Lecture 5 Basic Rendering

Transcription

1 Lecture 5 Basic Rendering Dr. Shuang LIANG School of Software Engineering Tongji University Spring 2013

2 At the end of this lecture, you will have enough weapons to generate the above cool figures!

3 Let s start this amazing and thorny journey!

4 Today s Topics Overview Vector operations Ray-casting framework Camera coordinate system Ray representation Ray-surface intersection Shading models Surface rendering

5 Today s Topics Overview Vector operations Ray-casting framework Camera coordinate system Ray representation Ray-surface intersection Shading models Surface rendering

6 Realism in computer graphics Goal of many graphics algorithms is to create results that are as physically realistic as possible Realism in graphics can be subdivided into different categories: geometric modeling, rendering, behavior, and interaction Techniques and algorithms used to achieve realism in category are dependent on several factors application or content (movies, scientific visualization, etc.) user type (novice vs. expert) resources (time, money, processing power, etc.) Degree of realism that can be achieved usually depends on amount of resources when resources are short, use approximations

7 Realism and media Early computer graphics focused on still images typically meant photorealism goal was to accurately reconstruct a scene at a specific time emphasized good approximations of geometry and light reflection properties of surfaces

8 Realism and media Early computer graphics focused on still images With the increased production of animated graphics (commercials, movies, special effects, cartoons) comes new standard of realism character animation natural phenomena: cloth, fur, hair, skin, grass, trees, smoke, water, clouds, wind Newtonian physics: objects which collide, fall, scatter, bend, shatter, etc.

9 Realism and media Always a tradeoff between realism and rendering time Applications that require real-time input/output could handle severely limited amount of realism Applications which do not require real-time interaction (films, static imagery), have unlimited time to achieve realism Real-time interaction during modeling (3DS Max) Rendered image

10 Tradeoff Consider a movie vs. scientific visualizations and CAD models movie will want to achieve as many special effects as possible (motion blur, depth of field, etc.) CAD (engineering) models will skip these special effects for clarity scientific visualizations will display the geometric model and/or simulation data as is, without any approximations, curve smoothing, etc., to avoid hiding artifacts, approximations, or errors

11 Polygonalization Realism in geometry represent objects as sets of polygons finite level of detail if we zoom in it doesn t look good easily hardware accelerated often want simplest mesh possible while preserving as much detail as possible fewer vertices needed at flat, simple portions of mesh more vertices needed at complex varying regions of mesh

12 Realism in geometry Mesh simplification / decimation take detailed model, reduce number of polygons while keeping as much shape as possible various algorithms exist active area of research Meshes can also be smoothed, sharpened, manipulated, etc. same concepts learned in filtering can be applied to 3d models instead of 2d images E.g., smoothing a mesh requires removing high frequencies

13 Realism in geometry Subdivision surfaces two-step process to add more detail to mesh: Topology refinement (add new vertices) Geometry refinement several schemes for subdividing continuously subdivide triangles into more triangles for more details as needed allows for editing of multi-resolution models (editing at various levels of detail)

14 Dynamic LOD Realism in geometry objects far from the camera don t need as many polygons, while objects closer should have more replace farther objects with low resolution meshes multiple versions of model in scene graph (differing only in level of detail) Terrain and water rendered at different LODs

15 Bump mapping Realism in geometry Achieved by perturbing the surface normals of the object and using the perturbed normal during lighting calculations Does not modify the underlying geometry

16 Realism in geometry Displacement mapping Displace vertices in a mesh by values stored in a 2D texture Actually modifies the geometry

17 Realism in rendering Multiple light bounces add more lighting to the scene by calculating lighting coming directly from light sources, as well as light reflected off of other objects appears more photorealistic, but much more expensive to compute Often hacked by using a constant ambient term instead Left: single ray cast via nonrecursive ray tracing (3 sec. render time) Right: multiple light bounces via photon mapping (57 sec.)

18 Realism in rendering Polygonal rendering and Gouraud shading fast, easily implemented in hardware (default lighting in OpenGL) per vertex lighting interpolates color between vertices lighting is only calculated at vertices from direct light sources no global illumination does not look very realistic (hard shadows, no light reflecting off other objects)

19 Raytracing Realism in rendering shoot rays from the eye into the scene at each intersection calculate amount of light hitting the object if the surface is reflective or refractive, reflect /refract the ray and continue tracing good for shiny, reflective surfaces expensive for global illumination hard to compute multi-bounce lighting

20 Radiosity Realism in rendering subdivide scene into patches, iteratively compute how much light each patch contributes to each other patch good for diffuse global illumination view independent, once solved, can move camera around A virtual scene rendered by radiosity algorithm

21 Realism in rendering Other rendering techniques: spherical harmonics path tracing point based rendering image based lighting (IBL)

