1. Last Time Midterm Shading Light source Shading Interpolation
Point light sources
Directional light sources
Spot light sources
Shading Interpolation
11/10/09
© NTUST

2. This Week Shading Interpolation Texture mapping Texture Anti-Aliasing
Barycentric coordinates for triangles
Texture Anti-Aliasing
Texture boundaries
Modeling introduction
Homework5 Due 11/17
11/10/09
© NTUST

3. Flat shading
Compute shading at a representative point and apply to whole polygon
OpenGL uses one of the vertices
Advantages:
Fast - one shading computation per polygon
Disadvantages:
Inaccurate
What are the artifacts?
11/10/09
© NTUST

4. Gouraud Shading Shade each vertex with it’s own location and normal
Linearly interpolate the color across the face
Advantages:
Fast: incremental calculations when rasterizing
Much smoother - use same normal every time a vertex is used for a face
Disadvantages:
What are the artifacts?
Is it accurate?
11/10/09
© NTUST

5. Phong Interpolation Interpolate normals across faces
Shade each pixel individually
Advantages:
High quality, narrow specularities
Disadvantages:
Expensive
Still an approximation for most surfaces
Not to be confused with Phong’s specularity model
11/10/09
© NTUST

6. 11/10/09
© NTUST

7. Shading and OpenGL OpenGL defines two particular shading models
Controls how colors are assigned to pixels
glShadeModel(GL_SMOOTH) interpolates between the colors at the vertices (the default, Gouraud shading)
glShadeModel(GL_FLAT) uses a constant color across the polygon
Phong shading requires a significantly greater programming effort – later we will try to have a short discussion if possible
Also requires fragment shaders on programmable graphics hardware
11/10/09
© NTUST

8. The Current Generation
Current hardware allows you to break from the standard illumination model
Programmable Vertex Shaders and Fragment Shaders allow you to write a small program that determines how the color of a vertex or pixel is computed
Your program has access to the surface normal and position, plus anything else you care to give it (like the light)
You can add, subtract, take dot products, and so on
Fragment shaders are most useful for lighting because they operate on every pixel
11/10/09
© NTUST

9. Original And Modified Pipeline
Replace transform and lighting with vertex shader
Vertex shader must now do transform and lighting
But can also do more
Replace texture stages with fragment (pixel) shader
Previously, texture stages were only per-pixel operations
Fragment shader must do texturing
11/10/09
© NTUST

10. The Full Story
We have only touched on the complexities of illuminating surfaces
The common model is hopelessly inadequate for accurate lighting (but it’s fast and simple)
Consider two sub-problems of illumination
Where does the light go? Light transport
What happens at surfaces? Reflectance models
Other algorithms address the transport or the reflectance problem, or both
Much later in class, or a separate course (CS 779)
11/10/09
© NTUST

11. Mapping Techniques Consider the problem of rendering a soup can
The geometry is very simple - a cylinder
But the color changes rapidly, with sharp edges
With the local shading model, so far, the only place to specify color is at the vertices
To do a soup can, would need thousands of polygons for a simple shape
Same thing for an orange: simple shape but complex normal vectors
Solution: Mapping techniques use simple geometry modified by a detail map of some type
11/10/09
© NTUST

12. Texture Mapping
The soup tin is easily described by pasting a label on the plain cylinder
Texture mapping associates the color of a point with the color in an image: the texture
Soup tin: Each point on the cylinder gets the label’s color
Question to address: Which point of the texture do we use for a given point on the surface?
Establish a mapping from surface points to image points
Different mappings are common for different shapes
We will, for now, just look at triangles (polygons)
11/10/09
© NTUST

13. Example Mappings
11/10/09
© NTUST

14. Basic Mapping The texture lives in a 2D space
Parameterize points in the texture with 2 coordinates: (s,t)
These are just what we would call (x,y) if we were talking about an image, but we wish to avoid confusion with the world (x,y,z)
Define the mapping from (x,y,z) in world space to (s,t) in texture space
To find the color in the texture, take an (x,y,z) point on the surface, map it into texture space, and use it to look up the color of the texture
Samples in a texture are called texels, to distinguish them from pixels in the final image
With polygons:
Specify (s,t) coordinates at vertices
Interpolate (s,t) for other points based on given vertices
11/10/09
© NTUST

15. Texture Interpolation
Specify where the vertices in world space are mapped to in texture space
A texture coordinate is the location in texture space that corresponds to the vertex
Linearly interpolate the mapping for other points in world space
Straight lines in world space go to straight lines in texture space
(s,t)
(x,y),(s,t) t s
Texture map – texture coordinates
Triangle in world space – vertex coordinates
11/10/09
© NTUST

