Geometry-Driven Diffusion in Computer Vision. 'Differential Invariant Signatures and Flows in Computer Vision: A Symmetry Group Approach'.

^ Olver, Peter Sapiro, Guillermo Tannenbaum, Allen (1994).Flow characteristics using computer vision.Noether's theorem implies the existence of differential invariants corresponding to every differentiable symmetry of a variational problem.Differential invariants can be applied to the study of systems of partial differential equations: seeking similarity solutions that are invariant under the action of a particular group can reduce the dimension of the problem (i.e.A differential invariant is a function on Y ( k) that is invariant under the prolongation of the group action. The group G acts, locally, on the space of such graphs, and induces an action on the k-th prolongation Y ( k) consisting of graphs passing through each point modulo the relation of k-th order contact. The graph of a mapping X → Y is a submanifold of X× Y that is everywhere transverse to the fibers over X. More generally, differential invariants can be considered for mappings from any smooth manifold X into another smooth manifold Y for a Lie group acting on the Cartesian product X× Y.

This method of computing the prolongation is impractical, however, and it is much simpler to work infinitesimally at the level of Lie algebras and the Lie derivative along the G action. Similar considerations apply for the computation of higher prolongations. Roughly speaking, a k-th order differential invariant is a function

Then G also acts, locally, on the space of all graphs of the form y = ƒ( x). The simplest case is for differential invariants for one independent variable x and one dependent variable y. Élie Cartan's method of moving frames is a refinement that, while less general than Lie's methods of differential invariants, always yields invariants of the geometrical kind. Whereas differential invariants can involve a distinguished choice of independent variables (or a parameterization), geometric invariants do not. Lie (1884) was the first general work on differential invariants, and established the relationship between differential invariants, invariant differential equations, and invariant differential operators.ĭifferential invariants are contrasted with geometric invariants. Differential invariants were introduced in special cases by Sophus Lie in the early 1880s and studied by Georges Henri Halphen at the same time. Differential invariants are fundamental in projective differential geometry, and the curvature is often studied from this point of view. In mathematics, a differential invariant is an invariant for the action of a Lie group on a space that involves the derivatives of graphs of functions in the space.

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