Watermarking 2D Vector Maps in the Mesh-Spectral Domain

Transcription

1 Watermarkng 2D Vector Maps n the Mesh-Spectral Doman 1 Ryutarou Ohbuch, 1 Hroo Ueda, 2 Shuh Endoh ohbuch@acm.org, k7026@kk.yamanash.ac.p, endoshu@p.bm.com 1 Computer Scence Department, Unversty of Yamanash Takeda, Kofu-sh, Yamanash-ken, , Japan. 2 GIS Busness Promoton, IBM Japan. Abstract Ths paper proposes a dgtal watermarkng algorthm for 2D vector dgtal maps. The watermark s a robust, nformed-detecton watermark to be used to prevent such abuses as an ntellectual property rghts volaton. The algorthm proposed n ths paper embeds watermarks n the frequency-doman representaton of a 2D vector dgtal map. Our method treats vertces n the map as a pont set, and mposes connectvty among the ponts by usng Delaunay trangulaton. The method then computes the mesh-spectral coeffcents [Karn00] from the mesh created. Modfcatons of the coeffcents accordng to the message bts, and nverse transformng the coeffcents back nto the coordnate doman produces the watermarked map. Our evaluaton experments showed that the watermark produced by the method s resstant aganst addtve random nose, smlarty transformaton, vertex nserton and removal. It s also resstant, to some extent, aganst croppng. Compared to our prevous algorthm [Ohbuch02], the algorthm descrbed n ths paper showed sgnfcantly mproved attack reslency. 1. Introducton Applcatons of dgtal maps have been ncreasng rapdly. They are used, for example, n car navgaton systems, locaton-based servces for cellular phones wth GPS (Global Postonng System) capablty, web-based map servces, and n geographcal nformaton systems (GIS) for cty plannng or dsaster management. Dgtal maps are easy to update, duplcate, and dstrbute. As a dgtal data, dgtal maps are very easy to update, duplcate, and dstrbute. They are also prone to such abuses as llegal duplcaton and llegal dstrbuton. Geographcal maps may be publshed by a government agency and shared (wth fees) among map producng companes. Map companes add value to the base maps. For example, a car-navgaton map should have up-to-date buldng postons, road and buldng outlnes, buldng ownershp, road sgns, busness and shop data (e.g., gas statons, hotels, and convenence stores), all of whch are the result of the work whch s often human-resource ntensve. Dgtal watermarkng s a possble approach to counter abuses of dgtal meda data, such as texts, audo data, mages, moves, as well as 2D dgtal maps [16, 6, 12]. Dgtal watermarkng adds a structure called watermark to the target data obect (mostly) mperceptbly to the users and nseparably from the obect. The nformaton encoded n the watermark can be used, for example, to dentfy the copyrght owner or to detect tamperng. Two-dmensonal (2D) dgtal maps can be classfed nto ether raster- or vector-dgtal maps (Fgure 1). A raster dgtal map represents a map as raster mage data,.e., an mage represented by a 2D array of pxels. A lmtaton of raster dgtal maps s qualty degradaton caused by rotaton, scalng, and other geometrcal transformatons. Many web-based map servces uses raster dgtal maps exactly for ths reason; raster dgtal maps havng lmted resoluton have a lmted value for reuse or redstrbuton. As an mage data, most of the watermarkng algorthm developed for dgtal mages [16, 6, 12] can be appled to the raster dgtal map. A vector dgtal map, on the other hand, employs geometrcal prmtves such as ponts, lnes, polylnes, and polygons to represent obects n the maps, such as buldng outlnes, roads, rvers, reference ponts for strngs, and contour CS dept. Fgure 1. Raster (left) and vector (rght) dgtal maps. 1

