
## 1. Reconstruction

A fundamental topic in algebraic vision is that of reconstructing a scene from images.
1.     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 ?
Numerical results suggest that there are 4 constraints but they only intersect the trifocal tensor variety in codimension three.
•     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?
Numerical methods show that each point correspondence only intersects the variety in codimension 3 and so will lead to a minimal problem. The degree is cut down at each step of imposing conditions leading to three complex solutions.
• #### 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 1.3.

What is the Hurwitz form of the essential variety?
• #### 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.
Two approaches were suggested: That neither of the cameras are contained in the convex hull of the world points; or that $P r_i = \lambda x_i$ and $Q r_i = \mu y_i$ where $\lambda>0$ and $\mu>0$.
• #### 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.