# Le Probleme Perspective-N-Point/The Perspective-N-Point Problem

## Presentation

Perspectice-N-point problem is an old problem. Given a set of N points well know in a local coordinate space and given a projection of this set of points on a plan in another local coordinate space, to find the transformation between the two coordinate space or to find the position of points in the second coordinate space. Many solutions already exist! We offer a new solution which is completely analytical and it is uncommon to have analytical among state of the art. Our method works with at least 3 aligned points, at least 4 coplanar points and at least 5 points in a general case. We illustrate our solution by solving the P4P.

## Principle

This problem is commonly solved using generalized cosines in perspective polyhedron. This lead to non linear systems. Our starting point is simpler as we use parametric representation:

Given 4 points (p1, p2, p3, p4) well know in a coordinate space. Given a projection of these 4 points (p'1, p'2, p'3, p'4). As our sensor is calibrated we know the center of projection F (the fovea of our sensor). We have a expression of all points:

p1 = F + l1 * v1 p2 = F + l2 * v2 p3 = F + l3 * v3 p4 = F + l4 * v4

with

vi=p'i-F

Now, locate points is equivalent to find values of l1, l2, l3 and l4. As we know points in space, it is possible to write relations between vectors (pipj), for example:

p1p3 = k1 p1p2+ k2 p1p4 (1)

k1 and k2 are known because points all pi are known and we have expression vectors (pipj) according to li:

p1p3 = p3-p1 = F + l3 * v3 - (F + l1 * v1) p1p2 = p2-p1 = F + l2 * v2 - (F + l1 * v1) p1p4 = p4-p1 = F + l4 * v4 - (F + l1 * v1)

Combined with equation (1) we get a system (3 equations - according to each axis). We can then express one li according all other.

We have now the orientation of points in space. Now we have to find the distance. We need another equation. To remove
ambiguity we use the norm Lpipj of one vector pipj, for example:

|| p1p3 || = || F + l3 * v3 - (F + l1 * v1) || = Lp1p3 (2)

Solution of equations (1) is obvious as it is a linear system! Solution of equation (2) is almost as simple. It is a quadratic solution (which can be simplified by using F as the center of coordinate space). Combination of solution of (2) in (1) give a solution for each li.

## Demonstration

## Conclusion

We offer a new simple, fast analytical solution for solving the perspective-N-point problem. We can find this analytical solution for N>=3 aligned points, N>=4 coplanar points and N>=5 in general case. This solution can also be used to initialize a minimization process.

## Bibliography and links

__To get more details:__

- "An Analytical Solution to the Perspective-N-Point Problem for Common Planar Camera and for Catadioptric Sensor", J. Fabrizio, J. Devars - International Journal of Image and Graphics (IJIG08)
- "The Perspective-N-Point Problem for Catadioptric Sensors: An Analytical Approach", J. Fabrizio, J. Devars, International Conference on Computer Vision and Graphics (ICCVG'04), Warsaw, september 2004

Published by Kluwer in the book series COMPUTATIONAL IMAGING AND VISION. - "Localisation d'obstacles coopératifs par systèmes de vision classiques et panoramiques", J. Fabrizio - PhD thesis - Université Pierre et Marie Curie - Paris VI - December 15, 2004
- "Une solution analytique au problème « Perspective-N-Points » et son extension aux capteurs catadioptriques", J. Fabrizio, J. Devars 19th GRETSI Symposium on Signal and Image Processing (GRETSI'03), Paris, 8 - 11 September 2003.

__A good review can be found in:__

- "Localisation d'obstacles coopératifs par systèmes de vision classiques et panoramiques", J. Fabrizio - PhD thesis - Université Pierre et Marie Curie - Paris VI - December 15, 2004

__Bibliography :__

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