WAVE STATISTICS AND SPECTRA VIA A VARIATIONAL WAVE ACQUISITION STEREO SYSTEM

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1 Proceedings of OMAE 008 7h Inernaional Conference on Offshore Mechanics and Arcic Engineering June 15-0, 008, Esoril, Porugal OMAE WAVE STATISTICS AND SPECTRA VIA A VARIATIONAL WAVE ACQUISITION STEREO SYSTEM Guillermo Gallego School of Elecrical & Compuer Engineering Georgia Insiue of Technology, Alana USA Alvise Beneazzo PROTECNO S.r.l. Padua ITALY Anhony Yezzi School of Elecrical & Compuer Engineering Georgia Insiue of Technology, Alana USA Francesco Fedele School of Civil & Environmenal Engineering Georgia Insiue of Technology, Savannah, USA ABSTRACT We propose a novel variaional Wave Acquisiion Sereo Sysem (WASS) ha explois new sereo reconsrucion echniques for accurae esimaes of he spaio-emporal dynamics of ocean waves. WASS has a significan advanage as a low-cos sysem in boh insallaion and mainenance. A sereo camera view provides hree-dimensional daa (boh in space and ime) whose saisical conen is richer han ha of a ime series rerieved from wave gauges, ulrasonic insrumens or buoys, he laer being expensive o insall and mainain. Indeed, wave specra can be easily esimaed from he muli-dimensional images obained wih WASS. The esimaed specra presen an inerial range ha decays as k -.5, k being he wave number, in agreemen wih wave urbulence heory (Zakharov 1999, Socque-Juglard e al. 005). Furher, he empirical probabiliy densiy funcions derived from he reconsruced surface daa compare very well wih heoreical models (Tayfun & Fedele 007, Fedele 008). The variaional WASS is a promising echnology wih broader impacs in offshore engineering since i will enrich he undersanding of he saisics of waves for an improved design of offshore srucures. STEREO VIDEO IMAGERY AN OVERVIEW The Wave Acquisiion Sereo Sysem (WASS) (Beneazzo 006) uilizes sereo vision based on wo calibraed views for providing ime series of scaered 3- D poins of he waer surface (see Figures 1, ). Waer surface opography can be measured from he convenional sereographic echnique algorihms (Ma e al., 004) used o survey geodeical surfaces or saic objecs. The major difference is ha he waer surface is a specular objec in rapid movemen. Thus, each sereopair is acquired simulaneously. Firs experimens wih cameras mouned on an ocean going ship are by Schumacher (1939) and Coe e al. (1960). Shemdin e al. (1988,199) proposed he direcional measuremens of shor ocean waves applying sereography. This experimen used a pair of cameras mouned on an oceanographic offshore ower near San Diego (USA) o creae 3D model of he sea surface and hen, via specral analysis, o exrac direcional informaion of waves. The mos recen inegraion of sereographic echniques ino he field of oceanography has been he WAVESCAN projec (Sanel e al. 004). The reconsrucion of he wave surface from sereo pairs of ocean wave images is a classical problem in compuer vision commonly known as he correspondence problem (Klee e al., 1998, Ma e al. 004). Is soluion is based on epipolar geomery echniques ha find corresponding poins in he wo images, from which one obains he esimae of he real poin in he hree dimensional erresrial coordinae sysem. WASS will process muli-dimensional image daa and obain a 4-D (space and ime) reconsrucion of 1 Copyrigh 008 by ASME