22 Realism in behavior Behavior realism is important we are very distracted by unrealistic behavior even if the rendering is realistic good behavior is very convincing even if the rendering is not realistic Keyframing most animations are created through keyframes animators spend hours specifying one key pose after another; in between frames interpolated based on the keyframes

23 Realism in behavior Motion capture (mocap) sample positions and orientations of real world markers over time once captured, apply these motions to a computer model usually better than keyframed animations extremely realistic

24 Physical simulation Realism in behavior can look extremely realistic usually very expensive to calculate (such as fluid dynamics) realism vs. time tradeoff Egg explosion simulation Simulating fracturing of stone

25 Today s Topics Overview Vector operations Ray-casting framework Camera coordinate system Ray representation Ray-surface intersection Shading models Surface rendering

26 Vector representation Vector operations a xi y j zk { x, y, z} Length (or norm) of a vector a x y z Normalized vector (unit vector) a { x, y, z } a a a a We say a 0, if and only if x 0, y 0, z 0

27 Vector operations if a ( x1, y1, z1), b x2 y2 z2 (,, ), then a b ( x x, y y, z z ), b Dot product (inner product) a a b a b cos x x y y z z Laws of dot product: Theorem a b b a, a ( b c)= a b+ a c a b 0 a b (why?)

28 Cross product Vector operations i j k ab x1 y1 z1 i j k x y z y z z x x y y z z x x y

29 Cross product c a b is also a vector, whose direction is determined by the right-hand law and c a, c b c a b sin c a b (easy to prove) represents the oriented area of the parallelogram taking and as two sides r1 r2 r2 r (why?) 1 Vector operations c b a

30 Vector operations Cross product Theorem Theorem same time, and they are not equal to zero at the (easy to understand) a b 0 (why?) Property r ( r r ) r r r r

31 Vector operations Mixed product (scalar triple product or box product) ( a, b, c) ( a b) c a a a b b Geometric Interpretation: it is the (signed) volume of the parallelepiped defined by the three vectors given b c c c

32 Mixed product (scalar triple product or box product) ( a, b, c) ( a b) c Vector operations a a a b b b c c c ( a b) c a b c cos a b sin c cos Base h c a

33 Mixed product (scalar triple product or box product) ( a, b, c) ( a b) c Property: Vector operations a a a b b b c c c ( a, b, c) ( b, c, a) ( c, a, b) ( a, b, c) ( b, a, c) ( a, c, b) why?

34 Vector operations Mixed product (scalar triple product or box product) Theorem a, b, c are coplanar ( a, b, c) 0 why? a, b, c are coplanar,, v, at the same time, and a b vc 0 they are not equal to zero

35 A question often met in applications Given two vectors v, v 1 2 v, making 3 v3 v1 Vector operations, how to construct another vector v 2 v 3 Solution: p v v cos normalized( v ) 2 1 v p v 1 v v v v v v v v1 v 2 v 1 v 1 then: v v v 3 2 p

36 Introduction to vector functions r( t) ( x( t), y( t), z( t)) where t is a variable, is a vector function Derivative of a vector function Properties (1) (2) dx dy dz r( t),, dt dt dt ( t) r( t) ( t) r( t) ( t) r ( t) r ( t) r ( t) r( t) r ( t) r ( t) r ( t) r ( t) r ( t) r( t) r ( t) r ( t) r ( t) (3)

37 Introduction to vector functions r( t) ( x( t), y( t), z( t)) where t is a variable, is a vector function Derivative of a vector function Properties (4) r ( t), r ( t), r ( t) dx dy dz r( t),, dt dt dt r 1 ( t), r2 ( t), r3 ( t) r1 ( t), r 2 ( t), r3 ( t) r1 ( t), r2 ( t), r 3 ( t)

38 Theorem Length of Proof: Introduction to vector functions r () t is const r( t) r( t) 0 r( t) const r( t) r( t) const r( t) r( t) r( t) r( t) 0 2 r( t) r( t) 0 r( t) r( t) 0 Geometric interpretation: sphere curve r() t is the trace of a (hyper) 2

39 Theorem Introduction to vector functions r( t) 0, and the direction of r() t is fixed r( t) r( t) 0 r() t Geometric interpretation: the trace of is a straight line passing the origin of the coordinate system

40 Theorem ( t) ( t), Introduction to vector functions r() t r r 0 and is parallel to a fixed plane r( t), r( t), r ( t) 0 Proof: Let n be the normal of that plane, then nr( t) 0, Then, nr( t) 0, nr( t) 0, Then, n is perpendicular to three vectors r( t), r( t), r ( t) Then, r( t), r( t), r ( t) should belong to same plane r( t), r( t), r ( t) 0 Then, Geometric interpretation: the origin r() t locates in a plane passing

41 Today s Topics Overview Vector operations Ray-casting framework Camera coordinate system Ray representation Ray-surface intersection Shading models Surface rendering

