View-Dependent Multiresolution Representation for a Height Map

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1 Internatonal Journal of Innovaton, Management and echnology, Vol. 4, No. 1, February 013 Vew-Deendent Multresoluton Reresentaton for a Heght Ma Yong H. Chung, Won K. Hwam, Dae S. Chang, Jung-Ju Cho, and Sang C. Park, Member, IACSI Abstract erran vsualzaton s a dffcult roblem for alcatons requrng accurate mage of large datasets at hgh frame rates. We roose new vew-deendent-mult-resoluton reresentaton method of large-scale terran, whch nherts the accuracy of 3D-methods as well as the memory effcent of.5d-methods. he roosed aroach mantans a full decmaton rocedure of a heght ma for a maxmum tolerance, and creates End-Start-tree, whch s used to reresent mult-resoluton trangular mesh. For the aroxmaton of a heght ma, the roosed algorthm emloys the QEM(quadrc error metrcs) method. We demonstrate new vsualzaton algorthm of large-scale terran. Index erms erran, Heght ma, Dgtal elevaton model, Vew-deendent LOD, Quadrc error metrcs, smlfcaton. I. INRODUCION o effcently vsualze a large-scale terran data, the use of a level-of-detal (LOD) has been wdely acceted. LOD technques ncrease the effcency of renderng by decreasng the workload on grahcs elne stages, usually vertex transformatons. In general, relatvely small obects ether dscrete LOD or contnuous LOD s usually suffcent, but terrans have some secal roertes we can exlot comared to generc LOD algorthms. Because the terran models are usually contnuous and large, the same model can smultaneously be both very close to the vewer and very far away from hm. hs large sze of the terran models makes t unfeasble to use a sngle LOD throughout the model and thus necesstates vew-deendent LOD.[1] Vew-deendent LOD can be thought as selectve refnement of contnuous LOD. Nearby regons of large obects can be rendered wth more detal than the regons n the dstance. Vew-deendent LOD meshes can be comuted automatcally usng mesh smlfcaton technques []. Most algorthms exect the terran to be stored as a heght ma whch s a raster mage used to store values n a unform grd for dslay n 3D comuter grahcs. Prevous results, whch can be aled to terran data smlfcaton, can be Manuscrt receved Setember 0, 01; revsed November 19, 01. hs work was suorted by the Defense Acquston Program Admnstraton (DAPA) and the Agency for Defense Develoment (ADD), Korea Research Foundaton under the Contract No. UD110006MD (DAPA), UD100009DD, UD10035JD, he authors wsh to exress sncere grattude for the fnancal suort. Y. H. Chung, W. K. Hwam, D. S. Chang and S. C. Park are wth the Deartment of Industral Engneerng, Aou Unversty, Suwon, Reublc of Korea (e-mal: yongho130@gmal.com, lunacy@aou.ac.kr, webmacome@aou.ac.kr, scark@aou.ac.kr). Jung Ju Cho s wth the Deartment of Meda, Aou Unversty, Suwon, Reublc of Korea (e-mal: ungu@aou.ac.kr). roughly classfed nto two man categores; 3D-methods [3]-[8], and.5d-methods [9]-[11]. Whle 3D-methods deal wth general 3D trangular meshes,.5d-methods are talored to heght felds whch are.5d surface. he smlfcaton methods for general 3D trangular meshes have extensvely receved ncreasng attenton. Several algorthms have been formulated for smlfyng surfaces. However, t s not ossble to drectly aly those algorthms to the smlfcaton of a heght ma. A tradtonal way s to aly those algorthms after convertng a heght ma nto a trangular mesh. However, the converson rocedure wll requre an addtonal comutatonal load as well as a large amount of memory, because the ntal trangular mesh may contan mllons of trangles. A tycal data structure for a trangular mesh has the least amount nformaton to reresent the toology and geometry of a mesh. Usually, a general trangular mesh requres at least 0 tmes of memory amount than a heght ma does. o avod above dffcultes, the smlfcaton methods for.5d heght ma are develoed. Most of.5d-methods for heght ma smlfcaton are based on quad-tree or bnary-tree.[1] Quadtrees and Bnary trees are commonly used herarchy for terran tessellaton. Both tree data structures roduce sem-regular meshes and thus smlfy the traversal algorthms and memory layout comared to the rregular meshes by 3D smlfcaton methods. Usually,.5D-methods are more smle and more effcent than 3D-methods, because they do not have to mantan the comlex toology nformaton of a trangular mesh. However, 3D-methods aroxmate the orgnal more accurately than.5d-methods by usng the same number of trangles, because 3D-methods have no constrants n the shae of aroxmatng trangles. he obectve of ths aer s to develo a new mult-resoluton reresentaton method of terrans called a VD-MR-heght-ma(Vew-deendent-Mult-resoluton-he ght-ma), suortng the hybrd smlfcaton rocedure for a heght ma, whch nherts the accuracy of 3D-methods as well as the memory effcency of.5d methods. he remander of ths aer s organzed as follows: Secton resents the overall aroach to the generaton of vew-deendent mult-resoluton models, and Secton 3 gves a detaled descrton of the roosed algorthm. Fnally, concludng remarks are resented n Secton 4. II. APPROACH As mentoned above, the roosed rocedure n ths aer suorts the vew-deendent mult-resoluton models of a heght ma by nhertng the accuracy of 3D-methods as well DOI: /IJIM.013.V

