Midterm Review. Wen-Chieh (Steve) Lin Department of Computer Science

Transcription

1 Midterm Review Wen-Chieh (Steve) Lin Department of Computer Science

2 Administration Assignment due on /3 :59 PM Midterm eam on /6 (Monda) Lecture slides Chapter 3 ecluding 3.6 & 3.8 Chapter 6, 7, 8 Chapter ecluding.3 &.4 Chapter Angel, Chapter , Chapter DCP456 Introduction to Computer Graphics 2

3 Topics we have learned so far Graphics pipeline Raster algorithms Intensities and colors Transformation Viewing and projection Shading Teture mapping Clipping Ma not be reviewed toda DCP456 Introduction to Computer Graphics 3

4 Graphics Pipeline OpenGL pipeline Primitives + material properties Rotate Translate Scale Is it visible? 3D to 2D Convert to piels Displa DCP456 Introduction to Computer Graphics 4

5 Rasteriation (Scan Conversion) Final step in pipeline From screen coordinates (float) to piels (int) Writing piels into frame buffer DCP456 Introduction to Computer Graphics 5

6 Raster Algorithms 2D graphics primitives Line drawing Circle drawing Area filling Polgons Antialiasing DCP456 Introduction to Computer Graphics 6

7 Line-Drawing Algorithms Start with line segment in window coordinates with integer values for endpoints = m + h m DCP456 Introduction to Computer Graphics 7

8 Good Discrete Lines No gaps in adjacent piels Piels close to ideal line Consistent choices; same piels in same situations Smooth looking Even brightness in all orientations Same line for P P as for P P DCP456 Introduction to Computer Graphics 8

9 Digital Differential Analer (DDA) Algorithm For each plot piel at closest Along scan line Δ = For(=; <=2,i++) { } +=m; write_piel(, round(), line_color) Problems for steep lines DCP456 Introduction to Computer Graphics 9

10 Using Smmetr Use for m For m >, swap roles of and For each, plot closest DCP456 Introduction to Computer Graphics

11 Bresenham s Algorithm DDA requires one floating point addition per step. Bresenham s algorithm eliminates all floating point operations Consider onl m Other cases b smmetr Assume piel centers are at half integers If we start at a piel that has been written, there are onl two candidates for the net piel DCP456 Introduction to Computer Graphics

12 Ke to Bresenham Algorithm Reasonable assumptions have reduced the problem to making a binar choice at each piel: NE (net) (Previous) E (net) DCP456 Introduction to Computer Graphics 2

13 Candidate Piels m candidates Last piel DCP456 Introduction to Computer Graphics 3

14 Go NE if M is below the line NE Q ideal line previous M midpoint (, E 2 ) DCP456 Introduction to Computer Graphics 4

15 Go E if M is above the line NE (, ) M midpoint (, 2 ) ideal line (, ) previous E Q (, ) DCP456 Introduction to Computer Graphics 5

16 Decision Variable d Define a logical decision variable d linear in form incrementall updated (with addition) tells us whether to go E or NE DCP456 Introduction to Computer Graphics 6

17 Recall that for m and > f(,) = ( ) + ( ) + = Above line: consider f(,+ ) f(,) > Below line f(,) < DCP456 Introduction to Computer Graphics 7

18 Bresenham s Algorithm = For = to do draw(,) If f(+, +.5) < then = + f(+,+.5) < DCP456 Introduction to Computer Graphics 8 f(+,+.5) >

19 Bresenham s Algorithm (cont.) Speed up b replacing function evaluation with incremental update f(,) = ( ) + ( ) + = f(+, +) = f(, ) + ( ) + ( ) f(+, ) = f(, ) + ( ) DCP456 Introduction to Computer Graphics 9

20 Bresenham s Algorithm (cont.) = d = f(+, +.5) For = to do draw(,) If d< then = + else d = d + (-) + (-) d = d + (-) DCP456 Introduction to Computer Graphics 2 f(+,+.5) < f(+,+.5) >