42 What is 3D rendering? Topics in computer graphics Imaging: representing 2D images Modeling: representing 3D objects Rendering: constructing 2D images from 3D models Animation: simulating changes over time

43 Interactive 3D rendering scenario I Images generated in fraction of a second (<1/10) as user controls rendering parameters (e.g., camera) Achieve highest quality possible in given time Useful for visualization, games, etc. Meshviewer

44 Batch 3D rendering scenario II One image generated with as much quality as possible for a particular set of rendering parameters Take as much time as is needed (minutes) Useful for photorealistism, movies, etc. Avatar

45 3D primitives 3D rendering pipeline Modeling transformation Lighting Viewing transformation Projection transformation Clipping Viewport transformation Scan conversion Transform into 3D world coordinate system Illuminate according to lighting and reflectance Transform into 3D camera coordinate system Transform into 2D camera coordinate system Clip primitives outside camera s view Transform into image coordinate system Draw pixels (includes texturing, hidden surface) 2D image

46 Ray-casting framework Ray-casting (a simple version of ray tracing) Generate an image by sending one ray per pixel It is a major rendering algorithm in CG

47 Ray-casting framework Image Raycast (Camera cam, Scene scene, int width, int height) { Image image = new Image (width, height) ; for (int i = 0 ; i < height ; i++) for (int j = 0 ; j < width ; j++) { Ray ray = RayThruPixel (cam, i, j) ; Intersection hit = Intersect (ray, scene) ; image[i][j] = FindColor (hit) ; } return image ; } We will focus on figuring out details of this framework!

48 Ray-casting framework illustration

49 Ray-casting framework illustration

50 Ray-casting framework illustration Image plane

51 Ray-casting framework illustration Image plane

52 Ray-casting framework illustration Each pixel corresponds to one ray. We need to figure out which scene point each ray hits.

53 Ray-casting framework illustration What s the color you put in each pixel?

54 Ray-casting framework illustration Pixel color determined by shading models

55 Ray-casting framework illustration Following terminologies are equivalent Eye = camera = viewpoint Eye coordinate system = camera coordinate system = viewpoint coordinate system View plane = virtual screen = image plane

56 Ray-casting framework Image Raycast (Camera cam, Scene scene, int width, int height) { Image image = new Image (width, height) ; for (int i = 0 ; i < height ; i++) for (int j = 0 ; j < width ; j++) { Ray ray = RayThruPixel (cam, i, j) ; Intersection hit = Intersect (ray, scene) ; image[i][j] = FindColor (hit) ; } return image ; } In what coordinate systems?

57 Overview Vector operations Ray-casting framework Camera coordinate system Ray representation Ray-surface intersection Shading models Surface rendering Outline

58 Camera coordinate system The scene and the camera are described in WCS However, ray-casting can be more conveniently described in the camera coordinate system Camera position

59 Camera coordinate system The scene and the camera are described in WCS However, ray-casting can be more conveniently described in the camera coordinate system Here comes the question: How to get the coordinates of objects in the camera coordinate system? Answer: we need a transformation matrix to transform the objects defined in WCS to the camera coordinate system

60 Camera coordinate system We assume the camera is at the origin of WCS. To specify the camera coordinate system, two vectors are given, N and V. (N and V are not necessarily perpendicular) n N N ( n, n, n ) x y z y view v u v Vn Vn nu ( u, u, u ) x y z ( vx, vy, vz ) z view n u x view

61 Camera coordinate system Then, in the camera coordinate system, the coordinate of the point (x, y, z, 1) defined in the WCS is x 0 c ux uy uz x y vx vy vz 0 c y z c nx ny nz 0 z

62 Camera coordinate system One step further, if the camera s position is (x 0, y 0, z 0, 1) x 0 c ux uy uz x0 x y vx vy vz 0 c y 0 y z c n z x ny n z 0 z The final transformation matrix from WCS to camera system is xc ux uy uz uxx0 uy y0 uzz0 x y v c x vy vz vxx0 vy y0 vzz 0 y z c n z x ny nz nxx0 ny y0 nzz

63 Camera coordinate system summary (1) Ray-casting is more convenient expressed in camera coordinate system The scene should be transformed from WCS to the camera coordinate system To specify the camera system, you need to provide the camera position P 0, the reference point P ref, and the up vector (V) (Note that: P 0 - P ref determines N) The center of view-plane resides on the z view ; normally the view-plane is perpendicular to z view ; So, only parameter needs to be provided to determine the viewplane, that is what?