2 lnes. Unlke the raster dgtal maps, the vector dgtal maps have an advantage of beng able to be scaled and rotated wthout loss of qualty. Ths advantage, on the other hand, makes a vector dgtal map a more valuable target for theft than an mage dgtal map. Ths paper presents a robust, nformed-detecton watermarkng algorthm for 2D vector dgtal maps. The watermarkng algorthm treats vertces n the map as a pont set, and creates a 2D mesh out of the pont set by usng Delaunay trangulaton. The algorthm then transforms the mesh shape nto a frequency doman representaton by usng the mesh spectral analyss technque proposed by Karn et al [14, 15]. The algorthm modfes the most sgnfcant, that s, the low-frequency coeffcents to encode watermark message bts. An nverse transformaton back nto the spatal doman creates a watermarked map. Experments showed that the watermark produced by the method s resstant aganst (1) addtve random nose, (2) global Affne transformaton, (3) vertex nserton and deleton, and (4) scramblng of obect orders n the fle. It s also resstant, to a certan extent, aganst (5) croppng. The method s mldly resstant to local deformatons as well. Compared to our prevous algorthm [22], the algorthm descrbed n ths paper showed sgnfcantly mproved attack reslency. The rest of the paper s organzed as follows. After revewng prevous work n the next secton, we wll present our watermarkng algorthm Secton 2. We then descrbe the results of evaluaton experments n Secton 3, followed by a concluson and future work n Secton Prevous Work We know of only a few publshed works on watermarkng vector dgtal maps [18, 17, 10]. Kurhara et al [18] encoded nformaton nto ndvdual vertex coordnate, and ther watermarks are qute fragle, among others, aganst addtve random nose. Endoh, Masuda, Kana, and Ohbuch collaboratvely developed nearly a dozen algorthms to watermark vector dgtal maps [10]. These watermarkng methods are avalable as a part of the GIS map development toolkt. These watermarkng methods targeted ether vertex coordnate or vertex connectvty for watermarkng. Ktamura et al reported, n detal, one of the algorthms developed by the collaboraton [17]. In the Ktamura s method, a vector dgtal map s converted nto a 2D array of scalar values,.e., a raster mage. They subdvded the dgtal map unformly nto a rectangular grd, and treated each rectangle of the grd as a pxel of a raster mage. As the pxel value of the mages, they used the average of the areas of buldngs defned by polygons that fall nsde each rectangular pxel. The mage s then watermarked by usng a method smlar to a wavelet-based magewatermarkng algorthm. We have prevously reported a watermarkng algorthm for 2D vector dgtal maps [22]. The algorthm used a smple dea of translatng a set of vertces n a unformly subdvded rectangular regon for embeddng a message bt. The drecton of translaton of the set of vertces n a rectangle encoded a bt of the watermark message. The algorthm employed modfed quad-tree [27] subdvson to create the rectangles adaptvely to the densty of vertces. A depth-frst traversal of the quad tree created an orderng among the pxels. By averagng the dsplacement of the vertces upon extracton, and by repeatedly embeddng the same message many tmes over a map, the watermarks produced by the method are resstant aganst addtve random nose, As a watermarkng target, a three-dmensonal (3D) polygonal mesh s somewhat smlar to a 2D vector map; both are defned as a set of vertces (wth ether 2D or 3D coordnate values) and ther connectvty. There are algorthms for watermarkng 3D meshes [19, 20, 13, 1, 29, 24, 28, 32, 30, 5, 23]. These algorthms alter ether vertex coordnate or vertex connectvty of the meshes for watermarkng. Many of the recent 3D mesh watermarkng methods employed transformed-doman approaches to watermarkng [13, 24, 21, 30, 5, 23]. A method n ths class would transform the mesh nto a doman that reflects the noton of frequency and modfy the most sgnfcant, low frequency components of the mesh shape to embed watermark messages. By modfyng the low frequency component, the watermarks embedded by usng these frequency-doman technques are resstant aganst addtve random nose, mesh smoothng (.e., low-pass flterng), and other attacks. Our ntal deas queston was f we could apply technques developed for 3D polygonal meshes to watermarkng 2D vector dgtal maps. Of course, there are dfferences between 2D vector dgtal maps and 3D polygonal meshes, one of whch s the relatve ease of pose normalzaton. One of the maor dffcultes n watermarkng 3D meshes s that of pose normalzaton, that s, normalzaton of the poston, sze, and orentaton of the orgnal and watermarked 3D meshes. Pose normalzaton of a 3D model s qute 2