Internatonal Journal of Innovaton, Management and echnology, Vol. 4, No. 1, February 013 as the memory effcency of.5d methods.

adatve trangle mesh for a frame. he overall scheme of the roosed algorthm conssts of two man stes, as shown Fg. 1: 1).

For a vertex, the Q matrx can be comuted by summng every KP of the trangles sharng the vertex.

2 Internatonal Journal of Innovaton, Management and echnology, Vol. 4, No. 1, February 013 as the memory effcency of.5d methods. All of the exstng algorthms that can nteractvely erform vew-deendent, locally-adatve terran meshng, ncludng ours, deend on a re-defned mult-resoluton terran reresentaton that s used to buld the adatve trangle mesh for a frame. he overall scheme of the roosed algorthm conssts of two man stes, as shown Fg. 1: 1). he constructon of VD-MR-heght-ma; ) he extracton of smlfed, vew-deendent trangular mesh from the heght ma usng VD-MR-heght-ma. entre set of lanes by a sngle matrx Q. For a vertex, the Q matrx can be comuted by summng every KP of the trangles sharng the vertex. After comutng the Q matrces for all vertces, t s ossble to erform the edge contractons n order to smlfy the geometry. Whle the orgnal algorthm of the QEM method fnds an otmal oston for the contracton target, the roosed aroach n ths aer chooses V or V for the contracton target. o do ths, the contracton cost n our aroach s defned as follows: Contracton cost of an edge [V, V ] If ( v ( Q Q ) v < v ( Q Q ) v ) cost = v ( Q + Q ) v ; // V : contracton target, [V, V ] V else cost = v ( Q Q ) v ; // V : contracton target, [V, V ] V Fg. 1. Pre-rocessng and trangular mesh extracton of the roosed algorthm For the frst ste, the roosed aroach mantan a full decmaton rocedure of a heght ma for a maxmum tolerance (ε m ), whch defned by user. For the aroxmaton of a heght ma, ths aer emloys the edge contracton aroach based on the QEM (quadrc error metrcs) method[1]. he QEM s algorthm for smlfcaton of olygonal models. Because the QEM assume that the model conssts of trangles only, t s necessary to nterret the heght ma as a trangular mesh. o do so, t s necessary to determne each grd cell n a heght ma. he shorter of the two dagonals between the onts s determned, and ths s used to dentfy the two trlets of onts that may become trangles. Δ( = = v ( lanes( ( v )( lanes( lanes( K ) v = v ( Q) v = v ( ) v (1) K PP a ab ac ad ab b bc bd ac bc c cd ad bd cd d () he QEM method s based on the observaton that each vertex s the soluton of the ntersecton of a set of lanes namely, the lanes of the trangles that meet at the vertex. It s ossble to assocate a set of lanes wth each vertex, and also to defne the error of the vertex wth resect to ths set as the sum of squared dstances to ts lanes, where = [a, b, c, d] reresents the lane defned by the equaton ax+by+cz+d = 0 where a +b +c =1. he error metrc can be rewrtten n quadratc form as equaton (1). he fundamental error quadrc K P can be used to fnd the squared dstance of any ont n sace to the lane. We can sum these fundamental quadrcs together, and reresent an Fg.. Sequence of edge contractons. hrough the decmaton rocedure, edge contractons cause the toology rregular as shown n Fg.. VD-MR-heght ma s effcent data structure to reresent the toology rregular of a heght ma. Whle a heght ma s an array of real numbers, a VD-MR-heght-ma s an array of contaners, each of whch can store a real number as well as the decmaton related nformaton. As a rerocessng ste, a tree of a VD-MR-heght-ma s bult. If a node s not a root 105

Internatonal Journal of Innovaton, Management and echnology, Vol. 4, No. 1, February 013 node, the node has a arent node. he arent node of chld node cannot be more than two.

