Final project report In addition to above, describe in detail the work you did, describe the results (experimental and/or theoretical), and discuss their relevance and applicability. The format and quality of the report should be similar to a short (6-10 pages) research paper. Study the style and conventions in the papers you read for the proposal.
Final report submission details
Example projects
Theory oriented projects include: 1. Early mathematical analysis predicted that a fixed wing would not have lift, and therefore manned flight would be unpractical. This is grounded in a macroscopic analysis using Navier-Stokes equations under what one thought at the time were reasonable assumptions. Yet ignoring these results the Wright brothers and others persisted and did fly. The apparent contradiction in the macroscopic mathematical analysis was formalized How can microscopic analysis of turbulence shed light on d'Alembert's paradox? Resources: Hoffman and Johnson "Computational Turbulent Incompressible Flow" 2. We are most commonly used to define equality as identical numerical values, i.e. x=y means the number x and y are the same. However, in important classes of mathematics, identity is defined up to a scale, ie two vectors x~y if lx=y, where l is an arbitrary scale factor. A common example of the above is the standard folding of Projective space P^n into R^n+1. Now consider how to adapt dimensionality reduction to this case. Ordinarily we can compute the best low rank subspace by PCA or SVD (X = USV^T). However under the "~" equivalency to a scale we like to find the smallest subspace basis (fewest rows in U and V) to best approximate LX = USV^T, where L is a diagonal matrix with the scale factor l_i for each equation (line) in the above equation. Consider how to modify standard SVD methods to compute the above. A trivial approach is an alternating fixed point iteration. Start with L=I the identity matrix, compute rank-reduced U,V, solve for L using the residual, repeat until convergence. However, a more interesting question is how to incorporate the above directy into the SVD computation based on either Givens rotations, or Householder transforms. See Ch3 and 4 in the textbook and also Golub and van Loan's book "Matrix computations" for tips. 2b Extension to above. Can you think of a similar method for equality w.r.t. SO^3 (Special Euclidean group in 3D -- the combination of translation and rotation). This problem is AFAIK unsolved. 3. Tri-linear tensor. 4. (For hon?) Optimization: Aligning 3D texture maps with geometry. Projects with a more substantial systems/programming aspect: 6. GP GPU 7. Optimizing numerical algorithms w.r.t. memory hierarchies (caches) Vision/graphics/robotics/games projects: 8. Dynamics simulation and rendering with phantom Robot Medical task 9. Video phone: track face and parameterize appearance in PCA/ICA or similar subspace. Consider using incremental PCA. 10. Path planning for animation: Plan constrained paths such as walking up stairs, reaching into drawer. 11. Physics simulation of rigid body interactions. Above subprojects + lab 3 can be combined into game