3 dffcult f mesh smplfcaton or remeshng has been appled n addton to geometrcal transformaton. In the case of the 2D vector maps, however, pose normalzaton s relatvely easy. A reference map to algn a watermarked map aganst can be found most of the tme, so the scale and the orentaton of the watermarked map can be normalzed easly. In the algorthm reported n ths paper, we adopted the frequency doman mesh watermarkng approach of [21, 23]. We also explot a characterstc of 2D vector dgtal maps, namely, the ease of pose normalzaton, n developng our robust, nformed-detecton watermarkng algorthm for 2D vector dgtal maps descrbed n ths paper. 2. The algorthm Our watermarkng algorthm embeds message bts nto a 2D vector dgtal map by modfyng a frequency doman representaton of the map. Fgure 2 shows an overvew of the embeddng and extracton steps. To compute the frequency doman representaton, the algorthm frst establshes connectvty among vertces of the map by usng Delaunay trangulaton, creatng a 2D mesh that covers every vertex n the map. The mesh s then transformed nto a frequency doman representaton by usng mesh spectral analyss proposed by Karn and others [14, 15]. Modfcaton of the frequency doman coeffcents accordng to the message bts embeds a watermark. Inverse transformng the modfed coeffcent back nto the coordnate doman produces a map wth the watermark embedded. The modfcaton of coeffcents n the frequency doman ultmately dsplaces vertex coordnates n the spatal doman. For computatonal effcency and for robustness aganst croppng, a map s frst dvded nto many rectangular subareas. We employed the k-d tree subdvson [9, 27] adaptvely to the densty of vertces n the map so that sub-areas have approxmately equal numbers of vertces. Aforementoned mesh spectral analyss and watermark embeddng s performed for each of the sub-areas. By embeddng the same watermark repeatedly n multple sub-areas, the watermark becomes reslent aganst croppng. The watermark s reslent aganst random nose and other attacks snce the watermark s embedded nto the low frequency component of the mesh verson of the map. The watermark s an nformed- (or non-blnd-) detecton watermark. Watermarks are extracted by Embeddng Reference map M Delaunay meshng Patch generaton Spectral analyss Modulaton Spectral synthess Watermarked map ˆM Watermark message Extracton Reference Watermarked map M map ˆM Pose normalzaton Vertex matchng Delaunay meshng Patch generaton Spectral analyss Demodulaton Watermark message Fgure 2. An overvew of the watermark embeddng and extracton steps. comparng the reference map (the map before watermarkng, whch may be escrowed) wth the watermarked and possbly attacked watermark map. To extract the watermark, the two maps are frst geometrcally regstered by usng an teratve optmzaton process to mnmze dstance among a set of landmarks. Ths regstraton could remove an Affne transformaton appled to the watermarked map. Then, the area subdvson equal to the one used for the embeddng s recreated on the reference map, and the subdvson s transferred to the watermarked map. For each correspondng sub-area, mesh spectral analyss and then comparson of spectral coeffcents recovers the embedded watermark. 3

4 2.1. Embeddng Meshng and area subdvson A vector dgtal map s a collecton of polygons and polylnes that are not connected to each other. We frst connect all the vertces nto a sngle mesh by usng Delaunay trangulaton [9]. Fgure 3a shows a vector dgtal map and ts Delaunay mesh s shown n Fgure 3b. The algorthm then creates multple rectangular submeshes, called watermarkng patches. The purpose of subdvson nto watermarkng patches are twofold; (1) to ncrease reslency aganst croppng attacks by repeatedly embeddng the same watermark nto multple patches of a map, and (2) to reduce computatonal cost of egenanalyss, the core of the mesh spectral analyss, by reducng the mesh sze. The patches generated should contan roughly equal number of vertces, and that the number of vertces must exceed certan threshold to ensure the payload (.e., the amount of watermark message bts embeddable.) We employed k-d tree [9, 27] to adaptvely subdvde the mesh nto patches of roughly equal vertex counts. An example of the patches generated by ths technque s shown n Fgure 3c, overlad on the orgnal map Spectral analyss (a) Orgnal map. (b) Vertces of the map to the left are Delaunay trangulated. Mesh