1. Lecture 9: Projection Li Zhang Spring 2008
CS559: Computer Graphics
Lecture 9: Projection
Li Zhang
Spring 2008

2. Today Projection Reading: Shirley ch 7
Last time, we talked how to sample a continuous light pattern into a discrete iamge.
How to resample an discrete at different resolution or sampling rate.
This lecture, we’ll look at what operation we can do on a discrete image.
We’ll start with convolution

3. Geometric Interpretation 3D Rotations
Can construct rotation matrix from 3 orthonormal vectors
Rows of matrix are 3 unit vectors of new coord frame
Effectively, projections of point into new coord frame
First row

4. The 3D synthetic camera model
3D model and virtual camera
Simulate how light goes through the camera to form an image
The synthetic camera model involves two components,
specified independently:
objects (a.k.a. geometry)
viewer (a.k.a. camera)

5. Imaging with the synthetic camera
In graphics, we don’t need to put film in the box, we can image putting it in front of the camera,
What’s the difference? Upside down.
Conceptually trivial. Connecting each point on the model, find the intersection, draw the color there.
For modeling, repeat shapes, waste of time to create pillar for every pillar. Having one and move it around.
In general, we have library of objects. That’s the modeling part. We’ll be looking at those later on.
The image is rendered onto an image plane or projection plane (usually
in front of the camera).
Rays emanate from the center of projection (COP) at the center of the lens (or pinhole).
The image of an object point P is at the intersection of the ray
through P and the image plane.

6. Specifying a viewer
Camera specification requires four kinds of parameters:
􀂊 Position: the COP.
􀂊 Orientation: rotations about axes with origin at the COP.
􀂊 Focal length: determines the size of the image on the film plane, or the field of view.
􀂊 Film plane: its width and height.
In this lecture, we’ll be focusing on the camera part.

7. 3D Geometry Pipeline Model Space (Object Space) World Space Rotation
Library of objects
Each one is in its’ own coord. Uint cube, unit sphere
Model Space
(Object Space)
World Space
Rotation
Translation
Resizing

8. 3D Geometry Pipeline Model Space (Object Space) Eye Space (View Space)
Setting up the frustrum, fov, cliping plane.
Occlusion, visibility changes
Converging is OK, but not most convenient to implement, instead…
Model Space
(Object Space)
Eye Space
(View Space)
World Space
Rotation
Translation
Projective,
Scale
Translation
Rotation
Translation
Resizing

9. 3D Geometry Pipeline (cont’d)
Every thing here is still in 3D
Continuous light pattern hitting the sensor plane
Sample the lighting pattern to form a digital image
Normalized Projection
Space
Image Space (pixels)
Raster Space
Screen Space (2D)
Project
Scale
Translation

10. World -> eye transformation
Let’s look at how we would compute the world->eye transformation.
This is a simple exercise of the last slide of Monday lecture or first slide of today’s lecture
This is a good test whether you really understand the geometric interpretation of rotation

11. World -> eye transformation
Let’s look at how we would compute the world->eye transformation.
This is a good test whether you really understand the geometric interpretation of rotation
P-e is a translation
Dot is rotation

12. How to specify eye coordinate?
Z negative convetion
Which is vertical direction, or up vector?
Advantage, no need to have u.ze=0
Give eye location e
Give target position p
Give upward direction u
OpenGL has a helper command:
gluLookAt (eyex, eyey, eyez, px, py, pz, upx, upy, upz )

13. Projections
Projections transform points in n-space to m-space, where m<n.
In 3-D, we map points from 3-space to the projection plane (PP) (a.k.a., image plane) along rays emanating from the center of projection (COP):
There are two basic types of projections:
Perspective – distance from COP to PP finite
Parallel – distance from COP to PP infinite

14. Parallel projections
For parallel projections, we specify a direction of projection (DOP) instead of a COP.
We can write orthographic projection onto the z=0 plane with a simple matrix, such that x’=x, y’=y.
Drop z value

15. Parallel projections
For parallel projections, we specify a direction of projection (DOP) instead of a COP.
We can write orthographic projection onto the z=0 plane with a simple matrix, such that x’=x, y’=y.
Normally, we do not drop the z value right away. Why not?
Drop z value
You may wonder, why bother to introduce so many zeros to have a matrix. You will see later the unification reason later.

16. Properties of parallel projection
Are actually a kind of affine transformation
Parallel lines remain parallel
Ratios are preserved
Angles not (in general) preserved
Not realistic looking
Good for exact measurements,
Most often used in
CAD,
architectural drawings,
etc.,
where taking exact measurement
is important
Scanalyze

17. Perspective effect
Distant objects are smaller
Paralle lines meet

18. Perspective Illusion

19. Perspective Illusion

20. Derivation of perspective projection
y/z = y’/-d
y’ = -y/z d
x’ = -x/z d
Distant things look smaller
Consider the projection of a point onto the projection plane:
By similar triangles, we can compute how much the x and y coordinates are scaled:

21. Homogeneous coordinates and perspective projection
Now we can re-write the perspective projection as a matrix equation:
x/z const, map to the same point
What’s important here is that we can represent nonlinear thing using homo coord
Only last row is different
(4,4) is no longer zero
Decompose complex tasks into simpler ones.
Lost z is not best choice.
Orthographic projection

22. Vanishing points
What happens to two parallel lines that are not parallel to the projection plane?
Think of train tracks receding into the horizon...
The equation for a line is:
After perspective transformation we get:
0  directional vector
P is one point here
V is the direction of the rail
T is the step

