Computer Graphics

1. Computer Graphics Sense from Symbol

2. Applications of Graphics • Entertainment • consumer enjoyment • Design • of physical objects to be built • Visualization • of data of any kind • Training • of reflexes for critical tasks

3. Entertainment • Games • interactive virtual worlds • graphics, sound, physics, AI, rules, networking • Movies • animation • totally computer generated imagery • special effects • mix of imagery of the physical and the virtual worlds • Television • cartoons - simple animation • advertisements - simple special effects

4. Design • Vehicles • cars, airplanes, etc. • Buildings • architecture & engineering • Appliances • Industrial Design • Electronics • integrated circuits (chips), boards • Coupled with • simulation & automated manufacturing

5. Visualization • Science & Engineering • simulations • mostly physics for engineering • real-world data • weather, oceans, astronomical, planetary • Medical • body sensing • MRI, CT, ultrasound sonograms • Data Mining • statistical analysis and decision support • economics, business, demographics, etc.

6. Training • Vehicle piloting • aircraft • ships, tanks • trucks, cars • Team • combat • emergency • Individual skill • soldiers • repair

7. Graphics Data Sources • Designed • computer aided design tool • used by artist or designer • specialized for each application • Sampled • cameras • lasers • magnetic, x-ray, ultra-sound • Algorithms • mathematical functions • e.g. sphere := x2 + y2 = r2 • algorithms • e.g. fractal terrain

8. Human Transducers Virtual World Sensory Input Cognition Motor Output Displays Speakers Force-feedback Rendering Modeling Animation Motion Tracking Speech Recognition Hand Control Graphics Systems

9. Geometric Modeling • Function • Represents the contents of space • the virtual world at a single moment in time • Constituents • Geometry – represents shape • Energy sources – light, sound and motion • Attributes – Appearance : properties that effect perception Mass : properties that effect motion • Foundations Mathematics : continuous functions Data structures : organizing data to facilitate algorithms

10. Rendering • Function • Generate perception of a virtual world at an instant (or small interval) of time • Form • Algorithms that take the geometric model as input and generate output that drives transducers • Displays, speakers, force-feedback • Based-on physics and perceptual physiology • only as needed to create perception

11. Motion • Function • updates state of geometric model from one moment in time to the next • Forms • Interactive input (possibly from a network) • Animation – designed or sampled • Physics - Newtonian dynamics • Behavior – Artificial Intelligence (AI)

12. Brain -> Geometry • Brain perception • pre-wired for perception of space (Kant :1724-1804) • presumably for most, if not all, animals • perception • objects, and motion of objects • relative location/orientation left/right of, front/behind of, above/below of • Cognition • experiential abstractions (right brain) • extension, shape, connectedness • location, spatial relations, and movement • linguistic abstractions (left brain) • space, time, ordering, change • geometry, topology, motion

13. Geometry • Foundations • space : extension • point : location • geometry : shape • dimension : spatial relation • Euclidian Geometry • points, lines, polygons • angles, length • coincidence, intersection, parallel • similarity, congruence

14. Geometry & Numbers • Geometry • sets of points • discrete vs. continuous • discrete means “separate and countable” • continuous means “no gaps” • Numbers • Integers • discrete • cardinals (counting) & ordinals (ordering) • Reals • continuous • integer + factional component : 3.14159

15. Analytic Geometry • Geometry + Numbers • introduced by Rene Descartes (1596-1650) • started the Northern Renaissance • Dimension • numbers : ordering <, =, > • space : left-of, coincident, right-of • multi-dimensional • multiple independent but simultaneous dimensions • Coordinates • tuples of values • one value per dimension : ( x,y ) or ( x,y,z )

16. Functions • Mappings • points -> points • domain -> range • can differ in dimension & coordinate systems • Equations • explicit : y = x*m + b • implicit : r2 = x2 + y2 • parametric : P = t*P1 + P0 • Uses • shape : all points that satisfy some equation • motion : change in position as a function of time • transforms : mapping a space onto itself, i.e. domain = range