spectral analyss has been ntroduced by Karn and Gotsman to analyze shapes of 3D polygonal mesh models for compressng ther geometry [14, 15]. Ohbuch et al. [21, 23], and later, Cayre et al [5], appled the technque for watermarkng 3D polygonal meshes n the frequency or mesh spectral doman. We borrow the technque for watermarkng 2D vector dgtal maps by convertng the maps nto 2D meshes pror to watermarkng. There are several dfferent mesh Laplacan matrces [4, 3, 8]. We employ Bggs defnton of mesh Laplacan R for the algorthm descrbed n ths paper. R = I HA (1) In the formula, I s the dentty matrx and H s a dagonal matrx whose dagonal element H = 1 d s the recprocal of the degree (or valence) of the vertex. A s the adacency matrx whose elements are defned as below; 1, f vertces and are adacent; a = (2) 0, otherwse. Fgure 4a shows a smple mesh and Fgure 4b shows ts Laplacan matrx. A polygonal mesh M havng n vertces yelds a Laplacan matrx R of sze n n. Egenanalyss of R produces n egenvalues λ and n n-dmensonal egenvectors w (1 n). Proectng each component of the vertex coordnate v = ( x, y) (1 n) separately onto the -th normalzed egenvectors e e = w w (1 n) (3) (c) Watermark patches are generated adaptvely to the local vertex counts. Fgure 3. Vertces are trangulated to create a mesh, whch s then subdvded nto watermark patches. 4

5 produces n mesh spectral coeffcent vectors r = ( rs,, rt, ) (1 n). The subscrpts s, and t denote orthogonal coordnate axes n the mesh-spectral doman correspondng to the spatal axes x and y. We form the matrx R for a watermark patch usng the connectvty wthn the patch. Edges connectng the patch wth other patches are not ncluded n our Laplacan matrx. The nverse transformaton, the mesh-spectral synthess, s smply a lnear combnaton of the bass e scaled by the mesh spectral coeffcents r. ( ) ( ) T 1 2 n = s,1e1 + s,2e2 + + s, nen T 1, 2,..., n = t,1e1 + t,2e2 + + t, nen. x, x,..., x r r r, y y y r r r The spectral coeffcents represent the noton of frequency (n the sense of the Fourer transformaton) of the shape of the mesh, especally f (1) the lengths of edges are unform over the mesh, and that (2) the degrees of vertces are unform over the mesh [14, 15]. The mesh produced by the Delaunay trangulaton has more unform edge length than by the other trangulaton methods gven a set of ponts. However, as t s obvous from the example shown n Fgure 3, the trangles n the mesh have a wde range of sze and varyng aspect rato that may nterfere wth the Frequency decomposton of the mesh shape by usng the mesh spectral analyss. A E B C D (a) A smple example mesh. A B C D E F A B C D E F (b) The Laplacan matrx for the mesh. Fgure 4. An example of the mesh Laplacan matrx. F (4) Modulaton After the spectral coeffcents are computed, a watermark s embedded nto the map by modfyng the spectral coeffcents accordng to the message bts. The algorthm employs a smple modulaton method smlar to Hartung s [11]. The data to be embedded s an m- dmensonal bt vector a = ( a1, a2,..., am ) n whch each bt takes values { 0,1 }. Each bt a s spread spatally over the map by duplcatng each symbol by chp rate c, producng a watermark symbol vector b = ( b1, b2,... b mc ), b { 0,1} of length mc. Repeatedly embeddng the same bt c tmes ncreases reslency of the watermark aganst addtve random nose. If a watermark patch contans more vertces than the specfed mnmum d, the maxmum value for the repetton s c = floor( L n) where n s the number of bts of the payload, the watermark message. For example, a mesh contans 480 vertces and the payload s 128 bt, chp rate c can be 1, 2, or 3. Each element b of the symbol vector b s then repeated or spread c tmes; b = a, c < ( + 1) c (5) After the spreadng, the bt vector b s converted to an embeddng symbol vector b = ( b 1, b 2,... b mc ), b { 1,1} by the followng mappng to create a zero-mean sgnal; 1, f b = 0; b = (6) 1, f b = 1. Assume that there are L usable rectangles and that th rectangle contans M vertces. Let v m, be the coordnate of mth vertex (1 < m< M) n the th rectangle pror to the watermarkng, p { 1,1} be the pseudo-random number sequence (PRNS) generated from a known key k w, and α ( α > 0) be the modulaton ampltude. The coordnate s ˆ. m of the vertex after watermarkng s computed by the followng formula; s = s + b p α (7) ˆ The extracton requres the key w k used for the embeddng. Key dstrbuton can be acheved by usng a publc-key cryptography scheme, for example. The modulaton ampltude α n the mesh spectral doman should be chosen so that the vertex dsplacement n the spatal doman won t affect vsual qualtes of maps. The geographcal map standard by the Geographcal Survey Insttute of Japan states that the maxmal error arrowed n a 1/2500-scale map s 0.3mm, whch corresponds to 75 cm n the real world. Perturbaton of 5

