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CS 551 / 645: Introductory Computer Graphics
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  1. CS 551 / 645: Introductory Computer Graphics David Luebke cs551@cs.virginia.edu http://www.cs.virginia.edu/~cs551 David Luebke 6/5/2014

  2. Administrivia • Assignment 6 notes • Viewport <= framebuffer (don’t stretch pixels) • Drawing lines: round to integer pixel values • Perspective projection: don’t need to preserve Z • Any other questions? David Luebke 6/5/2014

  3. Lighting: Transforming Normals • Irritatingly, the matrix for transforming a normal vector is not the same as the matrix for the corresponding transformation on points • In other words, don’t just treat normals as points: David Luebke 6/5/2014

  4. Lighting: Transforming Normals • What is homogeneous representation of a vector (as opposed to a point?) • Some not-too-complicated affine analysis shows : • If A is a matrix for transforming points,then (AT)-1 is the matrix for transforming normals • When is this the same matrix? • Can use this to simplify the problem: only upper 3x3 matrix matters, so use only it • More detail in F&vD Appendix A.5 David Luebke 6/5/2014

  5. Texture Mapping: Motivation • Scenes created with diffuse lighting look convincingly three-dimensional, but are flat, chalky, and “cartoonish” • Phong lighting lets us simulate materials like plastic and (to a lesser extent) metal, but scenes still seem very cartoonish and unreal • Big problem: polygons are too coarse-grained to usefully model fine surface detail • Solution: texture mapping David Luebke 6/5/2014

  6. Texture Mapping: Motivation • Adding surface detail helps keep CG images from looking simple and sterile • Explicitly modeling this detail in geometry can be very expensive • Zebra stripes, wood grain, writing on a whiteboard • Texture mapping pastes images onto the surfaces in the scene, adding realistic fine detail without exploding the geometry David Luebke 6/5/2014

  7. Texture Mapping: Examples David Luebke 6/5/2014

  8. Texture Mapping: Fundamentals • A texture is typically a 2-D image • Image elements are called texels • Value stored at a texel affects surface appearance in some way • Example: diffuse reflectance, shininess, transparency… • The mapping of the texture to the surface determines the correspondence, i.e., how the texture lies on the surface • Mapping a texture to a triangle is easy (why?) • Mapping a texture to an arbitrary 3-D shape is more complicated (why?) David Luebke 6/5/2014

  9. Texture Mapping: Rendering • Rendering uses the mapping: • Find the visible surface at a pixel • Find the point on that surface corresponding to that pixel • Find the point in the texture corresponding to that point on the surface • Use the parameters associated with that point on the texture to shade the pixel David Luebke 6/5/2014

  10. Texture Mapping: Basics • We typically parameterize the texture as a function in (u, v) • For simplicity, normalize u & v to [0, 1] • Associate each triangle with a texture • Give each vertex of the triangle a texture coordinate (u, v) • For other points on the triangle, interpolate texture coordinate from the vertices • Much like interpolating color or depth • But there’s a catch... David Luebke 6/5/2014

  11. Naïve Texture Mapping • A first cut at a texture-mapping rasterizer: • For each pixel: • Interpolate u & v down edges and across spans • Look up nearest texel in texture map • Color pixel according to texel color (possibly modulated by lighting calculations) • McMillan’s demo of this is at http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture21/Slide05.html • What artifacts do you see in this demo? David Luebke 6/5/2014

  12. Naïve Texturing Artifacts • Probably the most obvious artifact is the blocky pixelated look of the texture • Basic problem: using a single texel to color each pixel • If the pixel is larger than a texel, we should average the contribution from multiple texles somehow • If the pixel is smaller than a texel, we should interpolate between texel values somehow • Even if pixel size  texel size, a pixel will in general fall between four texels • An example of a general problem called aliasing • More on this later… David Luebke 6/5/2014

  13. Naïve Texturing Artifacts • Another serious artifact is warping at the edges of triangles making up the mesh • A more obvious example:http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture21/Slide06.html • To address this, need to consider the geometry of interpolating parameters more carefully David Luebke 6/5/2014

  14. Interpolating Parameters • The problem turns out to be fundamental to interpolating parameters in screen-space • Uniform steps in screen space  uniform steps in world coords David Luebke 6/5/2014

  15. Interpolating Parameters • Perspective foreshortening is not getting applied to our interpolated parameters • Parameters should be compressed with distance • Linearly interpolating them in screen-space doesn’t do this • Is this a problem with Gouraud shading? • A: It can be, but we usually don’t notice (why?) • http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture21/Slide17.html David Luebke 6/5/2014

  16. Perspective-Correct Interpolation • Skipping a bit of math to make a long story short… • Rather than interpolating u and v directly, interpolate u/z and v/z • These do interpolate correctly in screen space • Also need to interpolate z and multiply per-pixel • Problem: we don’t know z anymore • Solution: we do know w  1/z • So…interpolate uw and vw and w, and compute u = uw/w and v = vw/w for each pixel • This unfortunately involves a divide per pixel (Just 1?) David Luebke 6/5/2014

  17. Perspective-Correct Texturing • Known as perspective-correct texture mapping • Some APIs and game consoles don’t support it • So how can they avoid the warping problem? • As mentioned, other interpolation schemes really ought to use perspective correction • E.g., Gouraud shading • Generally get away without it because it is more important to be smooth than correct • Java code fragment from McMillan’s edge-equation triangle rasterizer: David Luebke 6/5/2014

  18. Perspective-Correct Texturing: Code Fragment ... PlaneEqn(uPlane, (u0*w0), (u1*w1), (u2*w2)); PlaneEqn(vPlane, (v0*w0), (v1*w1), (v2*w2)); PlaneEqn(wPlane, w0, w1, w2); ... for (y = yMin; y <= yMax; y += raster.width) { e0 = t0; e1 = t1; e2 = t2; u = tu; v = tv; w = tw; z = tz; boolean beenInside = false; for (x = xMin; x <= xMax; x++) { if ((e0 >= 0) && (e1 >= 0) && (e2 >= 0))) { int iz = (int) z; if (iz <= raster.zbuff[y+x]) { float denom = 1.0f / w; int uval = (int) (u * denom + 0.5f); uval = tile(uval, texture.width); int vval = (int) (v * denom + 0.5f); vval = tile(vval, texture.height); int pix = texture.getPixel(uval, vval); if ((pix & 0xff000000) != 0) { raster.pixel[y+x] = pix; raster.zbuff[y+x] = iz; } } beenInside = true; } else if (beenInside) break; e0 += A0; e1 += A1; e2 += A2; z += Az; u += Au; v += Av; w += Aw; } t0 += B0; t1 += B1; t2 += B2; tz += Bz; tu += Bu; tv += Bv; tw += Bw; }

  19. Texture Tiling • It is often handy to tile a repeating texture pattern onto a surface • The previous code does this via tile(): int uval = (int) (u * denom + 0.5f); uval = tile(uval, texture.width); int vval = (int) (v * denom + 0.5f); vval = tile(vval, texture.height); int pix = texture.getPixel(uval, vval); int tile(int val, int size) { while (val >= size) val -= size; while (val < 0) val += size; } See http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture21/Slide18.html David Luebke 6/5/2014

  20. Texture Transparency • McMillan’s code also includes a “quick fix” for handling transparent texture: if ((pix & 0xff000000) != 0) { raster.pixel[y+x] = pix; raster.zbuff[y+x] = iz; } • Note that this doesn’t handle partial transparency (How might such partial transparency arise?) • Demo at: http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture21/Slide19.html David Luebke 6/5/2014