64 Camera coordinate system summary (2) The camera coordinate system mentioned in our lecture is a right-handed system However, in some existing systems, camera system is left-handed (e.g., OpenGL); For more details about the 3D pipeline, you can refer to Chapter 7 of the recommended textbook In OpenGL, the following three functions are most related glulookat (x0, y0, z0, xref, yref, zref, Vx, Vy, Vz) glortho (xwmin, xwmax, ywmin, ywmax, dnear, dfar) gluperspective (theta, aspect, dnear, dfar) Note: in OpenGL, the view-plane coincides with near clipping plane

65 Ray-casting framework Image Raycast (Camera cam, Scene scene, int width, int height) { Image image = new Image (width, height) ; for (int i = 0 ; i < height ; i++) for (int j = 0 ; j < width ; j++) { Ray ray = RayThruPixel (cam, i, j) ; Intersection hit = Intersect (ray, scene) ; image[i][j] = FindColor (hit) ; } return image ; How? }

66 Overview Vector operations Ray-casting framework Camera coordinate system Ray representation Ray-surface intersection Shading models Surface rendering Outline

67 Ray representation Ray representation One point P 0 (if you select the camera position as P 0, in the camera coordinate system P 0 = (0, 0, 0)) Direction vector d (normalized is better) Parametric line equation: r P d 0 t d P 0

68 Ray representation Note in most cases, the ray is constructed by linking the eye and each pixel position in the view-plane. In this way, the projection of the scene to the view-plane is actually perspective projection (pinhole camera). If each ray takes z view as the direction, you get the orthographic (parallel) projection. Both of these two projection modes are supported in OpenGL

69 Ray representation Looking different?

70 Ray-casting framework Image Raycast (Camera cam, Scene scene, int width, int height) { Image image = new Image (width, height) ; for (int i = 0 ; i < height ; i++) for (int j = 0 ; j < width ; j++) { Ray ray = RayThruPixel (cam, i, j) ; Intersection hit = Intersect (ray, scene) ; image[i][j] = FindColor (hit) ; } return image ; How? }

71 Ray-plane intersection Actually, for each ray through, we need to find the closest intersection point Intersection Intersect (Ray ray, Scene scene) { min_t = infinity; min_primitive = NULL; foreach primitive in scene { t = intersect (ray, primitive) ; if (t>0 && t < min_t) { min_primitive = primitive; min_t = t; } } return Intersection(min_t, min_primitive) ; }

72 Overview Vector operations Ray-casting framework Camera coordinate system Ray representation Ray-surface intersection Ray-polygon intersection Ray-sphere intersection Ray-cylinder intersection Ray-cone intersection CSG Shading models Surface rendering Outline

73 Ray-polygon intersection At first, check whether the plane is back-facing; if so, you do not need to calculate intersection since the eye cannot see such a face at all! Each planar polygon is enclosed by boundary edges in a counterclockwise manner; its normal points outward ray n d n The eye can see this face d ray The eye cannot see this face

74 Ray-polygon intersection At first, check whether the plane is back-facing; if so, you do not need to calculate intersection since the eye cannot see such a face at all! Each planar polygon is enclosed by boundary edges in a counterclockwise manner; its normal points outward nd 0 The eye can see this face Then, you can calculate the intersection point of the ray and the planar polygon

75 Ray-polygon intersection The equation of the infinite plane The plane passes the known point r 0, and its normal is n Its equation is Why? r n nr n 0 0 r 0 We let D nr 0 rn D, then 0

76 Ray-polygon intersection Intersection of the infinite plane with the ray Plane equation rn D 0 Ray equation r P0 dt n Ray r 0 Then, we have ( P dt) n D 0 0 D P0 n t dn The intersection point is D P0 n r P0 d dn

77 Ray-polygon intersection Intersection of the infinite plane with the ray Plane equation rn D 0 Ray equation r P0 dt D P0 n The intersection point is r P0 d E.g. dn z Plane: passing (1, 0, 0), (0, 1, 0), and (0, 0, 1) y Ray: the Z-axis Calculate their intersection point x Can I work it out?

78 Ray-polygon intersection Intersection of the infinite plane with the ray Plane equation rn D 0 Ray equation r P0 dt D P0 n The intersection point is r P0 d dn It is straightforward to get the distance between the camera (eye) and the intersection point How to decide whether the intersection point is within a planar polygon mesh?

79 Ray-polygon intersection In or out checking B A P C E D B A C E D P Check the direction of the following cross products for the two cases PAPB, PB PC, PC PD, PDPE, PE PA If P locates within the polygon, all the cross products have the same direction; otherwise, the directions would be different. Any limitations?