6 vertces on the map by 2 or 3 nteger coordnate ponts, that are, 20 or 30 cm n the real world, should be acceptable as long as the dscontnuty artfacts ntroduced by the dsplacements are unnotceable. Snce the spatal doman dsplacement s determned by the formula (4), both the modulaton ampltude α as well as the number of coeffcents modfed, that s, the chp rate c, determnes the vertex dsplacement n the spatal doman. The vertex dsplacement s roughly proportonal to the product of α c Extracton Pose normalzaton Pror to the extracton, an affne transformaton appled to the watermarked map ˆM s removed. Ths s a rather smple case of so-called affne matchng problem (e.g., [6, 26]). In our algorthm, a set of correspondng pars of landmarks are selected n the maps M and ˆM. Then, the sum of Eucldan dstance between the landmark pars s mnmzed. Ths mnmzaton s performed by repettvely applyng mnscule rotaton, translaton, and scalng transformatons. Our algorthm uses the coordnate of the lower-left corner of a strng that s assocated wth a buldng (or any other land obect) for the matchng. Correspondence between a par of landmarks n the maps can be establshed easly by smply comparng the strngs attached to them. We typcally employ about 30 to 80 landmark pars for the pose normalzaton. In terms of watermarkng, the most sgnfcant dfference between 2D vector dgtal map and the 3D polygonal meshes les n the relatve dffculty of the pose normalzaton. For 2D vector dgtal maps, accurate pose normalzaton s possble even after affne transformaton usng (lterally) landmarks. For 3D mesh models, even wth reference mesh, such normalzaton can be qute dffcult f the mesh had gone through geometrcal transformaton combned wth mesh smplfcaton and other connectvty changes Vertex matchng After the pose normalzaton, vertces that are ether nserted or deleted due to attacks are found by comparng the coordnate values of the reference map M and the watermarked map ˆM. To deal wth the vertex-nserton attack, the algorthm chooses a vertex n ˆM that s nsde the crcle of dameter t of the vertex v r of M. If there s more than one such vertex, the vertex closest to v r s used. To deal wth the vertex-removal attack, f no vertex n ˆM s found nsde the crcle of dameter t of the vertex v r of M, a vertex v w s nserted nto ˆM. The nserted vertex v w has the coordnate of v r. The coordnate value of v r s of course ncorrect; t s smply treated as nose by the extracton algorthm. The dameter t s a user-defned parameter. We chose the value of t=100 cm for the experments descrbed below. The value s chosen based on the maxmal error of 75 cm (n the real-world scale) arrowed n a 1/2500-scale Japanese geographcal map Meshng and patch generaton The reference map M s Delaunay trangulated, and the watermarkng patches are created. The trangulaton and the patches on M are exactly the same as the ones used for embeddng. The trangulaton and the patchfcaton are then transferred to the watermarked map ˆM. Usng the vertex-to-vertex correspondence between the maps M and ˆM establshed above Spectral Analyss The spectral analyss s performed frst on the reference mesh, whch produces exactly the same set of egenvectors and mesh spectral coeffcents as the ones computed for the M durng embeddng. Note that the egenvectors computed for the M can be used to derve mesh spectral coeffcents for the ˆM, for they have the same connectvty. Expensve egenvalue decomposton computaton need to be performed only once per patch for the reference map M Demodulaton To extract a message bt, the algorthm compares a mesh spectral coeffcent of M wth the correspondng coeffcents of ˆM. Let s assume that the th coeffcents of ˆM and ˆM are s and s ˆ, respectvely, and that p s the same PRNS as s used for embeddng, generated from the same stego-key k w. Then, the sum of the products q can be computed as follows; ( + 1) c 1 ( + 1) c 1 2 = ( ˆ ) = α = c = c (8) q s s p b p If p s the same for embeddng and extracton, and f dsturbances of the vertex coordnates of ˆM (e.g., addtve random nose) are neglgble, q = c α b (9) 6