21 Bresenham s Algorithm (cont.) = d = f(+, +.5) For = to do draw(,) If d< then = + else d = d + (-) + (-) d = d + (-) Now, we want to remove the last floating point operation in the code! DCP456 Introduction to Computer Graphics 2

22 Bresenham s Algorithm (cont.) 2*d = 2*f(+, +.5) f( +, +.5) = ( ) ( +) + ( )( +.5) + = 2f( +,+.5) = 2( ) ( +) + ( )(2 +) = DCP456 Introduction to Computer Graphics 22

23 Code for Bresenham s Algorithm = d = 2*( )(+) + ( )(2*+) + 2*(* *) For = to do draw(,) If d< then = + else d = d + 2*(-) + 2*(-) d = d + 2*(-) DCP456 Introduction to Computer Graphics 23

24 Other cases for line drawing m=; m= trivial cases > m > - flip about -ais m > flip about = DCP456 Introduction to Computer Graphics 24

25 Circle-drawing Algorithm for each, if r 2 <= ε SetPiel (, ) for θ in [~36 degree ] = r cos(θ) = r sin(θ) SetPiel (, ) DCP456 Introduction to Computer Graphics 25

26 Midpoint Circle Algorithm Can we utilie the similar idea in Bresenham s line-drawing algorithm? Check onl the net candidates Use smmetr and simple decision rules DCP456 Introduction to Computer Graphics 26 Smmetr of a Circle

27 Midpoint Circle Algorithm (cont.) f(,) = R 2 f(,) > => point outside circle f(,) < => point inside circle P k = f circ ( k +, k ½) DCP456 Introduction to Computer Graphics 27

28 Midpoint Circle Algorithm (cont.) Given the starting point (,r), the computation is more efficient. P = 5/4 r At each position, if(pk < ) else the net point is ( k+, k ) p k+ = p k + 2 k+ + the net point is ( k+, k -) p k+ = p k + 2 k+ + 2 k+ DCP456 Introduction to Computer Graphics 28

29 2D Polgon Filling Inside or Outside are not obvious It s not obvious when the polgon intersects itself. DCP456 Introduction to Computer Graphics 29

30 Concave vs. Conve We prefer dealing with simpler polgons. Conve (eas to break into triangles) θ< 8 o θ > 8 o conve concave DCP456 Introduction to Computer Graphics 3

31 Filling Conve Polgons Find top and bottom vertices List edges along left and right sides For each scan line from top to bottom Find left and right endpoints of span, l and r Fill piels between l and r Can use Bresenham s algorithm to update l and r l r DCP456 Introduction to Computer Graphics 3

32 Concave Polgons: Odd-Even Test Approach : odd-even test For each scan line Find all scan line/polgon intersections Sort them left to right Fill the interior spans between intersections Parit rule: inside after an odd number of crossings DCP456 Introduction to Computer Graphics 32

33 Concave Polgons: Winding Rule Approach 2: winding rule Orient the lines in polgon For each test line (not passing a verte) C E Winding number = right-hdd left-hdd crossings Interior if winding number non-ero Test line A B D ++++=2 Starting from A: ++++= DCP456 Introduction to Computer Graphics 33 2

34 Even-odd Rule vs. Winding Rule Different onl for self-intersecting polgons 2 Even-odd rule Winding rule DCP456 Introduction to Computer Graphics 34

35 Aliasing Artifacts created during scan conversion Inevitable (going from continuous to discrete) Aliasing (name from digital signal processing): we sample a continuous image at grid points Effects Jagged edges Moiré patterns DCP456 Introduction to Computer Graphics 35

36 Sampling and Reconstruction An image is a 2D arra of discrete samples from real-world continuous signal DCP456 Introduction to Computer Graphics 36

37 Sampling and Aliasing Artifacts due to undersampling or poor reconstruction Formall, high frequencies masquerading as low e.g. high frequenc line as low freq jaggies DCP456 Introduction to Computer Graphics 37

38 (Spatial) Aliasing DCP456 Introduction to Computer Graphics 38

39 (Spatial) Aliasing Jaggies probabl biggest aliasing problem DCP456 Introduction to Computer Graphics 39

