Computer Graphics (CMU 15-462/662)

This page contains lecture slides and recommended additional readings for the Fall 2016 offering of 15-462/662.

(a bit of history on graphics + making a line drawing of a cube!)

(Vectors, vector spaces, linear maps, inner product, norm, L2 inner product, span, basis, orthonormal basis, Gram-Schmidt, frequency decomposition, systems of linear equations, bilinear and quadratic forms, matrices)

(Euclidean inner product, cross product, matrix representations, determinant, triple product formulas, differential operators, directional derivative, gradient, differentiating matrices, differentiating functions, divergence, curl, Laplacian, Hessian, (multivariable) Taylor series)

(drawing triangles, mathematical representation of sampling and reconstruction, spatial and frequency domain signals, aliasing, anti-aliasing, dinosaurs)

(linear maps, Cartesian space, homogenous coordinates, transforms, coordinate frames)

(3D rotations, commutativity of rotations, 2D rotation matrix, Euler angles, rotation from axis/angle, complex numbers, quaternions, quaternion rotation)

(view frustum, perspective transformations, teture mapping, mipmaps, attribute interpolation)

(occlusion, depth buffer, alpha compositing, graphics pipeline, modern GPUs)

(implicit and explicit representations, geometric data structures)

(smooth surfaces, manifold condition, manifold polygon mesh, surfaces with boundary, polygon soup, incidence matrices, halfedge data structure, local mesh operations, subdivision modeling)

(geometry processing pipeline, surface reconstruction, upsampling, downsampling, resampling, filtering, compression, shape analysis, remeshing, mesh quality, subdivision, Catmull-Clark scheme, Loop scheme, iterative edge collapse, quadric error metric, minimizing a quadratic form, Delaunay flipping, Laplacian smoothing, isotropic remeshing, signal degradation, geometric queries, closest point, ray-element intersection, element-element intersection)

(bounding volume hierarchies, space partitioning structures, geometric queries, closest point, ray-element intersection, element-element intersection)

(ray casting, ray tracing, rasterization-based global illumination, packet-tracing)

(numerical integration, quadrature, probability theory, monte carlo methods)

(light as EM radiation, black body radiation, emission and absoprtion spectra, light perception, rods and cones, metamers, color spaces and color models, additive and subtractive color models, video color compression, color spaces, gamut, chromaticity, gamma, acommodation effect)

(light in equilibrium, radiant energy, radiant flux, irradiance, radiometric units, spectral power distribution, Lambert's law, N-dot-L lighting, light models, directional lights, point lights, solid angle, area lights, dome light, ambient occlusion, screen-space ambient occlusion, goniometry, photometry, photometric units)

(review of radiance, the rendering equation, recursive ray tracing, radiance along a ray, reflection models, scattering phenomena, bidirectional reflectance distribution function (BRDF), Lambertian, specular, Snell's law, total internal reflection, anisotropic scattering, subsurface scattering, BSSRDF, the reflection equation, reivew of Monte carlo, path tracing, direct lighting, indirect lighting, Russian roulette)

(Monte Carlo integration, expected value, variance, continuous random variables, variance reduction, bias and consistency, path space formulation of light transport, importance sampling, bidirectional path tracing, Metropolis-Hastings algorithm, multiple importance sampling, sampling patterns, stratified sampling, low-discrepancy sampling, quasi Monte Carlo, Hammersley and Halton sequences, blue noise, Poisson disk sampling, Lloyd relaxation, alias table, photon mapping, finite element radiosity)

Further Reading:

- "Monte Carlo Methods for Light Transport Simulation" the Academy Award-winning PhD thesis by Eric Veach
- "Stratified Sampling of Spherical Triangles" by Jim Arvo
- "Computing the Discrepancy with Applications to Supersampling Patterns" by David Dobkin, David Eppstein, and Don P. Mitchell
- "Generating Antialiased Images at Low Sampling Densities" by Don P. Mitchell

(computer animation, keyframing, motion capture, procedural animation, splines, natural splines, cubic Hermite/Bezier/Catmull-Rom, B-splines, interpolation)

(computer animation, keyframing, motion capture, blend shapes, cage-based deformation, skeletal animation, forward kinematics, inverse kinematics, skinning)

(continuous vs. discrete optimization, standard form of an optimization problem, local vs. global minima, existence and uniqueness of solutions, convex optimization, descent methods, gradient descent, Newton descent, inverse kinematics)

(physically-based animation, Newton's 2nd law of motion, generalized coordinates, ordinary differential equations (ODE), Lagrangian mechanics, Euler-Lagrange equations, pendulum/double pendulum, n-body systems, mass-spring systems, particle systems, flocking, crowds, particle-based fluids, granular materials, molecular dynamics, hair simulation, numerical integration, forward/backward/symplectic Euler, stability analysis, numerical differentiation, automatic differentiation, symbolic differentiation)

((PDEs in computer graphics, defintion of a PDE, order and linearity, model equations (elliptic/parabolic/hyperbolic), Laplace equation, heat equation, wave equation, numerical solution of PDEs, Lagrangian and Eulerian discretization, the Laplace operator, discrete Laplacian, Dirichlet and Neumann boundary conditions, Jacobi method)

(Rigid Body Dynamics, forces and torques, mass and moment of inertia, linear and angular momenta, contacts and friction, articulated rigid body dynamics)