2 he sea surface. Beneazzo (006) successfully used WASS in experimens offshore he California Coas and a he Venice coas in Ialy. He was able o esimae wave specra from he exraced ime series of he surface flucuaions a one fixed poin from he daa images. The accuracy of such specral esimaes is as good as ha from ulrasonic ransducer measuremens, as shown in Figure 3. Figure 1: WASS physical se-up of wo calibraed cameras. THE STEREO RECONSTRUCTION OF OCEAN WAVES In his paper, a variaional approach is considered o address he reconsrucion problem in a differen way from he radiional correspondence problem based on epipolar geomeries. Consider he waer surface supporing a Lamberian radiance funcion. The laer is approximaely rue on a cloudy day because, a a given poin on he waer surface, he same amoun of ligh is received from all possible direcions and refleced owards he observer causing a visual blurring of he speculariies of he waer. Under hese condiions, he radiional sereographic algorihm based on epipolar geomeries may fail o provide a smooh surface reconsrucion because of holes corresponding o unmached image regions (Ma e al., 004, Beneazzo 006). In his paper, we address his problem by proposing a variaional WASS ha explois variaional parial differenial equaion (PDE) echniques for 3D dense and regular sereo reconsrucions of he waer surface. Finally, we presen resuls showing ha he variaional WASS yields accurae esimaes of he spaio-emporal ocean dynamics and of he associaed wave specra and saisics. THE VARIATIONAL GEOMETRIC METHOD Figure : Se-up of wo camera sereomeric inersecion (From Beneazzo, 006). Mahemaical pinhole camera model. PSD [m s] Sereo Ulrasonic 0. f Figure 3: Wave specra as funcion of he frequency f esimaed from boh sereo daa and ulrasonic measuremens. (From Beneazzo 006). Under he assumpions of a Lamberian surface, following he seminal work by (Faugeras e al. 1998), he 3-D reconsrucion of he waer surface is obained in he conex of acive surfaces by evolving an iniial surface hrough a PDE derived from he gradien descen flow of a cos funcional designed for he sereo reconsrucion problem. To be more specific, he energy being maximized is he normalized cross correlaion beween he image inensiies obained by projecing he same waer surface pach ono boh image planes of he cameras. I is clear ha such energy depends on he shape of he waer surface. Therefore, he acive surface esablishes an evolving correspondence beween he pixels in boh images. Hence, he correspondence will be obained by evolving a surface in 3-D insead of jus performing image-o-image inensiy comparisons wihou an explici 3-D model of he arge surface being reconsruced. Copyrigh 008 by ASME

3 Noaion Consider he same se-up of he WASS, in which wo calibraed cameras are viewing he waer surface. Quaniies corresponding o he lef (resp. righ) camera will be noed wih a 1 (resp. ) subscrip. If he projecion marices ha model he perspecive projecion of he cameras are known wih respec o he lef camera coordinae sysem, hen P 1 = K1( I,0) and P = K ( R, ), where K 1, K are he inrinsic parameer marices (conaining he focal lenghs, principal poins and skews) and g = ( R, ) defines he rigid ransformaion beween boh camera coordinae sysems. Le X = ( x, y, z) represen a generic 3-D poin in he scene, expressed wih respec o he same coordinae sysem as he projecion marices. As i is well known, he equaions ha model he geomery of he image formaion process are linear in homogeneous coordinaes form: a 3-D poin X = ( x, y, z,1) maps o he image poins xi = ( ui, vi, 1 ) ~ Pi X, (i=1,), where ~ means equaliy up o a non-zero scale facor. Therefore, x i = ( u i, v i ) are he pixel coordinaes of he image 3 poins. Le π i : R R noe he perspecive projecion maps: xi = π i (X ), (i = 1,). The primary objec of ineres will be a regular surface S in R 3 (wih area elemen da), represening he waer surface. As i will be shown laer in he implemenaion of he algorihm, he surface S will be represened as he zero level se of a smooh funcionu ˆ : R 3 R, i.e., S ={X U ˆ (X) = 0}. Cos funcional To infer he shape of he waer surface S, we se up a cos funcional on he discrepancy beween he projecion of he model surface and he image measuremens. As previously announced, such cos is based on a cross correlaion measure beween image inensiies, which will be noed