1. (c) 2002 University of Wisconsin, CS 559
Last Time
The end of Sampling and Filtering
Compositing
10/1/02
(c) 2002 University of Wisconsin, CS 559

2. (c) 2002 University of Wisconsin, CS 559
Today
Compositing Summary
Intro to 3D (and 2D) Graphics
Homework 2 due
10/1/02
(c) 2002 University of Wisconsin, CS 559

3. Compositing Operations
F and G describe how much of each input image survives, and cf and cg are pre-multiplied pixels, and all four channels are calculated
Operation F G
Over
Inside
Outside
Atop
Xor
Clear
Set
10/1/02
(c) 2002 University of Wisconsin, CS 559

4. (c) 2002 University of Wisconsin, CS 559
Unary Operators
Darken: Makes an image darker (or lighter) without affecting its opacity
Dissolve: Makes an image transparent without affecting its color
10/1/02
(c) 2002 University of Wisconsin, CS 559

5. (c) 2002 University of Wisconsin, CS 559
“PLUS” Operator
Computes composite by simply adding f and g, with no overlap rules
Useful for defining cross-dissolve in terms of compositing:
10/1/02
(c) 2002 University of Wisconsin, CS 559

6. (c) 2002 University of Wisconsin, CS 559
Obtaining  Values
Hand generate (paint a grayscale image)
Automatically create by segmenting an image into foreground background:
Blue-screening is the analog method
Remarkably complex to get right
“Lasso” is the Photoshop operation
With synthetic imagery, use a special background color that does not occur in the foreground
Brightest blue or green is common
10/1/02
(c) 2002 University of Wisconsin, CS 559

7. Compositing With Depth
Can store pixel “depth” instead of alpha
Then, compositing can truly take into account foreground and background
Generally only possible with synthetic imagery
Image Based Rendering is an area of graphics that, in part, tries to composite photographs taking into account depth
10/1/02
(c) 2002 University of Wisconsin, CS 559

8. (c) 2002 University of Wisconsin, CS 559
Where to now…
We are now done with images
We will spend several weeks on the mechanics of 3D graphics
Coordinate systems and Viewing
Clipping
Drawing lines and polygons
Lighting and shading
We will finish the semester with modeling and some additional topics
10/1/02
(c) 2002 University of Wisconsin, CS 559

9. (c) 2002 University of Wisconsin, CS 559
Graphics Toolkits
Graphics toolkits typically take care of the details of producing images from geometry
Input (via API functions):
Where the objects are located and what they look like
Where the camera is and how it behaves
Parameters for controlling the rendering
Functions (via API):
Perform well defined operations based on the input environment
Output: Pixel data in a framebuffer – an image in a special part of memory
Data can be put on the screen
Data can be read back for processing (part of toolkit)
10/1/02
(c) 2002 University of Wisconsin, CS 559

10. (c) 2002 University of Wisconsin, CS 559
OpenGL
OpenGL is an open standard graphics toolkit
Derived from SGI’s GL toolkit
Provides a range of functions for modeling, rendering and manipulating the framebuffer
What makes a good toolkit?
Alternatives: Direct3D, Java3D - more complex and less well supported
10/1/02
(c) 2002 University of Wisconsin, CS 559

11. (c) 2002 University of Wisconsin, CS 559
Coordinate Systems
The use of coordinate systems is fundamental to computer graphics
Coordinate systems are used to describe the locations of points in space
Multiple coordinate systems make graphics algorithms easier to understand and implement
10/1/02
(c) 2002 University of Wisconsin, CS 559

12. (c) 2002 University of Wisconsin, CS 559
Coordinate Systems (2)
Different coordinate systems represent the same point in different ways
Some operations are easier in one coordinate system than in another
We saw this in filtering: It’s easier to define the filter in one space, but transform it to another for use
(2,3)
(1,2) v v y y u u x x
10/1/02
(c) 2002 University of Wisconsin, CS 559

13. (c) 2002 University of Wisconsin, CS 559
Transformations
Transformations convert points between coordinate systems
(2,3)
u=x-1
v=y-1
(1,2) v v y y u
x=u+1
y=v+1 u x x
10/1/02
(c) 2002 University of Wisconsin, CS 559

14. Transformations (Alternate Interpretation)
Transformations modify an object’s shape and location in one coordinate system
The previous interpretation is better for some problems, this one is better for others
(2,3)
x’=x-1
y’=y-1 y y
(1,2)
x=x’+1
y=y’+1 x x
10/1/02
(c) 2002 University of Wisconsin, CS 559

15. 2D Affine Transformations
An affine transformation is one that can be written in the form:
10/1/02
(c) 2002 University of Wisconsin, CS 559

16. Why Affine Transformations?
Affine transformations are linear
Transforming all the individual points on a line gives the same set of points as transforming the endpoints and joining them
Interpolation is the same in either space: Find the halfway point in one space, and transform it. Will get the same result if the endpoints are transformed and then find the halfway point
10/1/02
(c) 2002 University of Wisconsin, CS 559

