CS 450: COMPUTER GRAPHICS RAY TRACING SPRING 2016 DR. MICHAEL J. REALE

Transcription

1 S 450: OMPUTR GRPHIS RY TRING SPRING 016 R. MIHL J. RL

2 RNRING IMGS Let s sa we have a scene of 3 objects We want to render this to a raster image We ve got this and we want this In general, there are two was we can do this: Object-order rendering For each OBJT find piels that object affects Often used for real-time rendering Image-ordering rendering For each PIXL find all objects that affect this piel Often used for non-real-time rendering

3 RY TRING We re going to talk about ra tracing Form of image-order rendering dvantages: High-qualit images as to compute: Shadows Reflections Refraction isadvantages: Generall prett slow

4 BSI RY-TRING LGORITHM

5 BSI I We have a virtual camera that starts at point Imagine that there are a grid of piels floating in front of the camera To get the color for each piel: Shoot a ra from the starting point to the piel position See what objects we hit with the ra Pick the nearest intersection point ompute color at that point Shading model looking at normal at that point, lighting direction, color, etc.

6 THR BSI STPS This means we have three basic steps: Ra generation ompute viewing ra for each piel Ra intersection Find closest intersection point among all objects in scene This step wh ratracers are generall slow Shading ompute piel color based on results of ra intersection

7 PRSPTIV

8 3 TO Our first question is how our camera will project the 3 world onto a image? Old problem: artists had been working on this one for centuries Simplest approach: istance from camera doesn t matter Onl care where it is in terms of left/right and up/down More realistic approach: Take perspective into account things farther awa should look smaller

9 PROJTIONS Two most commonl used projection methods: Orthographic or parallel Perspective Note: when defining a projection, also define: Near plane = closer than near plane don t render Far plane = beond far plane don t render oupled with other aspects of camera, defines view volume Objects in view volume will be rendered Orthographic Perspective Projection direction = direction our camera is pointed

10 ORTHOGRPHI PROJTION Orthographic or parallel Move object along projection direction until ou hit image plane istance from image plane doesn t change appearance Parallel lines remain parallel View volume = rectangular bo

11 ORTHOGRPHI PROJTION: PROS N ONS Orthographic or parallel dvantages: Good for mechanical/architectural drawings Sie of objects remains the same, no matter how far awa the are Parallel lines preserved isadvantages: Not realistic effectivel assumes multiple camera ee starting points In realit, just one or two, with the human vision sstem

12 PRSPTIV PROJTION Perspective Single starting point ee point all objects project towards ee point Things look smaller when farther awa View volume = truncated pramid with rectangular base called view frustum

13 SI: OBLIQU PROJTION If the image plane is NOT orthogonal to the project direction oblique projection Below: eample of an oblique parallel projection

14 OMPUTING VIWING RYS

15 RTING RYS To define our ras, we ll just use 3 parametric ras ssuming we have: Starting point also called ee point Point S on the image plane Then our ra is defined b: P t t S

16 BHIN BLU YS We know that: P0 P1 S We also know that, if: 0 t t 1 Then Pt 1 is closer to ee than Pt If t < 0 behind the ee shouldn t be able to see item

17 MR FRM We need to find the S points will be given to us, depending on the kind of projection To do this, we will look at things from the perspective of the camera frame amera frame = orthonormal coordinate frame defined b the ee point and three vectors: U points to camera s RIGHT V points upward W points BKWR Remember: RIGHT-HN RUL, so camera is actuall looking along the W ais!

18 GTTING TH IMG PLN Based on the camera frame, we ll construct our image plane We know: Right U and Up V directions for plane ssume we also know where the plane is centered How large is the plane? Usuall define plane in terms of the following coordinates: r right ma coordinate LONG U ais r > 0 l left min coordinate LONG U ais l < 0 t top ma coordinate LONG V ais t > 0 b bottom min coordinate LONG V ais b < 0

19 GTTING POINTS ON TH IMG PLN We then need to define what points on the image plane correspond to our piels If: n = # of piels in n = # of piels of To get the coordinates u,v for a given point assuming that piels are TULLY centered on the coordinates: u l v b r l i 0.5 / t b j 0.5 / n n

20 ORTHOGRPHI VIWS For an orthographic view, all ra IRTIONS = -W Image plane centered at ee point For each piel: ompute u and v previous slide Ra direction W Ra starting point uu vv For oblique views all ra directions = d some other direction

21 PRSPTIV VIWS For perspective views, all ra STRTING POINTS = Image plane centered at - dw d = image plane distance For each piel: Sometimes loosel called focal length changing d does change field of view ompute u and v two slides back Ra direction uu vv Ra starting point dw For oblique views use d instead of -dw

