Locating Lake and Enclosed Islands in SRTM DEM Data Using Boundaries Obtained from Landsat Imagery via Geodesic Active Contour Model

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1 Locatig Lake ad Eclosed slads i STM DEM Data Usig oudaries Obtaied from Ladsat magery via eodesic Active Cotour Model SHEN Chaomi Joit Laboratory for magig Sciece & Techology, ad Dept of Computer Sciece East Chia Normal Uiversity Shaghai 0006,.. Chia cmshe@cs.ecu.edu.c FAN Jisog School of Mathematics ad formatio Sciece, Wezhou Uiversity, Wezhou, Zhejiag rovice 35000,.. Chia ad Dept of Mathematics, East Chia Normal Uiversity Shaghai 0006,.. Chia fjs@wzu.edu.c Lig Dept of Mathematics, East Chia Normal Uiversity Shaghai 0006,.. Chia pligzh@eyou.com Abstract: The Shuttle adar Topography Missio STM Digital Elevatio Model DEM data over a lake with more tha 000 islads i Chia are studied. DEM data may have missig data or be iaccurate over waterbodies because of mirror effect. Thus, locatig the shorelies of the lake ad eclosed islads from DEM data is o-trivial. We achieve this by overlayig the boudary derived from Ladsat imagery of same area. the Ladsat image, the delieatio of iterior ad eterior boudaries of the lake is coducted by the method of modified geodesic cotour model. the Ladsat image, the boudary delieatio procedure begis by studyig the properties of piels the lake area. This is doe by choosig sample piels visually from the lake area. ricipal compoet aalysis is the applied to these sample piels ad the iterval estimatio of piel values of the lake is obtaied. The delieatio of shorelies is a evolutio process ad the evolutio equatio is derived accordig to the theory of active cotour model as well as the properties of the lake piels. We costruct a iitial cotour iside lake ad the start evolutio accordig to the evolutio equatio. The iitial cotour epads accordig to the gradiet ad color iformatio at the locatio of cotour. The active cotour procedure termiates whe it meets the iterior ad eterior boudaries of the lake, returig the shorelies. The result is satisfactory. The the shorelies are overlaid to DEM data accordig to latitude ad logitude. Keywords: STM, DEM, Ladsat, geodesic active cotour model.. troductio The Shuttle adar Topography Missio STM is a joit project betwee the Natioal eospatial-telligece Agecy NA ad the Natioal Aeroautics ad Space Admiistratio NASA []. The STM Digital Elevatio Model DEM data were calculated from the data acquired by the space shuttle Edeavour i February 000. The DEM data studied here are i a resolutio of 3 arc secods approimately 90 m. DEM data have may useful applicatios. From DEM data, we ca easily distiguish features such as moutai ad plai areas by elevatio values. this paper, we wat to locate shorelies from STM DEM data. Locatig shorelies would ot be a difficult task if the DEM data were absolutely accurate. Sice a lake surface is almost flat, the DEM data values of the lake surface ca be assumed to be same. Thus, i the DEM data plae, a area of costat elevatio should be easily foud ad the boudary of this area would be the shorelie. ut i the real case, DEM data may have errors accordig to NASA, the vertical accuracy is 6 m, ad i lake areas, a lot of data are missig because of mirror effect ad some DEM values would be higher tha the values of real lake surface due to the

