INF3320 Computer Graphics and Discrete Geometry

Transcription

1 INF3320 Computer Graphics and Discrete Geometry Texturing Christopher Dyken Martin Reimers Page 1

2 Texturing Linear interpolation Real Time Rendering: Chapter 5: Visual Appearance Chapter 6: Texturing The Texturing Pipeline Image Texturing Procedural Texturing The Red Book: Chapter 9: Texture Mapping Page 2

3 Interpolation Page 3

4 Linear interpolation Problem: approximate a func. f from samples f 0, f 1 at x 0, x 1. Typical case is data given in a table. f f f (x) 1 f 0 x 0 x x 1 x Solution: find f (x) = ax + b such that f (x 0 ) = f 0 and f (x 1 ) = f 1 Page 4

5 Linear interpolation Problem: approximate a func. f from samples f 0, f 1 at x 0, x 1. Typical case is data given in a table. f f f (x) 1 f 0 x 0 x x 1 x Solution: find f (x) = ax + b such that f (x 0 ) = f 0 and f (x 1 ) = f 1 Remember: affine functions f (x) = Ax + b satisfy f ((1 α)x 0 + αx 1 ) = (1 α)f (x 0 ) + αf (x 1 ) Yields a formula (Bernstein form) for linear interpolation: only need to find (unique) α s.t. x = (1 α)x 0 + αx 1 Note: x and f (x) are weighted averages (convex combinations) Page 5

6 Linear interpolation in 2D Similar problem in 2D: find affine f (x, y) = ax + by + c such that f (v i ) = f i for 3 points v 0, v 1, v 2, then evaluate f (x). Page 6

7 Linear interpolation in 2D Similar problem in 2D: find affine f (x, y) = ax + by + c such that f (v i ) = f i for 3 points v 0, v 1, v 2, then evaluate f (x). Alternative: f is affine and satisfies for all α 0 + α 1 + α 2 = 1 f (α 0 v 0 + α 1 v 1 + α 2 v 2 ) = α 0 f (v 0 ) + α 1 f (v 1 ) + α 2 f (v 2 ) For given v, the problem is then reduced to finding α i s.t. αi = 1 and v = α 0 v 0 + α 1 v 1 + α 2 v 2 Note: x and f (x) are convex combinations, with the same weights (α i ), called barycentric coordinates Page 7

8 Linear interpolation 2D - barycentric form Given a point v and a triangle T = [v 0, v 1, v 2 ], there are unique barycentric coordinates α i = α i (v) such that v = α 0 v 0 + α 1 v 1 + α 2 v 2, α 0 + α 1 + α 2 = 1 v is expressed as a convex combination (average) of the vertices. p b p b p a p c pa p pc α 0 = area(v, v 1, v 2 ) area(v 0, v 1, v 2 ) 0 α i 1 for v inside T α 1 = area(v, v 0, v 2 ) area(v 0, v 1, v 2 ) α 2 = area(v, v 0, v 1 ) area(v 0, v 1, v 2 ) One coordinate is redundant - e.g. α 2 = 1 α 0 α 1 Also called homogenuous coordinates Generalize to nd, important and useful Page 8

9 Linear interpolation 2D - barycentric coordinates II Given data (f i ) at the vertices of a triangle T = [v 0, v 1, v 2 ], we can interpolate at an arbitrary point v by f (v) = α 0 f (v 0 ) + α 1 f (v 1 ) + α 2 f (v 2 ) f (v) is a convex combination (average) of f at the vertices. f 1 f 0 f y v 1 f 2 v 0 x v 2 Page 9

10 Bi-linear interpolation Given tabulated data f ij on a square v ij = (i, j), with i, j = 0, 1, we can find a bi-linear function f such that f (i, j) = f ij : f (x, y) = (1 x)(1 y) f 00 + x(1 y) f 10 + xy f 11 + (1 x)y f 01 v 01 v 11 f 11 y f 01 f 00 f 10 v 00 x v 10 v 01 v 00 v 10 v 11 A linear function in each variable... but a quadratic on the diagonal x = y Generalizes to any dimension (tri-,..., n-linear interpolation) Used to interpolate gridded data, eg. textures... Page 10

11 Texturing Page 11

12 What is texturing? Texturing in computer graphics Texturing means modifying a surface s appearance using an image, function or other data source. Texturing may for example modify: Color (image mapping). The most common case. Normals (bump mapping). Geometry (displacement mapping). Page 12