17. Digital Numbers • Integers • base 2 string of binary digits : 98 = 1100010 • 8-bit (byte), 16-bit (short), 32-bit (int), 64-bit (long) • Reals • fixed point • 5.75 -> 101.110 • floating point • scientific notation -> 2.25 * 103 • fraction (mantissa) + exponent -> 5.75 -> .101110 * 23 • precision • single (float) : 32-bit -> 24-bit mantissa + 8-bit exponent • double : 64-bit -> 56-bit mantissa + 8-bit exponent

18. Digital Points • Coordinates • number of coordinates == dimension of space • need one number per coordinate • number can be any digital number (int, float, etc.) • Java2D Class Constructors • java.awt.Point( int x, int y ) - discrete space • java.awt.geom.Point2D.Float( float x, float y ) • java.awt.geom.Point2D.Double( double x, double y )

19. Digital Lines • Lines • 1D, continuous, straight (not curved) • infinite length, unbounded • Line Segments • finite subset of a line • definable by two end-points • Java2D Class • java.awt.geom.Line2D

20. Line Equations • Explicit • y = mx + b, slope/y-intercept form • Implicit • 0 = Ax + By + C, normal/distance form • Parametric • P = P1 * (1-t) + t * P2, weighted sum form • P = O + D*t, directed ray form

21. Transformations • Purpose • make global geometric changes • restricted to use of linear equations in CG • Elementary Transforms • Translation - change in position • Rotation - change in orientation • Scaling - change in size • Shearing - change in perpendicularity • Combinations • any sequence of elementary transformations can be used • order of transforms matters in most cases

22. Classes of Transformations • Rigid Body • preserves distances and angles • translation & rotation • Affine • preserves parallel lines • all 4 elementary transformations • Projective • preserves straightness/flatness • affine + perspective • Linear • rotation, scaling and shearing

23. Translation • Geometry • change only in position • relative to the origin or to other shapes • Vector/Linear Algebra • is vector addition • P’ = P + T, where T is the translation vector • Properties • identity, inverse, closure, commutative

24. Rotation • Geometry • change in orientation • by some angle A either CCW or CW • about a some point (in 2D) • Properties • identity, inverse, closure, commutative • fixed point • Algebraic form • x’ = x * cos A - y * sin A • y’ = x * sin A - y * cos A

25. Scaling • Geometry • change in size • change in aspect ratio • change in handedness • projection • Algebra • x’ = x * Sx • y’ = y * Sy • Sx == Sy => symmetric scaling • Sx != Sy => asymmetric scaling

26. Shearing • Geometry • change in perpendicularity • introduces slant • Algebra • in X • x’ = x + y * Shx • y’ = x * Shy + x

27. Display of Motion • Create visual perception of motion • movies, TV, interactive graphics • sequence of snap shots : frames • played back rapidly at a constant frame rate • Variety of frame rates • Film : 24 fps (frames per second) • S-TV: 30 fps • HD-TV: 60 fps • Computers : 60-120 fps

28. Interactive Animation • Demands • constant display rate • no pre-computed frames • all images synthesized dynamically • need fast interactive response time : < 50ms • Issues • rendering time varies per frame • depends on quantity of geometry and sophistication of rendering • display fps != rendering fps • distance moved per rendered frame should depend upon rendering time to create illusion of constant rate motion

29. Interactive Animation • Solutions • Frame Buffers • holds a single frame as a digital-image • used to drive the display at constant frame rate • decouples rendering rate from display rate • Problems • rendering process will be visible • buffering clearing • multi-frame rendering times

30. Double Buffering • Separate Rendering from Display • use two buffers, one “on-screen” and one “off-screen” • sequence • display from on-screen buffer while rendering into off-screen buffer • swap buffers quickly when rendering is completed • Methods • Blt - copy off-screen to on-screen • Flip - switch which buffer display processor reads

31. Displays • Monitors • CRT’s - cathode ray tube • electron beam hits phosphors • image is drawn left-to-right, top-to-bottom • LCD • light source : backlit or reflected • opacity of surface is modulated • Latest • LED - light emitting diode • Plasma - charged gas • DLP - laser + mirrors

