1. CS 325 Introduction to Computer Graphics 01 / 29 / 2010 Instructor: Michael Eckmann

2. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Today’s Topics Color CIE system chromaticity diagram –color gamut, complementary colors, dominant wavelength RGB, YIQ CMY, CMYK, HSV Line drawing algorithms

3. Homework READING: –If you haven't read Ch. 12 yet, do so. –For OpenGL info in our text see: sections 2.8 through 2.9 of Chapter 2 – deals with openGL (using C, but don't let that bother you, the information is directly applicable to what we'll be doing in Java with JOGL.)‏ –Put off the first 7 sections of chapter 2 for now. Michael Eckmann - Skidmore College - CS 325 - Spring 2010

4. Color Recall that –Physically, color is electromagnetic radiation in the visible light frequency range. –Psychologically, how humans perceive color is more than just the frequency. Hue (dominant wavelength), brightness (amount of light energy) and purity (how little white light is added) are what characterize what color is seen. –Chromaticity refers collectively to hue and purity (everything except brightness)‏

5. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color If we combine light with different dominant frequencies, we can create other colors. We combine primary colors in various proportions to make up a wide range of colors known as the color gamut. Complementary colors are any two colors that combine to produce white. No finite set of “real” light sources can produce all visible colors. The color gamut of three primary colors is typically sufficient for most purposes.

6. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color The tristimulus theory of color perception states that the retina in the human eye has three kinds of color sensors (cones) with peak sensitivities to “red”, “blue” and “green” light. –Cones (6-8 million in retina, concentrated in fovea) for color / bright light vision –Rods (over 100 million in retina, none in fovea) for achromatic brightness levels / night vision –fovea is in center of retina --- that's where most of the cones are

7. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color The tristimulus theory of color perception (continued)‏ –Experiments have shown that based on this theory that “blue”'s peak is 440 nm, “green”'s is 545 nm, and “red”'s is 580 nm. –Also, experiments have shown that the eye's response to blue light is much less strong than to red and green. –Given 3 primary colors, we can combine them with various proportions to produce new colors. Defining colors by mixtures of three primaries is an extremely useful approach.

8. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color Experiments were done on humans to determine what proportions of three primaries would produce certain test colors. The experiments were like this: –A test color was shown. –An adjustable color was also shown that contained a mixture of three primaries 700 nm (red), 546.1 nm (green) and 435.8 nm (blue). –The observer adjusted the brightness of each of the three primaries until the observer determined that it matched the test color. Some test colors were not matchable. In these cases, a primary was added to the test color and then attempted to be matched. This resulted in the original test color being represented by a certain mixture of the three primaries (one with a negative contribution.)‏

9. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color Public domain image – This shows how much of each of the three primaries are needed to generate each monochromatic color (the wavelength on the x-axis.)‏

10. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color This is a chromaticity diagram (shows colors at a single (highest) brightness). Only hue and saturation are represented. The bottom left circle is the blue primary, the top circle is the green primary and the bottom right circle is the red primary. These are the three primaries from the last slide. The triangle contains the color gamut of those primaries.

11. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color The next slide shows color matching functions f Z, f Y and f X for a standard set (CIE 1931) of primaries Z, Y and X. These three colors were defined so that there is no need for a negative contribution of one of them and so that the entire range of visible colors is producible. These three colors X, Y and Z correspond roughly to red, green, and blue respectively.

12. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color

13. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color http://cvision.ucsd.edu/cie.htm contains text files with the amounts of each of the color matching functions needed to produce colors at all the wavelengths between 360 nm and 830 nm.http://cvision.ucsd.edu/cie.htm The values in the charts correspond to the graph on the last slide. The values are used as weights thusly: Color = xX + yY + zZ, where x, y and z are the weights applied to the primaries X, Y and Z. x+y+z is the total light energy. We can normalize with respect to this amount.

14. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color Color = xX + yY + zZ x' = x / (x+y+z)‏ y' = y / (x+y+z)‏ z' = z / (x+y+z)‏ Since x' + y' + z' = 1, any color can be represented by just two of the x', y' and z'. By normalizing based on the total light energy (brightness) we are only now dealing with hue and purity (collectively known as Chromaticity)‏ The graph on the next slide is in 2d and it's x axis corresponds to x' here, and y axis to y' here.

15. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color

16. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color source: Wikipedia Where do the three colors X,Y and Z live on this diagram?

17. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color Chromaticity diagram on the last two slides –White is formally defined by the point C (which approximates sunlight)‏ –All perceivable colors with the same chromaticity but different brightness map to the same point. –The spectrally pure colors are on the outer curve.

18. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color

19. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color The chromaticity diagram is useful for –comparing color gamuts of different primaries is it obvious that no 3 colors within the diagram will produce a color gamut encompassing all the visible colors? 2 colors can combine to make any color along the straight line segment joining them (C 1 to C 2 )‏ –determining complementary colors how can this be done using the diagram?

20. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color The chromaticity diagram is useful for –comparing color gamuts of different primaries is it obvious that no 3 colors within the diagram will produce a color gamut encompassing all the visible colors? 2 colors can combine to make any color along the straight line segment joining them (C 1 to C 2 )‏ –determining complementary colors recall that complementary colors are two that combine in some way to make white so if the line segment between two colors contains the white point then these two are complements of each other.

21. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color

22. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color The chromaticity diagram is also useful for determining purity and hue for a color –purity (saturation): ratio of distance of line from color (eg. C 1 ) to C to the distance of line from C to spectral color C s on the outer curve. –hue (dominant wavelength): draw a line from C through the color C 1 to the spectral color curve (the intersection of this line with the outer curve is the hue)‏ why? because the color C 1 can be represented by some combination of white ( C ) and the spectral color C s on the outer curve. –if the dominant wavelength lands on the “purple line” then...

23. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color For the colors (called non-spectral colors) in the chromaticity diagram that lie such that when drawing a line starting at C through the color will intersect the “purple” line –the dominant wavelength is the complement of some wavelength draw a line from the color C 2 through C (white) to the outer spectral curve C sp the intersection is the wavelength that is subtracted from the spectral distribution of the color –purity (saturation): is still the closer to white, the less pure the further from white, the more pure

24. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color Is the Chromaticity diagram useful for determining the brightness of a color?

25. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Color Is the Chromaticity diagram useful for determining the brightness of a color? –No, the brightness (light energy) was normalized to 1 –recall, Color = xX + yY + zZ –x' = x / (x+y+z)‏ –y' = y / (x+y+z)‏ –z' = z / (x+y+z)‏ –Since x' + y' + z' = 1, any color can be represented by just two of the x', y' and z'. –By normalizing based on the total light energy (brightness) we are only now dealing with hue and purity (collectively known as Chromaticity)‏

26. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 RGB Color Model The color gamut for RGB colors appears on the next slide. Color = rR + gG + bB r, g and b range from 0.0 to 1.0 notice: white's (r,g,b) = (1,1,1)‏ black's (r,g,b) = (0,0,0)‏ complementary colors are those that add up to (1,1,1)‏ can anyone tell me 2?

27. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 RGB Color Model

28. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 YIQ Color Model Graphics monitors use RGB (separate signals for R, G and B)‏ NTSC television monitors use a composite signal called the YIQ model. Y is luminance (brightness)‏ I & Q together incorporate the hue and purity (collectively chromaticity) and some luminance RGB can be transformed into YIQ and vice versa which means, given RGB values, you can compute the equivalent color in YIQ and vice versa. See section 12.5 of textbook for transforms. One nice feature of YIQ is that since Y is the luminance, if that's all that is captured by the television then you end up with black & white television (grey-scale) Also since the human eye responds more to luminance than color (recall the number of rods vs. cones) more bandwidth is set aside for the Y value than I & Q.

29. Michael Eckmann - Skidmore College - CS 376 - Spring 2010 Additive vs. Subtractive Both RGB and YIQ color models discussed thus far are additive models. That is, colors are produced by adding light sources. When we view a monitor or television we are looking at the light sources themselves. Since hard copy devices print colors, the colors we see are reflected from the paper (when white light hits the paper). A subtractive process is required to print something that will reflect a certain color of light on paper. We are not viewing the light source this time, but instead what color light is reflected off the paper when white light is shown on it. How about CIE --- is it additive?

30. Michael Eckmann - Skidmore College - CS 376 - Spring 2010 Additive (left), Subtractive (right) source: wikipedia

31. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 CMY & CMYK Color Models C = Cyan, M = Magenta, Y = Yellow white's (c,m,y) = (0,0,0)‏ black's (c,m,y) = (1,1,1)‏ cyan absorbs red (and reflects green and blue)‏ magenta absorbs green yellow absorbs blue cyan & magenta ink combined will reflect blue light (b/c red and green are absorbed)‏ cyan & yellow ink combined will reflect green light magenta & yellow ink combined will reflect red light

32. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 CMY & CMYK Color Models Transformation from RGB to CMY is simple. [C, M, Y] T = [1, 1, 1] T - [R, G, B] T Transformation from CMY to RGB is: [R, G, B] T = [1, 1, 1] T - [C, M, Y] T CMYK C = Cyan, M = Magenta, Y = Yellow, K = Black instead of using cyan, magenta and yellow ink all combined to make black, typically a fourth ink, black is used to produce black and grey levels

33. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 Recap on Color Models RGB color model is an additive model. Amounts of red, green and blue are combined by adding light sources of these colors to produce new colors. CMY and CMYK are subtractive models. Amounts of cyan, magenta and yellow are combined by adding pigments of these colors to produce new colors that reflect certain wavelengths of light. –it is subtractive because adding pigments reduce what light is reflected, when all 3 primaries (C M and Y) are put together then the result is no light is reflected (black).

34. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 HSV Color Model

35. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 HSV Color Model Hue is an angle between 0 and 360 degrees Saturation (purity) is one axis and Value (luminance) is another axis You can picture this as a circular cone if you prefer or even as a cylinder because S is considered a ratio from the V axis (0) to max 1.

36. Michael Eckmann - Skidmore College - CS 325 - Spring 2010 HSV Color Model HSV color model is more intuitive/useful to people/artists Starting with the pure hue: Vary the saturation (add white) to get different tints Vary the value (add black) to get different shades

37. What's most important of Color theory for us in this course? Know the terminology – dominant wavelength, hue, brightness, luminance, purity, saturation, chromaticity, color gamut, idea of primary colors, etc. Know the relationship between wavelength and frequency Know what the CIE chromaticity digram is useful for and how –determining: color gamuts, complementary colors, purity –comparison of: color gamuts, purity Know how additive (e.g. RGB, YIQ) and subtractive color models (e.g. CMY) work to produce new colors Know about more intuitive models like HSV and why they're useful Know that different color models can be transformed into each other by a transform matrix. Michael Eckmann - Skidmore College - CS 325 - Spring 2010

38. Michael Eckmann - Skidmore College - CS 376 - Spring 2010 Lines When drawing lines we must convert the continuous line into discrete pieces --- we need to decide which pixels should be on. We need to calculate the integer coordinates. The goals of line drawing are several –A main one is that it should be fast, because lines are drawn often That is, we prefer integer arithmetic and additions and subtractions over floating point arithmetic and multiplies and divides. –Another is that the lines should look nice --- be straight and have constant thickness, independent of angle, length

39. Michael Eckmann - Skidmore College - CS 376 - Spring 2010 Lines Recall this information about lines Assume we have two known points (x1,y1) and (x2, y2) on a plane. We can calculate the equation of a line if we pick an unknown point (x, y) on the line and compute the slope. Slope = (y-y1)/(x-x1) = (y2-y1)/(x2-x1)‏ Multiply both sides by (x-x1) to get y-y1 = (y2-y1)(x-x1)/(x2-x1)‏ y = (y2-y1)(x-x1)/(x2-x1) + y1 y = ((y2-y1)/(x2-x1)) x + (-x1)/(x2-x1) + y1 y = mx + b

40. Michael Eckmann - Skidmore College - CS 376 - Spring 2010 Parametric Equation of a Line This information appears in section 6.7 of text but is appropriate now because it is used in the DDA algorithm (discussed next slide). Given 2 points on a line (x 0, y 0 ) and (x end, y end )‏ x = x 0 + u (x end - x 0 )‏ y = y 0 + u (y end - y 0 )‏ u is the parameter If we consider only the values 0 <= u <= 1, we get all the points on the line segment between (x 0, y 0 ) and (x end, y end ). When u = 0, x=x 0 and y = y 0 When u = 1, x=x end and y = y end

41. Michael Eckmann - Skidmore College - CS 376 - Spring 2010 DDA – digital differential analyzer Consider a line with positive slope and x 0 < x end If slope <=1 then we will sample the line at unit x intervals and compute y. If slope > 1 then we will sample the line at unit y intervals and compute x. What does that mean? In the DDA, the parameter u (from the parametric equation of a line) is 1/steps. Let's take a look at the code for this algorithm and “execute” it with actual values on the board.

42. Michael Eckmann - Skidmore College - CS 376 - Spring 2010 From Hearn & Baker's Computer Graphics with OpenGL, 3 rd Edition. void lineDDA (int x0, int y0, int xEnd, int yEnd)‏ { int dx = xEnd - x0, dy = yEnd - y0, steps, k; float xIncrement, yIncrement, x = x0, y = y0; if (fabs (dx) > fabs (dy))‏ steps = fabs (dx); else steps = fabs (dy); xIncrement = float (dx) / float (steps); yIncrement = float (dy) / float (steps); setPixel (round (x), round (y)); for (k = 0; k < steps; k++) { x += xIncrement; y += yIncrement; setPixel (round (x), round (y)); }

43. Michael Eckmann - Skidmore College - CS 376 - Spring 2010 DDA – digital differential analyzer It creates good lines. The lines aren't going to differ in “thickness” based on angle, etc. Can you comment on any efficiency issues with this algorithm?

44. Michael Eckmann - Skidmore College - CS 376 - Spring 2010 DDA – digital differential analyzer It creates good lines. The lines aren't going to differ in “thickness” based on angle, etc. Problems with this algorithm are –Rounding is expensive –Floating point arithmetic is expensive So, a goal is to do all integer arithmetic and reduce / eliminate divides.