# multiview¶

The multiview module consists of:

- a collection of solvers for 2 to n-view geometry constraints that arise in multiple view geometry,
- a generic framework “Kernel” that can embed these solvers for robust estimation.

First accessible solvers are listed and explained and the “Kernel” concept is documented.

## 2-view solvers (2d-2d correspondences)¶

openMVG provides solver for the following geometric estimation:

- affine,
- homographic,
- fundamental,
- 7 to n pt,
- 8 to n pt (Direct Linear Transform) [HZ].

- essential,
- 8 to n pt (Direct Linear Transform) [HZ],
- 5pt + intrinsic [Stewenius], [Nister].

## N-View geometry estimation¶

- Triangulation
- 2 to n view (Direct Linear Transform),
- 2 to n view (Iterated least square).

- Rotation averaging
- L2 (sparse) [Martinec],
- L1 (sparse) [Chatterjee].

- Translation averaging
- L2 Chordal [Kyle2014],
- SoftL1 ‘approximation of the LInf method of [GlobalACSfM]’.

### Homography matrix:¶

The homography matrix maps the relation between two projections of a plane: Figure.

\(H\) is a (3 x 3) matrix that links coordinates in left and right images with the following relation.

OpenMVG implementation follows the DLT (Direct Linear Transform) explained in [HZ] book: \(H\) can be estimated from 4 to \(n\) corresponding points.

### Fundamental matrix:¶

The fundamental matrix is a relation between two images viewing the same scene where those point’s projections are visible in the two images. Given a point correspondence between two views \((x_i,x'_i)\):

We obtain the following relation:

\(F\) is the (3 x 3) Fundamental matrix, it puts in relation a point x to a line where belong the projection of the 3D X point. \(l'_i = Fx_i\) designs the epipolar line on which the point \(x'_i\) could be. The relation \(x'^T_i Fx_i = 0\) exists for all corresponding point belonging to a stereo pair.

The fundamental matrix is sometime called bifocal-tensor, it is a 3 x 3 matrix of rank 2 with 7 degree of freedom. 8 ou 7 correspondences are sufficient to compute the \(F\) matrix. Implementation follows the DLT (Direct Linear Transform) explained in [HZ] book.

## Relative pose estimation (Essential matrix)¶

Adding intrinsic parameters to the fundamental matrix gives a metric “object” that provides the following relation \(E = K'^T FK\) , this is the Essential relation explained by Longuet-Higgins in 1981 [Longuet]. This essential matrix links the relative position of the camera to the fundamental matrix relation.

## Absolute pose estimation/Camera resection (Pose matrix)¶

Given a list of 3D-2D point correspondences it is possible to compute a camera pose estimation. It consists in estimating the camera parameters of the right camera that minimizes the residual error of the 3D points re-projections, it’s an optimization problem that trying to solve P parameters in order to minimize

openMVG provides 3 different solvers for this problem:

### Kernel concept¶

In order to use the solver in a generic robust estimation framework, we use them in conjuction with the Kernel class that allow to link:

- data points,
- the set of correspondences that are used for a robust estimation problem.

- a model solver/estimator,
- a metric to measure data fitting to a putative model.

- Solver:
- MINIMUM_SAMPLES: The minimal number of point required for the model estimation, MAX_MODELS: The number of models that the minimal solver could return, A Solve function that estimates a model from MINIMUM_SAMPLES to n vector data.
- ErrorModel:
- An metric function that return the error of a sample data to the provided model.
- Kernel:
- Embed data (putative), the model estimator and the error model. This kernel is core brick used in the openMVG robust estimator framework.