32. Pixels • Picture Element • smallest area of digital color • displays : light emitting • cameras : light sensing • uses Red-Green-Blue color components • analogue of cones in the retina

33. Displays & Cameras • Use 2D array of pixels • uniform distribution • Resolutions • displays • SDTV : 480 x 640 • HDTV : 1080 x 1920 • computers : adjustable • cameras • “high res” : 1200 x 1600 • Aspect ratio • height / width OR width / height • square pixels only • displays • SDTV : 3 / 4 • HDTV : 9 / 16

34. Digital Images • 2D Array of pixels • pixels are digitally represented in memory • each pixel has red-green-blue components • numbers -> brightness • Resolution • spatial resolution : width (X) & height (Y) • color resolution (bits per pixel) • 1-bit -> bi-level bitmap (white or black) • 3-bits -> corners of color cube • color -> 3 x component-resolution • 8-bit integer per component -> 24-bit RGB • 32-bit float per component -> 96-bit RGB

35. Frame Buffers • Display Image • used to drive display • holds a single digital image, i.e. a frame • used to map logical to physical pixels • Display Refresh • physical pixel output decays rapidly needed to see sequence of frames • refresh rates • 30 - 120 fps

36. Light • Photons • all light is comprised of photons • properties: velocity, wavelength, frequency, polarization wavelength = velocity / frequency • interact with matter : transmitted, reflected, absorbed • Spectral Distribution • # of photons at each frequency/wavelength • visible spectrum

37. Visual Perception • Retina • 2D array of photo receptors; 7 million • rods : low illumination, B&W (grey scale) • cones : medium-high illumination, color • Photo Receptors • measure light intensity (counts photons) • sensitivity is frequency dependent

38. Cones • Types • “red”, “green”, “blue”

39. Color Perception • Photo receptor activation level -> brightness • Combinations of cone activation • one wavelength -> “rainbow colors” • two wavelength -> rainbow + purples (hues) • all cones at equal activation -> white (grey) • arbitrary spectral distribution • hue • purity/saturation (complement of whiteness) • brightness/lightness/value

40. Color Cone • Hue • Hue, Saturation, Lightness/Value

41. RGB Color Cube

42. RGB Color Space Blue Cyan Magenta Black White Red Green Yellow

43. Retina Photoreceptor Distribution Cell Structure

44. 3D Graphics • 3D analogues of 2D • Geometry • Transformations • Set Operations • Animation • Interaction • 3D only • Viewers • Lights • Shading • Projection

45. 3D Geometry • Coordinates • points : (x,y,z) & vectors : [ x y z]T • X, Y, Z axes & analogous bases vectors • handedness • Shapes • interior is a solid which has volume • boundary is a surface • defined piecewise • surface algorithms more complicated than for curves • e.g. set operations, collision detection

46. 3D Curved Surfaces • Implicits • polynomials define interior, boundary and exterior • sphere : x2 + y2 + z2 = r2 • Parametrics • many Bezier curves used to define a single surface • recursive subdivision used to create triangle mesh approximation

47. Polyhedra • Boundary • piecewise linear • each piece is called a “face” or “facet” • if closed => solid • polyhedra proper are closed • polyhedral meshes may be open • Use • real-time graphics is all polyhedral • hardware only draws triangular meshes • can approximate any curved surface arbitrarily well • often combined with “smooth shading”

48. Polytopes • Dimension independent concept 3D polyhedra 2D polygons 1D line segments 0D point • Boundary : recursive definition • d-polytope bounded by set of (d-1)-polytope • d is the dimension, stops at d == 0

49. 3D Transformations • Categories : same as 2D • types • rigid body, affine, projective, linear • elementary • translations, rotation, scaling, shearing • Representations • change of vector basis P = x * [X] + y * [Y] + z * [Z] + T • matrices : 4x4

50. 3D Translation • Change in 3D position • T = [ Dx Dy Dz]T • P’ = P + T x’ = x + Tx 1 0 0 Tx y’ = y + Ty 0 1 0 Ty z’ = z + Tz 0 0 1 Tz w’ = w 0 0 0 1