13 Motivation Better appearance at lower cost. May use lower quality geometry, i.e. fewer vertices. May use less advanced lighting algorithms. Easy to add decals, text, images. Page 13

14 Example textures Example textures from Diffuse Bump Specular Page 14

15 Example rendering Page 15

16 Tradeoffs Basic rule: Fine details in textures, coarser features need geometry. The optimal boundary between the two may depend on a multitude of factors. Typical case 1. Far away: Image texturing is fine. 2. Closer: Illusion breaks down, bump mapping optimal solution. 3. View approaches tangential: Illusion breaks down again, displacement mapping or proper geometry necessary. Page 16

17 Defining textures in OpenGL Textures are managed through texture names: GLuint texture_names[n]; glgentextures(n, texture_names); gldeletetextures(n, texture_names); OpenGL supports several texture types (targets): GL_TEXTURE_1D, GL_TEXTURE_2D, GL_TEXTURE_3D,... We bind texture to use/manipulate it: glbindtexture( GL_TEXTURE_2D, texture_names[0]); Then we can specify texture, manipulate it, etc: glteximage2d( GL_TEXTURE_2D, level, iformat width, height, border, format, type, &image[0][0][0] ); Page 17

18 Defining textures in OpenGL - II Many parameters controls the bound texture (wrapping, filtering,...) gltexparameteri(gl_texture_2d,...); Texturing must be enabled: glenable(gl_texture_2d); Many functions for manipulating textures gltexsubimage2d();... glcopyteximage2d(); Page 18

19 Texture coordinates Texture coords t 1, t 2, t 3 are texture space positions that assign a position in the texture to each vertex v 1, v 2, v 3 (red arrows) t 3 v 2 t 2 t v 1 1 v 3 Specified arbitrarily at each vertex. Interpolated over each triangle. = texture coordinate for each pixel (blue arrow). Notice that pixels and texels rarely match Page 19

20 Parametric surfaces usually have a natural set of texcoords Parameterization Objects without parameterization needs to be parameterized. We must find a mapping φ from the domain T to the image S. S φ T Parameterization is not trivial and is an open research question. Sometimes we can obtain texcoords by projecting the vertex positions onto an intermediate geometry (projector funcs): Parallel projection onto a plane Along surface normal onto a sphere Along surface normal onto a cube Page 20

21 In OpenGL we use gltexcoord to specify the current texcoord, glvertex creates a vertex with the current texcoord associated. gltexcoord before glvertex Corresponder functions Corresponder functions specifies the mapping between texture coordinates and texels. Vanilla OpenGL corresponder function gltexcoord( 0.0, 0.0 ) my_texels[0][0] gltexcoord( 1.0, 0.0 ) my_texels[511][0] gltexcoord( 0.0, 1.0 ) my_texels[0][511] gltexcoord( 1.0, 1.0 ) my_texels[511][511] Page 21

22 Generalized Texturing Texturing works by modifying surface attributes over the triangle. 1. The fragment location in space is the starting point. 2. A projector function gives texture space values. 3. The corresponder function transforms texture space values to a location in texture image (not a pixel in image). 4. Sampling the texture at the image location gives a value. 5. The value transform function transforms the value The resulting value is used to modify a surface property 2. 1 e.g. gamma correction 2 e.g. surface color Page 22

23 The projector function Takes an object space pos. and gives a texture space pos. We can project onto a simple intermediate object and use that object s parameterization to determine the texture position: Parallel projection onto a plane Along surface normal onto a sphere Along surface normal onto a cube Parametric curved surfaces usually have an intrinsic parameter space, and we can use this directly. OpenGL s gltexgen provides some different projector functions. Page 23

24 Ideally, the texture is glued to the object in a way that: minimizes distortion good correspondence between object and parameter space. φ S T The problem of fitting a texture space (parameter space) onto an object is known as parameterization and is an open research question. Parameterization is a topic in INF Page 24

25 The corresponder function Maps from texture space position to image locations. Texture domains usually 2D with (u, v) in [0, 1] [0, 1]. Sometimes 3D where the third coordinate can represent depth = volumetric texture. And in some cases 4D (homogeneous coordinates) = e.g. spotlight effects. Often, the corresponder function simply scales the coordinates [0, 1] [0, 1] [0, 256] [0, 256] for a texture of size Page 25