16. Interpolating Coordinates
(x3, y3), (s3, t3)
(x2, y2), (s2, t2)
(x1, y1), (s1, t1)
11/10/09
© NTUST

17. Barycentric Coordinates
An alternate way of describing points in triangles
These can be used to interpolate texture coordinates
Gives the same result as previous slide
Method in textbook (Shirley) x2 x x3 x1
11/10/09
© NTUST

18. Steps in Texture Mapping
Polygons (triangles) are specified with texture coordinates at the vertices
A modeling step, but some ways to automate it for common shapes
When rasterizing, interpolate the texture coordinates to get the texture coordinate at the current pixel
Previous slides
Look up the texture map using those coordinates
Just round the texture coordinates to integers and index the image
Take the color from the map and put it in the pixel
Many ways to put it into a pixel (more later)
11/10/09
© NTUST

19. Pipelines and Texture Mapping
Texture mapping is done in canonical screen space as the polygon is rasterized
When describing a scene, you assume that texture interpolation will be done in world space
Something goes wrong…
Interpolation then projection
Projection then interpolation
Textured square
11/10/09
© NTUST

20. Perspective Correct Mapping
Which property of perspective projection means that the “wrong thing” will happen if we apply our simple interpolations from the previous slide?
Perspective correct texture mapping does the right thing, but at a cost
Interpolate homogeneous coordinate w and divide it out just before indexing texture
Is it a problem with orthographic viewing?
11/10/09
© NTUST

21. Basic OpenGL Texturing (cont)
Enable texturing: glEnable(GL_TEXTURE_2D)
State how the texture will be used:
glTexEnvf(…)
Texturing is done after lighting
You’re ready to go…
11/10/09
© NTUST

22. Nasty Details
There are a large range of functions that control the layout of texture data:
You must state how the data in your image is arranged
Eg: glPixelStorei(GL_UNPACK_ALIGNMENT, 1) tells OpenGL not to skip bytes at the end of a row
You must state how you want the texture to be put in memory: how many bits per “pixel”, which channels,…
Textures must be square with width/height a power of 2
Common sizes are 32x32, 64x64, 256x256
Smaller uses less memory, and there is a finite amount of texture memory on graphics cards
Some extensions to OpenGL allow arbitrary textures
11/10/09
© NTUST

23. Controlling Different Parameters
The “pixels” in the texture map may be interpreted as many different things. For example:
As colors in RGB or RGBA format
As grayscale intensity
As alpha values only
The data can be applied to the polygon in many different ways:
Replace: Replace the polygon color with the texture color
Modulate: Multiply the polygon color with the texture color or intensity
Similar to compositing: Composite texture with base color using operator
11/10/09
© NTUST

24. Example: Diffuse shading and texture
Say you want to have an object textured and have the texture appear to be diffusely lit
Problem: Texture is applied after lighting, so how do you adjust the texture’s brightness?
Solution:
Make the polygon white and light it normally
Use glTexEnvi(GL_TEXTURE_2D, GL_TEXTURE_ENV_MODE, GL_MODULATE)
Use GL_RGB for internal format
Then, texture color is multiplied by surface (fragment) color, and alpha is taken from fragment
11/10/09
© NTUST

25. Specular Color Typically, texture mapping happens after lighting
More useful in general
Recall plastic surfaces and specularities: the highlight should be the color of the light
But if texturing happens after the lighting, the color of the specularity will be modified by the texture – the wrong thing
OpenGL lets you do the specular lighting after the texture
11/10/09
© NTUST

26. Some Other Uses
There is a “decal” mode for textures, which replaces the surface color with the texture color, as if you stick on a decal
But texture happens after lighting, so the light info is lost
BUT, you can use the texture to store lighting info, and generate better looking lighting
You can put the color information in the polygon, and use the texture for the brightness information
Called “light maps”
Normally, use multiple texture layers, one for color, one for light
11/10/09
© NTUST

27. Texture Recap
Triangle in 8x8 Texture Map, I(s,t)
Triangle in world space
I(s,t)
Interpolated (s,t) – we need a texture sample from I(s,t) t s
We must reconstruct the texture image at the point (s,t)
Time to apply the theory of sampling and reconstruction
11/10/09
© NTUST

28. 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
11/10/09
© NTUST

29. Point Sampled Texture Aliasing
Texture map
Polygon far from the viewer in perspective projection
Rasterized and textured
Note that the back row is a very poor representation of the true image
11/10/09
© NTUST

30. 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 the storage of the original texture (in the limit)
* *0.25*0.25+…
11/10/09
© NTUST