7 where q takes one of the two values { αc, αc}. Snce α and c are always postve, smply testng for the sgns of q recovers the orgnal message bt sequence a, a = sgn( q ) (10) 3.1. Perceptblty Perceptblty of the watermark depends on varous The strng a can easly be converted to the orgnal message bt sequence b by applyng an nverse of the mappng as the embeddng. 3. Experments and results In all the experments descrbed below, we used the followng parameters. Mnmum patch sze d: d = 480 s used. The perceptblty experment used d = 128 as well. Payload: A message of sze 128 bt s embedded. Modulaton ampltude α : α = 1.0 and α = 1.5. Chp rate c: c=1 for the cases where d 128, and c=2 and c=3 for the cases where d 480. For the experments, we used the 6 maps lsted n (a) Orgnal map (enlarged.) (b) d=128, c=1, α =1.0 (c) d=128, c=1, α =1.5 Map A Map B Map C (d) d=480, c=2, α =1.0 (e) d=480, c=3, α =1.0 Map D Map E Map F Fgure 5. Sx maps used for the evaluaton experment. Fgure 5. As we target maps that manly represent houses and buldngs, we choose the maps from the urban and suburban resdental and commercal areas. (f) d=480, c=2, α =1.5 (g) d=480, c=3, α =1.5 Fgure 6. Perceptblty of the watermark usng varous watermarkng parameters. parameters, ncludng the payload m, patch sze d, the chp rate c, and the modulaton ampltude α. An ncrease n c or α mproves the attack reslency but t decreases the vsual qualty of the map. The effect of the patch sze d s 7

8 subtler. For example, a larger patch would have a hgher reslency aganst addtve random nose, but the resultng decrease n the number of patches per map reduces the reslency aganst croppng attacks. Fgure 6a-6g shows the effect on vsual qualty of the map of the watermarkng, usng 6 combnatons of watermarkng parameters. The fgure shows the map area of the sze approxmately 30m 30m n the real world. A hgher chp rate c and the hgher ampltude α (of the coeffcents modulaton) clearly degraded the vsual qualty of the map. In ths example, parameters d=480, c=2, α=1.0 showed the least amount of dstorton. It also showed acceptable performance n terms of attack reslency n the experments that follows Reslency aganst attacks We expermentally evaluated the attack reslency of the watermark usng the followng attacks; 1. Translaton: Translate all the vertces n the map by 1000 unts and 500 unts, respectvely, toward postve x and y drectons. 2. Upscalng: Unformly enlarge the map by the factor Downscalng: Unformly shrnks the map by 0.3 (attack 3a) and 0.6 (attack 3b) tmes the orgnal. Coordnate values are rounded to the nearest ntegers. 4. Rotate: Rotate the map by 45 degree about the upperleft corner (0,0) (attack 4a) or the center (3750, 2500) (attach 4b) of the map. 5. Smlarty transformaton: The map s rotated by 45 degree about the center of the map (3750, 2500), translated, and then downscaled by the factor of 0.6. Coordnate values are rounded to the nearest ntegers after the downscalng. 6. Local deformaton: The map s subdvded unformly nto rectangular grd of sze 10 10, and the vertces nsde each rectangle are rotated by 1 degree about the center of the rectangle. Gven the vertex coordnate of ( x, y ), the drecton of rotaton for the coordnate s clockwse f ( x+ y)mod2= 0, counterclockwse f otherwse. 7. Obect order scramblng: The order of appearance of obects (e.g., polygons of buldng outlnes) n a data fle s scrambled. 8. Vertex nserton: Vertces are added to target obects.e., polygons and polylnes, whle tryng to preserve the appearance of the map. In the experment, 3000 vertces are added to a map. 9. Addtve random nose: Random nose havng the ampltudes of ether α =10 cm, 30cm, or 50cm (attacks 9a, 9b, and 9c, respectvely) s added to the vertex coordnate. 