Mohammed El Hassouni DESTEC FLSHR, University of Mohammed V-Agdal- Rabat, Morocco

Transcription

1 (IJCSIS) Internatonal Journal of Computer Scence an Informaton Securty, Vol. 8, No., 00 3D-Mesh enosng usng an mprove vertex base ansotropc ffuson Mohamme El Hassoun DESTEC FLSHR, Unversty of Mohamme V-Agal- Rabat, Morocco Drss Aboutane LRIT, UA CNRST FSR, Unversty of Mohamme V-Agal- Rabat, Morocco Abstract Ths paper eals wth an mprovement of vertex base nonlnear ffuson for mesh enosng. Ths metho rectly flters the poston of the vertces usng Laplace, reuce centere Gaussan an Raylegh probablty ensty functons as ffusvtes. The use of these DFs mproves the performance of a vertex-base ffuson metho whch are aapte to the unerlyng mesh structure. We also compare the propose metho to other mesh enosng methos such as Laplacan flow, mean, mean, mn an the aaptve MMSE flterng. To evaluate these methos of flterng, we use two error metrcs. The frst s base on the vertces an the secon s base on the normals. Expermental results emonstrate the effectveness of our propose metho n comparson wth the exstng methos. Keywors- Mesh enosng, ffuson, vertex. I. INTRODUCTION The current graphc ata processng tools allow the esgn an the vsualzaton of realstc an precse 3D moels. These 3D moels are gtal representatons of ether the real worl or an magnary worl. The technques of acquston or esgn of the 3D moels (moellers, scanners, sensors) generally prouce sets of very ense ata contanng both geometrcal an appearance attrbutes. The geometrcal attrbutes escrbe the shape an mensons of the obect an nclue the ata relatng to a unt of ponts on the surface of the moelle obect. The attrbutes of appearance contan nformaton whch escrbes the appearance of the obect such as colours an textures. These 3D moels can be apple n varous fels such as the mecal magng, the veo games, the cultural hertage... etc []. These 3D ata are generally represente by polygonal meshes efne by a unt of vertex an faces. The most meshes use for the representaton of obects n 3D space are the trangular surface meshes. The presence of nose n surfaces of 3D obects s a problem that shoul not be gnore. The nose affectng these surfaces can be topologcal, therefore t woul be create by algorthms use to extract the meshes startng from groups of vertces; or geometrcal, an n ths case t woul be ue to the errors of measurements an samplng of the ata n the varous treatments []. To elmnate ths nose, a frst stuy was mae by Taubn [3] by applyng sgnal processng methos to surfaces of 3D obects. Ths stuy has encourage many researchers to evelop extensons of mage processng methos n orer to apply them to 3D obects. Among these methos, there are those base on Wener flter [4], Laplacan flow [5] whch austs smultaneously the place of each vertex of mesh on the geometrcal center of ts neghborng vertex, mean flter [5], an Alpha-Trmmng flter [6] whch s smlar to the nonlnear ffuson of the normals wth an automatc choce of threshol. The only fference s that nstea of usng the nonlnear average, t uses the lnear average an the non teratve metho base on robust statstcs an local prectve factors of frst orer of the surface to preserve the geometrc structure of the ata [7]. There are other approaches for enosng 3D obects such as aaptve flterng MMSE [8]. Ths flter epens on the form [9] whch can be consere n a specal case as an average flter [5], a mn flter [9], or a flter arrange between the two. Other approaches are base on blateral flterng by entfcaton of the characterstcs [0], the non local average [] an aaptve flterng by a transform n volumetrc stance for the conservaton of the characterstcs []. Recently, a new metho of ffuson base on the vertces [3] was propose by Zhang an Ben Hamza. It conssts n solvng a nonlnear screte partal fferental equaton by entrely preservng the geometrcal structure of the ata. In ths artcle, we propose an mprovement of the vertex base ffuson propose by Zhang an Ben Hamza. The only fference s to use of fferent ffusvtes such as the functons of Laplace, reuce centre Gaussan an Raylegh nstea of the functon of Cauchy. To estmate these varous methos of flterng, two error metrc L [3] are use. Ths artcle s organze as follows: Secton presents the problem formulaton. In Secton 3, we revew some 3D mesh ISSN

