Sketching Reaction-Diffusion Texture

Transcription

1 EUROGRAPHICS Workshop on Sketch-Bsed Interfces nd Modeling (006), pp. 1 8 Sketching Rection-Diffusion Texture Pper ID 100 Abstrct In this work, we present n interctive interfce for sketching synthesized textures. Rection-Diffusion (RD) is used s the bsis for texture synthesis. RD llows n unlimited mount of non-repeting texture nd offers gret flexibility for mpping textures to rbitrry surfces. However, it cn be difficult to find strting vlues of prmeters tht will produce interesting ptterns. We use mchine lerning to resolve the difficulty of determining pproprite initil vlues of the RD system. The system described here llows user to sketch pttern of spots or stripes with rbitrry orienttions, nd then utomticlly genertes pttern with the sme ttributes s the sketch. It lso llows the user to interctively crete more complex textures by dding nother lyer of pttern, s well s mnipulte the color of the resulting texture. We lso show tht this procedure cn be pplied to relistic 3D surfces. Ctegories nd Subject Descriptors (ccording to ACM CCS): I.3.7 [Computer Grphics]: Color, shding, shdowing, nd texture; I.3.8 [Computer Grphics]: Applictions Figure 1. () User sketches ellipses with different orienttion. Our system genertes dots with similr size, shpe, nd orienttion. (b) User dds smller ellipses to the sketch. The system genertes smller dots in between lrger dots. (c) User sketches lines with different slopes. The system genertes rndom stripes with similr orienttion. 1. Introduction Textures re two-dimensionl imges or three-dimensionl volumes tht cn be mpped onto n object s surfce. Very often, the object s surfce is lrger thn the texture smple vilble, hence the need for texture synthesis generting more of texture from given smple. There re two min types of textures: rster textures, nd procedurl textures. Rster textures cn be scnned pictures or pinted imges. They re redy-to-mp, but the cost of memory storge is very high nd resmpling nd mpping them to n rbitrry surfce cn be very difficult. Procedurl textures, on the other hnd, re defined mthemticlly, nd therefore tke little memory to store. They re lso esier to resmple nd mp to surfce. In this pper, we focus on textures generted from method clled Rection-Diffusion. In ddition to llowing n unlimited mount of non-repeting texture, this pproch lso offers greter flexibility for both controlling the ttributes of ptterns nd mpping textures to rbitrry surfces. Rection-Diffusion (RD) ws first introduced s model of morphogenesis, biologicl pttern-formtion process in which two or more chemicls clled morphogens diffuse over surfce nd rect with ech other. The differences in concentrtion of morphogens form certin niml cot ptterns, such s spots nd stripes [Tur5]. This process is defined by differentil equtions, which control the chnge in concentrtion of one morphogen reltive to other morphogens over time. A RD system consists of numerous prmeters, few of these being the rection rtes k/s, the diffusion rtes D, nd the rndom fctor β (Section 3). At present there is no intuitive wy to determine the vlues of these prmeters tht will produce interesting ptterns, much less pttern tht hs certin ttributes. Thus, lthough RD cn produce interesting nd vried textures, it is difficult for user to use RD system without hving deep knowledge of its internl workings. We solve this problem by developing n interctive interfce tht llows users to sketch rough version of their desired pttern. Our system then utomticlly genertes texture smple tht contins similr fetures (Figure 1). This is done by nlyzing the sketch s ttributes nd using mchine-lerning technique to mp these ttributes to the pproprite strting prmeters. Our pproch llows for more control over the ttributes of the ptterns produced by the RD system (in cse of Figure 1, the size nd orienttion of spots nd stripes). The system lso includes coloring interfce tht gives the user freedom to vry the color of the generted textures. Finlly, we show tht our pproch cn be pplied to generte ptterns, s defined by user, on 3D surfce. submitted to EUROGRAPHICS Workshop on Sketch-Bsed Interfces nd Modeling (006)

