CS452/552; EE465/505. Color Display Issues

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1 CS452/552; EE465/505 Color Display Issues

2 2

3 Outline! Color Display Issues Color Systems Dithering and Halftoning! Splines Hermite Splines Bezier Splines Catmull-Rom Splines Read: Angel, Chapter 8, section 8.13 Display Considerations Chapter 11, Curves and Surfaces Chapter 12, section Advanced Rendering: Ray-Tracing Lab#5 posted, due: April 22nd simple scene: platform & object; 1 light; shadow map; camera controls Project#2 posted due: April 23rd

4 Light mapping

5 Recursive Ray Tracer (Simple version) // starting point p, direction d, max # of steps; returns a color c (single light source) trace(p, d, step) { color local, reflected, transmitted; point q; normal n; if(step > max) return(background_color); q = intersect(p, d, status); if(status==light_source) return(light_source_color); if(status==no_intersection) return(background_color); n = normal(q); r = reflect(q, n); t = transmit(q,n); local = phong(q, n, r); reflected = trace(q, r, step+1); // recursive call transmitted = trace(q, t, step+1); // recursive call return(local + reflected + transmitted); }

6 Ray Tracing: Summary! Efficiency replace the recursion with iteration use bounding boxes to simplify the math used to compute intersections! Aliasing errors, due to sampling use a stochastic sampling method in which the decision on where to cast the next ray is based on the rays cast so far (used in Renderman)! Ray tracing is an inherently parallel process! There are many free ray tracers available

7 Next Topic: Displays & Color! Consider perceptual issues related to displays! Introduce chromaticity space Color systems Color transformations! Standard Color Systems Angel and Shreiner: Interactive Computer Graphics 7E Addison-Wesley 2015

8 Displays & Color! Problems that affect the quality of a display two different displays may have the same resolution, but display pixels at different sizes the colors displayed on two different monitors may differ map software-defined colors onto the display map brightness values onto the display RGB values are independent of display properties, but do not account for the full range of the human visual system! The range of displayable colors for a device is called its color gamut

9 Luminance and Color Images! Luminance Image Monochromatic Values are gray levels Analogous to working with black and white film or television! Color Image Has perceptional attributes of hue, saturation, and lightness Do we have to match every frequency in visible spectrum? No! E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

10 Perception Review! Light is the part of the electromagnetic spectrum between ~ nm! A color C(λ) is a distribution of energies within this range! Human visual system has two types of sensors Rods: monochromatic, night vision Cones: color sensitive Three types of cones! Consequently, only three values, the tristimulus values, are perceived by the brain E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

11 Color vision deficiency! red-green difficulty distinguishing between shades of red + green affects ~8% of males and ~0.5% of females in populations of North European ancestry! blue-yellow difficulty differentiating shades of blue and yellow affects males & females equally, < 1/10,000! achromatopsia cannot perceive any colors rare, < 1/30,000 for most populations

12 Tristimulus Values! The human visual center has three cones with sensitivity curves S 1 (λ), S 2 (λ), and S 3 (λ)! For a color C(λ), the cones output the tristimulus values T = S λ) C( λ) dλ 1 1 ( C(λ) T = S λ) C( λ) dλ 2 2 ( T = S λ) C( λ) dλ 3 3 ( cones optic nerve T 1, T 2, T 3 E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

13 Three Color Theory! Any two colors with the same tristimulus values are perceived to be identical! Thus a display (CRT, LCD, film) must only produce the correct tristimulus values to match a color! Is this possible? Not always Different primaries (different sensitivity curves) in different systems E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

14 The Problem! The sensitivity curves of the human are not the same as those of physical devices! Human: curves centered in blue, green, and green-yellow! CRT: RGB! Print media: CMY or CMYK! Which colors can we match and, if we cannot match, how close can we come? E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

15 Representing Colors! Consider a color C(λ)! It generates tristimulus values T 1, T 2, T 3 Write C = (T 1, T 2, T 3 ) Conventionally,we assume 1 T 1, T 2, T 3 0 because there is a maximum brightness we can produce and energy is nonnegative 1 C is a point in color solid C 1 T 3 T 2 T 1 1 E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

