Triangulation (computer vision)

In computer vision triangulation refers to the process of determining a point in 3D space given its projections onto two, or more, images. In order to solve this problem it is necessary to know the parameters of the camera projection function from 3D to 2D for the cameras involved, in the simplest case represented by the camera matrices. Triangulation is sometimes also referred to as reconstruction or intersection.
The triangulation problem is in principle trivial. Since each point in an image corresponds to a line in 3D space, all points on the line in 3D are projected to the point in the image. If a pair of corresponding points in two, or more images, can be found it must be the case that they are the projection of a common 3D point x. The set of lines generated by the image points must intersect at x (3D point) and the algebraic formulation of the coordinates of x (3D point) can be computed in a variety of ways, as is presented below.
In practice, however, the coordinates of image points cannot be measured with arbitrary accuracy. Instead, various types of noise, such as geometric noise from lens distortion or interest point detection error, lead to inaccuracies in the measured image coordinates. As a consequence, the lines generated by the corresponding image points do not always intersect in 3D space. The problem, then, is to find a 3D point which optimally fits the measured image points. In the literature there are multiple proposals for how to define optimality and how to find the optimal 3D point. Since they are based on different optimality criteria, the various methods produce different estimates of the 3D point x when noise is involved.
Introduction.
In the following, it is assumed that triangulation is made on corresponding image points from two views generated by pinhole cameras. Generalization from these assumptions are discussed here.
The image to the left illustrates the epipolar geometry of a pair of stereo cameras of pinhole model. A point x (3D point) in 3D space is projected onto the respective image plane along a line (green) which goes through the camera's focal point, formula_1 and formula_2, resulting in the two corresponding image points formula_3 and formula_4. If formula_3 and formula_4 are given and the geometry of the two cameras are known, the two projection lines (green lines) can be determined and it must be the case that they intersect at point x (3D point). Using basic linear algebra that intersection point can be determined in a straightforward way.
The image to the right shows the real case. The position of the image points formula_3 and formula_4 cannot be measured exactly. The reason is a combination of factors such as 
As a consequence, the measured image points are formula_9 and formula_10 instead of formula_3 and formula_4. However, their projection lines (blue) do not have to intersect in 3D space or come close to x. In fact, these lines intersect if and only if formula_9 and formula_10 satisfy the epipolar constraint defined by the fundamental matrix. Given the measurement noise in formula_9 and formula_10 it is rather likely that the epipolar constraint is not satisfied and the projection lines do not intersect.
This observation leads to the problem which is solved in triangulation. Which 3D point xest is the best estimate of x given formula_9 and formula_10 and the geometry of the cameras? The answer is often found by defining an error measure which depends on xest and then minimizing this error. In the following sections, some of the various methods for computing xest presented in the literature are briefly described.
All triangulation methods produce xest = x in the case that formula_19 and formula_20, that is, when the epipolar constraint is satisfied (except for singular points, see below). It is what happens when the constraint is not satisfied which differs between the methods.
Properties.
A triangulation method can be described in terms of a function formula_21 such that
where formula_23 are the homogeneous coordinates of the detected image points and formula_24 are the camera matrices. x (3D point) is the homogeneous representation of the resulting 3D point. The formula_25 sign implies that formula_21 is only required to produce a vector which is equal to x up to a multiplication by a non-zero scalar since homogeneous vectors are involved.
Before looking at the specific methods, that is, specific functions formula_21, there are some general concepts related to the methods that need to be explained. Which triangulation method is chosen for a particular problem depends to some extent on these characteristics.
Singularities.
Some of the methods fail to correctly compute an estimate of x (3D point) if it lies in a certain subset of the 3D space, corresponding to some combination of formula_28. A point in this subset is then a "singularity" of the triangulation method. The reason for the failure can be that some equation system to be solved is under-determined or that the projective representation of xest becomes the zero vector for the singular points.
Invariance.
In some applications, it is desirable that the triangulation is independent of the coordinate system used to represent 3D points; if the triangulation problem is formulated in one coordinate system and then transformed into another the resulting estimate xest should transform in the same way. This property is commonly referred to as "invariance". Not every triangulation method assures invariance, at least not for general types of coordinate transformations.
For a homogeneous representation of 3D coordinates, the most general transformation is a projective transformation, represented by a formula_29 matrix formula_30. If the homogeneous coordinates are transformed according to
then the camera matrices must transform as (Ck)
to produce the same homogeneous image coordinates (yk)
If the triangulation function formula_34 is invariant to formula_30 then the following relation must be valid
from which follows that
For each triangulation method, it can be determined if this last relation is valid. If it is, it may be satisfied only for a subset of the projective transformations, for example, rigid or affine transformations.
Computational complexity.
The function formula_34 is only an abstract representation of a computation which, in practice, may be relatively complex. Some methods result in a formula_34 which is a closed-form continuous function while others need to be decomposed into a series of computational steps involving, for example, SVD or finding the roots of a polynomial. Yet another class of methods results in formula_34 which must rely on iterative estimation of some parameters. This means that both the computation time and the complexity of the operations involved may vary between the different methods.
Methods.
Mid-point method.
Each of the two image points formula_9 and formula_10 has a corresponding projection line (blue in the right image above), here denoted as formula_44 and formula_45, which can be determined given the camera matrices formula_24. Let formula_47 be a distance function between a (3D line) L and a x (3D point) such that 
formula_48 is the Euclidean distance between formula_49 and formula_50.
The "midpoint method" finds the point xest which minimizes
It turns out that xest lies exactly at the middle of the shortest line segment which joins the two projection lines.
Via the essential matrix.
The problem to be solved there is how to compute formula_52 given corresponding normalized image coordinates formula_53 and formula_54. If the essential matrix is known and the corresponding rotation and translation transformations have been determined, this algorithm (described in Longuet-Higgins' paper) provides a solution.
Let formula_55 denote row "k" of the rotation matrix formula_56:
Combining the above relations between 3D coordinates in the two coordinate systems and the mapping between 3D and 2D points described earlier gives
or
Once formula_60 is determined, the other two coordinates can be computed as
The above derivation is not unique. It is also possible to start with an expression for formula_62 and derive an expression for formula_60 according to
In the ideal case, when the camera maps the 3D points according to a perfect pinhole camera and the resulting 2D points can be detected without any noise, the two expressions for formula_60 are equal. In practice, however, they are not and it may be advantageous to combine the two estimates of formula_60, for example, in terms of some sort of average.
There are also other types of extensions of the above computations which are possible. They started with an expression of the primed image coordinates and derived 3D coordinates in the unprimed system. It is also possible to start with unprimed image coordinates and obtain primed 3D coordinates, which finally can be transformed into unprimed 3D coordinates. Again, in the ideal case the result should be equal to the above expressions, but in practice they may deviate.
A final remark relates to the fact that if the essential matrix is determined from corresponding image coordinate, which often is the case when 3D points are determined in this way, the translation vector formula_67 is known only up to an unknown positive scaling. As a consequence, the reconstructed 3D points, too, are undetermined with respect to a positive scaling.