1. Attributes of Graphics Primitives
Sang Il Park
Sejong University

2. OpenGL State variables:
Color
Point attributes
Line attributes
Fill-Area attributes

3. Boundary Filling

4. Flood Filling
: Start a point inside the figure, replace a specified interior color only.

5. Scan Line Seed Fill scan line conversion seed filling +
Shani, U., “Filling Regions in Binary Raster Images:
A Graph-Theoretic Approach”, Computer Graphics,
14, (1981),

6. Boundary Filling Efficiency in space!
finish the scan line containing the starting position
process all lines below the start line
process all lines above the start line

7. Problems of Filling Algorithm
What happens if a vertex is shared by more than one polygon, e.g. three triangles?
What happens if the polygon intersects itself?
What happens for a “sliver”?
Solutions?
Redefine what it means to be inside of a triangle
Different routines for nasty little triangles

8. OpenGL Fill-Area Function
Shade model:
Wire-frame or point:
glShadeModel ( shadeModel );
GL_SMOOTH
GL_FLAT
glPolygonMode ( face, displayMode );
GL_FRONT
GL_BACK
GL_FRONT_AND_BACK
GL_LINE
GL_POINT
GL_FILL

9. Aliasing in CG
Which is the better?

10. Aliasing in CG
Digital technology can only approximate analog signals through a process known as sampling
The distortion of information due to low-frequency sampling (undersampling)
Choosing an appropriate sampling rate depends on data size restraints, need for accuracy, the cost per sample…
Errors caused by aliasing are called artifacts. Common aliasing artifacts in computer graphics include jagged profiles, disappearing or improperly rendered fine detail, and disintegrating textures.

11. The Nyquist Theorem
the sampling rate must be at least twice the frequency of the signal or aliasing occurs

12. Aliasing Effects

13. Artifacts - Jagged profiles
Jagged silhouettes are probably the most familiar effect caused by aliasing.
Jaggies are especially noticeable where there is a high contrast between the interior and the exterior of the silhouette

14. Artifacts - Improperly rendered detail

15. Artifacts - Disintegrating textures
The checkers should become smaller as the distance from the viewer increases.

16. Antialiasing
Antialiasing methods were developed to combat the effects of aliasing. The two major categories of antialiasing techniques are prefiltering and postfiltering.

17. Prefiltering
Eliminate high frequencies before sampling (Foley & van Dam p. 630)
Convert I(x) to F(u)
Apply a low-pass filter
A low-pass filter allows low frequencies through, but attenuates (or reduces) high frequencies
Then sample. Result: no aliasing!

18. High Frequency

19. Prefiltering

20. Prefiltering

21. Basis for Prefiltering Algorithms

22. Catmull’s Algorithm A2 A1 AB A3 Find fragment areas
Multiply by fragment colors
Sum for final pixel color

23. Prefiltering Example

24. Prefiltering So what’s the problem?
Problem: most rendering algorithms generate sampled function directly
e.g., Z-buffer, ray tracing

25. Supersampling
The simplest way to reduce aliasing artifacts is supersampling
Increase the resolution of the samples
Average the results down
Or sometimes, it is called “Postfiltering”.

26. Supersampling The process:
Create virtual image at higher resolution than the final image
Apply a low-pass filter
Resample filtered image

27. Supersampling: Limitations
Q: What practical consideration hampers super-sampling?
A: Storage goes up quadratically
Q: What theoretical problem does supersampling suffer?
A: Doesn’t eliminate aliasing! Supersampling simply shifts the Nyquist limit higher

28. Supersampling: Worst Case
Q: Give a simple scene containing infinite frequencies
A: A checkered ground plane receding into infinity
See next slide…

30. Supersampling
Despite these limitations, people still use super-sampling (why?)
So how can we best perform it?

31. Sampling in the Postfiltering method
Supersampling from a 4*3 image.
Sampling can be done randomly or regularly. The method of perturbing the sample positions is known as "jittering."

32. Stochastic Sampling
Stochastic: involving or containing a random variable
Sampling theory tells us that with a regular sampling grid, frequencies higher than the Nyquist limit will alias
Q: What about irregular sampling?
A: High frequencies appear as noise, not aliases
This turns out to bother our visual system less!