2 cotributio of higher elevatio of islads i the lake. Therefore, a area of costat elevatio correspodig to the lake areas does ot eist. Thus, other method ad iformatio are eeded to retrieve shorelies. this paper, we retrieve shorelies from a Ladsat image of same area ad overlay the Ladsat-retrieved shorelies oto the DEM data. The STM DEM data are from U.S. eological Survey USS [], ad the Ladsat data are from lobal Lad Cover Facility LCF, Uiversity of Marylad []. We will first delieate shorelies from the Ladsat image usig the method of modified geodesic cotour model, ad the overlay the Ladsat-derived shorelies to DEM data accordig to latitude / logitude. Thus we ca locate shorelies i DEM data. The descriptio ad implemetatio of the shorelie delieatio algorithm is the mai part of this paper. The study area is Qiadao Lake i Zhejiag rovice, Chia. Figure Fig.. Locatio of study area the red dot idicates the locatio of Qiadao Lake i Chia Sectio describes the geeral iformatio of Qiadao Lake area. The correspodig Ladsat image ad STM DEM data are show ad some features of these data are discussed. Sectio 3 has two parts. The first part describes how to delieate shorelies i Ladsat image. Sectio 3. discusses geeral active models ad why they are ot applicable to our case. Sectio 3. describes our improved geodesic active cotour model. The secod part describes how to overlay the derived cotour to DEM data. Sectio 4 shows the result of the delieated cotour from the Ladsat image ad the DEM data with cotour overlaid. Sectio 5 cocludes the paper ad discusses the limitatios as well as future work.. eeral iformatio of study area ad correspodig Ladsat ad DEM images Fig. Map of Qiadao Lake light blue idicates lake; the red arrow idicates the locatio of dam Our study area is Qiadao Lake. Chiese, Qiadao Lake meas a lake with 000 islads. Actually this is a reservoir built i the 960s. The area of the reservoir is aroud 580 km, the volume is aroud 7.84 billio m 3, ad the average depth is 34 m if the water level is 08 m. There are 078 islads each is bigger tha 500 m iside the reservoir [3]. The average water level of the reservoir is aroud 98 m. The map of this reservoir is show i Figure. The reservoir is show i light blue i the map, ad the red arrow idicates the locatio of dam. The dam top is at 5 m above sea level ad the legth of the dam is 465 m. elow the dam is a river called Xi a iver. Now this reservoir is usually called Qiadao Lake. Figure 3 shows the sceery of Qiadao Lake. Figure 3a shows the lake ad islads. The eposed soil yellow i the figure is caused by to the risig ad fallig of the water level. Thus there is o vegetatio i this area. Figure 3b shows the dam.

3 Fig. 3a Sceery of Qiadao Lake Fig. 3b Dam Height of top: 5 m; Legth: 465 m Figure 4 shows the correspodig Ladsat image Orbit umber: path 9, row 040; Date: ; ad combiatio: 74, Courtesy of USS. Figure 5 shows the DEM data of same area Courtesy of USS. Figure 5a shows the origial DEM data. Compared with Figure 4, it ca be observed that a lot of data are missig i lake area. The arrow i Figure 5a idicates the locatio of the dam. The DEM values of the Xi a iver below the dam are sigificatly lower tha the lake area. The reaso is that this is the river area below the dam. Sice Figure 5a cotais may missig data, various iterpolatio methods are used to fill those gaps. Figure 5b shows such a result. This was implemeted by remote sesig software ENV. Fig. 4 Ladsat image of Qiadao Lake area The DEM values i the lake area are ot costat i both Figure 5a ad Figure 5b, ad i some places those values are obviously wrog. Thus it is hard to get shorelies directly from this DEM data. o data m m Fig. 5a DEM data with missig values Fig. 5b DEM data after iterpolatio processed by ENV The red arrow idicates the locatio of the dam So our task is: with the help of Figure 4, locatig iterior ad eterior boudary of Lake i Figure 5b.