80 Special case: ray-triangle intersection Barycentric definition of a plane A (non-degenerate) triangle (a, b, c) defines a plane Any point P on this plane can be written as P(,, ) a b c, where 1 c a P b

81 Special case: ray-triangle intersection Barycentric definition of a plane Since 1, we have P(, ) (1 ) a b c a ( b a) ( c a) c a P b non-orthogonal coordinate system on the plane

82 Special case: ray-triangle intersection Barycentric definition of a plane Since 1, we have P(, ) (1 ) a b c a ( b a) ( c a) If we require that E.g., if 0,, 0, P must lines on the line c-a c, we get just the triangle! a P b

83 Special case: ray-triangle intersection Barycentric definition of a plane Given a, b, c, and P, how to compute,? P a ( b a) ( c a) e 1 e 2 a+ be 1 +ge 2 - P = 0 e 1 a e1 e2 P 0 e 2 a e1 e2 P 0 e1 e1 e1 e2 e1 ( P a) e e e e e ( P a) e1 e1 e1 e2 e1 ( P a) e1 e2 e2 e2 e2 ( P a)

84 Special case: ray-triangle intersection Intersection of a ray and barycentric triangle Ray equation r P0 dt Points on the barycentric triangle r a ( b a) ( c a) The ray intersects with the triangle if P0 dt a ( b a) ( c a) with 1, 0, 0 a c P d b P 0

85 Special case: ray-triangle intersection Intersection of a ray and barycentric triangle P0 dt a ( b a) ( c a) P0 x dxt ax ( bx ax) ( cx ax) P0 y d yt a y ( b y a y ) ( c y a y ) P0 z dzt az ( bz az ) ( cz az ) Re-group and write in the matrix form Ax=b ( a b ) ( a c ) d t a P ( a b ) ( a c ) d t a P ( a b ) ( a c ) d t a P x x x x x x 0x y y y y y y 0 y z z z z z z 0z

86 Special case: ray-triangle intersection Intersection of a ray and barycentric triangle a b a c d a P a b a c d a P a b a c d t a P A x x x x x x 0x y y y y y y 0 y z z z z z z 0z Can be easily solved by Cramer s rule, a P a c d a b a P d a b a c a P x 0x x x x x x x 0x x x x x x x 0x a P a c d a b a P d a b a c a P y 0 y y y y y y y 0 y y y y y y y 0 y a P a c d a b a P d a b a c a P,, t A A A z 0z z z z z z z 0z z z z z z z 0z Can be copied mechanically into code for implementation!

87 Special case: ray-triangle intersection Pros of barycentric represenation for intersection Efficient Stores no plane equation Get the barycentric coordinates for free Useful for interpolation, texture mapping, etc. c a P d b P 0

88 Special case: ray-triangle intersection Barycentric interpolation v, v, v a, b, c Values defined at They can be colors, normals, texture coordinates etc P(,, ) a b c v(,, ) v v v is the point Then, is the barycentric interpolation of v 1 ~v 3 at point P,, i.e., once you know, you can interpolate values using the same weights, very convenient a ( v ) 1 () v b c ( v ) 3 ( v2)

89 Overview Vector operations Ray-casting framework Camera coordinate system Ray representation Ray-surface intersection Ray-polygon intersection Ray-sphere intersection Ray-cylinder intersection Ray-cone intersection CSG Shading models Surface rendering Outline

90 Ray-sphere intersection Can be used when the object in the scene is described in explicit analytical form of sphere Suppose that the center of the sphere is S 0 and its radius is R. Its equation is 2 r S0 r S0 R (1) The ray equation is r P0 d t (d is a unit vector) (2) Ray

91 Ray-sphere intersection Can be used when the object in the scene is described in explicit analytical form of sphere Suppose that the center of the sphere is S 0 and its radius is R. Its equation is 2 r S0 r S0 R (1) The ray equation is r P0 d t (d is a unit vector) Combine (1) and (2), we can get t 2( P0 S0) dt ( P0 S0) R 0 b c 2 If b 4c0, the ray does not intersect with the sphere 2 b b 4c otherwise t 2 (2)

92 Overview Vector operations Ray-casting framework Camera coordinate system Ray representation Ray-surface intersection Ray-polygon intersection Ray-sphere intersection Ray-cylinder intersection Ray-cone intersection CSG Shading models Surface rendering Outline

93 Ray-cylinder intersection Can be used when the cylinder is analytically described Cylinder: the center of the bottom face is S 0, radius is R, height is h, the central symmetric axis is AXIS, Such a solid cylinder can be described as AXIS r S0 r S0 ( r S0) AXIS R r S0 AXIS 0 r S1 AXIS 0 Since S1 S0 h* AXIS 0 AXIS h (1) (2) rs (3)

94 Ray-cylinder intersection Can be used when the cylinder is analytically described Cylinder: r S0 r S0 ( r S0) AXIS r S0 AXIS 0 r S AXIS h 0 Ray: r P0 dt P dt S P dt S ( P dt S ) AXIS R Simplify it, we have: R (1) (2) (3)

95 Ray-cylinder intersection where, a 1 ( AXIS d) 2 at bt c b d( P d ) AXIS d AXIS ( P d ) c ( P d ) AXIS ( P d ) R Then, b b ac t1,2 a Then, you need to check whether the intersection points really reside on the cylindrical surface bounded by the bottom and top faces. That is, whether it satisfies (2) and (3) r S0 AXIS 0 (2) r S0 AXIS h 0 (3)