2 (IJCSIS) Internatonal Journal of Computer Scence an Informaton Securty, Vol. 8, No., 00 enosng technques; Secton 4 presents the propose T. Denote by N(T) the set of all mesh trangles that have a approaches; Secton 5 presents the use error metrcs. In common ege or vertex wth T (see Fg. )T Secton 6, we prove expermental results to emonstrate a much mprove performance of the propose methos n 3D mesh smoothng. Secton 7 eals wth some conclung remarks. II. ROBLEM FORMULATION 3D obects are usually represente as polygonal or trangle meshes. A trangle mesh s a trple M= (, ε, T) where = {,, n } s the set of vertces, ε = {e } s the set of eges an T = {T,,T n } s the set of trangles. Each ege connects a par of vertces (, ). The neghbourng of a vertex s the set * = { : ~ }. The egree of a vertex s the number of the neghbours. N( ) s the set of the neghbourng vertces of. N(T ) s the set of the neghbourng trangles of T. We enote by A(T ) an n(t ) the area an the unt normal of T, respectvely. The normal n at a vertex s obtane by averagng the normals of ts neghbourng trangles an s gven by n = n( T ) () * 5 T T The mean ege length l of the mesh s gven by l = e ε e ε Durng acquston of a 3D moel, the measurements are perturbe by an atve nose: () = ' +η (3) Where the vertex nclues the orgnal vertex an the ranom nose processη. Ths nose s generally consere as a Gaussan atve nose. For that, several methos of flterng of the meshes were propose to flter an ecrease the nose contamnatng the 3D moels. III. RELATED WORK In ths secton, we present the methos base on the normals such as the mean, the mean, the mn an the aaptve MMSE flters an the methos base on the vertces such as the laplacen flow an the vertex-base ffuson usng the functons of Cauchy, Laplace, Gaussan an Raylegh. A. Normal-base methos Conser an orente trangle mesh. Let T an U be a mesh trangles, n(t) an n(u ) be the unt normal of T an U respectvely, A(T) be the area of T, an C(T) be the centro of Fg. Left: Trangular mesh. Rght: upatng mesh vertex poston. ) Mean Flter: The mesh mean flterng scheme nclues three steps [5]: Step. For each mesh trangle T, compute the average normal m(t) : m T = U N ( T ) n( U ) Step. Normalze the average normal m(t) : (4) m T m( T ) (5) m T Step 3. Upate each vertex n the mesh: Wth ( T ) new ol + A T v( T ) A T ( T ) C m( T ) m( T ) v.. (6) = (7) v s the proecton of the vector C.. onto the recton of m(t), as shown by the rght mage of Fg.. ) Mn flter : The process of mn flterng ffers from the average flterng only at step. Instea of makng the average of the normals, we etermne the narrowest normal, η, for each face, by usng the followng steps [9]: - Compute of angle Φ between n(t) an n(u ). - Research of the mnmal angle: If Φ s the mnmal angle n N(T) then n(t) s replace by n(u ) ISSN

