Surface shading: lights and rasterization. Computer Graphics CSE 167 Lecture 6

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1 Surface shading: lights and rasterization Computer Graphics CSE 167 Lecture 6

2 CSE 167: Computer Graphics Surface shading Materials Lights Rasterization 2

3 Scene data Rendering pipeline Modeling and viewing transformation Shading Projection Scan conversion, visibility Position object in 3D Determine colors of vertices Per vertex shading Map triangles to 2D Draw triangles Per pixel shading Image 3

4 Types of light sources At each point on surfaces we need to know Direction of incoming light (the L vector) Intensity of incoming light (the c l values) Three light types: Directional: from a specific direction Point light: from a specific point Spotlight: from a specific point with intensity that depends on direction Based on slides courtesy of Jurgen Schulze 4

5 Directional light source Light from a distant source Light rays are parallel Direction and intensity are the same everywhere As if the source were infinitely far away Good approximation of sunlight Specified by a unit length direction vector, and a color Light source Receiving surface 5

6 Point light source Similar to light bulbs An infinitesimally small point that radiates light equally in all directions Light vector varies across receiving surface Intensity drops off proportionally to the inverse square of the distance from the light Reason for inverse square falloff: Surface area of sphere A = 4 r 2 6

7 Point light source p c src Light source At any point v on the surface: c l v c l Receiving surface v Unitize Attenuation: 7

8 Point light source, attenuation Adding constant factor to denominator for better control Quadratic attenuation Most computationally expensive, most physically accurate Linear attenuation Less expensive, less accurate Constant attenuation Least expensive, least accurate 8

9 Spotlight source Like point light source, but intensity depends on direction Parameters Position p: location of light source Cone direction d: center axis of light source Intensity falloff: Beam width (cone angle ) The way the light tapers off at the edges of the beam (cosine exponent f) 9

10 Spotlight source Light source Unitize Receiving surface 10

11 Rasterization 11

12 The graphics pipeline The standard approach to object order graphics Many versions exist Software (e.g., Pixar s REYES architecture) Many options for quality and flexibility Hardware (e.g., graphics cards in PCs) Amazing performance: millions of triangles per frame We ll focus on an abstract version of hardware pipeline Called a pipeline because of the many stages Very parallelizable Leads to remarkable performance of graphics cards (many times the flops of the CPU at ~1/5 the clock speed) Based on slides courtesy of Steve Marschner 12

13 Pipeline overview You are here APPLICATION COMMAND STREAM 3D transformations, shading VERTEX PROCESSING TRANSFORMED GEOMETRY Conversion of primitives to pixels RASTERIZATION FRAGMENTS blending, compositing, shading FRAGMENT PROCESSING FRAMEBUFFER IMAGE user sees this DISPLAY 13

14 Primitives Points Line segments (and chains of connected line segments) Triangles And that s all! Curves Approximate them with chains of line segments Polygons Break them up into triangles Curved surfaces Approximate them with triangles Trend over the decades towards minimal primitives Simple, uniform, repetitive = good for parallelism 14

15 Rasterization First job: enumerate the pixels covered by a primitive Simple, aliased definition: pixels whose centers fall inside Second job: interpolate values across the primitive e.g., colors computed at vertices e.g., normals at vertices e.g., texture coordinates 15

16 Rasterizing lines Define line as a rectangle Specify by two endpoints Ideal image: black inside, white outside 16

17 Point sampling Approximate rectangle by drawing all pixels whose centers fall within the line Problem: sometimes turns on adjacent pixels 17

18 Point sampling in action 18

19 Bresenham lines (midpoint algorithm) Point sampling unit width rectangle leads to uneven line width Define line width parallel to pixel grid That is, turn on the single nearest pixel in each column 19

20 Midpoint algorithm in action 20

21 Algorithms for drawing lines Line equation: y = b + mx Simple algorithm: evaluate line equation per column Without loss of generality, x 0 < x 1 and 0 m 1 for x = ceil(x0) to floor(x1) y = b + m*x output(x, round(y)) y = x 21

22 Optimizing line drawing Multiplying and rounding is slow At each pixel the only options are E and NE d = m(x + 1) + b y d > 0.5 decides between E and NE 22

23 Optimizing line drawing d = m(x + 1) + b y Only need to update d for integer steps in x and y Do that with addition Known as digital differential analyzer (DDA) 23

24 Midpoint line algorithm x = ceil(x0) y = round(m*x + b) d = m*(x + 1) + b y while x < floor(x1) if d > 0.5 y += 1 d = 1 x += 1 d += m output(x, y) 24

25 Linear interpolation We often attach attributes to vertices For example, computed diffuse color of a hair being drawn using lines Desire for color to vary smoothly along a chain of line segments Linear interpolation in 1D f(x) = (1 α) y 0 + α y 1, where α = (x x 0 ) / (x 1 x 0 ) In the 2D case of a line segment, α is just the fraction of the distance from (x 0, y 0 )to (x 1, y 1 ) 25

26 Linear interpolation Pixels are not exactly on the line Define 2D function by projection on line This is linear in 2D, so DDA can be used to interpolate 26