96 Overview Vector operations Ray-casting framework Camera coordinate system Ray representation Ray-surface intersection Ray-polygon intersection Ray-sphere intersection Ray-cylinder intersection Ray-cone intersection CSG Shading models Surface rendering Outline

97 AXIS Ray-cone intersection Can be used when the cone is analytically described Cone: the apex of the cone is S 0, half apex angle is, height is h, the central symmetric axis is AXIS, Try to describe such a cone in a mathematical formula and compute its intersection with the ray r P d 0 t

98 Overview Vector operations Ray-casting framework Camera coordinate system Ray representation Ray-surface intersection Ray-polygon intersection Ray-sphere intersection Ray-cylinder intersection Ray-cone intersection CSG Shading models Surface rendering Outline

99 CSG Constructive solid geometry Build a complex object based on a set of primitives Three basic boolean operations: union, intersection, and subtraction CSG can be naturally implemented in the ray-casting framework

100 Constructive solid geometry CSG implementation in ray-casting framework Step 1: compute the intersection points (actually ray parameters ts ) with left child node and right child node Step 2: sort the intersection points according to their corresponding parameters Step 3: classify the line segments in the final results as in or out Step 4: remove the redundant points for the in line segments; after this step, any two adjacent line segments belong to different categories ( in or out ) Above 4-steps procedure actually is a recursive process since each child node is also a CSG tree

101 Constructive solid geometry CSG implementation in ray-casting framework Rules for line segments classification Operator Left node Right node Result union intersection subtraction in in out out in in out out in in out out in out in out in out in out in out in out in in in out in out out out out in out out

102 Constructive solid geometry CSG implementation in ray-casting framework E.g., union of A and B B A t 4 t t 2 3 t 1 ray Step 1: intersections with A: t 1, t 2 ; intersections with B: t 3, t 4 Step 2: sort the points, t 1, t 3, t 2, t 4 Step 3: classify the line segments out in in in out Step 4: remove redundant points out in out

103 Constructive solid geometry CSG implementation in ray-casting framework E.g., intersection of A and B B A ray t 4 t t 2 3 t 1 Step 1: intersections with A: t 1, t 2 ; intersections with B: t 3, t 4 Step 2: sort the points, t 1, t 3, t 2, t 4 Step 3: classify the line segments out out in out out Step 4: remove redundant points out in out

104 Constructive solid geometry CSG implementation in ray-casting framework E.g., difference of A and B B A ray t 4 t t 2 3 t 1 Step 1: intersections with A: t 1, t 2 ; intersections with B: t 3, t 4 Step 2: sort the points, t 1, t 3, t 2, t 4 Step 3: classify the line segments out in out out out Step 4: remove redundant points out in out

105 Ray-casting framework Image Raycast (Camera cam, Scene scene, int width, int height) { Image image = new Image (width, height) ; for (int i = 0 ; i < height ; i++) for (int j = 0 ; j < width ; j++) { Ray ray = RayThruPixel (cam, i, j) ; Intersection hit = Intersect (ray, scene) ; image[i][j] = FindColor (hit) ; } return image ; }

106 Overview Vector operations Ray-casting framework Camera coordinate system Ray representation Ray-surface intersection Shading models Light sources Phong shading model Shadow Transparency Surface rendering Outline

107 Shading model Shading model, or illumination model, or lighting model is used to mimic the physical world to decide the color of a point in the visual scene The object s color is decided by The light The attributes of the object

108 The most simple one It is omni-directional Point light source Attributes to specify a point light source Position (p x, p y, p z ) Intensity I (if it is a chromatic light, three values representing R, G, and B are needed) Coefficients (a 0, a 1, a 2 ) to specify its attenuation property with distance d I d I d

109 The most simple one It is omni-directional Point light source Attributes to specify a point light source Position (p x, p y, p z ) Intensity I (if it is a chromatic light, three values representing R, G, and B are needed) Coefficients (a 0, a 1, a 2 ) to specify its attenuation property with distance d In this lecture, except otherwise specified, the light source is considered as monochromatic. For a practical chromatic light, simply repeat the theories to R, G, and B respectively, and combine the results together.