as E daa (S). We conjecure ha, o have a well-posed problem, a regularizaion erm ha imposes a geomeric prior mus also be included, E geom (S). We consider he cos funcional o be he (weighed) sum: E(S) = E daa (S) E geom (S). (1) In paricular, he geomeric erm favours surfaces of leas area: E geom (S) = da. () S The daa fideliy erm may be expressed as I1, I E = daa ( S) 1 da, (3) S I1 I where only he surface in a region wihin he field of views of boh cameras is considered. The unnormalized cross-correlaion beween I 1 and I corresponding o a common 3-D poin X ha projecs on he pixels x 1 = π X ), x = π ( X ), is 1 ( p q 1 = [ I1( π 1( X m)) I1( x1 )] I1, I ( S) 4 pq p q [ I ( π ( X m)) I ( x )]dm, where he inegraion is over a small pach in he angen plane T S o he waer surface S a X and, herefore, he averages I x ) are defined as We noe i ( i p q 1 I i ( xi ) = Ii( π i ( X m )) dm, (i = 1,). 4 pq p q I = I, I. Observe ha his definiion of I 1, I is symmeric and, herefore, he erm I, I is 1 already included in E daa (S). The erm 1 in (3) yields a posiive energy, which prevens anidiffusive erms from occurring in he following evoluion equaion. Evoluion equaion To find he surface S ha minimizes E(S), we sar from an iniial esimae of he surface a ime = 0, S 0, and se up a gradien flow based on he firs variaion of E(S) ha will make he surface evolve owards a minimizer of E(S), hopefully converging o he desired waer surface shape. Based on he heorem in (Faugeras e al. 1998) ha says ha for a funcion Φ : R 3 R 3 R and he energy E X, N da, (4) = S Φ( ) where N is he uni normal o S a X, he flow ha minimizes E is given by he evoluion PDE S = β N, (5) where S is he derivaive of S wih respec o ime and he speed funcion b is β = H ( Φ Φ N N) Φ X N race[ ( Φ XN ) T dn o ( Φ ] S NN ) T S All quaniies are evaluaed a he poin S=X wih normal N o he surface. H denoes he mean curvaure. Φ, Φ X N are he firs-order derivaives of Φ, while Φ, Φ are XN NN 3 Copyrigh 008 by ASME

4 he second-order derivaives. dn is he differenial of he Gauss map of he surface and ( ) T S means resricion o he angen plane T S o he surface a S=X. Noe ha our proposed energy (1) can be expressed in he form of (4) if I1, I Φ = 1 α, I1 I where α is jus a weigh for he geomeric prior. In pracice, we use he flow based on he firs-order derivaives of Φ because i provides similar resuls o hose of he complee expression, bu saves a significan amoun of compuaions, S = ( H( Φ Φ N N) Φ X N ) N. (6) The level se framework (Osher e al. 1988) has been adoped o numerically implemen (6). This formulaion requires an implici represenaion of he surface. A each ime, he surface S is he zero level se of he level se funcion U: R 4 R, i.e., S a ime is {X R 3 U(X, ) = 0}. The uni normal of he surface is N = U / U, where sands for he derivaives wih respec o he firs hree coordinaes of U. The evoluion equaion for he level se funcion U corresponding o he evoluion equaion for he surface (5) is U = β U. (7) In a numerical implemenaion, he level se funcion U is discreized in a 3-D grid enclosing he volume of ineres in he world where he waer surface is known o be. The derivaives of Φ, he mean curvaure H, he uni normal vecor N, ec. are approximaed by numerical finie difference formulas. Afer he evoluion, he surface iself is exraced using he marching cube algorihm, which is a very popular rouine, such as MATLAB s isosurface command. Iniializaion As is well known, a good iniializaion is srongly encouraged in gradien descen algorihms o miigae he problem of geing suck in a local minimum far away from he desired soluion. A sensible iniializaion for he proposed variaional sereo algorihm is o consider he surface a ime =0 o be he simples (lowes frequency) approximaion o he waer surface, i.e., a plane (fla surface). In he experimens, his approximaion is compued from a se of 