27 Alternate interpretation We are updating d and α as we step from pixel to pixel d tells us how far from the line we are α tells us how far along the line we are So d and α are coordinates in a coordinate system oriented to the line 27

28 Alternate interpretation View loop as visiting all pixels the line passes through Interpolate d and α for each pixel Only output fragment if pixel is in band This makes linear interpolation the primary operation 28

29 Pixel walk line rasterization x = ceil(x0) y = round(m*x + b) d = m*x + b y while x < floor(x1) if d > 0.5 y += 1; d = 1; else x += 1; d += m; if 0.5 < d 0.5 output(x, y) 29

30 Rasterizing triangles The most common case in most applications With good antialiasing can be the only case, as some systems render a line as two skinny triangles Triangle represented by three vertices Simple way to think of algorithm follows the pixel walk interpretation of line rasterization Walk from pixel to pixel over (at least) the polygon s area Evaluate linear functions as you go Use those functions to decide which pixels are inside 30

31 Rasterizing triangles Input: Three 2D points (the triangle s vertices in pixel space) (x0, y0); (x1, y1); (x2, y2) Parameter values at each vertex q00,, q0n; q10,, q1n; q20,, q2n Output: A list of fragments, each with The integer pixel coordinates (x, y) Interpolated parameter values q0,, qn 31

32 Rasterizing triangles Summary 1.Evaluation of linear functions on pixel grid 2.Functions defined by parameter values at vertices 3.Use of extra parameters to determine fragment set 32

33 Incremental linear evaluation A linear (affine, really) function on the plane is: Linear functions are efficient to evaluate on a grid: 33

34 Incremental linear evaluation lineval(xm, xm, ym, ym, cx, cy, ck) { // setup qrow = cx*xm + cy*ym + ck; } // traversal for y = ym to ym { qpix = qrow; for x = xm to xm { output(x, y, qpix); qpix += cx; } qrow += cy; } c x =.005; c y =.005; c k = 0 (image size 100x100) 34

35 Rasterizing triangles Summary 1.Evaluation of linear functions on pixel grid 2.Functions defined by parameter values at vertices 3.Use of extra parameters to determine fragment set 35

36 Defining parameter functions To interpolate parameters across a triangle we need to find the cx, cy, and ck that define the (unique) linear function that matches the given values at all 3 vertices 3 constraints on 3 unknown coefficients In matrix form 36

37 Defining parameter functions More efficient version: shift origin to (x0, y0) Which is a 2x2 linear system (since q 0 falls out): Solve using Cramer s rule 37

38 Defining parameter functions lininterp(xm, xm, ym, ym, x0, y0, q0, x1, y1, q1, x2, y2, q2) { // setup det = (x1 x0)*(y2 y0) (x2 x0)*(y1 y0); cx = ((q1 q0)*(y2 y0) (q2 q0)*(y1 y0)) / det; cy = ((q2 q0)*(x1 x0) (q1 q0)*(x2 x0)) / det; qrow = cx*(xm x0) + cy*(ym y0) + q0; q = 0 q = 1 } // traversal (same as before) for y = ym to ym { qpix = qrow; for x = xm to xm { output(x, y, qpix); qpix += cx; } qrow += cy; } q = 0 38

39 Interpolating several parameters lininterp(xm, xm, ym, ym, n, x0, y0, q0[], x1, y1, q1[], x2, y2, q2[]) { // setup for k = 0 to n 1 // compute cx[k], cy[k], qrow[k] // from q0[k], q1[k], q2[k] } // traversal for y = ym to ym { for k = 1 to n, qpix[k] = qrow[k]; for x = xm to xm { output(x, y, qpix); for k = 1 to n, qpix[k] += cx[k]; } for k = 1 to n, qrow[k] += cy[k]; } 39

40 Rasterizing triangles Summary 1.Evaluation of linear functions on pixel grid 2.Functions defined by parameter values at vertices 3.Use of extra parameters to determine fragment set 40

41 Barycentric coordinates A coordinate system for triangles Algebraic viewpoint Geometric viewpoint Areas of subtriangles Triangle interior test [Shirley 2000] 41

42 Barycentric coordinates A coordinate system for triangles Geometric viewpoint Distances Linear viewpoint Basis of edges 42

43 Barycentric coordinates Linear viewpoint Basis for the plane [Shirley 2000] Triangle interior test 43

44 Clipping to the triangle Interpolate three barycentric coordinates across the plane Each barycentric coordinate is 1 at one vertex and 0 at the other two Output fragments only when all three are > 0 44

45 Pixel walk (Pineda) rasterization Conservatively visit a superset of the pixels you want Interpolate linear functions Use those functions to determine when to emit a fragment 45

46 Rasterizing triangles with shared edge Exercise caution with rounding and arbitrary decisions Need to visit these pixels once But it is important not to visit them twice! 46

47 Clipping Rasterizer tends to assume entire triangles are on screen After projection transformation (in normalized device coordinate frame) Clip against the planes X, Y, Z= 1, 1 6 planes defining canonical view volume Primitive operation Clip triangle against axis aligned plane 47

48 Clipping a triangle against a plane 4 cases, based on sidedness of vertices All inside (keep) All outside (discard) One inside, two outside (one clipped triangle) Two inside, one outside (two clipped triangles) 48

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