustering methds. At NAL these methds have been cmbined with an ISODATA-like methd fr frestry analysis. ~. "" a: '" :e 10.0 c: 5.0 'iii' a: Fig Mean Number f Prbes Measurement f efficiency in basic hashing scheme. (d) 1981 Machine Prcessing f Remtely Sensed Data Sympsium 324

7 Table I. Perfrmance in MDH cnstructin fr a full scene f a LANDSAT (Basic Scheme I [Multi u yered Schemel Gathering APU Time Gathering APUTime Rati (%) (sec) Rati (%) (sec) N=6. R=O.4S 4-Neighburs <:): Islated Cell. 0 : Key Cell lr step 4. Fig. 2 Example f MDH clustering 1-12 shws the rder after step I. Step I Step Step Ttal N. f genetlited u yen: 46 Cnditin in Step I: Re~cted Samples' Rati" 1% Table length: 20,000 Fig. 3 Urban, Bare Sil Snw t'ld Grassy Place Lake Shade, Rejectin MDH clustering result f LANDSAT id.1145-oos42. The area is arund the Lake Yamanaka. Table 2 Perfrmance f MDH clustering fr synthetic data. "". Errr (S) "'. Errr (I ) , , Cluster Dist. Clustering Dist. : Separability Measure Clusterin9 Parameter: R 0.001, N 2 Cluster Dis t. Clusteri ng 1981 Machine Prcessing f Remtely Sensed Data Sympsium 325