2 Pper ID 100 The rest of this pper is orgnized s follows. In Section, we present our interfce for sketching RD textures. In Section 3, we provide brief bckground of RD systems. In Section 4, we summrize the previous work tht hs been done on generting textures with RD. Section 5 contins detiled description of the implementtion of this system, including our extension to existing techniques nd our ppliction of mchine lerning to solve the problem of finding the pproprite prmeter vlues. In Section 6, we illustrte how our method extends to 3D surfces. We conclude with discussion of the future work in Section 7. Figure. Size of the spots produced vries over the grid (right), s depicted in user s sketch (left).. Sketching RD textures using the interfce Initilly, the user sketches ellipses nd lines in the sketch window, which ct s inputs to the system. The system then genertes texture consisting of spots or stripes tht hs similr ttributes s the initil sketch. These ttributes re shpe, size, spcing, nd orienttion of the spots nd stripes tht form the textures. The sketch window is divided into smller regions to llow for pttern with vried ttributes cross the entire imge. For the purpose of experimenting, the sketch ws divided into four regions (Figure ), but this could be n by n regions for lrger texture. Averge vlues of ttributes in ech region re used in determining the finl ttributes of the output texture. Since ptterns in different regions cn vry in size, shpe, or orienttion, liner interpoltion of the ttribute vlues between neighboring regions gives the output texture smooth trnsition from one region to the next..1. Sketching spots.1.1. Controlling size of spots. Spots re described by their size, shpe nd orienttion. The user cn generte spots by sketching ellipses in the sketch window. The system clcultes the verge size of ellipses in ech region. A mpping function then mps these vlues to set of strting prmeters such tht the spots generted by this system will hve the sme size s those sketched. This mpping function is described lter in Section As shown in Figure, the spots re vried in size cross the imge with smooth trnsition in size from the top left to the bottom right corner..1.. Controlling orienttion of spots. Using the sme technique s tht for vrying the size of spots s explined bove, their orienttion cn lso be vried cross the imge (Figure 3). Implementtion detils for generting textures with rbitrrily oriented spots cn be found in Section Controlling spcing mong spots. In generl, RD system of spots would hve the spcing between spots determined by the size of spots. Implementing cscded RD system s described in [Tur9], our system llows the user to generte ptterns from sketch where rbitrry sized spots cn hve rbitrry spcing between them (Figure 4). This technique will be described in detil in Section 4. Figure 3. Orienttion of spots produced (right) vries over the grid s sketched (left). Figure 4. Smll spots with lrge spcing (right) s sketched in the sketch window (left). The red lines re produced when the sketch is nlyzed. These lines correspond to the distnces mong the spots. Figure 5. The first simultion produces yellow spots (right) bsed on the sketch of the lrger ellipses (left). Figure 5b. The second simultion produces red spots (right). Red lines between smller ellipses in the sketch re produced when the sketch is nlyzed (left). submitted to EUROGRAPHICS Workshop on Sketch-Bsed Interfces nd Modeling (006)

3 Pper ID Figure 6. The system genertes pttern of stripes (right) tht hve the sme orienttion s in user s sketch (left) Complex ptterns. The previous subsections described how spots with desired size, orienttion nd spcing cn be obtined. However, the spots generted in this wy hve only one size in ech region. In order to hve smll spots interspersed with lrger spots through out the imge (Figure 5b), we implement cscded RD system s described in [Tur9]. Using our interfce, the user first sketches only the lrge ellipses in the sketch window (Figure 5). The system genertes spots with spcing nd orienttion s sketched. These spots form the first lyer of ptterns, with coloring scheme s described in Section.3. To dd more spots into this texture, the user first freezes these lrger spots (set the morphogens concentrtion in these cells constnt). A second set of ellipses cn now be drwn in the sketch window (Figure 5b). The system produces second set of spots (the second lyer of ptterns), with orienttion s sketched, in between the first set. The outcome of this second set of spots depends on the spcing mong the first set nd the size of the second set. The size of the second set of spots should be smll enough for them to form in between the lrger spots... Sketching stripes Stripe pttern cn be generted by sketching stright lines in the sketch