4 3. Descriptio o how to delieate iterior ad eterior shorelies We first discuss the mathematical problem cocered i delieatig iterior ad eterior shorelies i Figure 4. Mathematically, our problem is to fid a solutio o how to delieate the boudaries of Objects Of terest OO. oughly speakig, there are two categories of methods for delieatig boudaries of objects: edge detectio methods ad active cotour methods. The first category is ot suitable for our case because edge detectio methods have two disadvatages: the detected edges may ot be cotiuous, ad the detected edges may cotai may u-wated poits. Therefore, we adapt a method from the secod category. Our method is a geodesic active cotour method developed from the sake algorithm. 3. From sake algorithm to the level set method The sake algorithm is a active cotour method to delieate the boudary of objects, proposed by Kass et al. [4]. The mai idea is as follows: ive a iitial cotour i a image, a correspodig eergy ca be defied. f the cotour chages, the eergy will chage accordigly. Whe the eergy reaches its miimum, the correspodig cotour is regarded as the boudary of objects. egardig our case, the sake algorithm has oe problem: it ca ot hadle chages i topology. For eample, if the iitial cotour is a circle, o matter how to evolve, it ca ot evolve to two circles. Thus, the sake algorithm is oly applicable to detectig the edges of objects with kow topology. The topology of the lake is ukow before processig, thus we ca ot assume the topology of the lake is the same as that of the iitial cotour. esides the problem of topology, for a give iitial cotour, say, the boudary of the image, it shriks durig the evolutio. Whe the cotour reaches the eterior shorelie, it will termiate evolutio. Thus, the sake algorithm ca ot the detect iterior shorelies. For these two reasos, the sake algorithm is ot applicable to our case. The level set method is a improved cotour evolutio method due to J. Sethia ad S. Osher [5]. etter tha the sake algorithm, it ca hadle the topology chage i evolutio. The mai idea of the level set method is to fid a family of surfaces z φ, y, t, where, y are image coordiates, t is time, such that the itersectios of these surfaces ad z 0 are the evolvig cotours. For a give time t, φ, y, t 0 is the cotour i the, y plae. Thus, φ, y,0 0 represets the iitial cotour ad φ, y, t 0 teds to the desired cotour of objects. The evolvig level set fuctio z φ, y, t ca be obtaied by solvig its correspodig partial differetial equatios. ut eve the level set method caot solve our problem directly. The evolved cotour may stop at some o-shorelie poits. The reaso is as follows. level set theory, the poits where the evolved curve should stop marchig o is cotrolled by a stoppig fuctio, a fuctio defied at every image poit. The stoppig fuctio idicates the speed of evolutio at that poit. f the value of the stoppig fuctio is very big, it meas the curve will pass that poit with that speed alog a certai directio. f the speed is reachig 0, it meas the curve will stop at that poit. So a good stoppig fuctio is crucial i the evolutio process. ut a usual stoppig fuctio may ot be good eough i our case. Usually, the stoppig fuctio is defied as the decreasig fuctio of the gradiet ad oly depeds o gradiet, i.e., at the locatio where the gradiet is big, the stoppig fuctio is small, which meas the evolutio teds to stop at that locatio. So the evolutio curve of level set method may stop at the place where the gradiet is big, but that place may ot be OO at all. So what we wat is: the curve should oly stop at the boudary of OO. At other places, o matter how big the gradiet is, the curve should ot stop. Thus the color iformatio of the OO should be cosidered, i.e., the properties of the OO piels should be studied. Sectio 3., we will discuss how to determie the properties of OO piels. There is aother problem i the level set method. t is similar to the problem metioed for the sake algorithm: whe the evolved curve meets the eterior boudary, it will termiate evolutio. Assume that the iitial curve is the image boudary, after several evolutios, it successfully arrives at the eterior boudary of the lake. We wat it shrik agai ad also fid the iterior shorelie. ut sice the stoppig fuctio reaches 0 at the eterior shorelie, the curve will ot march o. Thus, the iterior boudary ca ot be detected. The Cha-Vese model proposed by Cha ad Vese ca be used to solve this problem [6]. ut i our case, we ca avoid this by givig a iitial cotour iside the lake, the let it

5 epad. Durig the epasio, whe the cotour reaches the iterior boudaries islads, it will delieate the islads ad goes o, util it reaches the eterior boudary. 3. Our algorithm of delieatig boudaries of OO Our algorithm is preseted i Sectio 3.. order to delieatig iterior ad eterior shorelies i the Ladsat image, followig problems should be solved.. determie the properties of lake piels ad give out a criterio whether a piel is i lake area;. give the stoppig fuctio at each poit i the image. The stoppig fuctio should be small if ad oly if the piel is i lake area ad the gradiet is big; 3. give the evolutio equatio; 4. give the umerical solutio of the evolutio equatio. what follows, assume the image, is a color image uless stated. f,, the image also ca be epressed by, where is a vector deoted by,,,, roperties of the lake piels ad iterval estimatio to the lake piels As stated i Sectio 3., we will choose the epasio method, i.e., the iitial cotour is iside the lake ad will epad util reachig the shorelies. Thus whe the curve evolves to a ew poit, the speed ad directio at that poit must be give. side the lake area, the speed should be big so that the algorithm ca be efficiet, ad at the boudary the speed should be reachig zero. So we eed to give out a criterio to distiguish whether a poit is i lake area or ot. To achieve this, the properties of the lake piels should be studied. This is doe by choosig some sample poits from the lake, ad the the properties of lake populatio are estimated from these samples. Thus, we first discuss the properties of the piels i lake area. Sice the image is displayed i ed, ree, lue for short combiatio, we wat to set up a criterio that describes whether a poit is i lake area usig a discrimiatio fuctio of,, ad. For eample, oe simple criterio might be: a poit is regarded as i lake area if ad oly if [ a, b ], [ a, b ] ad 3 [ a 3, b3 ], where a i, b i, i,, 3 are itegers. The criterio should be as precise as possible, i.e., type error omissio ad type error commissio should be avoided. t is quite ituitive that the above metioed rectagular parallelepiped may ot best describe the lake area. For eample, poits i the lake area may cocetrate alog certai lie i co-ordiates. Thus, usig a rectagular parallelepiped criterio i co-ordiates will defiitely cause type error, which could be avoided by usig rectagular parallelepiped criterio i a ew co-ordiate system with the cocetratio lie as the ew X-ais. So if there eists a coordiate system better tha co-ordiate system, so that i that ew system the discrimiatio fuctio is simple ad precise, that co-ordiate system should be used. ricipal Compoet Aalysis CA is a way to fid out this kid of ew system. For the detail of CA, please refer to [7]. The mai procedures of CA are as follows.,, we choose sample piels,, 3, L,,, 3 from the lake. Therefore, X ij is a 3 matri. Let X ij 3 be the ormalized matri of X. f the * correlatio coefficiet matri of X is deoted by ρ ρij 3 3, the the correspodig eigevectors of ρ are aes of the ew co-ordiate system. Assume the eigevalues are λ,λ ad λ 3 λ λ λ3. Correspodig eigevectors are ω, ω, ω3, where ω i i,, 3 is a colum. The a poit i, i, i3 i the co-ordiate system ca be writte as p i, pi, pi3 i the ew co-ordiate system, i.e., if let p ij 3, the X ω, ω, ω3. f a vector is deoted by a a, a, a3 i the ew co-ordiate system, the a i is called the i-th pricipal compoet of a i,, 3. The percetage of cotributio of the i-th pricipal compoet to the total sample variace is λi ei,,, 3 λ λ λ i, 3 which measures the importace of the i-th pricipal compoet. f the percetage of cotributio of the first pricipal compoet e is big eough, the iformatio from,, ca be mostly eplaied by the first pricipal compoet. So