31. 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
11/10/09
© NTUST

32. Mipmaps For far objects For middle objects For near objects 11/10/09
© NTUST

33. Mipmap Math Define a scale factor, =texels/pixel Define =log2 
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 =log2 
 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
11/10/09
© NTUST

34. 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
11/10/09
© NTUST

35. Filtering Example s=0.12,t=0.1 =1.4 =0.49 Level 2 Level 1 Level 0
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
Level 1
NEAREST_MIPMAP_LINEAR:
level 0, combination of
pixels (0,0), (1,0), (1,1), (0,1)
Level 0
s=0.12,t=0.1
=1.4
=0.49
LINEAR_MIPMAP_LINEAR:
Combination of level 0 and level 1, 4 pixels from each level, using 8 pixels in all
11/10/09
© NTUST

36. 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)
11/10/09
© NTUST

37. Repeat Border
(1,1)
(0,0)
11/10/09
© NTUST

38. Clamp Border
(1,1)
(0,0)
11/10/09
© NTUST

39. Border Color
(1,1)
(0,0)
11/10/09
© NTUST

40. 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
11/10/09
© NTUST

41. 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
11/10/09
© NTUST

42. 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
11/10/09
© NTUST

43. 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
11/10/09
© NTUST

44. The Story So Far We’ve looked at images and image manipulation
We’ve looked at rendering from polygons
Next major section:
Modeling
11/10/09
© NTUST

45. Shape Modeling
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
Why?
Drawing, Sample, Analyze
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
Why is this hard
Shapes can be arbitrary and complex – hard to describe
Conflicting goals: Concise, Intuitive, Expressive, Analyzable, …
11/10/09
© NTUST 45

46. What is a Shape Mathematical definition is elusive Set of Points
Potentially (usually) infinite
“Lives” in some bigger space (e.g. 2D or 3D)
Many ways to describe sets
Set inclusion test (implicit representation)
Procedural for generating elements of the set
Explicit mapping from a known set
What are some things to think about when choosing a representation?
11/10/09
© NTUST 46

47. 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, …
11/10/09
© NTUST

48. Some kinds of Shapes Curves Surfaces / Areas Solids / Volumes
With 1D parameter space
Like what you draw with a pen
Surfaces / Areas
With 2D parameter space – the insides of things that take up surface
Bounded by a Curve
Solids / Volumes
With 3D parameter space – the insides of things that take up volume
Different definition: set with the same dimension as the embedded space (an area of 2D)
11/10/09
© NTUST 48

49. 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 x2 + y2 + z2- 1 = 0
11/10/09
© NTUST

50. Boundary vs. Solid Representations
B-rep: boundary representation
Sometimes we only care about the surface
Rendering opaque objects and geometric computations
Solid modeling
Some representations are best thought of defining the space filled, rather than the surface around the space
Medical data with information attached to the space
Transparent objects with internal structure
Taking cuts out of an object; “What will I see if I break this object?”
11/10/09
© NTUST

51. Parametric vs. Implicit vs. Procedural
Parametric generates all the points on a surface (volume) by “plugging in a parameter” eg
Implicit models use an equation that is 0 if the point is on the surface
Essentially a function to test the status of a point
Procedural: a procedure is used to answer any question you might have about the surface
eg Where does a given ray hit the surface?
11/10/09
© NTUST

52. 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
11/10/09
© NTUST

53. 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
Triangles in 2D can be parameterized by their barycentric coordinates
Triangles in 3D can be parameterized by 3 vertices and the barycentric coordinates (need both to locate a point in 3D space)
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; require (s,t) parameters for every point in a triangle
11/10/09
© NTUST

54. Techniques We Will Examine
Curves
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
11/10/09
© NTUST

55. What are Parametric Curves?
Define a parameter space
1D for curves
2D for surfaces
Define a mapping from parameter space to 3D points
A function that takes parameter values and gives back 3D points
The result is a parametric curve or surface
(Fx(t), Fy(t))
Mapping:F:t→(x,y) 1 1 t
11/10/09
© NTUST

56. Why Parametric Curves?
Parametric curves are intended to provide the generality of polygon meshes but with fewer parameters for smooth surfaces
Polygon meshes have as many parameters as there are vertices (at least)
Fewer parameters makes it faster to create a curve, and easier to edit an existing curve
Normal vectors and texture coordinates can be easily defined everywhere
Parametric curves are easier to animate than polygon meshes
11/10/09
© NTUST

57. Local vs. Global Properties
Local – what can you tell at a point
Position, Direction (tangent)
Global – need to look at the whole curve
Closed, consistent
Tangent vector – direction of motion
First derivative w.r.t. “time” (or parameter)
11/10/09
© NTUST 57