In ths aer, ths tree, called a ES-tree, s used to reresent mult-resoluton trangular mesh. As a rerocessng ste, heght ma conssts of several ES-tree whch has each tolerance.

VD-MR-heght-ma store the toology changes by usng the End_d, nstead of mantanng a heavy data structure such as a general trangular mesh.

ALGORIHM As mentoned above, a VD-MR-heght-ma s an array of contaners, each of whch can store a real number as well as the decmaton related nformaton.

contracton cost f the vertex s merged; 5) ree_tolerance, the contracton cost for ES-tree. Fg. 4 shows local areas of a heght ma. ES-tree conssts of four nodes, and ES-tree has tolerance 0.7.

Because V 1, V, V 5, V 6, V 8 do not have End_d, they are also root node. hus, there are sx ES-trees n Fg. 3(a). In the same way, Fg. 3(b) shows another ES-tree whch has tolerance 0.8. he tolerance 0.

3 Internatonal Journal of Innovaton, Management and echnology, Vol. 4, No. 1, February 013 node, the node has a arent node. he arent node of chld node cannot be more than two. If a tree conssts of a arent node and a chld node, the arent node s a end vertex, the chld node s a start vertex. In ths aer, ths tree, called a ES-tree, s used to reresent mult-resoluton trangular mesh. As a rerocessng ste, heght ma conssts of several ES-tree whch has each tolerance. ES-tree s used to buld contnuous adatve meshes. For vew-deendent vsualzaton of heght ma, each ES-tree has certan tolerance whch s roortonal to the dstance between root node and camera. VD-MR-heght-ma store the toology changes by usng the End_d, nstead of mantanng a heavy data structure such as a general trangular mesh. he constructon rocedure of the VD-MR-heght-ma can be descrbed as Fg. 4. III. ALGORIHM As mentoned above, a VD-MR-heght-ma s an array of contaners, each of whch can store a real number as well as the decmaton related nformaton. A VD-MR-heght-ma has the followng nformaton; 1) value, a real number ndcatng the heght; ) Q, a matrx for QEM; 3) End_d, the d of the end of ths vertex f the vertex s merged; 4) cost, the edge contracton cost f the vertex s merged; 5) ree_tolerance, the contracton cost for ES-tree. Fg. 4 shows local areas of a heght ma. ES-tree conssts of four nodes, and ES-tree has tolerance 0.7. For examle, V 4 of the VD-MR-heght-ma contans followng nformaton, V 4 [value = heght 4, Q = Q 4, End_d = d of V 7, cost = 0.65, ree_tolerance = 0.7]. Because V 1, V, V 5, V 6, V 8 do not have End_d, they are also root node. hus, there are sx ES-trees n Fg. 3(a). In the same way, Fg. 3(b) shows another ES-tree whch has tolerance 0.8. he tolerance 0.8 of ES_tree reresents that the dstance between the root node of ES-tree and camera s farther than the dstance between the root node of ES-tree 1 and camera. However, t s not that all node of ES-tree n comarson wth ES-tree 1 s far. Fg. 4. ree-tolerance comutaton rocedure Once the VD-MR-heght-ma s constructed for ε m, we can extract a smlfed trangular mesh for certan ree-tolerance (0 < ε t < ε m ) by alyng edge contractons havng costs less than ε. he ree tolerance ε t s decded by the dstance between the root node of ES-tree and camera, as follow : ree_tolerance = Dstance * Smlfcaton coeffcent - Smlfcaton coeffcent s statc. - Smlfcaton coeffcent s decded by some tests. - ree_tolerance s smaller than maxmum tolerance(ε m ) he rocedure to comute ree-tolerance can be descrbed as follow: Fg. 3. VD-MR-heght-ma and ES-tree for the tolerance An edge contracton makes the toology rregular, and the Fg. 5. ree_tolerance comutaton rocedure 106

Internatonal Journal of Innovaton, Management and echnology, Vol. 4, No. 1, February 013 As shown Fg. 5, the ree-tolerance of a certan vertex s decded by root node.

