Lecture 6 Rasterisation, Antialiasing, Texture Mapping, Computer Graphics Lecture 6 Rasterisation, Antialiasing, Texture Mapping,

Some Tutorial about the Project I have already covered all the topics needed to finish the 1st practical Today, I will briefly explain how to start working on it. I have already provided you a program to import an obj file.

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 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 }

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 = αz1 + β z2 + γ z3 c = αc1 + β c2 + γ c3 If (z < zbuf[pixel]) { framebuffer[pixel] = colour zbuf[pixel] = c }

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

Today Anti-aliasing Texture mapping Common texture coordinates mapping

Rasterisation 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

Anti-aliasing Aliasing: distortion artifacts produced when representing a high-resolution signal at a lower resolution. Anti-aliasing : techniques to remove aliasing Aliased polygons (jagged edges)‏ Anti-aliased polygons

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

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

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

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

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

The oversampling rate is 1 and 2 from left to right

Today Anti-aliasing Texture mapping Common texture coordinates mapping

Texture Mapping : Why needed? We don't want to represent all this detail with geometry

Texture mapping. Method of improving surface appearance by adding details on surface.

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. v u y x

Photo-textures

Texture Interpolation Specify a texture coordinate (u,v) at each vertex Can we just linearly interpolate the values in screen space? (0,0)‏ (1,0)‏ (0,1)‏

Interpolating the uv coordinates Again, we use baricentric coordinates u= α u1 + β u2 + γ u3 v = α v1 + β v2 + γ v3 u1 v1 u3 v3 u2 v2

Interpolation - What Goes Wrong? Linear interpolation in screen space: texture source what we get| what we want

Why does it happen? Uniform steps on the image plane does not correspond to uniform steps along the edge

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

Texture Mapping Examples Linear interpolation vs. Hyperbolic interpolation Two triangles per square

Computing the uv coordinates at the internal points For three points of the triangle, get Compute 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)‏

Common Texture Coordinate Mappings Orthogonal Cylindrical Spherical

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 Diffuse Texture Color Constant Diffuse Color Texture used as Label Texture used as Diffuse Color

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