High-Boost Mesh Filtering for 3-D Shape Enhancement

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1 Hgh-Boost Mesh Flterng for 3-D Shape Enhancement Hrokazu Yagou Λ Alexander Belyaev y Damng We z Λ y z ; ; Shape Modelng Laboratory, Unversty of Azu, Azu-Wakamatsu Japan y Computer Graphcs Group, Max-lanck-Insttut für Infomatk, Saarbrücken, Germany E-mal: fm , belyaev, dm-weg@u-azu.ac.jp, belyaev@mp-sb.mpg.de Abstract In ths paper, we propose a new technque for enhancng 3-D shapes approxmated by trangle meshes: hgh-boost mesh flterng. Concepts based on the hgh-boost flter n sgnal and mage processng are used for the development of the mesh flterng. An algorthm of the hgh-boost mesh flterng conssts of affectng the hgh-boost flter to face normals on a trangle mesh and updatng mesh vertex postons to make them adapt to the boosted normals. An enhancement operaton s acheved by a procedure whch subtract smoothed face normals from the orgnal ones. To apply the hgh-boost flter to face normals on a trangle mesh, we requre a low-pass flter for smoothng them. We use mean flterng appled to face normals n numercal experments. The results ndcate that the hgh-boost mesh flterng s a technque effectve for 3-D shape enhancement. Keywords: trangle meshes, mesh flterng, shape enhancement, hgh-boost flter, normal-based error mnmzaton. Techncal Area: 3-D Computer Graphcs 1 Introducton The hgh-boost flter s a smple sharpenng operator n sgnal and mage processng. It s used for amplfyng hgh frequency components of sgnals and mages. The amplfcaton s acheved va a procedure whch subtracts an smoothed verson of the meda data from the orgnal one. In mage processng, we can sharpen edges of a mage through the amplfcaton and obtan a more clear mage. If the flter s appled to a 3-D trangle mesh, ts surface textures are enhanced. Such the enhancement operaton gves us more attractve 3-D shape models. Several technques for enhancng 3-D shapes have been proposed n computer graphcs so far. Guskov et al. [1] apply a smlar algorthm drectly to vertces. They developed Fgure 1. A horse model. Top: orgnal one. Bottom: enhanced by the hgh-boost mesh flterng.

2 (a (b (c Fgure 2. (a Stanford dragon enhanced by vertexbased enhancement. (b Black areas are flppng trangles. (c A conceptual fgure of the trangle flppng. the algorthm from complex concepts based on multresoluton sgnal processng. Tolga et al. [6] appled hgh-boost flter to face normals on a surface represented as a level set of volumes. After the flterng, a level set method that s computatonally expensve s used for manpulatng the surface n order to ft t nto the processed normals. Our approach frst apples the hgh-boost flter to face normals on a enhanced 3-D trangle mesh. Second, mesh evoluton by updatng vertex postons s performed va mnmzng errors between the orgnal and fltered normals [3]. The purpose of the mesh evoluton s to adapt the mesh to the processed normals. When a flterng technque s drectly appled to mesh vertces, trangle flppng frequently occurs. We can prevent flterng results from the trangle flppng wth a combnaton of flterng face normals and evolvng mesh vertces va the normal-based error mnmzaton. Fgure 2 shows an example of the trangle flppng. An enhancement wth the hgh-boost mesh flterng s accomplshed by a procedure that subtract smoothed face normals from the orgnal ones. Ths means that we need a process for smoothng the face normal feld on a trangle mesh. Our development realzes the smoothng process by mean flterng for averagng face normals at local neghborhood on a trangle mesh. Expermental results show that the hgh-boost mesh flterng method stably works wthout nducng trangle flppng. Ths paper s organzed as follows. In secton 2, we ntroduce mplementaton detals of the hgh-boost mesh flterng. We demonstrate expermental results n secton 3, and they are dscussed n secton 4. Our conclusons are gven n secton 5. 2 Implementaton Detals An mplementaton of the hgh-boost mesh flterng s dscussed n ths secton. We frst prepare smoothed face normals wth mean flterng for them. Second, a new face normal feld on a trangle mesh s defned usng a boost threshold, the orgnal normals, and the smoothed normals. Fnally mesh vertex postons are updated va mnmzng errors between the boosted normals and the orgnal ones. A combnaton of the three steps acheves a 3-D shape enhancement. It s frst dscussed how to smooth face normals usng the mean flterng. Consder an orented trangle mesh. Let T be a mesh trangle, m(t be the unt normal of T, and A(T be the area of T. Denote by N (T a set of all mesh trangles sharng the same edge or vertex wth T. Fgure 3 vsually shows the N (T. One teraton of the mean flterng on the face normal feld s composed of the followng two steps. Step 1. For each mesh trangle T, we compute the face

