1. Lecture 8: Dynamic Range and Trichromacy Li Zhang Spring 2008
CS559: Computer Graphics
Lecture 8: Dynamic Range and Trichromacy
Li Zhang
Spring 2008
Most Slides from Stephen Chenney

2. Today Finish image morphing
Dynamic Range: Is 8 bits per channel enough?
Trichromacy: Is RGB 3 bands enough?
Start on 3D graphics
Reading:
Shirley ch
Shirley ch
Shirley ch 6, except 6.2.2
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. Dynamic Range
The dynamic range is the ratio between the maximum and minimum values of a physical measurement. Its definition depends on what the dynamic range refers to.
For a Scene: ratio between the brightest and darkest parts of the scene.
For a Display: ratio between the maximum and minimum intensities emitted from the screen.
For circuits, it means for voltage
For physics, temperature.
In our class, we mean the brightness of light.
If you look at this room, the brightest region is the ceiling light and the darkest is the blackboard.

4. Dynamic Range
Dynamic range is a ratio and as such a dimensionless quantity. In photography and imaging, the dynamic range represents the ratio of two luminance values, with the luminance expressed in candelas per square meter.
A scene showing the interior of a room with a sunlit view outside the window, for instance, will have a dynamic range of approximately 100,000:1.
The human eye can accommodate a dynamic range of approximately 10,000:1 in a single view.
If you look at this room, the brightest region is the ceiling light and the darkest is the blackboard.

5. Dynamic Range
Dynamic range is a ratio and as such a dimensionless quantity. In photography and imaging, the dynamic range represents the ratio of two luminance values, with the luminance expressed in candelas per square meter.
A scene showing the interior of a room with a sunlit view outside the window, for instance, will have a dynamic range of approximately 100,000:1.
The human eye can accommodate a dynamic range of approximately 10,000:1 in a single view.
Beyond that, use adaptation to adjust
If you look at this room, the brightest region is the ceiling light and the darkest is the blackboard.

6. Dynamic Range Image depth refers to the number of bits available.
We can use those bits to represent
a large range at low resolution, or
a small range at high resolution
Long exposure time, use 8 bit to represent a small range of brightness, higher resolution
Short exposure time, use 8 bits to represent a large range of brightness, lower resolution

7. High dynamic imaging High Dynamic Range Image Floating point pixel
Tone mapped image
8 bits, for standard monitors

8. Intensity Perception
Humans are actually tuned to the ratio of intensities, not their absolute difference
So going from a 50 to 100 Watt light bulb looks the same as going from 100 to 200
So, if we only have 4 intensities, between 0 and 1, we should choose to use 0, 0.25, 0.5 and 1
Ignoring this results in poorer perceptible intensity resolution at low light levels, and better resolution at high light levels
It would use 0, 0.33, 0.66, and 1

9. Display on a Monitor
When images are created, a linear mapping between pixels and intensity is assumed
For example, if you double the pixel value, the displayed intensity should double
Monitors, however, do not work that way
For analog monitors, the pixel value is converted to a voltage
The voltage is used to control the intensity of the monitor pixels
But the voltage to display intensity is not linear
Similar problem with other monitors, different causes
The outcome: A linear intensity scale in memory does not look linear on a monitor
Even worse, different monitors do different things

10. Gamma Control
The mapping from voltage to display is usually an exponential function:
To correct the problem, we pass the pixel values through a gamma function before converting them to the monitor
This process is called gamma correction
The parameter, , is controlled by the user
It should be matched to a particular monitor
Typical values are between 2.2 and 2.5
The mapping can be done in hardware or software

11. Color Vision
We have seen that humans have three sensors for color vision
What’s the implications for digital color representation…

12. Qualitative Response Sensitivity,  E 400 500 600 700 400 500 600 700
Multiply
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Area under curve?
#photons, E
Red
Big response!
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13. Qualitative Response Sensitivity,  E 400 500 600 700 400 500 600 700
Multiply
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Area under curve?
#photons, E
Blue
Tiny response!
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14. People think these two spectra look the same
Trichromacy means…
Spectrum
Representing color:
If you want people to “see” the continuous spectrum, you can just show the three primaries
Color Matching:
People think these two spectra look the same
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3 Primaries
Now you see the implication. No matter how complex the source spectrum is, you can always represent it using 3 colors to our vision.
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15. Trichromacy
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Spectrum
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3 primaries
For almost any given E_target(λ), we can solve for [r,g,b] so that the displayed color looks indistinguishable from the target color to our eyes.