23. Vanishing points (cont'd)
Dividing by w:
Letting t go to infinity:
We get a point!
What happens to the line l = q + tv?
Each set of parallel lines intersect at a vanishing point on the PP.
Q: How many vanishing points are there?
x’ = -vx/vz d
y’ = -vy/vz d
Inf number of vanishing points
Same v means the two lines are parapllel

24. Vanishing points

25. Vanishing points

26. Vanishing points

27. General Projective Transformation
The perspective projection is an example of a projective transformation.

28. Properties of perspective projections
Here are some properties of projective transformations:
Lines map to lines
Parallel lines do not necessarily remain parallel
Ratios are not preserved
One of the advantages of perspective projection is that size varies inversely with distance – looks realistic.
A disadvantage is that we can't judge distances as exactly as we can with parallel projections.
Stereo, occlusions, perspective (comparative size), focus, motion parallax
Auxiliary: sound

29. 3D Geometry Pipeline Model Space (Object Space) Eye Space (View Space)
Setting up the frustrum, fov, cliping plane.
Occlusion, visibility changes
Converging is OK, but not most convenient to implement, instead…
Model Space
(Object Space)
Eye Space
(View Space)
World Space
Rotation
Translation
Projective,
Scale
Translation
Rotation
Translation
Resizing

30. Clipping and the viewing frustum
The center of projection and the portion of the projection plane that map to the final image form an infinite pyramid. The sides of the pyramid are clipping planes.
Frequently, additional clipping planes are inserted to restrict the range of depths. These clipping planes are called the near and far or the hither and yon clipping planes.
All of the clipping planes bound the viewing frustum.
The basic equations we have seen give a flavor of what happens, but they are insufficient for all applications
They do not get us to a Canonical View Volume
They make assumptions about the viewing conditions
To get to a Canonical Volume, we need a Perspective Volume …

31. Clipping and the viewing frustum
Notice that it doesn’t really matter where the image plane is located, once you define the view volume
You can move it forward and backward along the z axis and still get the same image, only scaled
Usually d = near

32. Clipping Planes
The left/right/top/bottom planes are defined according to where they cut the near clip plane
Left Clip
Plane
Near Clip Plane xv
Far Clip Plane
View Volume l n
-zv f r
What is image size
Right Clip
Plane
Need 6 parameters,
Or, define the left/right and top/bottom clip planes by the field of view

33. Field of View Assumes a symmetric view volume Left Clip Plane
Near Clip Plane xv
Far Clip Plane
View Volume
FOV
-zv f
Image size
Right Clip
Plane
Need 4 parameters,
Or, define the near, far, fov, aspect ratio

34. Focal Distance to FOV
You must have the image size to do this conversion
Why? Same d, different image size, different FOV
height/2 d
FOV/2 d

35. Perspective Parameters
We have seen three different ways to describe a perspective camera
Clipping planes, Field of View, Focal distance,
The most general is clipping planes – they directly describe the region of space you are viewing
For most graphics applications, field of view is the most convenient
You can convert one thing to another

36. Perspective Projection Matrices
We want a matrix that will take points in our perspective view volume and transform them into the orthographic view volume
This matrix will go in our pipeline before an orthographic projection matrix
(r,t,f)
(r,t,f)
(r,t,n)
Parallel is eariser than perspective
Clipping is more efficient
(r,t,n)
(l,b,n)
(l,b,n)

37. Mapping Lines y -z
We want to map all the lines through the center of projection to parallel lines
This converts the perspective case to the orthographic case, we can use all our existing methods
The relative intersection points of lines with the near clip plane should not change
The matrix that does this looks like the matrix for our simple perspective case n f

38. How to get this mapping? y -z n f

39. Properties of this mappingy -z n f
If z = n, M(x,y,z,1) = [x,y,z,1]
near plane is unchanged
If x/z = c1, y/z = c2, then x’=n*c1, y’=n*c2
bending converging rays to parallel rays
If z1 < z2, z1’ < z2’
z ordering is preserved

40. General Perspective

41. To Canonical view volume
Perspective projection transforms a pyramid volume to an orthographic view volume
(r,t,f)
(r,t,f)
(r,t,n)
(r,t,n)
(l,b,n)
(l,b,n)
Rotate, translate
Parallel is eariser than perspective
Clipping is more efficient
(-1,1,-1)
(1,-1,1)
Canonical view volume [-1,1]^3

42. Orthographic View to Canonical Matrix

43. Orthographic View to Canonical Matrix

44. Complete Perspective Projection
After applying the perspective matrix, we map the orthographic view volume to the canonical view volume:

45. 3D Geometry Pipeline Model Space (Object Space) Eye Space (View Space)
Setting up the frustrum, fov, cliping plane.
Converging is OK, but not most convenient to implement, instead…
Model Space
(Object Space)
Eye Space
(View Space)
World Space
Rotation
Translation
Projective,
Scale
Translation
Rotation
Translation
Resizing

46. 3D Geometry Pipeline (cont’d)
Continuous light pattern hitting the sensor plane
Sample the lighting pattern to form a digital image
Normalized Projection
Space
Image Space (pixels)
Raster Space
Screen Space (2D)
Project
Scale
Translation

47. Canonical  Window Transform
(1,1)
(xmax,ymax)
(xmin,ymin)
(-1,-1)

48. Canonical  Window Transform
(1,1)
(xmax,ymax)
(xmin,ymin)
(-1,-1)

49. Mapping of Z is nonlinear
Many mappings proposed: all have nonlinearities
Advantage: handles range of depths (10cm – 100m)
Disadvantage: depth resolution not uniform
More close to near plane, less further away
Common mistake: set near = 0, far = infty. Don’t do this. Can’t set near = 0; lose depth resolution.