3 (IJCSIS) Internatonal Journal of Computer Scence an Informaton Securty, Vol. 8, No., 00 3) Angle Mean Flter: Ths metho s smlar to mn flterng; the only fference s that nstea of seekng the narrowest normal we etermne the mean normal by applyng the angle mean flter [5]: ( ( T ) n) θ = n, (8) U If θ s the mean angle n N(T) then n(t) s replace by n(u ). 4) Aaptve MMSE Flter: Ths flter ffers from the average flter only at step. The new normal m(t) for each trangle T s calculate by [8]: ( T) Ml σn > σlouσ l = 0 m T = σ n σn n + 0 T M l T σn σletσ l σl σl N n ( U ) = 0 (9) = = 0 M l T (0) N Conserng the followng expresson whch allows the upate of the mesh vertces [5] new ol ( ) + λd () Where D() s a splacement vector, an λ s a step-sze parameter. The Laplacan smoothng flow s obtane f the splacement vector D() s efne by the so-calle umbrella operator [4] (see Fg. ) : U ( ) = n N ( ) ol N() s the -rng of mesh vertces neghbourng on (3) ) Vertex-Base Dffuson usng the Functon of Cauchy: Ths metho [3] conssts n upatng the mesh vertces by solvng a nonlnear screte partal fferental equaton usng the functon of Cauchy. σ n s the varance of atve nose an σ l s the varance of neghbourng mesh normals whch s change accorng to elements of normal vector. Thus, σ l s calculate as follows: N n ( U ) = 0 σ l = M l ( T ) () N = 0 B. Vertex-base methos ) Laplacan Flow Fg 3. Illustraton of two neghbourng rngs. The upate of the vertces of mesh (see Fg. 3) s gven by + * ( g( ) + g( ) Where g s Cauchy weght functon gven by (4) g ( x) = (5) x + c Fg. Upatng vertex poston by umbrella operator. an c s a constant tunng parameter that nees to be estmate ISSN

4 (IJCSIS) Internatonal Journal of Computer Scence an Informaton Securty, Vol. 8, No., 00 The graent magntues are gven by Conser an orgnal moel M an the moel after ang nose or applyng several teratons smoothng M. s a vertex of M. Let set st (, M ) equal to the stance between an / a trangle of the eal mesh M closest to. Our L vertexposton error metrc s gven by = (6) An * / k = * (7) k k ε v = A () 3A ( M ) M ' ( ) st(, M ) Where A() s the summaton of areas of all trangles ncent on an A(M) s the total area of M. Note that the upate rule of the propose metho requres the use of two neghbourng rngs as epcte n Fg. 3. IV. ROOSED METHOD The metho of vertex-base ffuson [3] was propose by Zhang an Ben Hamza an whch conssts n solvng a nonlnear screte partal fferental equaton usng the functon of Cauchy. In ths secton, we propose an mprovement of the vertexbase ffuson propose by Zhang an Ben Hamza. The only fference s the use of other ffusvty functons nstea of Cauchy functon. These functons are presente as follows: - Reuce Centere Gaussan functon : x exp c g x = (8) p - Laplace functon : x exp abs c g x = (9) - Raylegh functon : x c x g x = exp (0) c c s a constant tunng parameter that nees to be estmate for each strbuton. V. L ERROR METRIC To quantfy the better performance of the propose approaches n comparson wth the metho base on the vertces usng the functon of Cauchy an the other methos, we compute the vertex-poston an the face-normal error metrcs L [3]. The face-normal error metrc s efne by ' ε f = A( T ) n( T ) n( T ) () A M T M Here T an T are trangles of the meshes M an M respectvely; n(t) an n(t ) are the unt normals of T an T respectvely an A(T) s the total area of T. VI. EXERIMENTAL RESULTS Ths secton presents smulaton results where the normal base methos, the vertex-base methos an the propose metho are apple to nosy 3D moels obtane by ang Gaussan nose as shown n Fgs 6 an 8. The stanar evaton of Gaussan nose s gven by σ = nose l (3) Where l s the mean ege length of the mesh. We also test the performance of the propose methos on orgnal nosy laser-scanne 3D moels shown n Fgs 4 an 0. The metho of vertex-base ffuson usng the propose ffusvty functons of Laplace, reuce centre Gaussan an Raylegh are a lttle bt more accurate than the metho of vertex-base ffuson usng the functon of Cauchy. Some features are better preserve wth the approaches of vertex base ffuson usng these functons (see Fgs 4 an 0). By comparng the four stnct methos (see Fgs 5 an ), we notce that the propose metho gves the smallest error metrcs comparng to metho of vertex-base ffuson usng the functon of Cauchy. The expermental results show clearly that vertex-base methos outperform the normal-base methos n term of vsual qualty. These results are llustrate by Fg 6. In Fg 7, the values of the two error metrcs show clearly that the vertex-base ffuson usng the functons of Laplace, reuce centre Gaussan an Raylegh gve the best results an they are more effectve than the methos base on the normals. Fg 7 also shows that the approaches base on the ISSN