16. Trichromacy
Many colors can be represented as a mixture of R, G, B: M=rR + gG + bB (Additive matching)
Gives a color description system - two people who agree on R, G, B need only supply (r, g, b) to describe a color
Some colors can’t be matched like this, instead, write: M+rR=gG+bB (Subtractive matching)
Interpret this as (-r, g, b)
Problem for reproducing colors – you can’t subtract light using a monitor

17. Primaries are Spectra Too
A primary can be a spectrum
Single wavelengths are just a special case
3 Primaries
3 Primaries or
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18. Color Matching
Given a spectrum, how do we determine how much each of R, G and B to use to match it?
Procedure:
For a light of unit intensity at each wavelength, ask people to match it using some combination of R, G and B primaries
Gives you, r(), g() and b(), the amount of each primary used for wavelength 
Defined for all visible wavelengths, r(), g() and b() are the RGB color matching functions

19. The RGB Color Matching Functions

20. Computing the Matching
Given a spectrum E(), how do we determine how much each of R, G and B to use to match it?
The spectrum function that we are trying to match, E(), gives the amount of energy at each wavelength
The RGB matching functions describe how much of each primary is needed to give one energy unit’s worth of response at each wavelength
R, G, B seem like a basis vector in a 3D space.
r,g,b, are coordinates of this space.
If we change r, g, b, we are changing the percieved color

21. Color Spaces
The principle of trichromacy means that the colors displayable are all the linear combination of primaries
Taking linear combinations of R, G and B defines the RGB color space
If R, G and B correspond to a monitor’s phosphors (monitor RGB), then the space is the range of colors displayable on the monitor

22. RGB Cube Demo Cyan (0,1,1) White(1,1,1) Yellow (1,1,0) Green(0,1,0)
Blue (0,0,1)
Magenta (0,1,1)
Black (0,0,0)
Red (1,0,0)
Demo

23. Problems with RGB
It isn’t easy for humans to say how much of RGB to use to make a given color
How much R, G and B is there in “brown”? (Answer: .64,.16, .16)

24. HSV Color Space (Alvy Ray Smith, 1978)
Hue: the color family: red, yellow, blue…
Saturation: The purity of a color: white is totally unsaturated
Value: The intensity of a color: white is intense, black isn’t
Space looks like a cone
Parts of the cone can be mapped to RGB space
Not a linear space, so no linear transform to take RGB to HSV
But there is an algorithmic transform
Alvy intronduced the concept of alpha channel.

25. RGB to HSV http://alvyray.com/Papers/hsv2rgb.htm
#define RETURN_HSV(h, s, v) {HSV.H = h; HSV.S = s; HSV.V = v; return HSV;}
#define RETURN_RGB(r, g, b) {RGB.R = r; RGB.G = g; RGB.B = b; return RGB;}
#define UNDEFINED -1
// Theoretically, hue 0 (pure red) is identical to hue 6 in these transforms. Pure
// red always maps to 6 in this implementation. Therefore UNDEFINED can be
// defined as 0 in situations where only unsigned numbers are desired.
typedef struct {float R, G, B;} RGBType;
typedef struct {float H, S, V;} HSVType;
HSVType RGB_to_HSV( RGBType RGB ) {
// RGB are each on [0, 1]. S and V are returned on [0, 1] and H is
// returned on [0, 6]. Exception: H is returned UNDEFINED if S==0.
float R = RGB.R, G = RGB.G, B = RGB.B, v, x, f;
int i;
HSVType HSV;
x = min(R, G, B);
v = max(R, G, B);
if(v == x) RETURN_HSV(UNDEFINED, 0, v);
f = (R == x) ? G - B : ((G == x) ? B - R : R - G);
i = (R == x) ? 3 : ((G == x) ? 5 : 1);
RETURN_HSV(i - f /(v - x), (v - x)/v, v); }

26. HSV to RGB http://alvyray.com/Papers/hsv2rgb.htm
RGBType HSV_to_RGB( HSVType HSV ) {
// H is given on [0, 6] or UNDEFINED. S and V are given on [0, 1].
// RGB are each returned on [0, 1].
float h = HSV.H, s = HSV.S, v = HSV.V, m, n, f;
int i;
RGBType RGB;
if (h == UNDEFINED) RETURN_RGB(v, v, v);
i = floor(h);
f = h - i;
if ( !(i&1) ) f = 1 - f; // if i is even
m = v * (1 - s);
n = v * (1 - s * f);
switch (i) {
case 6:
case 0: RETURN_RGB(v, n, m);
case 1: RETURN_RGB(n, v, m);
case 2: RETURN_RGB(m, v, n)
case 3: RETURN_RGB(m, n, v);
case 4: RETURN_RGB(n, m, v);
case 5: RETURN_RGB(v, m, n); }

27. L-A-B Color Space L-A-B L: luminance
A: position between magenta and green (negative values indicate green while positive values indicate magenta)
B: position between yellow and blue (negative values indicate blue and positive values indicate yellow)

28. Spatial resolution and colorG
In the next few slides, I’ll change these individual channels to you will see how the modification will change the final look of the color image. B
original

29. Blurring the G component
original
processed

30. Blurring the R component
original
processed

31. Blurring the B component
processed
original

32. Lab Color Component L
A rotation of the color coordinates into directions that are more perceptually meaningful:
L: luminance,
a: magenta-green,
b: blue-yellow a b

33. Bluring L L a b
processed
original

34. Bluring a L a b
processed
original

35. Bluring b L a b
processed
original

36. Application to image compression
(compression is about hiding differences from the true image where you can’t see them).

