Introduction Rasterization Z-buffering Shading. Graphics 2012/2013, 4th quarter. Lecture 09: graphics pipeline (rasterization and shading)

Transcription

1 Lecture 9 Graphics pipeline (rasterization and shading)

2 Graphics pipeline - part 1 (recap) Perspective projection by matrix multiplication: x pixel y pixel z canonical 1 x = M vpm per M cam y z 1 This is an important part of the graphics pipeline.

3 Graphics pipeline - part 2 (recap) Triangles that lie (partly) outside the view frustum need not be projected, and are clipped. The remaining triangles are projected if they are front facing. Projected triangles have to be shaded and/or textured.

4 Rasterizing triangles Introduction Scanline conversion Incremental updates Edge table and active edge table We know how to project the vertices of a triangle in our model onto pixel centers. To draw the complete triangle, we have to decide which pixels to turn on. For now, let s assume that all pixels of the triangle get the same color.

5 Scanline conversion Incremental updates Edge table and active edge table Limiting the number of pixels to consider Instead of testing for all pixels in the image if they belong to the triangle, our book suggests to test only pixels in the bounding box of the triangle. The bounding box of a polygone can be specified by two points b 0 and b 1 with b0 = (min{x i }, min{y j }) b1 = (max{x i }, max{y j }) for all i, j = 1,, n (n = number of vertices)

6 Scanline conversion Incremental updates Edge table and active edge table Limiting the number of pixels to consider However, for long and skinny triangles that are diagnonally oriented, considering all pixels in their bounding box may still be quite inefficient.

7 Rasterizing general polygons Scanline conversion Incremental updates Edge table and active edge table Let s look at a more efficient way to draw triangles. The method actually works for general polygons; for triangles, it becomes even more efficient.

8 Scanline conversion Incremental updates Edge table and active edge table From pixel centers to pixel coordinates First, let s move our attention from pixels to the underlying coordinate system where the pixel centers have integer coordinates.

9 Scanline conversion Incremental updates Edge table and active edge table From pixel centers to pixel coordinates First, let s move our attention from pixels to the underlying coordinate system where the pixel centers have integer coordinates.

10 Scanline conversion Incremental updates Edge table and active edge table From pixel centers to pixel coordinates First, let s move our attention from pixels to the underlying coordinate system where the pixel centers have integer coordinates.

11 Scanline conversion Introduction Scanline conversion Incremental updates Edge table and active edge table Idea of the scanline algorithm: Instead of looking at each pixel individually, we draw our polygon one scanline at a time, from bottom to top, thus taking advantage of scanline coherence.

12 Scanline conversion Introduction Scanline conversion Incremental updates Edge table and active edge table Scanline 3 intersects two edges; these are called active edges. If we know the intersections of the active edges with the current scanline, then we know which pixels to set.

13 Computing intersections Introduction Scanline conversion Incremental updates Edge table and active edge table The left active edge runs from (1, 1) to (3, 8). The slope intercept equation, i.e. y = yj yi x j x i x + d with j > i for the line through these points is y = 3.5x 2.5 The x-coordinate of the intersection of this edge with scanline 3 is 1 4 7, so we know that we have to set pixels starting from x = 2.

14 Computing intersections Introduction Scanline conversion Incremental updates Edge table and active edge table The left active edge runs from (1, 1) to (3, 8). The slope intercept equation, i.e. y = yj yi x j x i x + d with j > i for the line through these points is y = 3.5x 2.5 The x-coordinate of the intersection of this edge with scanline 3 is 1 4 7, so we know that we have to set pixels starting from x = 2.

15 Scanline conversion Incremental updates Edge table and active edge table Computing intersections incrementally Computing intersections of scanlines and edges requires a division. These are expensive, and we d like to avoid them. We use the fact that the intersection of an edge with scanline i is related to the intersection with scanline i 1. This is called vertical coherence.

16 Scanline conversion Incremental updates Edge table and active edge table Computing intersections incrementally The slope of our left active edge is y j y i x j x i = y x = 7 2 (with j < i) Hence, the increase in x-coordinate from one scanline to the next is x = 2 7

17 Scanline conversion Incremental updates Edge table and active edge table Computing intersections incrementally Example: The x-coordinate at scanline 1 is obviously 1. Hence, we have: scanline x-coordinate (= )

18 Data structures: edge table (ET) Scanline conversion Incremental updates Edge table and active edge table In the Edge Table (ET), we maintain for every scanline a list of edges that start at the scanline. Each edge record contains: The x-coordinate of the lowest vertex. The y-coordinate of the highest vertex The value of x for the edge.

19 Scanline conversion Incremental updates Edge table and active edge table Data structures: active edge table (AET) When we proceed from one scanline to the next, we also maintain an Active Edge Table (AET). This table contains the edge record from the ET that are intersected by the current scanline. Whenever we move to the next scanline, we add the x -values of the active edges to the x-value.

