Computer Graphics (CMU 15-462/662)

This page contains lecture slides, videos, and recommended readings.

(Overview of 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)

(coverage testing as sampling a 2D signals, challenges of aliasing, performing point-in-triangle tests)

Supplementary material:

(basic math of spatial transformations and coordinate spaces)

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

Supplementary material:

(understanding perspective projection, texture mapping using the mip-map)

(occlusion via the depth buffer, alpha composition, the graphics pipeline and 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)

(distance queries, point-to-triangle, definition of a ray, ray-sphere intersection, ray-triangle intersection, triangle-triangle intersection)

(acceleration via bounding volume hierarchies and space partitioning structures, rasterization and ray casting as solutions to the same visibility query problem)

(tristimulus nature of color perception, color matching experiments, XYZ primaries, luminance vs. brightness, color spaces, gamma correction, tone mapping)

(radiometric quantities and units, photometry, radiometry integrals, how real cameras work)

(the rendering equation, the importance of indirect illumination, path tracing, splitting, Russian roulette)

Supplementary material:

(quadrature, sampling distributions, basic Monte Carlo integration)

(ray tracing vs. rasterization, local vs. global illumination, Monte Carlo integration, expected value, variance, law of large numbers, importance sampling, direct lighting estimate, cosine weighting, path tracing, 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)

(history of (computer) animation, splines, natural splines, cubic Hermite/Bezier, B-splines, interpolation, keyframing, rigging, skeletal animation, inverse kinematics, blend shapes)

(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, continuous vs. discrete optimization, standard form of an optimization problem, local vs. global minima, existence and uniqueness of solutions, convex optimization)

(Hermite splines, Catmull-Rom Splines, forward and inverse kinematics, computing the Jacobian, linear blend skinning, simulating the wave equation)

(descent methods, gradient descent, Newton descent, kinematic chains, inverse kinematics, 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)