110 Directional light source Besides position and color, the apex angle of the lighting cone needs to be specified l f V obj angatten 0.0, when Vobj Vlight cos l Vobj Vlight, otherwise where is a constant V light can be combined

111 Directional light source Besides position and color, the apex angle of the lighting cone needs to be specified l V obj V light

112 Overview Vector operations Ray-casting framework Camera coordinate system Ray representation Ray-surface intersection Shading models Light sources Phong shading model Shadow Transparency Surface rendering Outline

113 Phong shading model (single light) Developed by Bui Tuong Phong ( 裴祥风 ) (Vietnamese, ) In 1973, he developed the Phong shading model He dies in 1975 for leukaemia By Phong model, the light energy emitted from a point on an object to the observer s eye comprises three components Ambient lighting Diffuse reflection Specular reflection

114 Phong shading model (single light) Ambient lighting Ambient lighting is a constant for a scene I a Different surface can have different ambient reflection coefficient k a (0<= k a <= 1) So, if only consider ambient lighting, the illumination at a point simply is I Iaka Rendering results when only ambient lighting is considered

115 Phong shading model (single light) Diffuse reflection Diffuse reflection is the reflection of light from a surface such that an incident ray is reflected at many angles rather than at just one angle as in the case of specular reflection (chalk, clay) An illuminated ideal diffuse reflecting surface will have equal luminance from all directions in the hemisphere surrounding the surface diffuse reflections specular reflection (big) surface reflected scatter distribution BRDF

116 Phong shading model (single light) Diffuse reflection Diffuse reflection is the reflection of light from a surface such that an incident ray is reflected at many angles rather than at just one angle as in the case of specular reflection (chalk, clay) An illuminated ideal diffuse reflecting surface will have equal luminance from all directions in the hemisphere surrounding the surface Lambert reflection law L N I diff Ik L d( NL), NL 0 0, NL 0 (Light is at the back of the surface) where, N is the normal of the surface at the examined position, L is the direction of the light, k d is the diffuse reflection coefficient of the surface, I L is the intensity of the incident light

117 Phong shading model (single light) Diffuse reflection Diffuse reflection is the reflection of light from a surface such that an incident ray is reflected at many angles rather than at just one angle as in the case of specular reflection (chalk, clay) An illuminated ideal diffuse reflecting surface will have equal luminance from all directions in the hemisphere surrounding the surface Lambert reflection law L N Or simply write as I I k diff L d max( NL,0)

118 Phong shading model (single light) When considering both the ambient lighting and the diffusion reflection, the observed intensity of a point can be described as I Iaka ILkd max( NL,0) Illuminated by ambient lighting and diffuse reflection Left: diffuse reflection only right: ambient + diffuse reflection

119 Phong shading model (single light) Specular reflection Specular reflection is the mirror-like reflection of light from a surface, in which light from a single incoming direction (a ray) is reflected into a single outgoing direction Used to model mirrors, metals, etc.

120 Phong shading model (single light) Specular reflection L Specular reflection is the mirror-like reflection of light from a surface, in which light from a single incoming direction (a ray) is reflected into a single outgoing direction Used to model mirrors, metals, etc. Surface normal Direction of reflection N R to viewpoint V The observed light is I spec L s Ik L s I k VR n cos ( ) R (2 NL) N L k s = specular reflection coefficient n = specular exponent n

121 Phong shading model (single light) L Specular reflection (a simplified version) Surface normal N Direction of reflection H R L V H L V to viewpoint V I I k spec L s I VR I k n Then, n spec L s More accurately, I HN approximated by I k spec L s max( HN,0) n NH (less computation)

122 Phong shading model (single light) When considering the ambient lighting, the diffusion reflection, and the specular reflection at the same time, the observed intensity of a point can be described as I I k I k max( NL,0) k max( NH,0) n a a L d s This is called as Phong shading model!!

123 Phong shading model (single light) When considering the ambient lighting, the diffusion reflection, and the specular reflection at the same time, the observed intensity of a point can be described as I I k I k max( NL,0) k max( NH,0) n a a L d s (two light sources) with increased n

124 Phong shading model (multi-lights) Multi-lights Single light case can be easily extended to multi-light case Suppose there are m lights in the scene, m N L N H I I k I k max(,0) k max(,0) a a Li d i s i i1 n

125 Overview Vector operations Ray-casting framework Camera coordinate system Ray representation Ray-surface intersection Shading models Light sources Phong shading model Shadow Transparency Surface rendering Outline

126 Shadows In previous discussions, we always assume that the light source can see the object Shadow term tells if a light source is blocked Cast a ray towards the light source L i s i = 0, if the ray is blocked; s i = 1, otherwise viewpoint X The light cannot illuminate the sphere; the light ray is blocked by the triangle View plane

127 Shadows In previous discussions, we always assume that the light source can see the object Shadow term tells if a light source is blocked Cast a ray towards the light source L i s i = 0, if the ray is blocked; s i = 1, otherwise Then, the Phong shading model becomes m N L N H I I k I s k max(,0) k max(,0) a a Li i d i s i i1 n Scene rendering results when considering shadow terms

128 Overview Vector operations Ray-casting framework Camera coordinate system Ray representation Ray-surface intersection Shading models Light sources Phong shading model Shadow Transparency Polygonal rendering Outline

129 Transparency If an object is transparent to some extent, the observer can see the objects behind it Accurate transparency modeling needs the raytracing algorithm, which is quite time-consuming