17. Composition of Affine Transforms
Any affine transformation can be composed as a sequence of simple transformations:
Translation
Scaling (possibly with negative values)
Rotation
See Shirley 1.3.6
10/1/02
(c) 2002 University of Wisconsin, CS 559

18. (c) 2002 University of Wisconsin, CS 559
2D Translation
Moves an object y y by x bx x
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(c) 2002 University of Wisconsin, CS 559

19. (c) 2002 University of Wisconsin, CS 559
2D Translation
Moves an object y y by x bx x
10/1/02
(c) 2002 University of Wisconsin, CS 559

20. (c) 2002 University of Wisconsin, CS 559
2D Scaling
Resizes an object in each dimension y y
syy y x x
sxx x
10/1/02
(c) 2002 University of Wisconsin, CS 559

21. (c) 2002 University of Wisconsin, CS 559
2D Scaling
Resizes an object in each dimension y y
syy y x x
sxx x
10/1/02
(c) 2002 University of Wisconsin, CS 559

22. (c) 2002 University of Wisconsin, CS 559
2D Rotation
Rotate counter-clockwise about the origin by an angle  y y  x x
10/1/02
(c) 2002 University of Wisconsin, CS 559

23. (c) 2002 University of Wisconsin, CS 559
2D Rotation
Rotate counter-clockwise about the origin by an angle  y y  x x
10/1/02
(c) 2002 University of Wisconsin, CS 559

24. (c) 2002 University of Wisconsin, CS 559
X-Axis Shear
Shear along x axis (What is the matrix for y axis shear?) y y x x
10/1/02
(c) 2002 University of Wisconsin, CS 559

25. (c) 2002 University of Wisconsin, CS 559
X-Axis Shear
Shear along x axis (What is the matrix for y axis shear?) y y x x
10/1/02
(c) 2002 University of Wisconsin, CS 559

26. (c) 2002 University of Wisconsin, CS 559
Reflect About X Axis x x
What is the matrix for reflect about Y axis?
10/1/02
(c) 2002 University of Wisconsin, CS 559

27. (c) 2002 University of Wisconsin, CS 559
Reflect About X Axis x x
What is the matrix for reflect about Y axis?
10/1/02
(c) 2002 University of Wisconsin, CS 559

28. Rotating About An Arbitrary Point
What happens when you apply a rotation transformation to an object that is not at the origin? y ? x
10/1/02
(c) 2002 University of Wisconsin, CS 559

29. Rotating About An Arbitrary Point
What happens when you apply a rotation transformation to an object that is not at the origin?
It translates as well y x x
10/1/02
(c) 2002 University of Wisconsin, CS 559

30. (c) 2002 University of Wisconsin, CS 559
How Do We Fix it?
How do we rotate an about an arbitrary point?
Hint: we know how to rotate about the origin of a coordinate system
10/1/02
(c) 2002 University of Wisconsin, CS 559

31. Rotating About An Arbitrary Pointx x y y x x
10/1/02
(c) 2002 University of Wisconsin, CS 559

32. Rotate About Arbitrary Point
Say you wish to rotate about the point (a,b)
You know how to rotate about (0,0)
Translate so that (a,b) is at (0,0)
x’=x–a, y’=y–b
Rotate
x”=(x-a)cos-(y-b)sin, y”=(x-a)sin+(y-b)cos
Translate back again
xf=x”+a, yf=y”+b
10/1/02
(c) 2002 University of Wisconsin, CS 559

33. Scaling an Object not at the Origin
What also happens if you apply the scaling transformation to an object not at the origin?
Based on the rotating about a point composition, what should you do to resize an object about its own center?
10/1/02
(c) 2002 University of Wisconsin, CS 559

34. Back to Rotation About a Pt
Say R is the rotation matrix to apply, and p is the point about which to rotate
Translation to Origin:
Rotation:
Translate back:
The translation component of the composite transformation involves the rotation matrix. What a mess!
10/1/02
(c) 2002 University of Wisconsin, CS 559

35. Homogeneous Coordinates
Use three numbers to represent a point
(x,y)=(wx,wy,w) for any constant w0
Typically, (x,y) becomes (x,y,1)
Translation can now be done with matrix multiplication!
10/1/02
(c) 2002 University of Wisconsin, CS 559

36. Basic Transformations
Translation: Rotation:
Scaling:
10/1/02
(c) 2002 University of Wisconsin, CS 559

37. Homogeneous Transform Advantages
Unified view of transformation as matrix multiplication
Easier in hardware and software
To compose transformations, simply multiply matrices
Order matters: AB is generally not the same as BA
Allows for non-affine transformations:
Perspective projections!
Bends, tapers, many others
10/1/02
(c) 2002 University of Wisconsin, CS 559