For each vertex of the two trangles, we can trace the end vertex n a recursvely way for a certan tolerance. he recursve tracng rocedure can be descrbed, as shown n Fg. 6.

Because maxmum tolerance for re-rocessng s equal, the number of ES-trees of each terran s equal.

4 Internatonal Journal of Innovaton, Management and echnology, Vol. 4, No. 1, February 013 As shown Fg. 5, the ree-tolerance of a certan vertex s decded by root node. hus, all nodes of a certan ES-tree have same ree-tolerance. he generaton rocedure of a smlfed trangular mesh from a VD-MR-heght-ma can be descrbed as follow. A certan cell conssts of two trangles. For each vertex of the two trangles, we can trace the end vertex n a recursvely way for a certan tolerance. he recursve tracng rocedure can be descrbed, as shown n Fg. 6. Fg. 6. he generaton rocedure of a smlfed trangular mesh Fg. 7 shows smlfed trangular meshes extracted wth vew-deendent LOD. Because maxmum tolerance for re-rocessng s equal, the number of ES-trees of each terran s equal. Nearby regons of terran data can be rendered wth more detal than the regons n the dstance, and vew-deendent LOD for terran data results from smlfcaton coeffcent, as shown n Fg. 7(b) and Fg. 7(c). Fg. 7. Smlfed trangular mesh wth vew-deendent LOD IV. CONCLUSION For vsualzaton of large-scale terran, vew-deendent LOD technque s essental. Vew-deendent LOD tres to construct the best reresentaton for the current vew, and crtcal comonent of t s a smlfcaton technque. Prevous results, whch can be aled to terran data smlfcaton, can be roughly classfed nto two man categores; 3D-methods, and.5d-methods. Usually,.5D-methods are more smle & memory effcent than 3D-methods, however, 3D-methods aroxmate the orgnal more accurately than.5d-methods by usng the same number of trangles. hs aer resents new vew-deendent-mult-resoluton reresentaton method of large-scale terran, whch nherts the accuracy of 3D-methods as well as the memory effcent of.5d-methods. For the aroxmaton of a heght ma, the roosed algorthm emloy the edge contracton aroach based on the QEM (quadrc error metrcs) method. he overall scheme of the roosed algorthm conssts of two 107

Internatonal Journal of Innovaton, Management and echnology, Vol. 4, No. 1, February 013 man stes: 1) he constructon of VD-MR-heght-ma; ) he extracton of smlfed, vew-deendent trangular mesh.

For the second ste, a smlfed trangular mesh can be traced to the end vertex n a recursvely way for a certan tolerance.

UD110006MD (DAPA), UD100009DD, UD10035JD, 01-0007496. he authors wsh to exress sncere grattude for the fnancal suort. [1] M. Garland, and P. S.

Chung was born n Korea, and receved hs BS degree from Aou Unversty n 011 He s currently ursung hs master s degree n the Deartment of Industral Engneerng at Aou Unversty, Reublc of Korea.

Hwam was born n Korea, and receved hs BS degree from Aou Unversty n 011. He s currently ursung hs master s degree n the Deartment of Industral Engneerng at Aou Unversty, Reublc of Korea.