40 Moiré Patterns due to Bad Downsampling Downsampled without filtering Original Image DCP456 Introduction to Computer Graphics 4 Downsampled after filtering

41 More Aliasing DCP456 Introduction to Computer Graphics 4

42 Antialiasing for Line Segments Use area averaging at boundar bottom is aliased, magnified top is antialiased, magnified DCP456 Introduction to Computer Graphics 42

43 Antialiasing b Supersampling Traditionall for off-line rendering Render, sa, 33 grid of mini-piels Average results using a filter Can be done adaptivel Stop if colors are similar Subdivide at discontinuities DCP456 Introduction to Computer Graphics 43

44 Supersampling Eample Other improvements Stochastic sampling (avoiding repetition) Jittering (perturb a regular grid) DCP456 Introduction to Computer Graphics 44

45 Temporal Aliasing Sampling rate is frame rate (3 H for video) Eample: spokes of wagon wheel in movie Possible to supersample and average Fast-moving objects are blurred Happens automaticall in video and movies Eposure time (shutter speed) Memor persistence (video camera) Effect is motion blur DCP456 Introduction to Computer Graphics 45

46 Intensities and Colors Gamma correction Color models RGB CMYK HSV Compositing RGBA Alpha blending DCP456 Introduction to Computer Graphics 46 2 green can blue 24. black. V ellow magenta H red S

47 Gamma Correction Monitors are nonlinear with respect to input Adjust intensities based on an approimate nonlinear model: input intensit displaed intensit = ( ma intensit ) a γ Gamma correct input a' a DCP456 Introduction to Computer Graphics 47

48 How to Determine Gamma? Adjust input intensit of gre piels until the intensities of checkerboard piels are halfwa between black and white.5 a adjust a ln.5 ln a DCP456 Introduction to Computer Graphics 48

49 RGB Color Space: Additive Color Form a color b adding amounts of three primaries: Red (R), Green (G), Blue (B) CRTs, projection sstems, positive film DCP456 Introduction to Computer Graphics 49

50 CMY Color Space: Subtractive Color Form a color b filtering white light with: Can (C), Magenta (M), and Yellow (Y) filters Printing, Negative film R + G = Y G + B = C B + R = M Y = W - B C = W - R M = W - G DCP456 Introduction to Computer Graphics 5

51 Complementar Colors Add to Gra blue magenta (,,) blue (,, ) white (,,) can (,,) red (,,) ellow (,,) green (,,) DCP456 Introduction to Computer Graphics 5

52 HSV Color Space Introduced b Albet Munsell, late 8s Hue: Color Saturation: strength of a color Neutral gra has saturation Value: Intensit of light emanating from image 2 green can blue 24. black. V ellow magenta H red S DCP456 Introduction to Computer Graphics 52

53 Compositing Frame buffer Simple color model: R, G, B; 8 bits each -channel A, another 8 bits Alpha determines opacit, piel-b-piel = : opaque = : transparent Blend translucent objects during rendering Achieve other effects (e.g., shadows) DCP456 Introduction to Computer Graphics 53 C = C f + (-C b

54 Transformations and Matrices Transformations are functions Matrices are function representations Matrices represent linear transformations 22 Matrices 2D Linear transformation Linear transformation Def : T ( a ) at ( ) T ( ), for scalar a and vectors and. 2 2 Matrices 2 D Linear Transf's DCP456 Introduction to Computer Graphics 54

55 DCP456 Introduction to Computer Graphics 55 Basic 2D Transformations Basic 2D Transformations Translation Scaling Rotation t t t t t s s S s s cos sin sin cos R cos sin sin cos

56 DCP456 Introduction to Computer Graphics 56 Basic 2D Transformations (cont.) Basic 2D Transformations (cont.) Shear a a Sh b b Sh Sh tan tan

57 Geometric View of Shear in Sh tan tan tanφ Φ DCP456 Introduction to Computer Graphics 57 57

58 Affine Transformation A function F is affine if it is linear plus a translation Thus the -D transformation = m + b is not linear, but affine Similarl for a translation and rotation of a coordinate sstem Affine transformations preserve lines = rigid transformation + shearing + scaling DCP456 Introduction to Computer Graphics 58