16 Producing Colors! Consider a device such as a CRT with RGB primaries and sensitivity curves! Tristimulus values T = R( λ) C( λ) dλ T = G( λ) C( λ) dλ T = B( λ) C( λ) dλ E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

17 Matching! This T 1, T 2, T 3 is dependent on the particular device! If we use another device, we will get different values and these values will not match those of the human cone curves! Need a way of matching and a way of normalizing E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

18 Color Systems! Various color systems are used Based on real primaries: RGB UVW CMYK HLS (aka HSB) NTSC YIQ (National Television System Committee) Theoretical XYZ! Prefer to separate brightness (luminance) from color (chromatic) information Reduce to two dimensions E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

19 Tristimulus Coordinates! For any set of primaries, define t = T T T Note: T t 3 = T T T t = T T T T 3 T + + 1,, 0 t t t = 1 t t 2 t E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

20 Maxwell Triangle 1 T 1 + T 2 +T 3 =1 1 1 color solid Project onto 2D: chromaticity space 1 t 1 + t 2 =1 t 1 possible colors t 2 1 E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

21 NTSC RGB 1 r+g+b=1 g r+g=1 r 1 E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

22 Producing Other Colors! However colors producible on one system (its color gamut) is not necessarily producible on any other! Note that if we produce all the pure spectral colors in the nm range, we can produce all others by adding spectral colors! With real systems (CRT, film), we cannot produce the pure spectral colors! We can project the color solid of each system into chromaticity space (of some system) to see how close we can get E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

23 Color Gamuts 600 nm spectral colors CRT colors printer colors producible color on CRT but not on printer unproducible color 350 nm producible color on both CRT and printer 750 nm E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

24 XYZ! Reference system in which all visible pure spectral colors can be produced! Theoretical systems, as there are no corresponding physical primaries! Standard reference system E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

25 Color Systems! Most correspond to real primaries National Television Systems Committee (NTSC) RGB matches phosphors in CRTs! Film both additive (RGB) and subtractive (CMY) for positive and negative film! Print industry CMYK (K = black) K used to produce sharp crisp blacks Example: ink jet printers E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

26 Color Transformations! Each additive color system is a linear transformation of another G G C = (T 1, T 2, T 3 ) = (T 1, T 2, T 3 ) B B R in R G B system R in RGB system E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

27 Additive and Subtractive Color! Additive color Form a color by adding amounts of three primaries CRTs, projection systems, positive film Primaries are Red (R), Green (G), Blue (B)! Subtractive color Form a color by filtering white light with cyan (C), Magenta (M), and Yellow (Y) filters Light-material interactions Printing Negative film E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

28 RGB: Red, Green, Blue! Primary colors: red, green, blue! Secondary colors: yellow = red+green cyan = green+blue magenta = blue+red! All colors white = red+green+blue black = no color C = T 1 R + T 2 G + T 3 B Color Cube

29 CMYK: Cyan, Magenta, Yellow, Black! Primary colors: cyan, magenta, yellow! Secondary colors: blue = cyan+magenta red = magenta+yellow green = yellow+cyan! All colors white = no color black = cyan+magenta+yellow for true black, add in black (+black) Also known as process color (used to print full-color images)

30 RGB, CMY, CMYK! Assuming 1 is max of a primary C = 1 R M = 1 G Y = 1 B! Convert CMY to CMYK by K = min(c, M, Y) C = C K M = M K Y = Y - K E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

31 Color Matrix! Exists a 3 x 3 matrix to convert from representation in one system to representation in another # &! $! = M$! $ " %! Example: XYZ to NTSC RGB find in colorimetry references & $ $ $ % T' T' T' T T T! Can take a color in XYZ and find out if it is producible by transforming and then checking if resulting tristimulus values lie in (0,1) #!!! " E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

32 YIQ! NTSC Transmission Colors! Here Y is the luminance arose from the need to separate brightness from chromatic information in TV broadcasting & Y# $! $ I! $ % Q! " = & $ $ $ % #& R# ! $!! $ G! 0.311!" $ % B! "! Note luminance shows high green sensitivity E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

33 Other Color Systems! UVW: equal numerical errors are closer to equal perceptual errors! HLS: perceptual color (hue, saturation, lightness) Polar representation of color space Single and double cone versions E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