22 RY-OBJT INTRSTION

23 FINING OBJTS We now have our ra: We need to find out what the ra intersects with in the 3 scene Want to pick the nearest intersection point with t in the interval: Usuall: [ 0, ] P t t [ t 0, t1] lthough there are man possible objects to intersect with, we will discuss ra intersections with: Spheres Triangles Polgons

24 RY INTRSTION WITH IMPLIIT SURF Here s our ra again: P t t Given an implicit surface: f P 0 Intersection points = points on ra that satisf the implicit equation: f P t 0 or f t 0 So we have to find values of t if an that satisf the above equation.

25 RY-SPHR INTRSTION

26 RY-SPHR INTRSTION Here s our implicit equation in vector form for a sphere: Let s plug in our ra for P: 0 r P P P f t t P 0 r t t P f Our Ra

27 RY-SPHR INTRSTION Rearranging terms: r t t r t t r t t t r t t t t t t r t t

28 RY-SPHR INTRSTION closer look at our result: reveal that this is a quadratic equation: where: 0 r t t 0 c bt at r c b a

29 RY-SPHR INTRSTION This means we can use the quadratic formula! a ac b b t 4 4 r t r c b a

30 RY-SPHR INTRSTION leaning things up a bit: t r If we look at the discriminant G: G > 0 TWO possible intersection points G = 0 ON possible intersection point graing surface of sphere G < 0 NO intersection In implementation, one should check the discriminant first before computing other terms In fact, if doing collision detection ma onl need to compute the discriminant!

31 RY-SPHR INTRSTION

32 RY-SPHR INTRSTION: GOMTRI INTRPRTTION t r Let s tr to break this down geometricall Vector going from center of circle TO ee point: Length of projection of - onto : cos However, if we just wanted the adjacent side and assumed was normalied: p cos THT SI, note that - and point in opposing directions, so the distance along we TULLY want is: p

33 RY-SPHR INTRSTION: GOMTRI INTRPRTTION That said, our original formula: becomes also converting certain dot products to the lengths-squared: r t r p p t

34 RY-SPHR INTRSTION: GOMTRI INTRPRTTION We can actuall move the length of outside of the square root: r p p t r p p t

35 RY-SPHR INTRSTION: GOMTRI INTRPRTTION Let s sa that: Then our formula becomes: r p p t q r q p p t

36 RY-SPHR INTRSTION: GOMTRI INTRPRTTION Let s look at the discriminant more closel: Rearranging it a bit: Because of the Pthagorean theorem: So, if we re tring to get s : SO, we are effectivel getting the difference between r and s : r q p q p r s p q p q s s r p q r q p r

37 RY-SPHR INTRSTION: GOMTRI INTRPRTTION SO, when we are computing the discriminant G: s r G on point no intersecti 0 on point ON intersecti 0 on points TWO intersecti 0 s r G s r G s r G

38 RY-SPHR INTRSTION: GOMTRI INTRPRTTION Let s assume we have TWO intersection points G > 0 It turns out that, if we look at NOTHR right triangle, then: Which means that the square root of our discriminant gives us: k k s r G s r k r k s

39 RY-SPHR INTRSTION: GOMTRI INTRPRTTION t SO, plugging this value into our original formula and canceling out the etra : In other words, in terms of TUL geometric distance, we need to go along b: -p + k -p k p t to get our two intersection points. p q p k HOWVR, because ma not be normalied, we have to divide b the length of r

40 RY-TRINGL INTRSTION

41 INTRSTION WITH TRINGL While there are man was to do ra-triangle intersections, we will use: Barcentric coordinates parametric plane containing the triangle Basicall, all we need to store for this are the vertices of the triangle

42 RY INTRSTION WITH PRMTRI SURF To intersect a ra with a parametric surface set up the following equations: Three unknowns t,u,v three equations Hopefull, we can solve this analticall, but in the worst case we can throw numerical methods at it Remember: parametric form plug in parameters return a POINT Parametric line: Parametric surface:,,,, v u f t v u f t v u f t v u f t 3 : R R t P 3 :, R R v u f

43 RY INTRSTION WITH PRMTRI PLN If our surface = parametric plane we can use the vector form with barcentric coordinates! β, γ parameters for surface Returns a 3 point Three points of triangle define a plane So, our equation to solve becomes: t B Solve for t, β, γ

44 RY INTRSTION WITH TRINGL INSI the triangle IF: β > 0 γ > 0 t B β + γ < 1 OTHRWIS, misses triangle although it OS hit the plane If there IS no solution either: Triangle is degenerate OR Ra is parallel to plane