130 Transparency A simple transparency model I k I k I 1 t refl t trans Where I refl is the intensity of the examined point by using the Phong shading mode; I trans is the intensity of the intersection point (with the viewpoint-ray) behind the transparent object; 0<=k t <=1 is transparency factor. Transparent object viewpoint I refl Itrans View plane

131 Overview Vector operations Ray-casting framework Camera coordinate system Ray representation Ray-surface intersection Shading models Polygon rendering Flat surface rendering Gouraud surface rendering Phong surface rendering Outline

132 Polygon rendering How do we choose a color for each filled pixel if the object is represented in a set of polygonal meshes? It is called as polygon shading problem Simplest shading approach is to perform independent lighting calculation for every pixel

133 Polygon rendering Actually, we can take advantage of spatial coherence Illumination calculations for pixels covered by the same primitive are related to each other Common used polygon algorithms, flat shading, Gouraud shading, and Phong shading

134 Flat surface rendering The most simplest one, also called as constant intensity surface rendering One illumination calculation per polygon Assign all pixels inside each polygon the same color No interpolation, pick a single representative intensity and propagate it over entire primitive loses almost all depth cues Objects look like they are composed of polygons OK for polyhedral objects Not so good for smooth surfaces

135 Flat surface rendering Some examples with flat surface rendering

136 Flat surface rendering Some examples with flat surface rendering

137 Flat surface rendering Some examples with flat surface rendering

138 Overview Vector operations Ray-casting framework Camera coordinate system Ray representation Ray-surface intersection Shading models Polygon rendering Flat surface rendering Gouraud surface rendering Phong surface rendering Outline

139 Gouraud surface rendering Was proposed by Henri Gouraud in 1971 H. Gouraud, Computer Display of Curved Surfaces, Doctoral Thesis, University of Utah, USA, Properties produce continuous shading of surfaces represented by polygon meshes It is often used to achieve continuous lighting on triangle surfaces by computing the lighting at the corners of each triangle and linearly interpolating the resulting colors for each pixel covered by the triangle

140 Gouraud surface rendering Working principles Step 1: estimate the normal of each vertex by averaging the surface normals of the polygons that meet at each vertex N 4 N v n k 1 n k 1 N N k k N 1 N 2 N v N 3

141 Gouraud surface rendering Working principles Step 1: estimate the normal of each vertex by averaging the surface normals of the polygons that meet at each vertex Step 2: compute the color for each vertex based on a shading model (e.g., Phong model) Step 3: for each screen pixel that is covered by the polygonal mesh, color intensities can then be interpolated from the color values calculated at the vertices (more on next page)

142 Gouraud surface rendering Working principles Bilinear interpolation At first, interpolate illumination along polygon edges (I a, I b ) Then, interpolate illumination along scan lines (I p ) scan line I 1 I 2 I 3 a I b I I p y 1 y s y 2 y y y y y I y y y y I I s s a y y y y I y y y y I I s s b a b a p b a b p b a p x x x x I x x x x I I

143 Gouraud surface rendering Working principles For interpolation of triangle, we can use barycentric coordinates, still remember? Very convenient and straightforward!! (Previous page of this slide)

144 Cons Gouraud surface rendering Gouraud shading can miss specular highlights because it interpolates vertex colors instead of calculating intensity directly at each point, or interpolating vertex normals

145 Gouraud surface rendering Some examples with Gouraud surface rendering Flat (left) vs Gouraud (right)

146 Gouraud surface rendering Some examples with Gouraud surface rendering Flat (left) vs Gouraud (right)

147 Gouraud surface rendering Some examples with Gouraud surface rendering Flat (left) vs Gouraud (right)

148 Gouraud surface rendering Some examples with Gouraud surface rendering Flat (left) vs Gouraud (right)

149 Overview Vector operations Ray-casting framework Camera coordinate system Ray representation Ray-surface intersection Shading models Polygon rendering Flat surface rendering Gouraud surface rendering Phong surface rendering Outline

150 Phong surface rendering Phong surface rendering refers to an interpolation technique for surface rendering in 3D computer graphics It is also called Phong interpolation or normal-vector interpolation surface rendering Specifically, it interpolates surface normals across rasterized polygons and computes pixel colors based on the interpolated normals and a reflection model

151 Phong surface rendering Phong model: normal vector interpolation Similar interpolation process as Gouraud shading; however, it interpolates N rather than I Always captures specular highlights, but computationally expensive At each pixel, N is recomputed and normalized Then I is computed at each pixel (lighting model is more expensive than interpolation algorithms) This is now implemented in hardware, very fast

152 Phong surface rendering Some examples with Phong surface rendering Flat (left), Gouraud (middle) and Phong shading models

153 Phong surface rendering Some examples with Phong surface rendering Flat (left), Gouraud (middle) and Phong shading models

154 Phong surface rendering Some examples with Phong surface rendering Flat (left), Gouraud (middle) and Phong shading models

155 Thanks everybody!