59 DCP456 Introduction to Computer Graphics 59 Homogeneous Coordinates Homogeneous Coordinates Translation is not linear--how to represent it as a matri? Trick: add etra coordinate to each vector This etra coordinate is the homogeneous coordinate, or w t t

60 Homogeneous Coordinates When etra coordinate is used, vector is said to be represented in homogeneous coordinates The associated transformation matrices Homogeneous Transformations Drop etra coordinate after transformation (project to w=) f f f f t t, for DCP456 Introduction to Computer Graphics 6

61 DCP456 Introduction to Computer Graphics 6 Homogeneous 2D Transformations Homogeneous 2D Transformations The basic 2D transformations become Translate: Scale: Rotate: An affine transformation can be epressed as a combination of these. We can combine homogeneous transforms b multiplication. Now an sequence of translate/scale/rotate operations can be collapsed into a single homogeneous matri t t s s cos sin sin cos

62 Composite Transformations Matri multiplication isn t commutative! DCP456 Introduction to Computer Graphics 62

63 DCP456 Introduction to Computer Graphics 63 Windowing (Viewport) Transformation Windowing (Viewport) Transformation [a, A] [b, B] [c, C] [d, D] Translate(-a, -b) Scale Translate(c, d) b a b B d D a A c C d c M

64 DCP456 Introduction to Computer Graphics 64 Basic 3D Transformations Basic 3D Transformations 3-D transformations are ver similar to the 2-D case Homogeneous coordinate transforms require 44 matrices Scaling and translation matrices are simpl: t t t S S S

65 3D Rotation Matrices Rotation is a bit more complicated in 3-D More possible aes of rotation e.g., rotate about -ais: onl and coordinate change Rotate-(Φ) = cos sin sin cos DCP456 Introduction to Computer Graphics 65

66 DCP456 Introduction to Computer Graphics 66 3D Rotation Matrices about - and - aes 3D Rotation Matrices about - and - aes cos sin sin cos Rotate-(Φ) = cos sin sin cos Rotate-(Φ) =

67 Properties of Rotation Matrices Rotation matrices are orthonormal Rows/Columns are mutuall orthonormal Rows/Columns form a Cartesian coordinate Geometricall, R - (θ) = R(-θ) Algebraicall, R - (θ) = R T (θ) Prove b ourself that R T (θ)r(θ) = R(θ)R T (θ) = I using orthonormalit of R R - (θ) = R(-θ) = R T (θ) DCP456 Introduction to Computer Graphics 67

68 Arbitrar 3D Rotations Can we construct an arbitrar 3D rotation using basic rotation matrices R() R() R()? +θ a α DCP456 Introduction to Computer Graphics 68

69 Transforming Normal Vectors If we transform normals like points We need a different rule to transform normals. DCP456 Introduction to Computer Graphics 69

70 Normals Transform Like Planes A plane n p n ( a,b,c ) If To p n p is find n n T T n p ( n T T ( n Mp a b T is Ip ( M T p the M the M to, equation equation transform ) T c plane transform ed, answer, M ) p )( Mp ) M d where how do for T for point point point to DCP456 Introduction to Computer Graphics 7 n n normal, can some T T a b c d, p d is should on on be the n magic plane plane transform written offset. transform? : in in plane original transforme space d space

71 Global vs. Local Coordinate Sstem Need to handle multiple coordinate sstems A canonical coordinate sstem is usuall designated as the frame-of-reference for all other coordinates Global coordinate World coordinate Frame of reference The others are called local coordinates DCP456 Introduction to Computer Graphics 7

72 DCP456 Introduction to Computer Graphics 72 Global vs. Local Coordinate Sstem Global vs. Local Coordinate Sstem,, u, v, o, e are all vectors in global sstem o p p p p p ), ( v u e p p p p p v u v u ), ( p p e e p p v u v u p p e e T T p p v u v u In global coord. ( p, p ) In local coord. (u p,v p )

73 DCP456 Introduction to Computer Graphics 73 Think about 3D case. Think about 3D case. Given two coordinate frames, how do ou represent a vector specified in one frame in the other frame? e u v w X Y Z P p p p p p p w v u p p p p p p w v u??

