1. 11/11/04© University of Wisconsin, CS559 Fall 2004 Last Time Shading Interpolation Texture mapping –Barycentric coordinates for triangles

2. 11/11/04© University of Wisconsin, CS559 Fall 2004 Today Texture Anti-Aliasing Texture boundaries Modeling introduction

3. 11/11/04© University of Wisconsin, CS559 Fall 2004 Texture Recap We must reconstruct the texture image at the point (s,t) Time to apply the theory of sampling and reconstruction Triangle in 8x8 Texture Map, I(s,t) s t Triangle in world space Interpolated (s,t) – we need a texture sample from I(s,t) I(s,t)

4. 11/11/04© University of Wisconsin, CS559 Fall 2004 Textures and Aliasing Textures are subject to aliasing: –A polygon pixel maps into a texture image, essentially sampling the texture at a point –The situation is essentially an image warp, with the warp defined by the mapping and projection Standard approaches: –Pre-filtering: Filter the texture down before applying it Useful when the texture has multiple texels per output image pixel –Post-filtering: Take multiple pixels from the texture and filter them before applying to the polygon fragment Useful in all situations

5. 11/11/04© University of Wisconsin, CS559 Fall 2004 Point Sampled Texture Aliasing Note that the back row is a very poor representation of the true image Texture map Polygon far from the viewer in perspective projection Rasterized and textured

6. 11/11/04© University of Wisconsin, CS559 Fall 2004 Mipmapping (Pre-filtering) If a textured object is far away, one screen pixel (on an object) may map to many texture pixels –The problem is: how to combine them A mipmap is a low resolution version of a texture –Texture is filtered down as a pre-processing step: gluBuild2DMipmaps(…) –When the textured object is far away, use the mipmap chosen so that one image pixel maps to at most four mipmap pixels –Full set of mipmaps requires at most 1.3333 the storage of the original texture (in the limit) 1+0.25+.25*.25+0.25*0.25*0.25+…

7. 11/11/04© University of Wisconsin, CS559 Fall 2004 Many Texels for Each Pixel Texture map with pixels drawn on it. Some pixels cover many texture elements (texels) Polygon far from the viewer in perspective projection

8. 11/11/04© University of Wisconsin, CS559 Fall 2004 Mipmaps For near objects For far objects For middle objects

9. 11/11/04© University of Wisconsin, CS559 Fall 2004 Mipmap Math Define a scale factor,  =texels/pixel –A texel is a pixel from a texture –  is actually the maximum from x and y –The scale factor may vary over a polygon –It can be derived from the transformation matrices Define =log 2  tells you which mipmap level to use –Level 0 is the original texture, level 1 is the next smallest texture, and so on –If <0, then multiple pixels map to one texel: magnification

10. 11/11/04© University of Wisconsin, CS559 Fall 2004 Post-Filtering You tell OpenGL what sort of post-filtering to do Magnification: When <0 the image pixel is smaller than the texel: –glTexParameteri(GL_TEXTURE_2D, GL_TEXTURE_MAG_FILTER, type) –Type is GL_LINEAR or GL_NEAREST Minification: When >0 the image pixel is bigger than the texel: –GL_TEX_MIN_FILTER –Can choose to: Take nearest point in base texture, GL_NEAREST Linearly interpolate nearest 4 pixels in base texture, GL_LINEAR Take the nearest mipmap and then take nearest or interpolate in that mipmap, GL_NEAREST_MIPMAP_LINEAR Interpolate between the two nearest mipmaps using nearest or interpolated points from each, GL_LINEAR_MIPMAP_LINEAR

11. 11/11/04© University of Wisconsin, CS559 Fall 2004 Filtering Example Level 0 Level 2 Level 1 s=0.12,t=0.1  =1.4 =0.49 NEAREST_MIPMAP_NEAREST: level 0, pixel (0,0) LINEAR_MIPMAP_NEAREST: level 0, pixel (0,0) * 0.51 + level 1, pixel (0,0) * 0.49 NEAREST_MIPMAP_LINEAR: level 0, combination of pixels (0,0), (1,0), (1,1), (0,1) LINEAR_MIPMAP_LINEAR: Combination of level 0 and level 1, 4 pixels from each level, using 8 pixels in all

12. 11/11/04© University of Wisconsin, CS559 Fall 2004 Boundaries You can control what happens if a point maps to a texture coordinate outside of the texture image –All texture images are assumed to go from (0,0) to (1,1) in texture space – means that the mapping is independent of the texture size –The problem is how to extend the image to make an infinite space Repeat: Assume the texture is tiled –glTexParameteri(GL_TEXTURE_2D, GL_TEXTURE_WRAP_S, GL_REPEAT) Clamp to Edge: the texture coordinates are truncated to valid values, and then used –glTexParameteri(GL_TEXTURE_2D, GL_TEXTURE_WRAP_S, GL_CLAMP) Can specify a special border color: –glTexParameterfv(GL_TEXTURE_2D, GL_TEXTURE_BORDER_COLOR, R,G,B,A)