window (Figure 6). Stripes re described by their width nd orienttion. The user needs to drw t lest one line representing stripes in ech region of the sketch window. The system clcultes verge slopes of lines in ech region. Bsed on these vlues, the system then genertes stripes with orienttions similr to those in the sketch. The detiled description of this implementtion cn be found in Section 5.1. The user cn choose one of four different sizes to drw the line, resulting in four different thicknesses of the stripes. texture by chnging the mpping of the morphogen concentrtions to colors. The color control interfce is bsed on the work presented in [KTBG03]. The interfce provides two sets of colors illustrted by the two long spcing brs in Figures 7 nd 8. The upper br displys the set of colors used for the first lyer of ptterns while the lower br displys the set of colors used for the second lyer (this second color set is used only in the cse of complex ptterns, s in Figure 8). A user cn use the Red/Green/Blue slide brs to define the colors to be used for either of the two sets. Colors cn be dded, modified, removed or the order of colors cn be chnged for ech of the two sets. A user cn lso djust the spcing between colors in one set by moving the buttons (one for ech color) on the spcing br. Aprt from the spcing br, the user cn use Bezier curve to blend between colors. The blending of colors in ech set is bsed on the distnce between two djcent colors on the spcing br, nd by the shpe of the corresponding Bezier curve. The concentrtion of given morphogen is mpped to the continuous set of colors; the lowest concentrtion (vlley) corresponds to the first color in the set, nd the highest concentrtion (pek) corresponds to the lst color (Figure 7). Figure 7. Concentrtion of morphogen A is mpped to the upper set of colors, with the lowest concentrtion mpped to violet nd the highest concentrtion mpped to pink. Trnsition between the vlley to the pek of the morphogen s concentrtions is visulized by chnge in color, from violet to yellow, to green, nd then to pink..3. Color control Our system genertes textures by imitting the RD process between morphogens. The concentrtion of ech morphogen is mpped to color. The ptterns of spots nd stripes re formed through the vritions in concentrtion of the morphogens s result of the RD process. After the texture is produced, the user cn use the color control interfce provided by the system to vry the color of the Figure 8. Spots belonging to the first lyer re mpped to the first color set (pink to yellow), while spots belonging to the second lyer re mpped to the second color set (pink to red). submitted to EUROGRAPHICS Workshop on Sketch-Bsed Interfces nd Modeling (006)

4 4 Pper ID Bckground of Rection-Diffusion A Rection-Diffusion (RD) system consists of two (or more) morphogens diffusing over surfce nd recting with one nother to form stble pttern. This pttern is visulized by mpping ech morphogen s concentrtion to color. In generl, this process cn be modeled by the following differentil equtions [Tur91]: = F(, b) + D b = G(, b) + Db b (Eq. 1) where nd b represent the concentrtions of morphogens A nd B, nd D nd D b re diffusion rtes of A nd B, respectively. At ny point in time, the concentrtions nd b re determined by the bove functions. F(,b) nd G(,b) describe the chnge in the concentrtion of A nd of B, respectively, s result of the rection processes between A nd B. These functions will decide if one morphogen inhibits the other, or sustins the other t the sme position. Lplcins nd b re the mesures of how high the concentrtions of A nd B re reltive to the concentrtions of the sme morphogens, respectively, in the surrounding cells. In the cse of grid (Figure 9), the formul for is s follows: = i 1, j + i+ 1, j + i, j 1 + i, j+ 1 4i, j (Eq. ) In generl, for n rbitrry surfce, e.g. mesh (Figure 9b), the formul will be: n = ik n (Eq. 3) k =1 i For given morphogen A, if the concentrtion of A in the current cell is higher thn tht of A in the surrounding cells, Lplcin is negtive. In the next time step, A diffuses wy from this cell, thus the concentrtion of A reduces. The reverse occurs if the concentrtion of A is lower in the current cell reltive to the surrounding cells. D nd D b specify how fst A nd B diffuse cross the surfce. At ny time t, the concentrtion of A is the sum of rection process between A nd B denoted by F(,b) nd diffusion process of A denoted by D (see Appendix). 