10. Croppng: Each map s cropped accordng to the 8 croppng patterns shown n Fgure 7. Areas of the cropped maps vared from 1/2 to 1/16 of the orgnal. Table 2 shows the results of the experments. In Table 2, all the bt error rates are the average over the sx maps used for the experment. Table 2 also ncludes the results obtaned usng our prevous watermarkng method [22] for comparson. Some of the boxes are left vacant, as these fgures are not avalable for our prevous method. Overall, the new frequency doman watermarkng method descrbed n ths paper sgnfcantly outperformed our prevous algorthm n terms of attack reslency. Comparng between the chp rates of 2 and 3, the results produced by the hgher chp rate of c=3 produced somewhat less error than the lower chp rate of c=2. The new watermark s robust aganst translaton, upscalng, or rotaton n whch case no error occurred. The new watermark s mmune to vertex nserton and scramblng of obect order n the map data fle as expected. Pattern 1 Pattern 2 Pattern 3 Pattern 4 Pattern 5 Pattern 6 Pattern 7 Pattern 8 Fgure 7. Eght croppng patterns used for the experments. Downscalng by the factor of 0.3 caused errors due to the round off error of the coordnate values. A combnaton of rotaton, translaton, and downscalng also caused errors, most lkely due to the round off error due to downscalng. 8

9 The watermark showed sgnfcant error after the local deformaton. Unlke a global geometrcal transformaton (e.g., smlarty transformaton), local random deformaton can t be compensated for by our pose normalzaton method. The watermarks also showed sgnfcant errors after the croppng that reduced the area of the map down to 1/2~1/16 of the orgnal. In both attacks, the error rates of the new algorthm are agan much lower than those usng our prevous method. 4. Concluson and future work In ths paper, we presented a frequency-doman watermarkng algorthm for vector dgtal maps. The algorthm embeds bts nto a map by modfyng frequency doman representaton of the map. The mesh spectral coeffcents are computed by frst convertng the Experments showed that the watermarks produced by our new method descrbed n ths paper are more reslent than our prevous algorthm [22]. The watermark produced by new algorthm s reslent, to some extent, aganst such attacks as addtve random nose, addton of vertces, rotaton, scalng, and croppng of the map. Our future work ncludes mprovements n attack reslency. For example, reslency aganst addtve random nose or other frequency dependent attacks can be mproved by computng a better frequency doman representaton of the shape. We also need to fnd a quanttatve measure of dstorton that reflects human percepton, as well as standard sets of attacks and maps that can be used for obectvely evaluatng and comparng the proposed map watermarkng methods. Table 2. Bt error rates due to varous attacks. The payload was 128 bt. The N/M and N/A n the boxes ndcate, respectvely, not measured and not avalable. New method Prevous method Mnmum vertex counts per patch d Modulaton ampltude α Chp rate c 2 3 (1) Translaton 0.0 % 0.0 % 0.0 % (2) Upscalng ( 5.5) 0.0 % 0.0 % 0.0 % (3) Downscalng (3a) % 0.0 % 0.7 % (3b) % 0.0 % --- (4) Rotaton (4a) Center at (0, 0), 45 degree 0.0 % 0.0 % 0.0 % (4b) Center at (3750, 2500), 45 degree 0.0 % 0.0 % 0.0 % (5) Smlarty transformaton ((4b)+(1)+(3a)) 0.1 % 0.0 % 1.0 % (6) Local deformaton 9.2 % 8.1 % 45.2 % (7) Obect order scramblng 0.0 % 0.0 % 0.0 % (8) Vertex nserton (3000 vertces) 0.0 % 0.0 % 0.0 % (9a) α=10cm 0.0 % 0.0 % 0.0 % (9) Addtve (9b) α=30cm 0.1 % 0.0 % --- random nose (9c) α=50cm 5.9 % 3.4 % 8.5 % (10a) Pattern % 0.0 % 1.4 % (10b) Pattern % 0.4 % 1.8 % (10c) Pattern % 1.3 % 7.0 % (10) Croppng (10d) Pattern % 0.0 % 6.8 % (10e) Pattern % 0.0 % 16.7 % (10f) Pattern % 0.4 % 15.0 % (10g) Pattern % 0.4 % 11.3 % (10h) Pattern % 15.1 % 35.9 % map nto a mesh by usng Delaunay trangulaton, and then applyng mesh spectral analyss proposed by Karn et al [14, 15]. References [1] O. Benedens, Geometry-Based Watermarkng of 3D Models, IEEE CG&A, pp , January/February