20 Scanline conversion Incremental updates Edge table and active edge table Data structures: active edge table (AET) AET: scanline: edge records:

21 Active edge table Introduction Scanline conversion Incremental updates Edge table and active edge table An edge record is removed from the AET when the current scanline reaches the top of the edge. Recall that the 2nd entry of the edge record stores the y-coordinate of the top vertex.

22 Scanline conversion Introduction Scanline conversion Incremental updates Edge table and active edge table Summary of the whole algorithm: AET = for i = 0 to n 1 do update(aet,i) append(aet,et[i]) sort(aet) JoinLines(AET,i) for each scanline delete edge records for which y == i add x to x-values add edges starting here sort by x-value set pixels between pairs of edges

23 Active edge table Introduction Scanline conversion Incremental updates Edge table and active edge table AET: 1: (1, 8, 2 ), (1, 2, 8) 7 2: (1 2, 8, 2 1 ), (9, 2, 8), (9, 6, ) : (1 4, 8, 2 ), (8 1 1, 6, ) : (1 6, 8, 2 1 ), (4, 8, ), (4, 6, ), (8, 6, ) : (2 1, 8, 2 ), (3 3 1, 8, ), (5 1, 6, 1 1 ), (7 1 1, 6, : (2 3, 8, 2 ), ( : (2 5, 8, 2 ), (3 1 1, 8, : (3, 8, 2 1 ), (3, 8, ) 7 4, 8, 1 4 ), (7, 6, ) ), (7, 6, 1 2 ) 2 ) ET: 1: (1, 8, 2/7), (1, 2, 8) 2:(9, 6, 1/2) 4:(4, 8, 1/4), (4, 6, 1.5)

24 Interpolating z-values with scanline conversion Recall that hidden surface elimination can be done efficiently with z-buffer algorithm. But how do we get the z-values for each pixel?

25 Interpolating z-values with scanline conversion Again, we can compute these efficiently with bilinear interpolation, maintaining a z value for every edge, and computing values for the z-increment along every scanline. Instead of y 1 y 0 x 1 x 0 = y x = m x we use the slope y 1 y 0 z 1 z 0 = y z = m z and for y = 1, we get z = 1/m z.

26 Further comments on Interpolating z-values Disadvantages: Large memory requirements Individual pixels might get painted several times (can be critical if computing the color of a pixel is expensive) Do not support transparency

27 Bilinear interpolation to color triangles With this we can linearly interpolate color 1 between two vertices 2 between two edges Q: How to do this efficiently? What about phong shading? We will learn this in a later lecture

28 Bilinear interpolation to color triangles With this we can linearly interpolate color 1 between two vertices 2 between two edges Q: How to do this efficiently? What about phong shading? We will learn this in a later lecture

29 For every scanline, color-values at the edges of the triangle are computed by linearly interpolating the color values at the vertices. This can be done incrementaly, by precomputing r, g, and b values, and adding these to the r, g and b values of the previous scanline.

30 Colors along each scanline are also determined by linear interpolation. Again, this is done incrementally, by precomputing values. For each triangle, we do four devisions for each of its edges, to compute the x, r, g, and b values, and three divisions for every scanline it spans, to compute the color increments in horizontal direction.

31 Gouraud shading produces diffuse lighting that we usually see on matte surfaces. We had diffuse reflection before, remember? We called it Lambertian shading. Notice that there are different notations used in the book chapters 4.5 and 10. Here we use the latter.

32 Lambertian shading Introduction The direction of the incident light influences the intensity of the reflected light. Lambert s cosine law: c cos θ l or c n l with c being the color at this particular point Remember: a b = cos θ a b

33 Lambertian shading Introduction Notice that we have to do this calculations in world coordinates (because of the normals). Remember that the light source is only described by the vector l, i.e. by its direction and not its location. We call such light sources directional light (in contrast to, e.g., a spotlight that is defined by direction and location).

34 Lambertian shading Lambertian shading: Introduction with c = c r c l max(0, n l) c = the perceived color value c r = diffuse reflectance (fraction of light reflected by surface) c l = RGB intensity term (intensity of light source) Note: this is the notation from chapter 10 in the book, otherwise it s the same as in lecture 3 / chapter 4)

35 Lambertian shading Introduction Lambertian shading: c = c r c l max(0, n l) Characteristics: Proportional to cos θ l, hence max. for l = n can get negative

36 But how do we compute diffuse lighting at the vertices of the triangles/surfaces? It depends. We may specify that normals are indeed perpendicular to the surfaces. We may also make surfaces have their vertices shared, and somehow interpolate vertex normals.

37 But how do we compute diffuse lighting at the vertices of the triangles/surfaces? It depends. We may specify that normals are indeed perpendicular to the surfaces. We may also make surfaces have their vertices shared, and somehow interpolate vertex normals.

