Triangulation (computer vision)

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, O 1 and O 2 , resulting in the two corresponding image points y 1 and y 2 . If y 1 and y 2 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 y 1 and y 2 cannot be measured exactly. The reason is a combination of factors such as

Geometric distortion, for example lens distortion, which means that the 3D to 2D mapping of the camera deviates from the pinhole camera model. To some extent these errors can be compensated for, leaving a residual geometric error.

A single ray of light from x (3D point) is dispersed in the lens system of the cameras according to a point spread function. The recovery of the corresponding image point from measurements of the dispersed intensity function in the images gives errors.

In digital camera the image intensity function is only measured in discrete sensor elements. Inexact interpolation of the discrete intensity function have to be used to recover the true one.

The image points y₁^' and y₂' used for triangulation are often found using various types of feature extractors, for example of corners or interest points in general. There is an inherent localization error for any type of feature extraction based on neighborhood operations.

As a consequence, the measured image points are y 1 ′ and y 2 ′ instead of y 1 and y 2 . 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 y 1 ′ and y 2 ′ satisfy the epipolar constraint defined by the fundamental matrix. Given the measurement noise in y 1 ′ and y 2 ′ 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 x_est is the best estimate of x given y 1 ′ and y 2 ′ and the geometry of the cameras? The answer is often found by defining an error measure which depends on x_est and then minimize this error. In the following some of the various methods for computing x_est presented in the literature are briefly described.

All triangulation methods produce x_est = x in the case that y 1 = y 1 ′ and y 2 = y 2 ′ , 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 of triangulation methods

A triangulation method can be described in terms of a function τ such that

x ∼ τ ( y 1 ′ , y 2 ′ , C 1 , C 2 )

where y 1 ′ , y 2 ′ are the homogeneous coordinates of the detected image points and C 1 , C 2 are the camera matrices. x (3D point) is the homogeneous representation of the resulting 3D point. The ∼ sign implies that τ 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 τ , 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 y 1 ′ , y 2 ′ , C 1 , C 2 . 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 x_est 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 x_est 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 4 × 4 matrix T . If the homogeneous coordinates are transformed according to

x ¯ ∼ T x

then the camera matrices must transform as (C_k)

C ¯ k ∼ C k T − 1

to produce the same homogeneous image coordinates (y_k)

y k ∼ C ¯ k x ¯ = C k x

If the triangulation function τ is invariant to T then the following relation must be valid

x ¯ e s t ∼ T x e s t

from which follows that

τ ( y 1 ′ , y 2 ′ , C 1 , C 2 ) ∼ T − 1 τ ( y 1 ′ , y 2 ′ , C 1 T − 1 , C 2 T − 1 ) ,   for all y 1 ′ , y 2 ′

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 τ is only an abstract representation of a computation which, in practice, may be relatively complex. Some methods result in a τ 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 τ 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.

Mid-point method

Each of the two image points y 1 ′ and y 2 ′ has a corresponding projection line (blue in the right image above), here denoted as L 1 ′ and L 2 ′ , which can be determined given the camera matrices C 1 , C 2 . Let d be a distance function between a (3D line) L₁' and a x (3D point) such that

d ( L , x ) =   the Euclidean distance between L and x .

The midpoint method finds the point x_est which minimizes

d ( L 1 ′ , x ) 2 + d ( L 2 ′ , x ) 2

It turns out that x_est lies exactly at the middle of the shortest line segment which joins the two projection lines.