10 o 15 manually mached poin k k n correspondences { x1 x } k= 1 beween boh images. Since he cameras are calibraed, a riangulaion (backprojecion) algorihm may be used o find he n approximae locaion of he 3-D poins { X k } k = 1 ha k k projec ono x1, x because (,, k, k X k = f P1 P x1 x ). Then, a plane can be fied o his se of 3-D poins by minimizing a leas-square crierion such as he orhogonal disance o he poins. Once his plane is known, he level se funcion U(X,0) is iniialized as he signed disance funcion o he plane and he algorihm can be run o evolve he surface. Observe ha if he plane is known, one may also back-projec he four corners of each image and inersec hose opical rays wih he plane o obain an esimae of he quadrilaeral field of view of he cameras on he plane. This is also used o se up he volume where he level se funcion is discreized. Camera o erresrial coordinaes Now, suppose ha he variaional sereo algorihm has converged. Nex, he surface mus be exraced from he level se and prepared for pos-processing (saisical analysis, ec.). So far, he coordinae sysem used has been ha of he (lef) camera, bu under he assumpion ha he waer surface can be expressed in he form of a graph z = f ( x, y ) in some erresrial coordinae sysem (prime noaion) relaed o he former by a rigid ransformaion, we would like o obain such a represenaion of he reconsruced surface so ha relevan informaion conained in he heighs of he poins on he surface may be analyzed. The problem now can be posed as he join esimaion of he average plane hrough he surface and he rigid ransformaion (R,T) ha moves he former o is canonical locaion z = 0. We follow a decoupling sraegy: firs, he bes plane hrough he daa is esimaed; second his plane is reoriened by a rigid ransformaion o become he plane z = 0. We now presen a mehod ha esimaes he mean plane hrough a se of 3-D poins by minimizing he geomeric orhogonal disance from he poins o he plane. This is a sensible crierion and provides a plane ha has zero mean signed disance o he given daa poins. The signed perpendicular disance from a 3-D poin X = ( x, y, z,1) o a plane whose coefficiens (or homogeneous coordinaes) are v = ( a, b, c, d) is X v ax by cz d sd( X, v) = = (8) a b c a b c and he perpendicular disance is is absolue value, 4 Copyrigh 008 by ASME

5 d ( X, v) = sd ( X, v). (9) cross produc w = n n. Rodrigues roaion formula Given m poins, he plane ha minimizes can be used o compue R: m R = I sin θ[ wˆ ] [ ] (1 cosθ ) wˆ, (13) d ( X i, v) (10) where wˆ = w/ w and []is a i= 1 he anisymmeric marix is defined up o a scaling of he coefficiens. This degree such ha [ a ] b = a b "b. If n = 1, hen he expression of freedom disappears afer enforcing for R in (13) simplifies o n = a b c = 1 on he magniude of he normal 1 a q abq a o he plane, n = ( a, b,c). I can be shown ha he R = abq 1 b q b, minimizer of (10) saisfies a generalized eigenvalue equaion. However, he following simpler procedure a b c gives he same resuls. where 1 c Given m³3 poins X i = ( xi, yi, zi ), q =. 1) Normalizaion. Apply a similariy ransformaion H a b S (roaion, ranslaion and isoropic scaling) o he poins The second equaion, T Rn = d, has an infinie such ha he new cenroid of he poins is placed a he number of soluions since only he hird coordinae of T origin and he average disance of he ransformed poins is relevan. The firs wo coordinaes of T may be used o o he origin is 3. specify he origin of erresrial coordinaes on he ) Esimaion. Compue he Singular Value plane z = 0. We may choose T = (0,0, d). Decomposiion of he m 3 marix A whose i-h row Finally, conver he waer surface poins from he is x, y, z ), i.e., A = UDV. Se v ( a, b, c,0) camera coordinae sysem o he erresrial coordinae ( i i i n ~ = ( a, b, c) is he uni righ singular vecor of A associaed o is smalles singular value. 