38 Considering ambient light Problem: points with normals facing away from the light will be black. We need some global illumination. Some heuristics to do this: Put dim light source at eye position or Add a constant color term to the color of each object (simulating a global light ) We already know that the latter is called ambient shading: c = c r (c a + c l max(0, n l))

39 Glossy reflection Introduction Diffuse reflection vs. specular reflection (source: shading)

40 Glossy reflection Introduction For glossy reflection, the viewing direction matters. Reflection is maximized when the angle of the reflected light equals the angle of the incident light, i.e., when e = r with r being the reflection vector. The scalar product (= cosine) offers a good heuristic that gradually decreases brightness when moving away from this maximum: c = c l ( e r)

41 Glossy reflection: Phong shading Using the Phone exponent p: c = c l max(0, e r) p we can influence size of our reflectance spot. To deal with negative values of the scalar product, we use max(0, e r) (cf. textbook, fig. 10.6, page 237 (9.6, p. 195 in 2nd ed.))

42 Glossy reflection: Phong shading Using the Phone exponent p: c = c l max(0, e r) p we can influence size of our reflectance spot. To deal with negative values of the scalar product, we use max(0, e r)

43 Glossy reflection: Phong shading Using the Phone exponent p: c = c l max(0, e r) p we can influence size of our reflectance spot. To deal with negative values of the scalar product, we use max(0, e r)

44 Glossy reflection Introduction In the Blinn-Phong model, we use the vector h that lies halfway between e and l: with c = c l ( n h) p h = e + l / e + l (note: e, l must be unit vectors) c l = RGB intensity term (intensity of light source) p = Phong exponent Note: again we are using the notation of chapter 10 here

45 Glossy reflection Introduction In the Blinn-Phong model, we use the vector h that lies halfway between e and l: with c = c l ( n h) p h = e + l / e + l (note: e, l must be unit vectors) c l = RGB intensity term (intensity of light source) p = Phong exponent Note: again we are using the notation of chapter 10 here

46 Phong normal interpolation Interpolating color values doesn t combine well with the Phong model: highlights in the interior of triangles are completely lost. However, we can interpolate normals instead, and do the computations of the Phong model using the interpolated normals. We can use barycentric coordinates for that: n = α n 0 + β n 1 + γ n 2

47 Phong normal interpolation Notice that we use a slightly different notation of barycentric coordinates here than before, i.e.: n(α, β, γ) = α n 0 + β n 1 + γ n 2 whereas in relation to triangles, we used: n(β, γ) = n 0 + β( n 1 n 0 ) + γ( n 2 n 0 ) which is the same if we set: α = 1 β γ

48 Phong normal interpolation Barycentric coordinates: p(β, γ) = a + β( b a) + γ( c a) We can rewrite this as p(β, γ) = a + β b β a + γ c γ a = (1 β γ) a + β b + γ c Which is the same as p(α, β, γ) = α a + β b + γ c if α = (1 β γ). The conditions for point-triangle intersection for this notation is 0 < α, β, γ < 1

49 Phong normal interpolation Barycentric coordinates: p(β, γ) = a + β( b a) + γ( c a) We can rewrite this as p(β, γ) = a + β b β a + γ c γ a = (1 β γ) a + β b + γ c Which is the same as p(α, β, γ) = α a + β b + γ c if α = (1 β γ). The conditions for point-triangle intersection for this notation is 0 < α, β, γ < 1

50 Bringing it all together... Combining ambient, Lambertian, and Blinn-Phong shading we get c = c r (c a + c l max(0, n l)) + c l ( h n) p Variations exist, e.g. we can add a control term c p that allows us to dim the highlight of the Phong shading: c = c r (c a + c l max(0, n l)) + c l c p ( h n) p Or we can add multiple light sources (cf. lecture 3).

51 Course overview Introduction Various other approaches exist for light calculations (e.g., Cook-Torrance BRDF, spotlights,... ). First part of the course: Introduction Vectors and curves Curves, surfaces, and shading Matrices and determinants Linear and affine transformations Texture mapping Second part: Pipeline 1: Perspective projection Pipeline 2: Clipping, culling, and z-buffer Pipeline 3: and shading Radiosity and shadows Ray tracing (basics) Ray tracing (advanced)

52 Conclusion Introduction Notice that we already covered almost all of the basic stuff i.e. chapters 1-11 of the book: We will not cover chapter 9, signal processing (2nd edition: chapter 4) We will not look into chapter 3: raster images (2nd edition: chapter 3, raster algorithms) We will however look into Chapter 4, ray tracing and 13, more ray tracing (2nd edition: chapter 10, ray tracing) And next time, we ll cover two topics not in the textbook, i.e. Radiosity (global illuminations) Shadows

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