CS329 winter 2001 course webpage

Multilinear Systems and Invariant Theory in the Context of Computer Vision and Graphics

CS329

The course is designed to teach the basic ideas behind linear multi-dimensional mappings from high to lower dimensional spaces. These techniques are widely used in 3D modeling of rigid scenes in computer vision and graphics, some aspects of 3D visual recognition, and modeling of non-rigid point configurations applicable to modeling ``actions'' and ``behaviors''. The lectures cover three parts:

P^2 -> P^2: (3 lectures) Two dimensional alignment problems (image-to-image) of rigid and non-rigid configurations. This includes the homography matrix, infinitesimal direct estimation and homography tensor, as well as some introductory material on projective geometry and tensor notations.
P^3 -> P^2: (3 lectures) The reconstruction of three dimensions from multiple 2D projections. This includes the plane+parallax model, fundamental matrix, trifocal tensor, and shape tensors.
P^k -> P^2: (1 lecture) Non-rigid configurations re-casted as rigid configurations in higher dimensional spaces. We will cover only the case of P^4->P^2.

Prerequisites:

Minimal background of basic Linear Algebra is recommended (you need to know what a matrix is and some basic properties), all the rest would be covered in class. Students will be given 3 home assignments (theoretical problems) and will have the option to choose between a final exam or project.

Teaching Staff:

Principal Instructor: Amnon Shashua (shashua@cs.stanford.edu)
Send in your questions to shashua@cs.stanford.edu

The class will meet Mondays 9:00 - 11:00 in Gates 159.
The first class will be on Monday, January 14.

All announcements will be made through this class website. All students enrolling in the course are expected to sign up to the cs329@lists.stanford.edu mailing list by sending a mail to majordomo@lists.stanford.edu with a blank subject field and the body being subscribe cs329.

Course Content by Lecture (Syllabus)

Part I: P^2 -> P^2
- Class 1: Projection matrices, basics of projective geometry of the plane, the 2D collineation (homography matrix) and its properties (derivation, plane at infinity, the rank-4 constraint).
- Class 2: Tensor Product, covariant-contravariant tensor notations, Dynamic P^2 -> P^2 alignment (Homography Tensor). Further reading material: Shashua-Wolf:ECCV'00.
- Class 3: Infinitesimal motion model of the plane, the constant brightness equation and gradient-based estimation, direct estimation, factorization principle.
Part II: P^3 -> P^2
- Class 4: The plane+parallax model, fundamental matrix, constraints across 3 views (why F-mats are not sufficient?), the trifocal tensor, slicing properties.
- Class 5: Self Calibration: the basic equations, absolute conic, Kruppa's euqations, recovering internal parameters. Further reading material: Hartley-Zisserman pages 441--479.
- Class 6: Duality and Shape tensors. Further reading material: Levin-Shashua:ECCV02.
Part III: P^k -> P^2
- Non-rigid configurations re-casted as rigid configurations in higher dimensional spaces where k=4,5,6. We will cover in detail the geometry of P^4->P^2.