# Tagged: Riemannian Geometry

## The space of essential matrices as a Riemannian manifold

The images of 3-D points in two views are related by the so-called _essential matrix_.

There have been attempts to characterize the space of valid essential matrices as a Riemannian manifold. These approaches either put an unnatural emphasis on one of the two cameras, or do not accurately take into account the geometric meaning of the representation.

We addressed these limitations[^1] by proposing a new parametrization which aligns the global reference frame with the baseline between the two cameras. This provides a symmetric, geometrically meaningful representation which can be naturally derived as a quotient manifold. This not only provides a principled way to define distances between essential matrices, but it also sheds new light on older results (such as the well-known twisted pair ambiguity).

We provide an implementation of the basic function for working with the essential manifold integrated with the Matlab toolbox MANOPT. Download link: Manopt 1.06b with essential manifold.

## Distributed localization algorithms

Imagine a wireless camera network, where each camera has a piece of local information, e.g., the pose of the object from a specific viewpoint or the relative poses with respect to the neighboring cameras.

It is natural to look for distributed algorithms which merge all these local measurements into a single, globally consistent estimate. I derived such algorithms by formulating a global optimization problem over the space of poses, and shown their convergence from a large set of initial conditions using the aforementioned theoretical tools.

## Distributed optimization on Riemannian manifolds

I worked on distributed optimization problems involving variables lying on non-linear spaces (that is, Riemannian manifolds) using extensions of gradient descent algorithms with fixed step size. I developed novel theoretical tools which significantly broadened the state of the art for determining sufficient conditions for global behaviors (algorithm convergence) using only local information. These tools have been used in consensus algorithms, camera localization and formation control.