4. Previous work The work by Turing in [Tur5] describes RD system of two morphogens tht forms spots (Turing spot system). Other reserchers hve since shown tht simple ptterns, from spots to stripes, cn lso be generted with rectiondiffusion mechnisms using different systems of morphogens nd equtions [Br81, Mur8, Mei8]. In [Mei8], Meinhrdt described RD system of two morphogens tht produces spots (Meinhrdt spot system), nd nother RD system of five morphogens tht produces stripes (Meinhrdt stripe system). In 1991, Witkin nd Kss dpted RD for texture synthesis nd dded nisotropy by vrying diffusion cross the surfce [WK91]. It ws lso shown tht complex ptterns could be generted using double simultions of cscded RD systems [Tur91]. Using the three RD systems (Turing spot, Meinhrdt spot, nd Meinhrdt stripe), we cn generte rndom spots or stripes by pplying rndom vritions to the concentrtions of morphogens in the system. We cn lso generte regulr stripes by rising the initil concentrtion of one morphogen t certin cells, clled inititor cells. During RD simultion, stripes rdite from these cells (Figure 10). Cscded systems built upon these three bsic forms cn lso generte more complex ptterns, such s leoprd spots or mixed spots of different sizes [Tur9]. We will explin the genertion of some of these complex ptterns in the following prgrphs. In the following section, we will describe our extensions tht give the user more flexibility in generting RD textures. Mixed spots of different sizes re obtined by first generting lrge size spots. These lrge spots re then frozen the concentrtions of morphogens in those cells tht form the spots re kept constnt. The second simultion then cretes smll spots in between the lrge spots (Figure 11). Leoprd spots, or rosettes, s shown in Figure 11b, re generted using the Turing spot system in similr wy, but with one extr step. After freezing the lrge spots, the concentrtions of morphogens in cells tht form these spots cells with pek concentrtion of given morphogen A nd vlley concentrtion of the other re reset to the initil level [Tur9]. During the second simultion, since the concentrtion of morphogen A in cells within the lrge spots is lower thn its pek, A tries to diffuse to cells surrounding the lrge spots, which leds to inhibition of morphogen B from collecting in these cells. As result, the smll spots tend to form circle round the lrge spots insted of in between them. In Figure 11b, A is represented by blck color nd B by yellow color. Figure 9. Morphogen A diffuses to nd from four djcent cells. Figure 9b. Morphogen A diffuses to nd from ll djcent cells. Figure 10. Regulr stripes rdite from inititor cells during RD process. submitted to EUROGRAPHICS Workshop on Sketch-Bsed Interfces nd Modeling (006)

5 Pper ID Implementtion 5.1. High-level description for spots Figure 11. Mixed spots generted by the Meinhrdt spot system. Figure 11b. Leoprd spots produced by Turing spot system. The min objective of this work is to develop n intuitive nd esy wy for user to utomticlly generte desired texture. The first step is the intuitive sketch interfce described in Section. However, the RD systems which re used to generte the textures require vlues of vrious prmeters s inputs. In order to extrct the prmeter vlues from the sketch, we devised high-level description for the spots in terms of the three prmeters: size, orienttion nd spcing. Figure 1. Two-step genertion of smll spots with lrge spcing using cscded Turing spot system. First, lrge spots with lrge spcing re produced (left). Next, smll spots re produced t the sme sites where lrge spots hd been (right). When single RD simultion is employed to generte spots, the distnce between spots, or spcing, is proportionl to the size of the spots. The lrger the spots re, the wider the spcing is. However, using cscded Turing spot systems s described in [Tur9], smll spots with lrge spcing cn be generted (Figure 1). The first simultion yields lrge spots with lrge spcing. The concentrtion of morphogens from the first simultion is then used to set up the second simultion such tht smller spots re produced t the sme positions where the lrge spots hd been, thus mintining lrge spcing. In other words, the first simultion produces spots with the specified spcing. The second simultion yields spots with the desired size while mintining the previous spcing. The existing systems of spots nd stripes described bove hve limited flexibility. For exmple, the spots nd stripes described before cnnot be oriented in ny rbitrry