3) Denormalizaion. Se v = H S v ~, where H S is he 4x4 marix of he similariy ransformaion (in homogeneous coordinaes). I can be shown ha he plane esimaed his way minimizes (10) and is one of he mean planes wih respec o signed disance, i.e., saisfies m i= 1 sd ( X i, v) = 0. (11) Once he plane v is known, he goal is o find he Euclidean ransformaion (R,T) ha moves he plane v o is canonical locaion in he erresrial coordinae sysem. Le v = ( n, d), if poins X ransform as R T X = H E X, where H E = (1) 0 1 is he 4 4 marix of he Euclidean ransformaion beween coordinae sysems, hen planes ransform as v = H E v. The goal is o find R and T such ha v = ( n, d ) = (0,0,1,0). This yields wo equaions: n = (0,0,1) = Rn and T Rn = d. Observe ha R is he mapping beween normal vecors. The angle and axis of roaion are given by he magniude and direcion of he sysem by X = RX T. The reconsrucion process described above is repeaed for each pair of images acquired by he calibraed cameras. So far, no dynamics of he physics of he waves have been included in he model o make a join spaio-emporal reconsrucion of he waer surface. This approach is posponed for fuure research. EXPERIMENTS The daa used o es he variaional reconsrucion algorihm is a se of images acquired by he WASS developed by Beneazzo (006); in paricular, he San Diego experimen described herein. The images of waves, on waer deph of 8 meers, were cropped o 504x336 pixels o decrease he deph of he field of view of he cameras, i.e., o focus in he region close o he cameras. The baseline beween he cameras is 3.04 meers and he reconsruced surface occupies a recangle of approximaely 8x8.7 m. Figure 4 shows he domain of he reconsruced surface on each pair of inpu images o he algorihm. The level se funcion was discreized on a 3-D grid of 56 3 poins and evolved during 100 ieraions for every pair of images, wih a ime sep of Copyrigh 008 by ASME

6 Figure 4: Inpu sereo pair images o he algorihm (lef and cener columns). The recangular domain (8 m 8.7 m) of he reconsruced surface or elevaion map (righ column) has been superimposed. The heigh of he waves is in he range ±0. cm. The disance beween adjacen grid poins is 8. cm, which is relaed o he maximum quanizaion error. The disance beween adjacen grid poins is 8. cm, which is relaed o he maximum quanizaion error. The surface is hen exraced and reoriened for pos-processing, as explained in he previous secion. The reconsruced surface, as elevaion maps is also displayed in Figure 4 (righ column). WAVE SPECTRA & STATISTICS Herein, we elaborae he naure of he reconsruced wave surface shown in Figure 4. We poin ou ha he following saisical analysis only explois daa from he reconsruced spaial snapsho of he ocean surface a a fixed insan of he ime. We show ha informaion conen of a single spaial snapsho of he ocean surface is richer han ha of radiional ime series measuremens. Indeed, we compue esimaes of he wave specra and provide evidences ha recen probabilisic wave models (Tayfun & Fedele 007, Fedele 008) well explain realisic oceanic condiions. In paricular, he wave specrum S(k) as funcion of he wave number k is given in Figure 5. Is ail shows an inerial range ha decays as k -.5 in agreemen wih wave urbulence heory (Zakharov 1999, Janssen 003, Socque-Juglard e al. 005). Deviaions of large surface heighs from he Gaussian condiions are expeced since waer waves presen seep cress and shallow roughs (Longue-Higgins 1963, Tayfun 1980;1986, Tayfun & Fedele 007, Fedele 008). This is also confirmed by our analysis. Indeed, Figure 6 shows ha he empirical probabiliy densiy funcion p η derived from he reconsruced surface η compare very well wih he Gram-Charlier model (Tayfun & Fedele, 007) given by exp( x / ) λ3 pη ( x) = [1 x( x 3) π 6 λ3 6 4 λ4 4 ( x 15x 45x 15) ( x 6x 7 4 3)], (13) where λ 3 and λ 4 are he skewness and kurosis of he wave