5 Internatonal Journal of Innovaton, Management and echnology, Vol. 4, No. 1, February 013 man stes: 1) he constructon of VD-MR-heght-ma; ) he extracton of smlfed, vew-deendent trangular mesh. For the frst ste, the roosed algorthm mantans a full decmaton rocedure of a heght ma for a maxmum tolerance, and creates ES-tree, whch s used to reresent mult-resoluton trangular mesh. For the second ste, a smlfed trangular mesh can be traced to the end vertex n a recursvely way for a certan tolerance. ACKNOWLEDGMEN hs work was suorted by the Defense Acquston Program Admnstraton (DAPA) and the Agency for Defense Develoment (ADD), Korea Research Foundaton under the Contract No. UD110006MD (DAPA), UD100009DD, UD10035JD, he authors wsh to exress sncere grattude for the fnancal suort. [1] M. Garland, and P. S. Heckbert, Surface Smlfcaton Usng Quadrc Error Metrcs, In Proceedng SIGGRAPH, , Yong H. Chung was born n Korea, and receved hs BS degree from Aou Unversty n 011 He s currently ursung hs master s degree n the Deartment of Industral Engneerng at Aou Unversty, Reublc of Korea. He s nterested n CAD/CAM, Robotcs, Modelng & Smulaton. Mr. Chung s conductng research n the felds of geometrc algorthms n CAD/CAM. Won K. Hwam was born n Korea, and receved hs BS degree from Aou Unversty n 011. He s currently ursung hs master s degree n the Deartment of Industral Engneerng at Aou Unversty, Reublc of Korea. He s nterested n Dstrbuted System, Synthetc envronment, Modelng & Smulaton. Mr. Hwam s conductng research n the felds of Defense Modelng & Smulaton. REFERENCES [1] M. A. Duchaneau, M. Wolnsky, Davd E. Sget, M. C. Mller, C. Aldrch, and M. B. Mneev-Wensten, ROAMng terran: real-tme otmally adatng meshes, 97 roceedngs of the 8th conference on Vsualzaton, 1997, [] H. Hoe, Vew-deendent refnement of rogressve meshes, 97 roceedngs of the 4th annual conference on Comuter grahcs and nteractve technques, 1997, [3] R. Svan, and H. Samet, Algorthms for Constructng Quadtree Surface Mas, In Proceedng 5th Internatonal Symosum on Satal Data Handlng, 199, [4] H. S. Km, H. K. Cho and K. H. Lee, Novel Evaluaton of the Edge Cost for Mesh Smlfcaton wth Vertex Color Attrbute, Proceedngs of the Socety of CAD/CAM Engneers Conference, 006, [5] H. S. Km, H. K. Cho, and K. H. Lee, Mesh Smlfcaton for Preservng Geometrc and Color Feature, Proceedngs of the Socety of CAD/CAM Engneers Conference, 007, [6] W. D. L, Y. L. Ca, and W. F. Lu, A 3D Smlfcaton Algorthm for Dstrbuted Vsualzaton, Comuters n Industry, vol. 58, ssue. 3,. 11-6, 007. [7] K. H. Lee, H. Woo, and. Suk, Data Reducton Methods for Reverse Engneerng, he Internatonal ournal of advanced manufacturng technology, vol. 17, no. 10, , 001. [8] H. Hoe, Progressve Meshes, In Proceedng SIGGRAPH, , [9] D. Hebert and H. Km, Image encodng wth trangulaton wavelets, In Proceedng SPIE, , [10] P. Lndstrom and D. Koller, Real-tme Contnuous Level of Detal Renderng of Heght Felds, In Proceedng SIGGRAPH, , [11] R. Svan and H. Samet, Algorthms for Constructng Quadtree Surface Mas, In Proceedng 5th Internatonal Symosum on Satal Data Handlng, , 199. Dae S. Chang was born n Korea, and receved hs BS degree from Aou Unversty n 01. He s currently ursung hs master s degree n the Deartment of Industral Engneerng at Aou Unversty, Reublc of Korea. He s nterested n Dscrete event system, Modelng & Smulaton. Mr. Chang s conductng research n the felds of CAD/CAM/CAPP software system. Jung Ju. Cho was born n Korea. He s an Assocate Professor of Dgtal Meda at Aou Unversty, Korea. He receved BS degree from KAIS n 1990, and receved hs MS, Ph.D. degrees n Comuter Scence from POSECH n 199, Hs man research nterests nclude non-hotorealstc anmaton and renderng, mage and vdeo rocessng, and moble grahcs systems. Sang C. Park s an assstant rofessor n the Deartment of Industral & Informaton Systems Engneerng at Aou Unversty. Before onng Aou, he worked for DamlerChrysler Cor. and Cubck-ek Co., develong commercal and nhouse CAD/CAM/CAPP/smulaton software systems. He receved hs BS, MS, and PhD degrees from KAIS n 1994, 1996, and 000, resectvely, all n ndustral engneerng. Hs research nterests nclude geometrc algorthms n CAD/CAM, rocess lannng, engneerng knowledge management, and dscrete event system smulaton. 108

A new Algorithm for Lossless Compression applied to two-dimensional Static Images

A new Algorithm for Lossless Compression applied to two-dimensional Static Images A new Algorthm for Lossless Comresson aled to two-dmensonal Statc Images JUAN IGNACIO LARRAURI Deartment of Technology Industral Unversty of Deusto Avda. Unversdades, 4. 48007 Blbao SPAIN larrau@deusto.es