6 we could oly cosider the first pricipal compoet. what follows, we costruct a iterval i ω ais by the method of iterval estimatio. The iterval should cover almost all samples by iterval estimatio. Lettig i i i i p S p,, we ca use a studet's t-distributio with degrees of freedom to costruct a cofidece iterval, α α t S t S with degree of cofidece α, where α t is the t-value with degrees of freedom ad 0 < < α. The above procedures ca be summarized as: Every piel i the image is trasformed from co-ordiates to a ew co-ordiates via CA. The first compoet i the ew co-ordiates is called first pricipal compoet ad cotais most importat iformatio. f the value of first pricipal compoet is withi the cofidece iterval, the piel is regarded i the lake area with a cofidece of α. Thus for a give piel, we could defie a fuctio γ, which describes whether this poit is i the area or ot. Supposig that the cofidece iterval obtaied by CA is [ ] b a,, ad the value of the first pricipal compoet is ~ u, the γ is defied by others. 0, ~ b u a γ This iformatio is used i the costructio of stoppig fuctio below. 3.. Stoppig fuctio We shall defie our stoppig fuctio ow. A stoppig fuctio is a decreasig fuctio of gradiet. Usually, it oly depeds o gradiet, regardless of other image iformatio. For eample, for a grey image, K g σ is a typical stoppig fuctio, where 0 > K is a cotrast factor, σ σ σ 4 ep is a aussia filter with a parameter σ, is the gradiet operator, ad * is the covolutio operator. This kid of stoppig fuctio has disadvatages metioed i 3. ad it is oly for grey images. We wat to defie a ew stoppig fuctio for color images. A good stoppig fuctio should be small if ad oly if two coditios are satisfied: the poit is i lake area, ad that poit has a big gradiet. color images, the cocept of gradiet is replaced by the largest eigevalue Λ of the followig matri, where, ad represets the piel s value of red, gree ad blue after aussia covolutio respectively; i i, etc. For the detailed defiitio of Λ, please refer to [8, 9]. Our stoppig fuctio is Λ g γ.