direction tht the user my desire. More importntly, the user needs to directly twek prmeters of the RD system in order to get desired pttern. It cn be difficult to find initil vlues of morphogens tht produce interesting ptterns, much less pttern tht hs certin desired ttributes. In the following section we describe our interfce with which user cn sketch RD texture. This is n intuitive method for user to generte ptterns utomticlly, without the need for knowledgeble tweking of RD prmeters. In order to llow more flexibility to the textures tht cn be generted, we lso provide extensions tht support rbitrrily oriented spots nd stripes. We lso show how we give dditionl control to the ttributes of the ptterns, nd how we pply mchine lerning to mp from our high-level spot description to RD prmeters. When the user drws spots in the sketch window, the system fits ellipses to the spots, nd extrcts their centers, orienttions nd rdii. The re of spot is used to describe its size; the rtio of minor rdius to mjor rdius to describe its shpe, nd the mjor xis to describe its orienttion. We then use Deluny tringultion method to determine the closest neighbors of spot, nd then clculte the distnce from this spot to its neighbors. In Figure 4, red lines produced during this process correspond to the distnces between spots. The verge distnce between djcent spots in the user s sketch is then used s the spcing between ll the spots for the purpose of generting the texture. 5.. Appliction of Mchine lerning for mpping user's sketch to RD prmeters In the previous section, we describe how physicl ttributes of the sketch re clculted. Once these vlues hve been extrcted from the user s sketch, they need to be mpped to the prmeters. In order to resolve the mpping between physicl ttributes nd RD prmeters, we use mchine lerning. This step is described in the following prgrphs Prepring trining dt for mchine lerning. The purpose of our mchine lerning function is to mp the spot ttributes to prmeter vlues. The first step is to lern how set of initil prmeter vlues mp to spot ttributes. Since the prmeters used to determine orienttion nd size with spcing re independent, lerning the mpping for the orienttion nd for the size with spcing cn be done seprtely. To reduce the totl number of dt points, we generted four different sets of trining dt, two for Turing nd Meinhrdt spots orienttion nd two for Turing nd Meinhrdt spots size nd spcing. The RD system ws run 15,600 times with different prmeter settings to generte textures with vrious ptterns tht serve s trining dt for this lerning process. For ech RD spot system, we use different threshold vlue for morphogen A or B to trce the boundry of the spot. We then fit ellipses to these spots. Bd textures (e.g. when the ptterns re not well formed, or if the size of the spots is too smll to perform ellipse fitting) re excluded. Using the sme method s for extrcting spot ttributes from user s sketch, we clculte the physicl ttributes of spots generted by the RD system (Figure 13). submitted to EUROGRAPHICS Workshop on Sketch-Bsed Interfces nd Modeling (006)

6 6 Pper ID 100 D nd D b shown in Eqution 1, we lso expnd the formul clculting Lplcins (Eqution ) nd dd direction coefficients to the flow of morphogens to nd from neighbor cells (Figure 14). These coefficients specify preferentil directions in which morphogens diffuse fster, thus the spots re elongted in these directions. As result, spots cn hve rbitrry orienttion. Figure 13. Ellipses (left) re fitted to RD spots (right). Yellow lines connect ech ellipse to their closest neighbors s determined by Deluny tringultion. During the texture genertion process, the trining dt were fed to mchine lerning function to mp the spot ttributes from the user s sketch to corresponding RD prmeters Non-homogeneous ptterns. In RD system, there re severl prmeters such s the rection rtes, diffusion rtes, nd direction coefficients tht determine the size, shpe, spcing, nd orienttion of the spots. In order to llow non-homogeneous ptterns, insted of using the sme prmeter vlues everywhere, we blend the prmeter vlues cross the grid. The result is ptterns of spots tht hve size, orienttion, nd spcing chnging cross the imge (Figure 15) Loclly Weighted Regression. To implement mchine lerning, we choose Loclly Weighted Regression (LWR) becuse it does not require n extremely lrge collection of trining dt, nd yet it is highly efficient for lerning complex mppings from rel-vlued input vectors to rel-vlued output