10 [2] O. Benedens, C. Busch, Towards Blnd Detecton of Robust Watermarks n Polygonal Models, Proc. EUROGRAPHICS 2000 (Computer Graphcs Forum, Volume 19(2000), No. 3) [3] N. Bggs, Algebrac Graph Theory (2 nd Ed.). Cambrdge Unversty Press, [4] B. Bollobás, Modern Graph Theory, Sprnger, [5] F. Cayre, P. Rondao Alface & H. Maître, Compresson and watermarkng of 3D trangle meshes, SPIE 47th Annual Meetng, Jul. 2002, Seattle, WA (2002) [6] G. S. Cox, G. DeJager, A survey of pont pattern matchng technques and a new approach to pont pattern recognton, Proc. South Afrcan Symposum on Communcatons and Sgnal Processng 1993, pp [7] Ingemar J. Cox, Matthew L. Mller, Jeffrey A. Bloom, DIGITAL WATERMARKING, Morgan Kaufmann Publshers, [8] F. R. K. Chung, Spectral Graph Theory, Number. 92 n Regonal Conference Seres n Mathematcs, Amercan Mathematcal Socety, [9] M. de Berg, M. van Kreveld, M. Overmars, O. Schwarzkopf, Computatonal Geometry: Algorthms and Applcatons, 2 nd edton, Sprnger, [10] Shuh Endoh, Hrosh Masuda, Ryutarou Ohbuch, Satosh Kana, Development of Dgtal Watermarkng Technology for Vector Dgtal Maps, IPA Technology Expo 2001 Reports, 2001 (Japanese). [11] F. Hartung, P. Esert, and B. Grod, Dgtal Watermarkng of MPEG-4 Facal Anmaton Parameters, Computers and Graphcs, Vol. 22, No. 4, pp , Elsever, [12] Nel F. Johnson, Zoran Durc, and Sushl Jaoda, Informaton Hdng: Steganography and Watermarkng Attacks and Countermeasures, Kluwer Academc Publshers, [13] S. Kana, H. Date, and T. Kshnam, Dgtal Watermarkng for 3D Polygons usng Multresoluton Wavelet Decomposton, Proc. Sxth IFIP WG 5.2 GEO-6, pp , Tokyo, Japan, December ( kana/wm1-geo6.pdf) [14] Zach Karn, Crag Gotsman, Spectral Compresson of Mesh Geometry, Proc. SIGGRAPH 2000, pp , [15] Zach Karn, Crag Gotsman, 3D Mesh Compresson Usng Fxed Spectral Bases, Proc. Graphcs Interface 2001, pp. 1-8, [16] S. Katzenbesser, F. A. P. Pettcolas, Dgtal Watermarkng, Artech House, London, [17] I. Ktamura, S. Kana, and T. Kshnam, Watermarkng Vector Dgtal Map usng Wavelet Transformaton, Proc. Annual Conference of the Geographcal Informaton Systems Assocaton (GISA) 2000, Vol. 9, pp , 2000 (Japanese). [18] M. Kurhara, N. Komatsu, H. Arta, Watermarkng Vector Dgtal Maps, Specal Interest Group Report Vol. 2000, No. 36, Informaton Processng Socety of Japan (IPSJ), (Computer Securty No , 2000) (Japanese). [19] R. Ohbuch, H. Masuda, and M. Aono, Watermarkng Three-Dmensonal Polygonal Models, Proc. ACM Multmeda 97, pp , Seattle, Washngton, USA, November [20] R. Ohbuch, H. Masuda, and M. Aono, Watermarkng Three-Dmensonal Polygonal Models Through Geometrc and Topologcal Modfcatons, pp , IEEE JSAC, May [21] Ryutarou Ohbuch, Shgeo Takahash, Takahko Myazawa, and Ako Mukayama, Watermarkng 3D Polygonal Meshes n the Mesh Spectral Doman, n Proc. Graphcs Interface 2001, pp. 9-17, Ontaro, Canada, June, [22] Ryutarou Ohbuch, Hro Ueda, Shu Endoh, Robust Watermarkng of Vector Dgtal Maps, n Proc. IEEE Conference on Multmeda and Expo 2002 (ICME 2002), Lausanne, Swstzerland, August 26-29, [23] Ryutarou Ohbuch, Ako Mukayama, Shgeo Takahash, A Frequency-Doman Approach to Watermarkng 3D Shapes, Computer Graphcs Forum 21(3), pp , 2002 (Proc. EUROGRAPHICS 2002) [24] Eml Praun, Hugues Hoppe, Adam Fnkelsten, Robust Mesh Watermarkng, Proc. SIGGRAPH 99, pp , Aug [25] W. H. Press et al., Numercal Recpes n C-The Art of Scentfc Programmng, 2 nd Ed., Cambrdge Unversty Press, Cambrdge, UK, [26] I. Rothe, H. Suesse, K. Voss, The method of normalzaton to determne nvarants, IEEE Trans. on PAMI, Vol. 18, No. 4 (1996), pp [27] H. Samet, The Desgn and Analyss of Spatal Data Structures, Addson-Wesley, Readng, MA, [28] M. G. Wagner, Robust Watermarkng of Polygonal Meshes, Proc. Geometrc Modelng & Processng 2000, pp , Hong Kong, Aprl 10-12, [29] B-L. Yeo and M. M. Yeung, Watermarkng 3D Obects for Verfcaton, IEEE CG&A, pp , January/February [30] Kangkang Yn, Zhgeng Pan, Jaoyng Sh, Davd Zhang, Robust mesh watermarkng based on multresoluton processng, Computers & Graphcs, Vol. 25 (2001), pp Acknowledgements We thank Zenrn Co. Ltd. for lendng us the map data for the experments. We d lke to thank Prof. Hrosh Masuda of the Unversty of Tokyo and Prof. Satosh Kana of Hokkado Unversty for valuable dscussons and comments. We also thank Prof. Shgeo Takahash for useful comments. A part of ths research s funded by the grant No from the Mnstry of Educaton, Culture, Sports, Scences, and Technology of Japan, as well as the Okawa Foundaton of Informaton and Telecommuncatons. 10