A Polynomial Formula for the Perspective Four Points Problem

We present a fast and accurate solution to the perspective n-points problem, by way of a new approach to the \(n=4\) case. Our solution hinges on a novel separation of variables: given four 3D points and four corresponding 2D points on the camera canvas, we start by finding another set of 3D points, sitting on the rays connecting the camera to the 2D canvas points, so that the six pair-wise distances between these 3D points are as close as possible to the six distances between the original 3D points. This step reduces the perspective problem to an absolute orientation problem, which has a solution via explicit formula. To solve the first problem we set coordinates which are as orientation-free as possible: on the 3D points side our coordinates are the squared distances between the points. On the 2D canvas-points side our coordinates are the dot products of the points after rotating one of them to sit on the optical axis. We then derive the solution with the help of a computer algebra system. Our solution is an order of magnitude faster than state of the art algorithms, while offering similar accuracy under realistic noise. Moreover, our reduction to the absolute orientation problem runs two orders of magnitude faster than other perspective problem solvers, allowing extremely efficient seed rejection when implementing RANSAC.

1. allowing for non-strict inequalities since we have not required the \(P_i\) to be distinct.

2. We did this calculation by using Singular to evaluate the coefficients of each \(Q_i\) and each \(\tau _{j,k} Q_i\), and then to calculate the gcd of \(Q_i\) and \(\tau _{j,k} Q_i\).

Authors and Affiliations

1. Tel Aviv University, Tel-Aviv, Israel

   David Lehavi

2. Davis, CA, USA

   Brian Osserman

Corresponding author

Correspondence to David Lehavi.

Communicated by Luca Magri.

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Singular calculations

In order to obtain the quadratic polynomials \(Q_i\) referenced in Theorem 5.1, we begin by simply coding the equations in (4.1) of Proposition 4.4 into Singular, and letting I be the ideal generated by all six polynomials.

The chosen polynomial ordering is natural due to our desire to eliminate the last four variables from the ideal.

To compute \(Q_0\), the first two steps are straightforward: eliminate \(z_3\) and then \(z_2\) from the ideal I.

However, trying to directly eliminate \(z_1\) from the above ideal hit a computational brick wall; instead, after some experimentation, we defined a certain subideal, and were able to make the final elimination starting from it:

In defining J0_2, our ultimate choices are in fact rather natural: the three polynomials in the definition of J0_2 are simply the shortest three generators of I0_2, and the reason for using three of them is that this is the minimal number which would be expected to support eliminating \(z_1\) while still having a nontrivial equation in \(z_0\).

In any case, the final line of the above Singular code shows that we have produced an element of I of the desired form, so we thus define \(Q_0\):

Definition A.1

Let \(Q_0(x)\) be the polynomial obtained by replacing z0\(^{2}\) with x in the above J0_1[1], considered as a quadratic polynomial in x with coefficients being polynomials in the \(a_\bullet \), \(b_\bullet \), \(c_\bullet \), \(d_\bullet \).

To obtain \(Q_3\) we run the following code in the same spirit as our derivation of \(Q_0\):

We then get that J3_0[1] is of degree 2 in z3\(^{2}\), so we can define \(Q_3\):

Definition A.2

Let \(Q_3(x)\) be the polynomial obtained by replacing z3\(^{2}\) with x in the above J3_0[1], considered as a quadratic polynomial in x with coefficients being polynomials in the \(a_\bullet , b_\bullet , c_\bullet , d_\bullet \).

The coefficients of the \(Q_i\)

Below are the formulas for the coefficients of \(Q_0\) and \(Q_3\). The output of our Singular code was post processed to remove the * symbol:

In order to obtain the formulas for \(X_{1,j}\) and \(X_{2,j}\), as described in the proof of Theorem 5.1 one simply starts with the formulas for the \(X_{0,j}\), and applies a transposition to every variable index occurring in the formula. For \(Q_1\), all of the 0s and 1s are swapped, while for \(Q_2\) all of the 0s and 2s are swapped.

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