vectors using noisy dt. Loclly Weighted Regression is memory-bsed lgorithm for lerning continuous curves tht uses only trining dt close to the prticulr point X. Points nerby re weighted by their distnce to point X. A nonliner regression n estimtion technique using interpoltion to predict one vrible from one or more other vribles is then clculted using these weighted points. Using memory-bsed lgorithm for mchine lerning lso mens tht we need to store the trining dt nd run LWR function every time to generte the prmeters. We use the LWR Mtlb function presented in [SA94]. Figure 14. Morphogens from one cell cn diffuse from nd to ll eight of its neighbors. Diffusion coefficients give the spots shpe nd orienttion. Figure 14b. Anisotropic Turing spots with the usersupplied direction indicted by rrow Mpping functions. The results we get from mpping size nd spcing of spots to RD prmeters re not lwys relible becuse of the errtic nture of trining dt. Therefore, we build mpping function for size nd spcing of spots using liner interpoltion. The function tkes spcing or re s input nd returns rection rtes nd direction coefficients s outputs. For orienttion of spots, the mpping function is LWR function tht tkes the rtio of minor rdius to mjor rdius s well s the mjor xis vector of the sketched ellipses s inputs, nd returns direction coefficients s outputs. () (b) Figure 15. Exmples of non-homogeneous textures of spots with size () nd orienttion (b) chnging cross the imge Extensions to existing RD spots Arbitrrily oriented spots. The existing RD systems described in the Section 4 cn generte only nisotropic spots, which re ellipticl spots in verticl or horizontl directions. This is becuse these systems llow the morphogens to diffuse to four neighbors only, long the directions North, South, Est, nd West (Figure 9). In this work, prt from using nisotropic diffusion rtes, we lso llow the morphogens to diffuse from nd to ech of the eight neighbors (Figure 14). In ddition to diffusion rtes Figure 16. Seeding the stripe system with line segments (left) results in texture with similr orienttion (right). Note tht slopes of the stripes vry cross the imge. submitted to EUROGRAPHICS Workshop on Sketch-Bsed Interfces nd Modeling (006)

7 Pper ID Oriented rndom stripes Direction coefficients cn be used to generte oriented spots (s explined in Section 5.3.1), but they do not help in generting oriented stripes. Inititor cells tht form oriented rndom line segments, however, cn crete oriented rndom stripes. The system clcultes the verge slopes of the lines sketched by the user in ech of the regions of the sketch window. These vlues re used to seed rndom lines with slight vritions of these verge slopes. The Meinhrdt stripes system then genertes rndom stripes tht hve similr orienttions (Figure 16). originl mesh to generte the desired pttern. As shown in Figure 0, the method produces spots with desired orienttion on cube. However, due to irregulr distribution of vertices, the orienttion of spots is not s cler when generted on the bunny. 6. Sketching RD texture on 3D mesh We demonstrte our interfce for creting RD textures on 3D surfces. Insted of drwing on seprte sketching window, the user cn drw spots directly on mesh. The system then fits ellipses to the spots, computes their ttributes, mps these to pproprite prmeters, which re then ssocited with vertices of the mesh tht lie within these ellipses. The remining vertices re ssigned prmeter vlues tht re liner combintion of the vlues t vertices lying within the closest ellipses. To generte spots of different sizes on the bunny in Figure 19, the diffusion rtes of morphogens A nd B re propgted nd blended over the mesh bsed on the size nd position of spots sketched by the user. The reltive proportions of morphogen flowing to ech of the neighboring vertices re clculted s explined below. For ech vertex i in the mesh nd its neighboring vertex ik (mong its n neighboring vertices numbered i1 through in), the term d ik denotes the distnce between centroids of the two tringles on either side of the edge connecting vertex i to vertex ik (Figure 17). The percentge coefficient P ik for the flow of morphogens between these two vertices is clculted s the rtio: d ik Pik = (Eq. 4) n d k = 1 ik And the Lplcin for vertex i is given by Eqution 5, where i is the morphogen concentrtion in vertex i. n = 4 Pik ik i (Eq. 5) k = 1 To generte oriented spots, the vector showing the orienttion of the ellipse is projected onto the