45 SOLVING FOR PRMTRS First, we ll rearrange things so that all our constants are on the right-hand side and everthing else is on the left: We re going to turn this into a standard linear sstem so we can use matrices to solve the equations hold tight B t t B t B B t

46 SOLVING FOR PRMTRS Then, we ll epand this out into the individual coordinates: Remember: the,, coordinates are known values B B B t t t t B

47 SOLVING FOR PRMTRS Finall, we ll turn this into a standard linear sstem of the form B B B t MV N B B B t t t

48 QUIK RVIW: MTRIX MULTIPLITION Matri = effectivel, a arra of numbers M = 33 matri Vector = basicall a matri with one of the dimensions equaling 1 V and N = 31 matrices = 3 column vectors When multipling a matri M b a vector V IN THT ORR: Result = vector N with: Same # of rows as M Same # of columns as V For each value n R in N: ot product of Rth row of M and 1 st column of V MV N m m m n n n m m m m m m m m m v v 0 v v v v m m m v v 1 v 1 1 n n n m m 1 m v v v

49 BK TO OUR LINR SYSTM So, for eample, if I wanted the first equation back B B B t B t

50 QUIK RVIW: TRMINNT OF MTRIX The determinant of a matri is calculated for a and 33 matri in the following fashion: NOT: The determinant is related to the cross product, scalar triple product, and the area/volume of parallelograms/parallelepipeds formed b the vectors, but we ll get to that later M *4 *3 i j k M 1 3 i*6 3*5 j3*4 1*6 k1*5 *

51 RMR S RUL For 33 sstems, a fast wa to solve them is with ramer s Rule: gain, we ll talk about how and wh this works later Given a sstem: MV N For each coordinate of V v i : Replace the i th column of matri M with N Get the determinant ivide b the determinant of the original matri M WRNING: If M = 0 no unique solution! Ma be no solution ma be INFINIT solutions

52 PPLYING RMR S RUL So, in determinant form, to solve for our parameters: M t M M B B B B B B B B B M

53 PPLYING RMR S RUL If we use dumm variables: We can show eactl what we need to calculate and we can reduce the number of operations b calculating things like ei hf once: l k j t i f c h e b g d a eg dh c di gf b hf ei a p p kc bl d al jc e jb ak f t p kc bl g al jc h jb ak i p eg dh l di gf k hf ei j

54 OING TH FUNTION When coding the intersection function, we would want to terminate earl if we can More efficient Let s assume t has to be in the interval [t 0, t 1 ] SO, our algorithm might look like: ompute t If t < t 0 OR t > t 1 return false ompute γ if γ < 0 OR γ > 1 return false ompute β If β < 0 OR β > 1 γ return false Return true

55 RY-POLYGON INTRSTION

56 WHT IS POLYGON, XTLY? The official definition varies, but at bare minimum: Polgon = a figure with 3 or more vertices connected in sequence b straight-line segments edges or sides of the polgon Most loose definition an closed-polline boundar More finick definitions contained in single plane, edges have no common points other than their endpoints, no three successive points collinear Standard polgon or simple polgon = closed-polline with no crossing edges

57 OS POLYGON LI IN SINGL PLN? In computer graphics, polgon not alwas in same plane: Round-off error I.e., points originall in single plane, but, after transformation, the points ma be slightl off Fitting to surface makes non-planar polgons.g., if using quads, approimation ma bend the quad in half Thus, we usuall use triangles to avoid this problem

58 GNRT POLYGONS egenerate polgons = often used to describe polgon with: 3 or more collinear vertices generates a line segment.g., in etreme case, triangle with no area Repeated verte positions generates shape with: traneous lines Overlapping edges dges with length 0

59 INTRSTING WITH TH PLN OF TH POLYGON If LL of the vertices P 1 through P m of a polgon lie in the same plane first get intersection with plane Point-normal form of plane we ll just use the first point: Plugging in our ra equation for P: 0 1 N P P N N P t N P N t N t N P N P t

60 R W INSI TH POLYGON? Once we have our intersection point P, the question is: are we inside the polgon? One trick: Project point P and polgon to the XY plane Shoot a ra out from P Usuall easiest to use X ais heck what we cross with this ra Number of edges OR direction of edges Two common approaches for the last step: Odd-ven Rule Nonero Winding-Number Rule

61 O-VN RUL lso called odd-parit or even-odd rule raw ra starting from position P ount how man edges we cross O inside polgon VN outside polgon

