Surface Details ✫ Incorp orate ne details in the scene. ✫ Mo deling with p olygons is impractical. ✫ Map an image texture/pattern on the surface Catmull, 1974; Blin & Newell, 1976. ✫ Texture map Mo dels patterns, rough surfaces, 3D e ects. ✫ Solid textures 3D textures to mo del wood grain, stains, marble, etc. ✫ Bump mapping Displace normals to create shading e ects. ✫ Environment mapping Re ections of environment on shiny surfaces. ✫ Displacement mapping Perturb the p osition of some pixels. CPS124, 296: Computer Graphics Surface Details Page 1 Texture Maps ✫ Maps an image on a surface. ✫ Each element is called texel. ✫ Textures are xed patterns, pro cedurally gen- erated, or digitized images. CPS124, 296: Computer Graphics Surface Details Page 2 Texture Maps ✫ Texture map has its own co ordinate system; st-co ordinate system. ✫ Surface has its own co ordinate system; uv -co ordinates. ✫ Pixels are referenced in the window co ordinate system Cartesian co ordinates. Texture Space Object Space Image Space (s,t) (u,v) (x,y) CPS124, 296: Computer Graphics Surface Details Page 3 Texture Maps v u y xs t s x ys z y xs t s x ys z CPS124, 296: Computer Graphics Surface Details Page 4 Texture Mapping CPS124, 296: Computer Graphics Surface Details Page 5 Texture Mapping Forward mapping: Texture scanning ✫ Map texture pattern to the ob ject space. u = f s; t = a s + b t + c; u u u v = f s; t = a s + b t + c: v v v ✫ Map ob ject space to window co ordinate sys- tem. Use mo delview/pro jection transformations. Drawback: Selected texture patch usual do es not match with pixel b oundaries. ✫ Requires fractional pixel calculations. CPS124, 296: Computer Graphics Surface Details Page 6 Inverse Mapping y xs t -1 -1 M M PV T s x ys z ✫ Map screen co ordinate system to ob ject space. ✫ Map ob ject co ordinate system to texture space. ✫ Avoids fractional pixel calculations. ✫ Allows anti-aliasing. ✫ Requires calculating inverse transformations; 1 1 M ;M . PV T 1 M can be computed from pro jection and PV mo delview matrices gluUnproject 1 Computing M is not easy. T CPS124, 296: Computer Graphics Surface Details Page 7 Parametric Representation Curves: Co ordinates are represented as functions of one parameter. t = xt;yt; t 2 [a; b] Line: ` : a + b t; a + b t, t 2 R 1 1 2 2 Surfaces: Co ordinates of each p oint is represented as a function of u and v . S u; v = xu; v ;yu; v ;zu; v . ✫ Sphere: S u; v = r cos u cos v; r cos u sin v; r sin u =2 u =2; 0 v 2. ✫Cylinder: S u; v = r cos u; r sin u; v 0 u 2; 0 v h. CPS124, 296: Computer Graphics Surface Details Page 8 Inverse Mapping: An Example y t (u,v) s x z u = ; v = y 0 =2; 0 y 1 x = sin ; z = cos ; y = v: 1 1 M : u = sin x; v = y: PV Map texture origin to left b ottom corner of the surface u = s =2; v = t: 1 Pro jected pixels are mapp ed to texture with M : T 2u 1 M : s = t = v: T 2 1 s = sin x t = z: CPS124, 296: Computer Graphics Surface Details Page 9 Object to Texture Mapping 1 How do es one de ne a reasonable M ? T Bier & Sloan, 1986: Two-step pro cess: S -mapping: Mapping from a 2D texture space to a simple 3D surface, e.g., cylinder. 0 T u; v ! T x ;y ;z : i i i O-mapping: Mapping from the 3D texture pat- tern onto the ob ject surface. 0 T x ;y ;z ! Ox ;y ;z : i i i w w w CPS124, 296: Computer Graphics Surface Details Page 10 Two-Step Inverse Mapping Inverse mapping: Apply them in the reverse or- der! -1 S -1 O M-1 CPS124, 296: Computer Graphics Surface Details Page 11 S -Mapping Four p ossible surfaces: Plane, cylinder, sphere, box. Cylinder: ; z 1 r ; z z S : ; z ! 0 0 c d ✫c; d: Scaling factors. ✫ Texture origin is mapp ed to the p oint ;z 0 0 on the cylinder. Sphere: ; ', stereographic pro jection. θ (α,β) φ (θ,φ) 2 S : ; ' ! ; 2 2 1=2 1 + 1 + + 2 2 2 1=2 1 + 1 + + = tan ' cos , = tan ' sin . CPS124, 296: Computer Graphics Surface Details Page 12 S -Mapping t s t Back Left Bottom Right Top s Front CPS124, 296: Computer Graphics Surface Details Page 13 O -Mapping P' P' P P (ii) (i) P' P' P P (iii) C (iv) ✫ : Intermediate surface 0 ✫ P : Intersection of with a ray emanating from P . ✫ Direction of : i Re ection direction ii Normal of O at P iii CP ; C: centroid of O 0 iv : Normal of at P ✫ ii, iii are bad if is cylinder. CPS124, 296: Computer Graphics Surface Details Page 14 