37. 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

38. Recall: All 2D Linear Transformations
Linear transformations are combinations of …
Scale,
Rotation,
Shear, and
Mirror

39. 2D Rotation Rotate counter-clockwise about the origin by an angle  yx x

40. Rotating About An Arbitrary Point
What happens when you apply a rotation transformation to an object that is not at the origin? y ? x

41. 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

42. 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

43. Rotating About An Arbitrary Pointx x y y x x

44. 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

45. Scaling an Object not at the Origin
What 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?

46. 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!

47. 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)
To go backwards, divide by w
Translation can now be done with matrix multiplication!

48. Basic Transformations
Translation: Rotation:
Scaling:

49. 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

50. Directions vs. Points We have been talking about transforming points
Directions are also important in graphics
Viewing directions
Normal vectors
Ray directions
Directions are represented by vectors, like points, and can be transformed, but not like points
(1,1)
(-2,-1)
Direction is translation invariant

51. Transforming Directions
Say I define a direction as the difference of two points: d=a–b
This represents the direction of the line between two points
Now I translate the points by the same amount: a’=a+t, b’=b+t
How should I transform d?
10/5/04
© University of Wisconsin, CS559 Fall 2004

52. Homogeneous Directions
Translation does not affect directions!
Homogeneous coordinates give us a very clean way of handling this
The direction (x,y) becomes the homogeneous direction (x,y,0)
The correct thing happens for rotation and scaling also
Scaling changes the length of the vector, but not the direction
Normal vectors are slightly different – we’ll see more later

53. 3D Transformations Homogeneous coordinates: (x,y,z)=(wx,wy,wz,w)
Transformations are now represented as 4x4 matrices

54. 3D Translation

55. 3D Rotation
Rotation in 3D is about an axis in 3D space passing through the origin
Using a matrix representation, any matrix with an orthonormal top-left 3x3 sub-matrix is a rotation
Rows are mutually orthogonal (0 dot product)
Determinant is 1
Implies columns are also orthogonal, and that the transpose is equal to the inverse

56. 3D Rotation Why so? Consider r1, r2, r3 as reseult of unit vectors
This implies how many degree of freedom does a rotation matrix have?

57. Problems with Rotation Matrices
Rotation matrix has a large amount of redundancy
Orthonormal constraints reduce degrees of freedom back down to 3
Keeping a matrix orthonormal is difficult when transformations are combined
Rotations are a very complex subject, and a detailed discussion is way beyond the scope of this course

58. Alternative Representations
Euler angles: Specify how much to rotate about X, then how much about Y, then how much about Z
Any three axes will do e.g. X,Y,X
Hard to think about, and hard to compose

59. Alternative Representations
Specify the axis and the angle (OpenGL method)
Hard to compose multiple rotations
A rotation by an angle around axis specified by the unit vector is given by

60. Other Rotation Issues Rotation is about an axis at the origin
For rotation about an arbitrary axis, use the same trick as in 2D: Translate the axis to the origin, rotate, and translate back again
Rotation is not commutative
Rotation order matters
Experiment to convince yourself of this

61. Transformation Leftovers
Scale, shear etc extend naturally from 2D to 3D
Rotation and Translation are the rigid-body transformations:
Do not change lengths or angles, so a body does not deform when transformed

62. Modeling 101
For the moment assume that all geometry consists of points, lines and faces
Line: A segment between two endpoints
Face: A planar area bounded by line segments
Any face can be triangulated (broken into triangles)

63. Modeling and OpenGL
In OpenGL, all geometry is specified by stating which type of object and then giving the vertices that define it
glBegin(…) …glEnd()
glVertex[34][fdv]
Three or four components (regular or homogeneous)
Float, double or vector (eg float[3])
Chapter 2 of the red book

64. Rendering
Generate an image showing the contents of some region of space
The region is called the view volume, and it is defined by the user
Determine where each object should go in the image
Viewing, Projection
Determine which pixels should be filled
Rasterization
Determine which object is in front at each pixel
Hidden surface elimination, Hidden surface removal, Visibility
Determine what color it is
Lighting, Shading

65. Graphics Pipeline
Graphics hardware employs a sequence of coordinate systems
The location of the geometry is expressed in each coordinate system in turn, and modified along the way
The movement of geometry through these spaces is considered a pipeline
Local Coordinate Space
World Coordinate Space
View Space
Canonical View Volume
Display Space