62 SI: OTHR USS OF O-VN RUL an use to check other kinds of regions.g., area between two concentric circles

63 NONZRO WINING-NUMBR RUL ount number of times boundar of object winds around a particular point in the clockwise direction gain, draw ra starting at position P Start winding number W at 0 If intersection with segment that crosses the line going from: Right-to-left counterclockwise add 1 to W Left-to-right clockwise subtract 1 from W Remember: think right-hand-rule heck value of W when done: If W == 0 XTRIOR If W!= 0 INTRIOR

64 NONZRO WINING-NUMBR RUL To check the direction of the edge relative to the line from P: Make vectors for each edge in the correct winding order vector i Make vector from P to distant point vector U Two options: Use cross product: If U i = +Z ais right-to-left add 1 to winding number If U i = Z ais left-to-right subtract 1 from winding number Use dot product: Get vector V that is 1 perpendicular to U, and goes right-to-left -u, u If V i > 0 right-to-left add 1 to winding number Otherwise left-to-right subtract 1 from winding number

65 O-VN VS. NONZRO WINING-NUMBR For simple objects polgons that don t self-intersect, circles, etc., both approaches give same result However, more comple objects ma give different results Usuall, nonero winding-number rule classifies some regions as interior that odd-even sas are eterior Odd-ven Rule Nonero Winding-Number Rule

66 INSI-OUTSI PROBLM aveat: need to check we don t cross endpoints vertices Otherwise, ambiguous we ll talk in more detail how to deal with this when we get to filling polgons later

67 SI: URV PTHS? For curved paths, need to computer intersection points with underling mathematical curve For nonero winding-number also have to get tangent vectors

68 SI: VRITIONS OF NONZRO WINING-NUMBR RUL an be used to define Boolean operations: Positive W onl, both counterclockwise union of and B W > 1, both counterclockwise intersection of and B Positive W onl, B clockwise - B

69 PROJTING TH POLYGON PROBLMS What if the projection of the polgon into XY is a LIN?.g., the polgon is in the XZ plane? Solution: choose best among XY, YZ, or ZX planes if abs elseif n abs abs abs else use YZ plane n n & & abs n n abs use ZX plane n use XY plane

70 SHING

71 SHING MOL Let s assume we ve: Intersected our view ra with ever object in the scene Found the nearest intersection point and associated object The piel value is computed b evaluating a shading model Shading model or lighting model Used to calculate the color of an illuminated position on the surface of an object In the slides that follow, we will concentrate on shading models that use point lights

72 POINT LIGHTS Point light Located at a single point in space mit light in all directions

73 IFFUS RFLTION iffuse reflection = when white light hits an object, what we see as the color of an object ample: apple absorbs all frequencies ecept red has red diffuse color Underling phsics: surfaces with microfacets bump, grain, matte reflects light in lots of different directions Ideal diffuse reflectors or Lambertian reflectors Incident light scattered with equal intensit in LL directions, INPNNT of viewing angle epends on angle between NORML and IRTION-to-LIGHT angle of incidence

74 LMBRTIN SHING Lambertian shading model Simplest of all shading models mount of light energ that hits a surface proportional to angle of surface to light View-independent the viewer s position doesn t matter If: N = NORMLIZ normal L = NORMLIZ vector from point on surface to light source Then dot product is proportional to angle between L and N: N L cos

75 LMBRTIN SHING If the light is BHIN the surface angle between L and N > 90 degrees dot product will be NGTIV So, we onl want the ma of 0 and the dot product: That said, our piel color will be: where: Piel color N = NORMLIZ normal L = NORMLIZ vector from point on surface to light source Subtract intersection point P from light position then normalie vector I = intensit of light k d I ma0, N L k d = diffuse coefficient / diffuse color of surface ma 0, N L

76 LMBRTIN SHING Piel color k d I ma0, N L Keep in mind that we basicall have THR equations one for red, one for green, and one for blue.g., for the red piel color have a red light intensit and a red diffuse coefficient N and L are the same in all cases

77 PROBLM WITH LMBRTIN SHING Lambertian model view independent Provides diffuse component of light HOWVR, in real life have shin spots/specular reflections that O depend on view position Without this, models look kind of matte and chalk So, we need some wa to model these specular highlights

78 BLINN-PHONG SHING Blinn-Phong shading model Given: V = view direction NORMLIZ vector from intersection point to ee point Produce brightest reflection of light when V and L are smmetric about normal N Specular reflection = when all or almost all of the incident light is reflected back shin or reflective spot

79 BLINN-PHONG SHING First, compute half-vector H H V V L L Bisector of angle between V and L Both V and L are the same length can just add together to get average direction vector, then normalie! If perfect reflection H will be perfectl in line with normal N So, we want to look at: ma 0, N H