surface, respecively. These parameers characerize he nonlinear feaures of ocean waves and can be esimaed from he video daa as follows. Drawing on Tayfun (1994, 008), we firs evaluae he probabiliy of posiive wave elevaions given by 1 λ3 P = Pr{ η > 0} =. (14) 6 π Noe ha P < 1/ reflecing he asymmery of wave cres and roughs. Esimaes for λ 3 follow from (14) as λ 3 = 3 π ( 1 P ), (15) where P is esimaed from he surface daa. The kurosis λ 4 is esimaed from he average posiive wave heigh, given by, 1 λ3 λ 4 η η > 0 = Pr( 0) 1 η > h η > dh =, 4 0 π P which yields λ = λ 4( π P η η > 0 1).. (16) 4 3 Furher, in Figure 7 i is ploed he condiional probabiliy p(η>z η>0) which also agrees well wih he corresponden Gram-Charlier model given by 6 Copyrigh 008 by ASME

7 Pr{ η > z} = p λ3 ( z 3 z η λ4 1) z( z 1 1 dx = erfc z 1 z exp π λ3 3) z( z z 15). (17) We hus have scienific evidences ha he variaional WASS provides accurae esimaes of he spaioemporal ocean dynamics. probabiliy p(η>z η>0) daa Gaussian model New model (Fedele 008, Bone e al. 008) 10-3 z Figure 7. Condiional probabiliy densiy p(η>z η>0) of he reconsruced wave surface η in Figure 4: comparisons wih heoreical sochasic models (Fedele & Tayfun 007, Fedele 008). Figure 5. Wave specrum S(k) as funcion of he wave number k compued from he reconsruced wave surface η in Figure 4. The specrum ail decays as k -.5 in agreemen wih wave urbulence heory (Zakharov 1999, Socque-Juglard e al. 005). probabiliy densiy p(η) daa Gaussian model New model (Fedele 008, Bone e al. 008)) η Figure 6. Probabiliy densiy p(η) of he reconsruced wave surface η in Figure 4: comparisons wih heoreical sochasic models for wave heigh probabiliies (Tayfun & Fedele 007, Fedele 008). DISCUSSION AND FUTURE WORK In his paper we have explored he possibiliies of applying a variaional algorihm for he sereo reconsrucion of ocean waves. In paricular, we have proposed a variaional WASS which has he following main advanages: i) i provides a dense reconsrucion of he waer surface (here are no holes corresponding o unmached image regions); ii) i has buil in regulariy due o he mean curvaure componen of he flow (6); iii) i yields reliable saisics of ocean waves due o he rich informaion conen of video daa. However, his mehod is sensiive o specular reflecions and ime consuming (non real-ime). These issues are subjec of fuure work ha aims improving he reconsrucion algorihm by considering he dynamics of he waves and he esimaion of he surface radiance funcion (Yezzi e al. 001). These may be included in he cos funcional and will ulimaely provide a robus generaive model of he images. Furher, he proposed variaional geomeric algorihm is very general and allows for he reconsrucion of complicaed surfaces. I has no been paricularized ye for he case of waer surface, which admis a simplified represenaion in he form of a graph, i.e., a poin on he surface may represened as X = ( x, y, z( x, y)). This is a opic of fuure research. We hus srongly believe ha he variaional WASS echnology and is generalizaion are beneficial o offshore indusries. More accurae predicions of wave 7 Copyrigh 008 by ASME

8 specra and large waves in sea sorms around offshore srucures can be provided. In paricular, reliable esimaes of he highes wave expeced over an area (Forrisall 005) can be compued by processing wave surface daa exraced from video-imagery. Furher, video daa will also suppor he validaion of heoreical models for cres or wave heigh exceedances (Tayfun 1986,006, Fedele 006, Forrisall 000, Tayfun & Fedele 007). We poin ou ha he highes expeced wave over an area is naurally higher han he expeced wave heigh a one poin (Forrisall 005). This reflecs he common sense of he surfers ha wander around a sie, and always find heir big waves. Thus, he probabiliy o encouner a big wave wihin an area of he ocean increases wih is