Lectures and Readings : Computer Graphics : 15-462/662 Fall 2017 (original) (raw)

(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)

(basic math of spatial transformations and coordinate spaces)

Further Reading:

• Real Time Rendering -- Chapter 4. by T. Akenine Moller, E. Haines, N. Hoffman
• 3D Rotations (exerpt from Ch. 15 of Advanced Animation and Rendering Techniques. by A. Watt, M. Watt)

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

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

(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)

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

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

(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)

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

(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, numerical differentiation, automatic differentiation, symbolic differentiation)

Further Reading:

• Physically Based Modeling: Principles and Practice (Differential Equation Basics) by Andrew Witkin and David Baraff.
• Physically Based Modeling: Principles and Practice (Particle System Dynamics) by Andrew Witkin.
• Geometric Numerical Integration by Ernst Hairer, Christian Lubich, and Gerhard Wanner.
• Structure and Interpretation of Classical Mechanics by Gerald Jay Sussman and Jack Wisdom.

(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, 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)