tngent plne t ech vertex (Figure 18). During RD simultion, the fces surrounding ech vertex re rotted so tht this projected vector becomes prllel to the x xis. These fces re then scled down long the x xis ccording to the rtio of the minor rdius to the mjor rdius of the ellipse sketched by the user. Finlly, the reltive proportions of morphogen flowing to ech of the neighboring vertices re determined bsed on these trnsformed fces surrounding the vertex in the sme mnner s described bove. Once the differentil flow of morphogens hs been clculted, it is pplied to the Figure 17. Proportionl flow of morphogens long ech edge between the current vertex nd its neighboring vertices is determined by the normlized distnces between the centroids of the two fces on either side of tht edge. () (b) Figure 18. () The originl mesh showing the current vertex nd its surrounding vertices. The rrow shows the projection of the sketched ellipse s mjor xis vector onto the tngent plne t the current vertex. (b) Illustrtion of the vertices fter rottion followed by scling down the fces long the x-xis. Figure 19. Spots of different sizes generted on 3D mesh. Figure 0. Oriented spots generted on cube. submitted to EUROGRAPHICS Workshop on Sketch-Bsed Interfces nd Modeling (006)

8 8 Pper ID Summry nd future work The system described here llows user to mke simple sketch of desired pttern nd then utomticlly genertes lrge mount of texture tht hs similr pttern. This interfce llows the user to produce desired texture with ese nd without the need for ny internl knowledge of the Rection-Diffusion process. Insted of hving to run the system severl times to find the right set of prmeters tht will produce desired texture, the user cn sketch the ptterns they wnt nd hve the system utomticlly generte texture with similr ttributes. This is much more nturl nd intuitive method. As prt of future work, the dimension nd resolution of the grid cn be dynmiclly determined bsed on user s sketch, such s number nd size of ellipses or stripes, in order to mke the texture synthesis more flexible. As demonstrted here, this system cn be extended to support utomtic texture synthesis on cnonicl 3D surfces. The next step will be the development of stble nd robust system for utomtic texture synthesis on 3D model. 8. Appendix: 8.1. Formule for Turing spot system b [ k ( 16 b) + D ] [ k ( b b β ) + D b] 8.. Formule for Meinhrdt spot system 0.01 β s p t b b [ s ( 0.01 β bp + p ) + D b] 1 b + p3 + D 8.3. Formule for Meinhrdt stripe system 0.01 eβ k c b 0.01b c c d e d bk [ 0.01 eβ b d ck ] [( d ) k + D d] [( b e) k + D e] de de b e d b 3 + D + Db b b c 9. References: [Br81] BARD J. B. L.: A Model for Generting Aspects of Zebr nd Other Mmmlin Cot Ptterns. Journl of Theoreticl Biology, Vol. 93, No., pp (November 1981). [EMP*0] EBERT D. S., MUSGRAVE F. K., PEACHEY D., PERLIN K., WORLEY S.: Texturing & Modeling: A Procedurl Approch, Morgn Kufmnn, 3 rd edition, (December 00). [KTBG03] KULLA C. D., TUCEK D. J.., BAILEY R., GRIMM C. M.: Using Texture Synthesis for Non-Photorelistic Shding from Pint Smples. 11th Pcific Grphics Conference on Computer Grphics nd Applictions, pp. 477 (October 003). [Ml8] MALDELBROT B. B.: The Frctl Geometry of Nture, W. H. Freemn nd Compny, New York, 198. [Mei8] MEINHARDT H.: Models of Biologicl Pttern Formtion, Acdemic Press, London, 198. [Mur81] MURRAY J. D.: On Pttern Formtion Mechnisms for Lepidoptern Wing Ptterns nd Mmmlin Cot Mrkings. Philosophicl Trnsctions of the Royl Society B, Vol. 95, pp (October 1981). [Per85] PERLIN K.: An Imge Synthesizer. Computer Grphics, Vol. 19, No. 3 (SIGGRAPH 85), pp (July 1985). [PH89] PERLIN K., HOFFERT E. M.: Hypertexture, Computer Grphics, Vol. 3, No. 3 (SIGGRAPH 89), pp (July 1989). [SA94] SCHAAL S., ATKESON C. G.: Assessing the qulity of lerned locl models. Advnces in Neurl Informtion Processing Systems 6 Morgn Kufmnn, Sn Mteo, CA, pp , [Tur5] TURING A.: The Chemicl Bsis of Morphogenesis. Philosophicl Trnsctions of the Royl Society B, Vol. 37, pp (August 14, 195). [Tur91] TURK G.: Generting Textures on Arbitrry Surfces Using Rection-Diffusion. Computer Grphics, Vol. 5, No. 4 (SIGGRAPH 91), pp (July 1991). [Tur9] TURK G.: Texturing Surfces Using Rection- Diffusion. PhD Disserttion, The University of North Crolin t Chpel Hill, 199. [WK91] WITKIN A., KASS M.: Rection-Diffusion Textures. Computer Grphics, Vol. 5, No. 4 (SIGGRAPH 91), pp (July 1991). submitted to EUROGRAPHICS Workshop on Sketch-Bsed Interfces nd Modeling (006)