74 Viewing Given geometr in the world coordinate sstem, how do we get it to the displa? Transform to camera coordinate sstem Transform (warp) into canonical view volume Clip Project to displa coordinates Rasterie DCP456 Introduction to Computer Graphics 74

75 Viewing and Projection Our ees collapse 3-D world to 2-D retinal image (brain then has to reconstruct 3D) In CG, this process occurs b projection Projection has two parts: Viewing transformations: camera position and direction Perspective/orthographic transformation: reduces 3-D to 2-D Use homogeneous transformations DCP456 Introduction to Computer Graphics 75

76 Viewing Transformation: Camera Control All we need is a single translation and angleais rotation (orientation), but... Good animation requires good camera control--we need better control knobs Translation knob - move to the lookfrom point Orientation can be specified in several was: specif camera rotations specif a lookat point (solve for camera rotations) DCP456 Introduction to Computer Graphics 76

77 A Popular View Specification Approach Focal length, image sie/shape and clipping planes are in the perspective transformation In addition: lookfrom: where the focal point (camera) is lookat: the world point to be centered in the image Also specif camera orientation about the lookfromlookfat ais DCP456 Introduction to Computer Graphics 77

78 Implementing lookat/lookfrom/vup viewing scheme Translate b lookfrom, bring focal point to origin Rotate lookfrom - lookat to the -ais with matri R: w = (lookfrom-lookat) (normalied) and = [,,] rotation ais: a = (w )/ w rotation angle: cosθ= w and sinθ= w Rotate about -ais to get vup parallel to the -ais DCP456 Introduction to Computer Graphics 78

79 It s not so complicated vup lookfrom START HERE w Translate LOOKFROM to the origin Rotate the view vector (lookfrom - lookat) onto the -ais. Multipl b the projection matri and everthing will be in the canonical camera position DCP456 Introduction to Computer Graphics 79 Rotate about to bring vup to -ais

80 DCP456 Introduction to Computer Graphics 8 Rotate Camera Frame Rotate Camera Frame Alternativel, we can view these two rotations as a single rotation that aligns u-vw-aes to ---aes! w v Ru Rv Rw w v u w v u w v u R w w w v v v u u u w v u w v u w v u R w: lookat lookfrom v: view up direction u = v w w v u w v u w v u w v u w v u w v u

81 DCP456 Introduction to Computer Graphics 8 Viewing Transformation Viewing Transformation Put translation and rotation together e w w w e v v v e u u u e e e w w w v v v u u u v RM t M X Y Z e u v w P p p p p p p w v u M v world coordinate local coordinate

82 Orthographic Projection When the focal point is at infinit the ras are parallel and orthogonal to the image plane When -plane is the image plane (,,) -> (,,) front orthographic view F Image World DCP456 Introduction to Computer Graphics 82

83 Orthographic Projection Map orthographic viewing cube to the canonical view volume 3D window transformation [l, r] [b, t] [f, n] [-, ][-, ][-,] DCP456 Introduction to Computer Graphics 83

84 DCP456 Introduction to Computer Graphics 84 Orthographic Projection (cont.) Orthographic Projection (cont.) 3D window transform (last class) [l, r] [b, t] [f, n] [-, ][-, ][-,] canonical is ignored f n t b r l f n b t l r canonical canonical canonical

85 DCP456 Introduction to Computer Graphics 85 Canonical View Volume to Screen Canonical View Volume to Screen If -ais of screen coord. points downward [-,] [-, ] [-.5, n -.5] [n -.5, -.5] canonical canonical piel piel piel n n n n reflection scale translation

86 DCP456 Introduction to Computer Graphics 86 Orthographic Projection Matri Orthographic Projection Matri Put everthing together f n t b r l f n b t l r n n n n M o canonical piel piel M o

