## 1. Reconstruction

A fundamental topic in algebraic vision is that of reconstructing a scene from images.-
The essential variety is a 5-dimensional variety of degree 10. A point correspondence gives a linear constraint so a five point correspondence should give 10 complex solutions.
#### Problem 1.05.

Of the ten possibilities, how many of the solutions can be real? -
#### Problem 1.1.

How many linear constraints does a point correspondence in three views determine on the trifocal tensor variety ? -
It is known that the trifocal tensor variety is dimension 18 with degree 297 inside of $\mathbb{P}^{28}$. It is also known that a six point correspondence in three views is a minimal problem.
#### Problem 1.15.

How do the 24 linear constraints lead a minimal problem? How many complex solutions are there? -
#### Problem 1.2.

In between the calibrated and uncalibrated cases there are partially calibrated cameras, say with both focal lengths known or with the center position known. What can we say about these situations? -
#### Problem 1.25.

Given an algebraic curve in two views, when are they compatible? This is interesting when the cameras are fixed beforehand. -
#### Problem 1.35.

Can we use conic correspondence between two images to do reconstruction? Can we use point and curve correspondences to do reconstruction? Can we use cubic spline correspondences to do reconstruction? -
#### Problem 1.4.

Fix two cameras and two image curves. Is there is a curve in $\mathbb{P}^3$ that projects onto these two curves? What if we put restriction on degrees or smoothness? -
#### Problem 1.45.

For a reconstruction what does it mean for all the points to be “in front” of the cameras? Can this easily be detected. -
#### Problem 1.5.

The focal lengths of a pair of cameras can be written as a function of their fundamental matrix. During reconstruction fundamental matrices are constructed for which these focal lengths are complex. Is there an efficient way to detect if the this will occur before computing the fundamental matrix? -
#### Problem 1.55.

How can we account for uncertainty, given the geometric structure of these problems? What if we replace points with disks?

Cite this as: *AimPL: Algebraic vision, available at http://aimpl.org/algvision.
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