80 BLINN-PHONG SHING: ISSUS WITH TH HLF- VTOR Problem: if V, L, and N are not coplanar slightl off nother problem: need to check if V and L are on same side of N L V > L N if so, don t use specular effect at all

81 BLINN-PHONG SHING: SHININSS ifferent surfaces reflect light over finite range of viewing positions Shin surfaces narrow range uller surfaces wider range So, we take the dot product to a power to make it decrease faster use Phong eponent or shininess s: ma 0, N H s

82 BLINN-PHONG SHING: SHININSS Tpical values of s: 10 eggshell 100 mildl shin 1000 reall gloss 10,000 nearl mirror-like

83 BLINN-PHONG SHING H V V L L So, our final Blinn-Phong shading model will be: Piel color k d I ma0, N L k s I ma0, N H S where: k s = specular coefficient / specular color of surface Usuall set this to gra or white

84 NORMLIZ YOUR VTORS!!! IMPORTNT: ON T FORGT TO NORMLIZ N, V, L, and H!!! Otherwise, we have lengths that we don t want creeping into the dot products

85 MBINT SHING Using either of the two previous models, if the surface is facing awa from the light completel black In real-life indirect reflections of other surfaces will cause SOM light to fall on those surfaces HORRIBL HK: dd a little bit of light to LL surfaces ambient shading

86 MBINT SHING mbient shading Light all surfaces with ambient light that comes equall from everwhere ombined with Blinn-Phong, we get: Piel color k a I a k d I ma0, N L k s I ma0, N H S where I a = ambient light intensit k a = surface s ambient coefficient an be kept separate to tune ambient light per surface OR can set the same as diffuse light coefficient

87 FINL MOL: IFFUS, SPULR, N MBINT Piel color k a I a k d I ma0, N L k s I ma0, N H S

88 MULTIPL LIGHTS Light has the propert of superposition I.e., effect caused b more than one light source = sum of effects from individual light sources So, to deal with multiple light sources, just sum all values up for diffuse and specular components ambient light is onl added once, however: Piel color k a I a n i1 I i k d ma 0, N L i k s ma 0, N H i s

89 PROGRMMING RY TRR

90 BSI LGORITHM For the ra-tracer, the basic algorithm is as follows: For each piel ompute viewing ra If ra hits an object with t >= 0 ompute N valuate shading model and set piel to that color lse Set piel color to background color

91 OBJT-ORINT PPROH s we ve seen, there are a lot of different kinds of objects ou could intersect with: Spheres, triangles, planes, polgons, etc. Good OOP-based approach: Make generic class Surface Give Surface two functions: hit returns whether a given ra hit the object lso hands back some kind of hit-record intersection point, normal, etc. bounding-bo returns ma etends of surface Useful for optimiation Have all other objects inherit from Surface and implement those two functions Then, ou can just loop through one list of Surface instances Instead of separatel going through our spheres, then our triangles, then

92 MTRILS Material class is a good idea as well Have list of Materials that all objects share Store a pointer in Surface class to appropriate Material

93 SHOWS N RFLTIONS

94 VNTGS OF RY TRRS One reall neat thing about ra tracers can add two additional effects ver easil: Shadows Ideal Specular Reflections These are generall more difficult with object-order rendering

95 SHOWS Once ou get our intersection point: Shoot ra shadow ra from intersection point to light Loop through all objects to see if ou hit something If es in shadow Otherwise NOT in shadow Basicall, ou are repeating what ou alread have to do to fill in each piel s color

96 SHOWS Two things to note: 1 You should add ambient light to the output piel color, irrespective of whether the point is in shadow When shooting shadow ra usuall start with t > e where e is some tin value Start at 0 might intersect with TH SM SURF ou re alread on! round-off error

97 IL SPULR RFLTIONS In the same wa, ideal specular reflections or mirror reflections are straightforward to add: ompute reflection vector R: Project onto N subtract twice that vector to get R heck to see what object ou hit with R and compute shading for that object assume it s a function called racolor Use racolor as part of output color: where k m = mirror reflection coefficient, since some surfaces will reflect some colors better than others.g., gold reflect ellow best R N N color c c kmracolor P sr

98 IL SPULR RFLTIONS Three things to watch out for: 1 Same issue with shadows don t start at t = 0 Need to include some kind of stopping criteria otherwise, keep bouncing around forever.g., limit number of times ou can recursivel call racolor 3 For efficienc don t call racolor if k m = 0 Not going to reflect anthing anwa