87 DCP456 Introduction to Computer Graphics 87 Orthographic Projection Summar Orthographic Projection Summar Given 3D geometr (a set of points a) Compute view transformation M v Compute orthographic projection M o Compute M = M o M v For each point a i, compute p = Ma i f n t b r l f n b t l r n n n n M e w w w e v v v e u u u

88 Perspective Projection of a Point View plane or image plane - a plane behind the pinhole on which the image is formed sees anthing on the line (ra) through the pinhole F a point W projects along the ra through F to appear at I (intersection of WF with image plane) I Image F World W DCP456 Introduction to Computer Graphics 88

89 A Simple Perspective Camera Canonical case: camera looks along the -ais (toward negative -ais) focal point is the origin image plane is parallel to the -plane at distance d We call d the focal length, mainl for historical reasons Image plane Center of projection DCP456 Introduction to Computer Graphics 89

90 Geometr Eq. for Perspective Projection Diagram shows -coordinate, -coordinate is similar Point (,,) projects to d s d d (,, d ) view plane e: ee position g: gae direction e d g s DCP456 Introduction to Computer Graphics 9

91 DCP456 Introduction to Computer Graphics 9 Perspective Projection Matri Perspective Projection Matri Projection using homogeneous coordinates: transform (,,) to 2-D image point: discard third coordinate appl viewport transformation to obtain phsical piel coordinates ),, ( d d d d d d d d d ),, ( d d d Divide b 4 th coordinate (the w coordinate)

92 DCP456 Introduction to Computer Graphics 92 Perspective Projection Matri Perspective Projection Matri Projection using homogeneous coordinates: transform (,,) to 2-D image point: discard third coordinate appl viewport transformation to obtain phsical piel coordinates ),, ( d d d d d d d d d ),, ( d d d Divide b 4 th coordinate (the w coordinate)

93 View Volume and View Frustum Pramid in space defined b focal point and window in the image plane Defines visible region of space Pramid edges are clipping planes Near and far clipping planes Field of view DCP456 Introduction to Computer Graphics 93

94 DCP456 Introduction to Computer Graphics 94 Map Perspective View Volume to Orthographic View Volume Map Perspective View Volume to Orthographic View Volume Map [,,n] to [n/, n/, n] Map [,,f] to [n/, n/, f] n f n f n M p

95 DCP456 Introduction to Computer Graphics 95 Projection Matri is not unique Projection Matri is not unique M p is not unique ' ' ' h h h h h M p M p nf f n n n n f n f n n n p M p M

96 Perspective Projection Summar Given 3D geometr (a set of points a) Compute view transformation M v Map perspective to orthographic M p Compute orthographic projection M o Compute M = M o M p M v For each point a i, compute p = Ma i DCP456 Introduction to Computer Graphics 96

97 Hidden Surface Elimination Backface culling BSP tree Z-buffer (net class) DCP456 Introduction to Computer Graphics 97

98 Back Face Culling: Object Space v n v n n DCP456 Introduction to Computer Graphics 98 n v v

99 Back Face Culling: Object Space Visible: vn v n n v Utah School of Computing 99

100 Creating a BSP tree b 4b 4f 4b 5f 5b 4f 5f 4 -> 4f,4b DCP456 Introduction to Computer Graphics

101 Back-to-Front Render 2 5b 4f 3 5f 4b b 4b 4f 4b 5f 3 5b 4f 5f DCP456 Introduction to Computer Graphics

102 -Buffer Algorithm The or depth buffer stores the depth of the closest object at each piel found so far As we render each polgon, compare the depth of each piel to depth in buffer If less, place the shade of piel in the color buffer and update buffer DCP456 Introduction to Computer Graphics 2

103 Integer Z-buffer If B bins are used for depth range (n, f) Bin sie = (n-f)/b Smaller is desirable Depth of two objects are indistinguishable if is larger than the difference between their real - values Increase B Move n and f closer DCP456 Introduction to Computer Graphics 3

104 Bin sie for -buffer in world space Bin sie in the world space increases as depth increases Objects farther awa is more likel to be indistinguishable in depth (same integer - values) w 2 w fn w =f ma w f n Maimie n and minimie f for small w DCP456 Introduction to Computer Graphics 4