5 (IJCSIS) Internatonal Journal of Computer Scence an Informaton Securty, Vol. 8, No., 00 vertces such as Laplacen flow an the vertex-base ffuson usng the functons of Cauchy, Laplace, reuce centre Gaussan an Raylegh gve results whose varaton s remarkably small. In all the experments, we observe that the vertex-base ffuson usng fferent laws s smple an easy to mplement, an requre only some teratons to remove the nose. The ncrease n the number of teraton nvolves a problem of over smoothng (see Fg 8). In Fg 9, we see that the metho of vertex-base ffuson usng the functon of Cauchy leas more quckly to an over smoothng than the methos of vertex-base ffuson usng the functons of Laplace, reuce centere Gaussan an Raylegh. VII. CONCLUSION In ths paper, we ntrouce a vertex-base ansotropc ffuson for 3D mesh enosng by solvng a nonlnear screte partal fferental equaton usng the ffusvty functons of Laplace, reuce centere Gaussan an Raylegh. These metho s effcent for 3D mesh enosng strategy to fully preserve the geometrc structure of the 3D mesh ata. The expermental results clearly show a slght mprovement of the performance of the propose approaches usng the functons of Laplace, reuce centere Gaussan an Raylegh n comparson wth the methos of the laplacen flow an the vertex-base ffuson usng the functon of Cauchy. The Experments also emonstrate that our metho s more effcent than the methos base on the normals to mesh smoothng. Fg 4. (a) Statue moel gtze by a Rolan LX-50 laser range scanner (3344 vertces an 453 faces); smoothng moel by metho base on the vertces usng the functons of (b) Cauchy (c = ), (c) Laplace (c = ), () Gaussan (c = ) an (e) Raylegh (c = ). The number of teraton tmes s 7 for each case. Fg 5. Top: L vertex-poston error metrc of 3D moel n Fg 4 Bottom: L face-normal error metrc of 3D moel n Fg ISSN

6 (IJCSIS) Internatonal Journal of Computer Scence an Informaton Securty, Vol. 8, No., 00 Fg7. Left: L vertex-poston error metrc of 3D moel n Fg 6. Rght: L face-normal error metrc of 3D moel n Fg 6. Fg 6. (a) Orgnal moel(4349 vertces an 60 faces); (b) Ang Gaussan nose (ε v = , ε f = an σ = 0.8 l ); (c) Mn flter (7 teratons); () Mean flter (3 teratons); (e) Aaptatf MMSE flter (3 teratons); (f) Mean flter (4 teratons); (g) Laplacen flow ( teratons an λ=0.45); smoothng moel by metho base on the vertces usng the functons of (h) Cauchy (3 teratons an c =.3849), () Laplace (6 teratons an c = ), () Gaussan (6 teratons an c= ) an (k) Raylegh (6 teratons an c = 0.3). Fg 8. (a) Orgnal moel (08 vertces an 46 faces); (b) Ang Gaussan nose (σ = 0.7 l ); smoothng moel by metho base on the vertces usng the functons of (c) Cauchy (c =.3849), () Laplace (c = ), (e) Gaussan (c = ) an (f) Raylegh (c = ). The number of teraton tmes s 0 for each case ISSN

7 (IJCSIS) Internatonal Journal of Computer Scence an Informaton Securty, Vol. 8, No., 00 Fg 9. Left: L vertex-poston error metrc of 3D moel n Fg 8. Rght: L face-normal error metrc of 3D moel n Fg8. Fg 0. (a) Statue moel gtze by mpact 3D scanner (59666 vertces an 0955 faces); smoothng moel by metho base on the vertces usng the functons of (b) Cauchy (c = ), (c) Laplace (c = ), () reuce centere Gaussan (c = ) an (e) Raylegh (c = ).The number of teraton tmes s for each case ISSN

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