7 t ca be verified that satisfies the two coditios metioed Evolutio equatio Let C r t be the evolvig curve at time t, i.e., φ, t 0 be the implicit represetatio of C r t. For our model i its level set formulatio, the partial differetial equatio of φ, t should satisfy: φ φ φ div g ν g φ, t φ φ,0 φ0, where g is defied i, ν is a costat called balloo force, ad φ 0 is the iitial curve i our case, the iitial curve is iside the lake. The balloo force idicates whether the cotour will shrik or epad. Whe the balloo force is less tha 0, the cotour will epad Numerical solutio of the evolutio equatio order to solve umerically, the Additive Operator Splittig AOS umerical method is used. The AOS scheme is a ucoditioally stable umerical scheme for oliear diffusio i image processig see [0] for detail. The iterated formula of is k k k φ [ E τ Al φ ] φ τνg 3 l {, } where τ is the time step, E is a idetity matri, ad A k l φ is a matri correspodig to derivatives alog the l-th coordiate ais for the detailed defiitio of A φ k please refer to [0]. l The eact solutios {, } of φ, 0 should be real umbers. ut, because of the discreteess of grid poits, we eed to fid the iteger solutios of φ, 0. Fidig the iteger solutios of φ, 0 ca be doe by comparig every piel, with its 8-eighborhood piels. f we defie if φ, 0 sig,, the else., is regarded as the solutio of φ, 0 if ad oly if 8 sig, sig y, y 0, y, y 8eighbor of,, i.e., the sig of φ, chages i the 8-eighborhood of Thus, the cotour φ, 0 is foud. For the detailed descriptio for our algorithm of modified geodesic active cotour model, please refer to []. 3.3 Overlayig delieated shorelies to DEM data Sectio 3. describes the method of delieatig shorelies i the Ladsat image. this sectio the delieated shorelies are overlaid to DEM data. Sice the Ladsat image has geographical iformatio, the latitude/logitude of every piel i the delieated shorelies is kow. Therefore the delieated shorelies ca be overlaid to DEM data accordig to latitude/ logitude.

8 4. esults Figure 5 shows the delieatio result i the Ladsat image. Fig. 5 Delieated terior ad eterior boudaries show i red Figure 6 shows the delieatio result overlaid to the STM DEM data. Fig. 6 DEM data with Ladsat-delieated shorelies overlaid The result is accurate. The cotour epads ad stops eactly at the locatio of real shorelies. This could be verified by the fact that the cotour is stopped at the locatio of dam.

9 5. Coclusio The Shuttle adar Topography Missio STM Digital Elevatio Model DEM data over a Qiadao Lake area i Chia are studied. Our aim is to locate shorelies from the DEM data. order to achieve this, firstly the iterior ad eterior shorelies of the lake are delieated from Ladsat data of the same area via the method of modified geodesic cotour model. The the shorelies are overlaid to DEM data accordig to latitude/logitude. The mai part of this paper describes the algorithm of delieatig iterior ad eterior boudaries of the lake. The properties of lake piels are studied ad a criterio whether a piel is i lake area is give; Our stoppig fuctio for the evolutio curve at each poit i the image is costructed. The evolutio equatio ad the umerical solutio of the equatio are give. The delieatio result is satisfactory. The the shorelies are overlaid to DEM data accordig to latitude ad logitude. Ackowledgemet The authors would like to thak U.S. eological Survey USS for providig the STM DEM data ad lobal Lad Cover Facility LCF, Uiversity of Marylad for providig the Ladsat imagery. The authors also would like to thak Christia Melsheimer for discussio. efereces [] UL: Shuttle adar Topography Missio Homepage. Available at: [] UL: lobal Lad Cover Facility Homepage. Available at: [3] UL: formatio of Qiadao Lake. Available at: [4] Kass, M., Witki, A., ad Terzopoulos, D., 988. Sakes: Active cotour models, teratioal Joural of Computer Visio, vol., [5] Osher, S., Sethia, J.A., 988. Frots propagatig with curvature-depedet speed: Algorithms based o Hamilto Jacobi formulatios. Joural of Computatioal hysics, 79, -49. [6] Cha, T., Vese, L., 00. Active cotour without edges for vector-valued image, Joural of Visual Commuicatio ad mage epresetatio, Vol., [7] Aderso, T. W., 984. A itroductio to multivariate statistical aalysis d Editio, Wiley, New York. [8] Soche, N., Kimmel,., ad Malladi,., 988. A geeral framework for low level visio, EEE Tras. mage rocessig, vol. 7, [9] oldeberg,., Kimmel,., ivli, E., ad udzsky, M., 00. Fast geodesic active cotours, EEE Tras. mage rocessig, vol. 0, [0] Kühe,., Weickert, J., eier, M., ad Effelsberg, W., 00. Fast implicit active cotour models, DAM 00, LNCS 449, [] i, L., Fa, J.S., ad She, C.M., 005. Segmetatio for objects of iterest with modified geodesic active cotour, preprit.