3D Textures Intro duced by Peachey and Perlin in 1985. ✫ De ne texture to be a 3D image. Carving out an ob ject from a 3D solid ma- terial. ✫ Ignoring scaling, M is identity. T ✫ Distortion is minimized. ✫ Three dimensional vector elds can be mapp ed coherently. ✫ Texture is generated by a pro cedure. ✫ Example: Wood grain can be mapp ed as a set of cylinders with resp ect to a presp eci ed axis. CPS124, 296: Computer Graphics Surface Details Page 15 3D Textures Basic texture 3D noise textured object 3D texture Textures object Object CPS124, 296: Computer Graphics Surface Details Page 16 Anti-aliasing & Texture Mapping Aliasing is particularly visible in p erio dic and co- herent textures. With Without Antialiasing Antialiasing CPS124, 296: Computer Graphics Surface Details Page 17 Anti-aliasing & Texture Mapping ✫ A pixel is mapp ed to a curvilinear quadrilat- eral. ✫ A single pixel may cover many texels. ✫ Compute a weighted sum of texel values cov- ered by the pixel. ✫ Summation is called ltering. Two-step pro cess: ✫ De ne and approximate the texture over which ltering is p erformed. ✫ Integrate by weighing and summing the texel values within the ltering area. CPS124, 296: Computer Graphics Surface Details Page 18 Anti-aliasing: Weight Functions Blinn & Newell, 1976: Pyramid weight function Preimage of Approximation Preimage of the pixel center Greene & Heckb ert, 1986: Elliptical weighted av- erage ✫ Approximate a pixel by circle. ✫ Circle always maps to an ellipse in the texture space. ✫ Find the texels that lie inside the ellipse. ✫ Use a lo ok-up table to determine the weighted value of each texel. CPS124, 296: Computer Graphics Surface Details Page 19 Anti-aliasing & Texture Mapping Mip-mapping: multum in parvo Williams, 1983 ✫ Store many texture images. ✫ i-th image is obtained by scaling down the pre- vious image by half along each axis. ✫ E ectively a 3D database. ✫ Given a pixel, search in the image with an ap- propriate resolution. CPS124, 296: Computer Graphics Surface Details Page 20 Texture Mapping in Op enGL Geometric Vertices Rasterization Display processing Pixel Pixels operations ✫ Relies on the pip eline architecture ✫ Texture mapping is done at the rasterization stage. glTexImage2D GL TEXTURE 2D, level, comp, w, h, b order, format, typ e, tarray GLubyte my texel[512][512]; glTexImage2D GL TEXTURE 2D, 0, 3, 512, 512, 0 GL RGB, GLuint, tarray; glEnableGL TEXTURE 2D; ✫ level: Multiple levels of texture maps; 0 for one level. ✫ comp: integer between 1 and 4; sp eci es how many of R, G, B, and A comp onents sp eci ed. ✫ format: Format of the texture map. CPS124, 296: Computer Graphics Surface Details Page 21 Texture Mapping in Op enGL glTexCoord2f s, t Assigns the two dimensional texture co ordinates to a vertex. (a) (b) glBeginGL QUAD; glTexCoord2f 0.0, 0.0; glVertex3f x1, y1, z1; glTexCoord2f 1.0, 0.0; glVertex3f x2, y2, z2; glTexCoord2f 1.0, 1.0; glVertex3f x3, y3, z3; glTexCoord2f 0.0, 1.0; glVertex3f x4, y4, z4; glEnd CPS124, 296: Computer Graphics Surface Details Page 22 CPS124, ✫ Texture ob jects 296: T exture glGenTexturesn, *names; glBindTexturetarget, name; Computer ✫ Rep eating the texture pattern Mapping glTexParameteriGL TEXTURE WRAP S,GL REPEAT Graphics glTexParameteri GL TEXTURE WRAP S,GL CLAMP Use GL TEXTURE WRAP T for t-co ordinates ✫ Assign a pixel color to the nearest texel in Surf glTexParameterf GL TEXTURE2D, penGL Op GL TEXTURE MAG FILTER, GL NEAREST a ce glTexParameterf GL TEXTURE2D, Det GL TEXTURE MIN FILTER, GL NEAREST ails ✫ GL LINEAR: Interp olates the color using a 2 2 average. P age 23 Bump Mapping Perturb normals in the illumination mo del calcu- lations. ✫ S u; v : Parameterized surface ✫ S = @Su; v =@ u, S = @Su; v =@ v . u v ✫ N u; v = S S ; n = N=jN j u v ✫bu; v : Bump function 0 0 0 S u; v = S u; v + bu; v n N u; v = S S v u @ 0 S = Su; v + bu; v n u @u = S + b n + bn u u u S + b n u u 0 S S + b n v v v 0 N S S + b S n + b n S + u v u u v v b b n n u v = S S + b S n + b n S u v u u v v CPS124, 296: Computer Graphics Surface Details Page 24 Bump Mapping S(u) b(u) S'(u) N'(u) CPS124, 296: Computer Graphics Surface Details Page 25 Bump Mapping ✫ De ne bump functions analytically. ✫ Use lo ok-up tables for bump functions. ✫ Approximate b ;b with nite di erences.