26 Corresponder func determines behavior outside [0, 1] [0, 1] repeat mirror clamp clamp to border This is called wrap mode and is specified with gltexparameter We can make the texture repeat itself [0, 1] [1, 1] [0, 4] [2, 4] [0, 3/4] [3/4, 3/4] [0, 2] [2, 2] [0,4] [4, 4] [0, 0] [1, 0] [0, 0] [2, 0] [0, 0] [3/4, 0] [0, 0] [2, 0] [0,0] [4, 0] Page 26

27 Texture filtering Texels and pixels rarely match: Use filtering techniques (interpolation) Mipmapping, anisotropic filtering, antialiasing,... Tradeoff: quality vs speed Page 27

28 Texture magnification In this case, a texel covers several pixels. OpenGL has two strategies to deal with this: Nearest neighbour. Bilinear interpolation. The texture magnification filter is specified using one of gltexparameter(gl_texture_2d, GL_TEXTURE_MAG_FILTER, GL_LINEAR); gltexparameter(gl_texture_2d, GL_TEXTURE_MAG_FILTER, GL_NEAREST); Page 28

29 Texture minification If a pixel covers several texels, the texture is under-sampled. We get aliasing artifacts. Mipmaps are pre-filtered textures, halving the size for each level: Then, use the mipmap level s.t. pixel and texel size matches and Choose the nearest mipmap level. Interpolate between two adjacent mipmap levels. The texture minification filter is specified using gltexparameter(gl_texture_2d, GL_TEXTURE_MIN_FILTER, filter); Page 29

30 Texture combine function The texture combine function specifies how the texture color is combined with the color of the fragment. OpenGL fixed function texture combine functions Replace replaces the texel color with the texture color. Decal lets texture α blend between pixel color and texture color. Blend uses texture color to blend texel and a pre-specified color. Modulate multiplies the surface colour with texture colour. Example: Lighted textures Render white shapes with lighting enabled and use the modulate texture combine. In OpenGL, gltexenv* to specifies the texture combine function. Page 30

31 Nate robbins tutorial Page 31

32 Texturing Methods Page 32

33 Environmental maps Shiny surfaces, such as mirrors, produce specular reflections that mirror the whole environment (not just the light sources). E.g., if a shiny metal ball is placed in a room, we see the contents of the room, albeit distorted, in the ball. Environmental maps is a simple yet powerful method of generating approximations of reflections in curved surfaces. We assume that reflected objects and lights are far away and the reflector will not reflect itself = only the reflected view direction matters, not the position on the surface. We will look at two environment mapping techniques: sphere mapping cube mapping Page 33

34 Sphere mapping (first environment mapping technique supported in hardware) view direction ray surface normal reflected ray Image plane Used for mapping spherical environment onto reflective objects The environment is modeled as a sphere For each pixel, the reflection vector is calculated The reflection corresponds to a texture coordinate (A texel corresponds to a reflection direction) Get color and other data from texture Page 34

35 Sphere mapping - example Page 35

36 Sphere mapping characteristics Sphere mapping pros: Can be implemented on any graphics hardware that supports texture mapping. Sphere map cons: The path between two points on the sphere map, therefore linear interpolation on the texture is just an approximation. The sphere map is only valid in one direction. Singularity located around the edge Page 36

37 Cube mapping Similar to spherical environment mapping Each of the six images of a cube map captures the environment seen from the center of a cube through one of the cube s faces. box around view point face frustum face image view point Since regular perspective is used, these images can be rendered at runtime with render-to-texture (dynamic environment maps) Page 37

38 Cube mapping characteristics Cube mapping pros: Has no singularities Has a quite uniform sampling characteristic Is view-independent Cube map cons: Requires special hardware Page 38

39 Bump and normal maps Geometry Idea: = Phong shading Normal map Parallax occlusion map Use textures to modulate the interpolated surface normal. the surface looks more geometric detailed! Page 39

40 Normal mapping Phong: interpolate normal vector and normalize varying vec3 normal; void main() { vec3 n = normalize( normal ); gl_fragcolor = max( dot(l,n),0.0 ) * gl_color + pow( max( dot(h,n),0.0), 40 )*vec4(1.0); } Normal map: fetch normal vector from texture uniform sampler2d nm; uniform mat3 nmat; // normal transform usually done by vertex shader void main() { vec3 n = 2*nmat*texture2D( ntex, gl_texcoord[0].xy )-vec(1); gl_fragcolor = max( dot(l,n),0.0 ) * gl_color + pow( max( dot(h,n),0.0), 40 )*vec4(1.0); } Excellent results, but difficult to animate (must update texture)! Page 40

41 The last slide Next time: The opengl pipeline Page 41