3 n(u (k n(t n (T U U T n( m( +1 Fgure 3. The center trangle T and neghborng trangles U. Fgure 4. The 1-rng neghborhood of the. normal n(t and perform the followng area-weghted averagng operaton: n (k (T = A(U 1 X 2N (T A(U n(u (1 where n (k (T s a face normal smoothed by k teratons. Step 2. For each mesh trangle T, we normalze the averaged normals n (k (T : n (k (T ψ n(k (T kn (k (T k : as X = X A( 1 n( (4 where S s a trangle whch s obtaned by projectng onto a plane defned by m(. The summaton s taken over all trangles n the F 1 (. The gradent of a trangle We can obtan the smoothed normals through the two steps. Second, let us consder a hgh-boost flterng operaton appled to face normals. Let n(t be the orgnal face normals and n (k (T be face normals smoothed by k teratons. A operaton of the hgh-boost flterng on the face normal feld s defned by m( n( m(t = (1 + ffn(t ffn(k (T k(1 + ffn(t ffn (k (T k (2 S where m(t s the boosted normals, and ff s the boost threshold. Ths operaton defnes a new unt vector feld fmg. Fnally, detals of optmzng mesh vertex postons va normal-based error mnmzaton [3] s presented. Let be a updated vertex and be a mesh trangle adjacent to the. The normal-based error at the s defned as the areaweghted summaton of squared dfferences of n( and m( : E n X = A( n( m( 2: (3 2F1( where F 1 ( s a set of all trangles at the 1-rng neghborhood of the. Its partal dervatve wth respect s gven Fgure 5. S s a projecton of onto a plane defned by a new face normal m(. area s computed by the = 1 2 ( 1 cotff +( 2 cot f (5 where A s an area of a trangle whose three vertces are f; 1 ; 2 g. Ths dervaton of Equaton (5 s descrbed n [4]. The optmal vertex poston 0 s gotten by the followng standard gradent descent method for mnmzng E n : X ψ @

4 1 β α 2 Fgure 6. A trangle whose vertces are f; 1 ; 2 g and ts angles. Accordng to our experments, Equaton (6 stably works by settng =0:1. When face normals on a trangle mesh are enhanced usng the normals smoothed by k teratons, the updatng operaton should be performed by 10 k teratons. The evoluton of vertex postons va normal-based error mnmzaton requres many enough teratons. (a 3 Expermental esults We check performance of the hgh-boost mesh flterng n ths secton. It s prmary verfed the flter s resstance to the trangle flppng. The Stanford dragon model n Fgure 7 s enhanced usng the orgnal face normals and ones smoothed by 100 teratons. The boost threshold s set to be 1.5. Its eyes and scales are well enhanced. Nevertheless, the trangle flppng does not occurs. The Stanford bunny model n Fgure 8 s enhanced by the orgnal face normals and ones smoothed by 100 teratons. The boost threshold s set to be 1.5. In ths case, mesh rregularzaton appears on the mesh model, as shown n Fgure 8-(d. Further, alasng occurs at edge parts, as seen n Fgure 8-(f. We can suppress the alasng and the mesh rregularzaton usng the Laplacan smoothng [7, 2], as demonstrated n Fgure 8-(e and (g. In Fgure 8-(e, a promnent alasng part s marked by an ellpse. (b 4 Dscusson When we apply the hgh-boost mesh flterng to some mesh models, the alasng and the mesh rregularzaton occurs. Those problems are suppressed by the Laplacan smoothng, but the hgh-boost mesh flter should have functons preventng ts flterng results from the alasng and the mesh rregularzaton. In ths paper, we use the mean flterng appled to face normals to obtan the hgh frequency component of the normal feld. For enhancng mesh models, we can consder (c Fgure 7. (a Stanford dragon model. (b Enhanced by the hgh-boost mesh flterng. (c Trangle flppng does not occur.

5 (a (b (c (d (e (f (g Fgure 8. Stanford bunny model. (a Orgnal one. (b Enhanced by the hgh-boost mesh flterng. (c (b smoothed by the Laplacan smoothng. (d Mesh of (b. (e Mesh of (c. (f Alasng. (g Ant-alasng by the Laplacan smoothng.

6 another approach such as the use of Laplacan on the face normal feld. Taubn proposes Laplacan smoothng appled to face normals n [8]. In future research, we nsert the smoothng method nto the development of an mproved hgh-boost mesh flterng algorthm. 5 Concluson In ths paper, we appled the hgh-boost mesh flterng to 3-D mesh enhancement. The hgh-boost mesh flter s constructed by a combnaton of 1 the applcaton of the hgh-boost flter to face normals on a trangle mesh and 2 the evoluton of mesh vertex postons to ft them to the fltered normals. Numercal experments show that the hghboost mesh flterng enhances 3-D trangle meshes wth no sgnfcant trangle flppng. eferences [1] I. Guskov, W. Sweldens, and. Schröder. Multresoluton sgnal processng for meshes. Computer Graphcs (roceedngs of SIGGAH99, pages , [2] L. Kobbelt, S. Zamperor, J. Vorsatz, and H.-. Sedel. Interactve multresoluton modelng on arbtrary meshes. Computer Graphcs (roceedngs of SIG- GAH98, pages , [3] Y. Ohtake. Mesh optmzaton and feature extracton. h.d Thess, Unversty of Azu, March [4] Y. Ohtake, A. Belyaev, and I. Bogaevsk. olyhedral surface smoothng wth modfed Laplacan and curvature flow. The Journal of Three Dmensonal Images, Vol. 13, No. 3, September 1999, pages [5] Y. Ohtake, A. Belyaev, and A. asko. Dynamc meshes for accurate polygonzaton of mplct surfaces wth sharp features. Internatonal Conference on Shape Modelng and Applcatons, Genova, Italy, May 2001, pages [6] T. Tasdzen,. Whtaker,. Burchard, and S. Osher. Geometrc surface processng va normal maps. Techncal eport UUCS , Unversty of Utah, January [7] G. Taubn. A sgnal processng approach to far surface desgn. Computer Graphcs (roceedngs of SIG- GAH95, pages , [8] G. Taubn. Lnear Ansotropc Mesh Flterng. IBM esearch eport C22213 (W , IBM, October 2001.