size. The proposed variaional WASS will allow a beer undersanding of he saisics of large waves over an area, and i may yield new insighs for an improved design of plaforms ha avoid he localized damages someime observed on heir lower decks afer sorms. ACKNOWLEDGMENTS The auhors would like o acknowledge Hailin Jin for providing source code ha was parly used o obain he resuls shown in his paper. We also hank boh Aziz M. Tayfun & Harald Krogsad for useful discussions and suggesions. REFERENCES Beneazzo, A. 006a. Measuremens of shor waer waves using sereo mached image sequences Coasal Engineering, 53: Beneazzo, A., 006b. Measuremen of waer surface wave fields using a non-inrusive mehod based on a sereomaching algorihm. Ph.D. Diss., Universiy of Padua. Coe, LJ, Davis, JO, Marks, W., McGough, RJ, Mehr, E., Pierson, WJ, Ropek, JF, Sephenson, G., Veer, RC The direcional specrum of winde generaed sea as deermined from daa obained by he sereo wave observaion projec. Meereological paper, (6), New York Universiy, College of Engineering. Faugeras O. & Keriven R. Variaional principles, surface evoluion, pde s, level se mehods and he sereo problem. IEEE Trans. on Image Processing, 7(3): , Fedele, F Exreme evens in nonlinear random seas. ASME Journal Offshore Mechanics and Arcic Engineering 18(1), Fedele, F Rogue Waves in Oceanic Turbulence. Physica D (o appear). Forrisall, GZ 000. Wave cres disribuions: observaions and second order heory. Journal of Physical Oceanography 30(8), Forrisall, GZ 005. Undersanding rogue waves: Are new physics really necessary? 14h Aha Huliko'a Hawaiian Winer Workshop, Universiy of Hawaii Janssen, P.A.E.M Nonlinear four-wave ineracions and freak waves. J. Phys. Oceanogr. 33(4): Klee, R., Schluns, K, Koschan A Three-dimensional daa from images. Springer, Singapore Krogsad H Maximum Likelihood Esimaion of Ocean Wave Specra from General Arrays of Wave Gauges. Modeling, Idenificaion, and Conrol, 9(): Longue-Higgins, M.S The effecs of non-lineariies on saisical disribuions in he heory of sea waves. J. Fluid Mech. 17, Ma, Y. Soao, S., Kosecka, J., Shankar Sasry, S., 004. An inviaion o 3-D vision: from images o geomeric models. Springer-Verlag New York Osher S. & Sehian J Frons propagaing wih curvauredependen speed: algorihms based on Hamilon-Jacobi equaions. J. of Comp. Physics, 79:1-49. Socque-Juglard, H., Dyshe, K., Trulsen, K., Krogsad, H. E. & Liu, J Probabiliy disribuions of surface graviy waves during specral changes. Journal of Fluid Mechanics 54: Sanel F., Linder W., Heipke C. 004 Sereoscopic 3D-Image Sequence Analysis of Sea Surfaces. Proceedings of he ISPRS Commission V Symposium, vol. XXXV, par 5: pp Schumacher, A Sereophoogrammerische Wellenaufnahmen. Wiss Ergeb. Dsch. Alan. Exped. Forschungs Vermessung. Meeor , Ozeanographische Sonderunersuchungen, Erse Lieferung, Berlin. Shemdin, OH, Tran HM, Wu SC Direcional measuremens of shor ocean waves wih sereography. J. Geophys. Res., 93: Shemdin, OH, Tran, HM, 199. Measuring Shor Surface Waves wih Sereography. Phoogrammeric Engineering & Remoe Sensing, 58: Shankar Sasry, S., 004. An inviaion o 3-D vision: from images o geomeric models. Springer-Verlag New York Tayfun, MA On narrow-band represenaion of ocean waves. Par I: Theory. J. Geophys. Res., (1(C6): Tayfun, MA Disribuions of envelope and phase in weakly nonlinear random waves. ASCE Journal of ngineering Mechanics, 10(4): Tayfun, MA 006. Saisics of nonlinear wave cress and groups. Ocean Engineering, 33(11-1): Tayfun, A., Fedele F Wave heigh disribuions and nonlinear effecs. Ocean Engineering 34(11-1):1631,1649. Yezzi A. & Soao S. 001 Sereoscopic Segmenaion. Inernaional Conference on Compuer Vision, Zakharov VE Saisical heory of graviy and capillary waves on he surface of a finie-deph fluid. European Journal of Mechanics B-fluids 18(3): Copyrigh 008 by ASME