33. Stochastic Sampling An intuitive argument:
In stochastic sampling, every region of the image has a finite probability of being sampled
Thus small features that fall between uniform sample points tend to be detected by non-uniform samples

34. Stochastic Sampling
Idea: randomizing distribution of samples scatters aliases into noise
Problem: what type of random distribution to adopt?
Reason: type of randomness used affects spectral characteristics of noise into which high frequencies are converted

35. Stochastic Sampling
Problem: given a pixel, how to distribute points (samples) within it?
Grid Random Poisson Disc Jitter

36. Stochastic Sampling Poisson distribution: Completely random
Add points at random until area is full.
Uniform distribution: some neighboring samples close together, some distant

37. Stochastic Sampling Poisson disc distribution:
Poisson distribution, with minimum- distance constraint between samples
Add points at random, removing again if they are too close to any previous points
Very even-looking distribution

38. Stochastic Sampling Jittered distribution
Start with regular grid of samples
Perturb each sample slightly in a random direction
More “clumpy” or granular in appearance

39. Nonuniform Supersampling
Adaptive Sampling

40. Adaptive Sampling Problem: Final Samples
Many more blue samples than white samples
But final pixel actually more white than purple!
Simple filtering will not handle this correctly
Final Samples

41. Filters

42. Antialiasing

43. Geometric Transformations

44. Transformations Linear transformations Rigid transformations
Local moving frame
Linear transformations
Rigid transformations
Affine transformations
Projective transformations T
Global reference frame

45. Linear Transformations
A linear transformation T is a mapping between vector spaces
T maps vectors to vectors
linear combination is invariant under T
In 3-spaces, T can be represented by a 3x3 matrix

46. Examples of Linear Transformations
2D rotation
2D scaling

47. Examples of Linear Transformations
2D shear
Along X-axis
Along Y-axis

48. Examples of Linear Transformations
2D reflection
Along X-axis
Along Y-axis

49. Properties of Linear Transformations
Any linear transformation between 3D spaces can be represented by a 3x3 matrix
Any linear transformation between 3D spaces can be represented as a combination of rotation, shear, and scaling

50. Composing Transformations
Suppose we want,
We have to compose two transformations

51. Composing Transformations
Matrix multiplication is not commutative
Translation
followed by
rotation
Translation
followed by
rotation

52. Composing Transformations
R-to-L : interpret operations w.r.t. fixed coordinates
L-to-R : interpret operations w.r.t local coordinates
(Column major convention)

53. Changing Bases
Linear transformations as a change of bases

54. Changing Bases
Linear transformations as a change of bases

55. Affine Transformations
An affine transformation T is an mapping between affine spaces
T maps vectors to vectors, and points to points
T is a linear transformation on vectors
affine combination is invariant under T
In 3-spaces, T can be represented by a 3x3 matrix together with a 3x1 translation vector

56. Homogeneous Coordinates
Any affine transformation between 3D spaces can be represented by a 4x4 matrix
Affine transformation is linear in homogeneous coordinates

57. Examples of Affine Transformations
2D rotation
2D scaling

58. Examples of Affine Transformations
2D shear
2D reflection

59. Examples of Affine Transformations
2D translation

60. Examples of Affine Transformations
2D transformation for vectors
Translation is simply ignored

61. Examples of Affine Transformations
3D rotation

62. Pivot-Point Rotation
Rotation with respect to a pivot point (x,y)

63. Fixed-Point Scaling
Scaling with respect to a fixed point (x,y)

64. Scaling Direction
Scaling along an arbitrary axis

65. Properties of Affine Transformations
Any affine transformation between 3D spaces can be represented as a combination of a linear transformation followed by translation
An affine transf. maps lines to lines
An affine transf. maps parallel lines to parallel lines
An affine transf. preserves ratios of distance along a line
An affine transf. does not preserve absolute distances and angles

66. Review of Affine Frames
A frame is defined as a set of vectors {vi | i=1, …, N} and a point o
Set of vectors {vi} are bases of the associate vector space
o is an origin of the frame
N is the dimension of the affine space
Any point p can be written as
Any vector v can be written as