13. 11/11/04© University of Wisconsin, CS559 Fall 2004 Repeat Border (0,0) (1,1)

14. 11/11/04© University of Wisconsin, CS559 Fall 2004 Clamp Border (0,0) (1,1)

15. 11/11/04© University of Wisconsin, CS559 Fall 2004 Border Color (0,0) (1,1)

16. 11/11/04© University of Wisconsin, CS559 Fall 2004 Other Texture Stuff Texture must be in fast memory - it is accessed for every pixel drawn –If you exceed it, performance will degrade horribly –Skilled artists can pack textures for different objects into one image Texture memory is typically limited, so a range of functions are available to manage it Specifying texture coordinates can be annoying, so there are functions to automate it Sometimes you want to apply multiple textures to the same point: Multitexturing is now in most new hardware

17. 11/11/04© University of Wisconsin, CS559 Fall 2004 Yet More Texture Stuff There is a texture matrix: apply a matrix transformation to texture coordinates before indexing texture There are “image processing” operations that can be applied to the pixels coming out of the texture There are 1D and 3D textures –Mapping works essentially the same –3D textures are very memory intensive, and how they are used is very application dependent –1D saves memory if the texture is inherently 1D, like stripes

18. 11/11/04© University of Wisconsin, CS559 Fall 2004 Procedural Texture Mapping Instead of looking up an image, pass the texture coordinates to a function that computes the texture value on the fly –Renderman, the Pixar rendering language, does this –Available in a limited form with fragment shaders on current generation hardware Advantages: –Near-infinite resolution with small storage cost –Idea works for many other things Has the disadvantage of being slow in most cases

19. 11/11/04© University of Wisconsin, CS559 Fall 2004 Other Types of Mapping Environment mapping looks up incoming illumination in a map –Simulates reflections from shiny surfaces Bump-mapping computes an offset to the normal vector at each rendered pixel –No need to put bumps in geometry, but silhouette looks wrong Displacement mapping adds an offset to the surface at each point –Like putting bumps on geometry, but simpler to model All are available in software renderers like RenderMan compliant renderers All these are becoming available in hardware

20. 11/11/04© University of Wisconsin, CS559 Fall 2004 The Story So Far We’ve looked at images and image manipulation We’ve looked at rendering from polygons Next major section: –Modeling

21. 11/11/04© University of Wisconsin, CS559 Fall 2004 Modeling Overview Modeling is the process of describing an object Sometimes the description is an end in itself –eg: Computer aided design (CAD), Computer Aided Manufacturing (CAM) –The model is an exact description More typically in graphics, the model is then used for rendering (we will work on this assumption) –The model only exists to produce a picture –It can be an approximation, as long as the visual result is good The computer graphics motto: “If it looks right it is right” –Doesn’t work for CAD

22. 11/11/04© University of Wisconsin, CS559 Fall 2004 Issues in Modeling There are many ways to represent the shape of an object What are some things to think about when choosing a representation?

23. 11/11/04© University of Wisconsin, CS559 Fall 2004 Choosing a Representation How well does it represents the objects of interest? How easy is it to render (or convert to polygons)? How compact is it (how cheap to store and transmit)? How easy is it to create? –By hand, procedurally, by fitting to measurements, … How easy is it to interact with? –Modifying it, animating it How easy is it to perform geometric computations? –Distance, intersection, normal vectors, curvature, …

24. 11/11/04© University of Wisconsin, CS559 Fall 2004 Categorizing Modeling Techniques Surface vs. Volume –Sometimes we only care about the surface Rendering and geometric computations –Sometimes we want to know about the volume Medical data with information attached to the space Some representations are best thought of defining the space filled, rather than the surface around the space Parametric vs. Implicit –Parametric generates all the points on a surface (volume) by “plugging in a parameter” eg ( sin  cos , sin  sin , cos  ) –Implicit models tell you if a point in on (in) the surface (volume) eg x 2 + y 2 + z 2 - 1 = 0

25. 11/11/04© University of Wisconsin, CS559 Fall 2004 Parameterization Parameterization is the process of associating a set of parameters with every point on an object –For instance, a line is easily parameterized by a single value Actually, the barycentric parameterization for a line –Triangles can be parameterized by their barycentric coordinates –Polygons can be parameterized by defining a 2D space in the plane of the polygon, and using the 2D coordinates to give you 3D Several properties of a parameterization are important: –The smoothness of the mapping from parameter space to 3D points –The ease with which the parameter mapping can be inverted –Many more We care about parameterizations for several reasons –Texture mapping is the most obvious one you have seen so far

26. 11/11/04© University of Wisconsin, CS559 Fall 2004 Techniques We Will Examine Polygon meshes –Surface representation, Parametric representation Prototype instancing and hierarchical modeling –Surface or Volume, Parametric Volume enumeration schemes –Volume, Parametric or Implicit Parametric curves and surfaces –Surface, Parametric Subdivision curves and surfaces Procedural models