**HARMONIC COMPUTATIONAL GEOMETRY:**

**The geometry of visual space-time**

Yiannis Aloimonos

Computer Vision Lab

**Abstracts:**

The process of recovering 3D models of the world
from images (views, cameras) is central to many of the problems relevant to
this Workshop. The state-of-the-art theory for solving this problem proceeds by
first finding features (points and lines) in the images and then matching them;
using the matches, the camera geometry is obtained and after the cameras are "placed",
a model of the scene emerges. The current theory appears to have reached its
limitations. This workshop is indirect evidence for that. Direct evidence comes
from the quality of the reconstructions (too many distortions, holes and the
need for manual intervention). Although we have a well developed theory for
matched points and lines in different views, we do not have the insight into
how to generate the input to the process (i.e. correspondence).

Looking at the state of the art, something
doesn't seem quite right. Geometry is totally separated from Statistics (Signal
Processing). Using points and lines a 3d mesh is developed and then the image
texture is mapped onto the mesh. This is pretty artificial. Texture (image
signal) contains a lot of information and the key to the correspondence
problem. I this talk I will explain in mathematical terms why points and lines
cannot be found very accurately in images. In so doing, I will explain a large
number of visual illusions. Then, I will introduce Harmonic Computational
Geometry, a new framework for 3D recovery from multiple views using the outputs
of filters applied to image patches. I will show new geometric constraints,
such as the harmonic epipolar and harmonic trilinear. I will also briefly discuss the relevance of the
new framework and results to the problem of building action descriptions.

This is joint work with Cornelia Fermuller and Patrick Baker

**Brief
Biography:**

Yiannis Aloimonos
studied Mathematics in