67. Changing Frames
Affine transformations as a change of frame

68. Changing Frames
Affine transformations as a change of frame

69. Changing Frames
In case the xyz system has standard bases

70. Rigid Transformations
A rigid transformation T is a mapping between affine spaces
T maps vectors to vectors, and points to points
T preserves distances between all points
T preserves cross product for all vectors (to avoid reflection)
In 3-spaces, T can be represented as

71. Rigid Body Rotation
Rigid body transformations allow only rotation and translation
Rotation matrices form SO(3)
Special orthogonal group
(Distance preserving)
(No reflection)

72. Rigid Body Rotation R is normalized R is orthogonal
The squares of the elements in any row or column sum to 1
R is orthogonal
The dot product of any pair of rows or any pair columns is 0
The rows (columns) of R correspond to the vectors of the principle axes of the rotated coordinate frame

73. 3D Rotation About Arbitrary Axis

74. 3D Rotation About Arbitrary Axis
Translation : rotation axis passes through the origin
2. Make the rotation axis on the z-axis
3. Do rotation
4. Rotation & translation

75. 3D Rotation About Arbitrary Axis
Rotate u onto the z-axis

76. 3D Rotation About Arbitrary Axis
Rotate u onto the z-axis
u’: Project u onto the yz-plane to compute angle a
u’’: Rotate u about the x-axis by angle a
Rotate u’’ onto the z-asis

77. 3D Rotation About Arbitrary Axis
Rotate u’ about the x-axis onto the z-axis
Let u=(a,b,c) and thus u’=(0,b,c)
Let uz=(0,0,1)

78. 3D Rotation About Arbitrary Axis
Rotate u’ about the x-axis onto the z-axis
Since we know both cos a and sin a, the rotation matrix can be obtained
Or, we can compute the signed angle a
Do not use acos() since its domain is limited to [-1,1]

79. Euler angles
Arbitrary rotation can be represented by three rotation along x,y,z axis

80. Gimble
Hardware implementation of Euler angles
Aircraft, Camera

81. Euler Angles Rotation about three orthogonal axes 12 combinations
XYZ, XYX, XZY, XZX
YZX, YZY, YXZ, YXY
ZXY, ZXZ, ZYX, ZYZ
Gimble lock
Coincidence of inner most and outmost gimbles’ rotation axes
Loss of degree of freedom

82. Euler Angles Euler angles are ambiguous
Two different Euler angles can represent the same orientation
This ambiguity brings unexpected results of animation where frames are generated by interpolation.

83. Taxonomy of Transformations
Linear transformations
3x3 matrix
Rotation + scaling + shear
Rigid transformations
SO(3) for rotation
3D vector for translation
Affine transformation
3x3 matrix + 3D vector or 4x4 homogenous matrix
Linear transformation + translation
Projective transformation
4x4 matrix
Affine transformation + perspective projection

84. Taxonomy of Transformations
Rigid
Affine
Projective

85. Composite Transformations
Composite 2D Translation

86. Composite Transformations
Composite 2D Scaling

87. Composite Transformations
Composite 2D Rotation

88. OpenGL Geometric Transformations
glMatrixMode(GL_MODELVIEW);

89. OpenGL Geometric Transformations
Construction
glLoadIdentity();
glTranslatef(tx, ty, tz);
glRotatef(theta, vx, vy, vz);
(vx, vy, vz) is automatically normalized
glScalef(sx, sy, sz);
glLoadMatrixf(Glfloat elems[16]);
Multiplication
glMultMatrixf(Glfloat elems[16]);
The current matrix is postmultiplied by the matrix
Row major

90. Hierarchical Modeling
A hierarchical model is created by nesting the descriptions of subparts into one another to form a tree organization

91. OpenGL Matrix Stacks Four matrix modes Stack processing
Modelview, projection, texture, and color
glGetIntegerv(GL_MAX_MODELVIEW_STACK_DEPTH, stackSize);
Stack processing
The top of the stack is the “current” matrix
glPushMatrix(); // Duplicate the current matrix at the top
glPopMatrix(); // Remove the matrix at the top