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Computer Graphics Inf4/MSc Computer Graphics Lecture 6 Rasterisation, Antialiasing, Texture Mapping, Computer Graphics Inf4/MSc Some Tutorial about the Project • I have alreadystcovered all the topics needed to finish the 1 practical • Today, I will briefly explain how to start working on it. • I have already provided you a program to import an obj file. 2 Computer Graphics Inf4/MSc Steps •Apply transformations to all vertices •Prepare the frame buffer and Z-buffer •For each triangle • Project it to the screen space • Find the 2D bounding box • For each pixel in the bounding box • Check if it is inside the triangle by computing its barycentric coordinates • If yes, use barycentric coordinates to compute the depth and colour at the pixel • If (depth < zbuf[pixel]) { framebuffer[pixel] = colour zbuf[pixel] = depth } •Export the frame buffer data into a PPM file 3 Computer Graphics Inf4/MSc • For each pixel in the bounding box • Check if it is inside the triangle by computing its barycentric coordinates • If yes, use barycentric coordinates to compute the depth and colour at the pixel z = αz + β z2 + γ z3 1 c = αc + β c2 + γ c3 1 • If (z < zbuf[pixel]) { framebuffer[pixel] = colour zbuf[pixel] = c } Computer Graphics Inf4/MSc Computing the baricentric coordinates of the interior pixels • The triangle is composed of 3 points p0 (x0,y0), p1 (x1, y1), p2(x2,y2) (α,β,γ) : barycentric coordinates Only if 0<α,β,γ<1, (x,y) is inside the triangle Depth can be computed by αZ0 + βZ1 +γZ2 Can do the same thing for color, normals, textures 5 Computer Graphics Inf4/MSc Today • Anti-aliasing • Texture mapping – Common texture coordinates mapping – Texture coordinates 6 Computer Graphics Rasterisation Inf4/MSc • Converts the vertex information output by the geometry pipeline into pixel information needed by the video display – Anti-aliasing – Z-buffer – Texture mapping – Bump mapping – Transparent objects – Drawing lines 7 Computer Graphics Anti-aliasing Inf4/MSc • Aliasing: distortion artifacts produced when representing a high-resolution signal at a lower resolution. • Anti-aliasing : techniques to remove aliasing Aliased polygons Anti-aliased polygons 8 (jagged edges) Computer Graphics Inf4/MSc Nyquist Limit • The signal frequency (fsignal) should be no greater than half the sample frequency (fsample) • fsignal <= 0.5 fsample • In the top, fsignal = 0.8 fsample -> cannot reconstruct the original signal • In the bottom fsignal =0.5 fsample -> the original signal can be reconstructed by slightly increasing the sampling rate Computer Graphics Inf4/MSc Screen-based Anti-aliasing Each pixel is subdivided (sub-sampled) into n regions, and each sub-pixel has a color; Compute the average color value n p x y) ic i, x y) (, w( , i1 (, color y) p x y):pixel at(x, ( , color c i, x y):sample w : weight i 10 Computer Graphics Inf4/MSc Accumulation Buffer (A-Buffer) • Use a buffer that has the same resolution as the original image • To obtain a 2x2 sampling of a scene, 4 images are made by shifting the buffer horizontally/vertically for half a pixel • The results are accumulated and the final results are obtained by averaging • Various sampling schemes are available Pixel center Subsampled point Different Sampling Schemes Computer Graphics Inf4/MSc Computer Graphics Inf4/MSc Computer Graphics Inf4/MSc Accumulation Buffer (A-Buffer) • The lighting computation is usually done only once per vertex • Not doing the lighting computation at each sample point • The A-buffer‟s focus is on the edge anti- aliasing • Also useful for rendering transparent objects, motion blur (will be covered later in the course) Edges Computer Graphics Inf4/MSc Stochastic Sampling • A scene can be produced of objects that are arbitrarily small • A regular pattern of sampling will always exhibit some sort of aliasing • One approach to solve this is to randomly sample over the pixel – Jittering : subdivide into n regions of equal size and randomly sample inside each region Computer Graphics Inf4/MSc The oversampling rate is 1 and 2 from left to right Computer Graphics Inf4/MSc Today • Anti-aliasing • Texture mapping – Common texture coordinates mapping – Texture coordinates 17 Computer Graphics Inf4/MSc Texture Mapping : Why needed? • We don't want to represent all this detail with geometry Computer Graphics Inf4/MSc Texture mapping. • Method of improving surface appearance by adding details on surface. 19 Computer Graphics Inf4/MSc Texture mapping. • Image is „pasted‟ onto a polygon. • Image is called a Texture map, it‟s pixels are often referred as a Texels and have coordinates (u,v) • Texture coordinates are defined for each vertex of the polygon and interpolated across the polygon. y v v u u x 20 Computer Graphics Inf4/MSc Photo-textures Computer Graphics Inf4/MSc Texture Interpolation • Specify a texture coordinate (u,v) at each vertex • Can we just linearly interpolate the values in screen space? (0,1) (0,0) (1,0) Computer Graphics Inf4/MSc Interpolating the uv coordinates Again, we use baricentric u1 v1 coordinates u3 v3 u= α u + β u2 + γ u3 1 v = α v + β v2 + γ v3 1 u2 v2 Computer Graphics Inf4/MSc Interpolation - What Goes Wrong? • Linear interpolation in screen space: texture source what we get| what we want Computer Graphics Inf4/MSc Why does it happen? • Uniform steps on the image plane does not correspond to uniform steps along the edge Computer Graphics Inf4/MSc How do we deal with it? • Use hyperbolic interpolation – (u,v) cannot be linearly interpolated, but 1/w and (u/w, v/w) can – w is the last component after the canonical view transformation 2n r l 0 0 pw x r l r l x w t y 0 2n b p 0 y tb t b z z pw f fn n 2 0 0 w f 1 n f n 0 0 0 1 Computer Graphics Inf4/MSc Texture Mapping Examples • Linear interpolation vs. Hyperbolic interpolation • Two triangles per square Computer Graphics Inf4/MSc Computing the uv coordinates at the internal points • For three points of the triangle, get 1u v i r [ i, , i , i ] p www i i i 1 u v [ , i, i] • Compute w w w i i i at the internal points of the triangle by interpolation • Compute wi by inverting 1/wi, and multiply them to (ui /wi, ,vi /wi) to compute (ui,vi) Computer Graphics Inf4/MSc Common Texture Coordinate Mappings • Orthogonal • Cylindrical • Spherical Computer Graphics Inf4/MSc Texture Mapping & Illumination • Texture mapping can be used to alter some or all of the constants in the illumination equation: – pixel color, diffuse color …. Phong’s Illumination Model Constant Diffuse Color Diffuse Texture Color Texture used as Label Texture used as Diffuse Color Computer Graphics Inf4/MSc Readings • Chapter 5.1-2 of Real-Time Rendering Second edition – http://books.google.co.uk/books?id=mOKEfBT w4x0C&printsec=frontcover&source=gbs_v2_ summary_r&cad=0#v=onepage&q=&f=false • “Hyperbolic Interpolation” IEEE Computer Graphics and Applications, vol12, no.4, 89- 94, 1992 • Demoed Software http://www-ui.is.s.u-tokyo.ac.jp/~takeo/java/smoothteddy/index.html