34 HSB: Hue, Saturation, Brightness! Hue is the actual color. It is measured in angular degrees counter-clockwise around the cone starting and ending at red=0 or 360 (yellow = 60, green = 120, etc.).! Saturation is the purity of the color, measured in percent from the center of the cone (0) to the surface (100).! Brightness is measured in percent from black (0%) to white (100%). grayscale axis from 0% (white) to 100% (black)

35 Gamma Correction! Intensity vs CRT voltage is nonlinear I = cv γ! Gamma correction (encoding) can code/decode luminance or tristimulus values! Can use a lookup table to correct! Human brightness response is logarithmic Equal steps in gray levels are not perceived equally Can use lookup table! CRTs cannot produce a full black Limits contrast ratio E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012

36 Dithering (digital halftoning)

37 Black/White Dithering

38 Color Dithering

39 Halftoning

40 Modeling Complex Shapes

41 What do we need from curves in Computer Graphics?! Local control of shape so that it is easy to build and modify! Stability! Smoothness and continuity! Ability to evaluate derivatives! Ease of rendering

42 Utah Teapot! Most famous data set in computer graphics! Widely available as a list of 306 3D vertices and the indices that define 32 Bezier patches

43 Curve Representations

44 Parameterization of a Curve

45 Polynomial Interpolation

46 Splines: Piecewise Polynomials

47 Piecewise Polynomials

48 Splines

49 Cubic Curves in 3D

50 Cubic Hermite Splines

51 Deriving Hermite Splines

52 Deriving Hermite Splines

53 Deriving Hermite Splines

54 The Cubic Hermite Spline Equation

55 Four Basis Functions for Hermite Splines

56 Piecing together Hermite Splines

57 Hermite Splines in Adobe Illustrator

58 Bezier s Idea! In graphics and CAD, we do not usually have derivative data! Bezier suggested using the same 4 data points as with the cubic interpolating curve to approximate the derivatives in the Hermite form

59 Bezier Splines

60 Approximating Derivatives p 1 located at u=1/3 p 1 p 2 p 2 located at u=2/3 p p0 p'(0) 1/ 3 1 p p2 p'(1) 1/ 3 3 slope p (0) slope p (1) p 0 u p 3

61 The Bezier Spline Matrix

62 Bezier Blending Functions

63 DeCasteljau Construction

64 Bezier Patches Using same data array P=[p ij ] as with interpolating form p( u, v) 3 3 = i= 0 j= 0 bi( u) b j ( v) p ij = u T M B P M T B v Patch lies in convex hull

65 Analysis! Although the Bezier form is much better than the interpolating form, we have the derivatives are not continuous at join points! Can we do better? Go to higher order Bezier More work Derivative continuity still only approximate Supported by OpenGL Apply different conditions Tricky without letting order increase

66 Catmull-Rom Splines

67 Constructing the Catmull-Rom Spline

68 Catmull-Rom Spline Matrix

69 Splines with More Continuity?

70 Comparison of Basic Cubic Splines

71 Natural Cubic Splines

72 B-Splines

73 B-Splines! Basis splines: use the data at p=[p i-2 p i-1 p i p i-1 ] T to define curve only between p i-1 and p i! Allows us to apply more continuity conditions to each segment! For cubics, we can have continuity of function, first and second derivatives at join points! Cost is 3 times as much work for curves Add one new point each time rather than three! For surfaces, we do 9 times as much work

74 B-Spline Basis

75 Other Common Types of Splines

76 Generalizing Splines! We can extend to splines of any degree! Data and conditions to not have to given at equally spaced values (the knots) Nonuniform and uniform splines Can have repeated knots Can force spline to interpolate points! Cox-deBoor recursion gives method of evaluation

77 NURBS! Nonuniform Rational B-Spline curves and surfaces add a fourth variable w to x,y,z Can interpret as weight to give more importance to some control data Can also interpret as moving to homogeneous coordinate! Requires a perspective division NURBS act correctly for perspective viewing! Quadrics are a special case of NURBS

78 How to Draw Spline Curves

79 Drawing Splines, continued

80 Summary

81 Project: Roller Coaster! use Catmull-Rom splines along with lighting and texture mapping to create a roller coaster simulation! runs in a first-person view, allowing the user to ride the coaster in an immersive environment

82 Animation

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