Convex Global 3D Registration with Lagrangian Duality

Abstract

The registration of 3D models by a Euclidean transformation is a fundamental task at the core of many application in computer vision. This problem is non-convex due to the presence of rotational constraints, making traditional local optimization methods prone to getting stuck in local minima. This paper addresses finding the globally optimal transformation in various 3D registration problems by a unified formulation that integrates common geometric registration modalities (namely point-to-point, point-to-line and point-to-plane). This formulation renders the optimization problem independent of both the number and nature of the correspondences.The main novelty of our proposal is the introduction of a strengthened Lagrangian dual relaxation for this problem, which surpasses previous similar approaches [32] in effectiveness. In fact, even though with no theoretical guarantees, exhaustive empirical evaluation in both synthetic and real experiments always resulted on a tight relaxation that allowed to recover a guaranteed globally optimal solution by exploiting duality theory.Thus, our approach allows for effectively solving the 3D registration with global optimality guarantees while running at a fraction of the time for the state-of-the-art alternative [34], based on a more computationally intensive Branch and Bound method.

Modified Rodrigues Parameters: An Efficient Representation of Orientation in 3D Vision and Graphics

“…This, however, does not render the representation entirely unattractive. For instance, Carlone et al [7], Olsson and Eriksson [43] as well as Briales and Jimenez [6] make explicit use of matrix orthonormality constraints to formulate the Lagrangian of the camera pose registration problem. The advantage of this approach is that it provides measures for the optimality of solutions of relaxations by monitoring the duality gap in the original problem.…”

“…denotes the Frobenius norm for matrices. 6 In our experimental setup, the dataset comprises 100 correspondences, i.e., matrices X and Y have size 3 × 100. The unrotated points X were sampled from a 3D Gaussian with a covariance matrix 10 2 I 3 , thus producing a "spread" of roughly 10 metric units.…”

Convex Relaxations for Consensus and Non-Minimal Problems in 3D Vision

“…This however entails the risk of falling into local minima traps of the global cost function resulting in inadequate solutions, which has been demonstrated for various problems [8,32,22,9,39,18,23]. More specifically, non-minimal solvers have been devised separately for 3D rotations [18], 3D rigid body transformation in [8,32,39,23], perspective-n-point in [22], and two-view relative pose in [9]. On the one hand, these solvers are very specific to their own addressed problems, and only [32] provides the theoretical optimality guarantee owing to the used Branch-and-Bound (BnB) search paradigm.…”

“…On the one hand, these solvers are very specific to their own addressed problems, and only [32] provides the theoretical optimality guarantee owing to the used Branch-and-Bound (BnB) search paradigm. On the other hand, methods in [22,8,9] are fast and also achieve aposteriori certificate, with an open question for theoretical proof, of the global optimality. One notable work [21], provides convex relaxations for multiple computer vision problems while primarily focusing on the task of geometric error minimization, where the addressed problems are often linear with respect to the algebraic error minimization.…”

“…The first term of Lasserre's relaxations is indeed the so-called Shor's relaxation, a well known result from the 80's [40], which is also known to be tight for quadratic polynomial optimization [6]. Therefore, it is not very surprising that the non-minimal solvers of [8,9], which use Shor's relaxation with additional problem specific constraints, are fast and accurate. Moreover, in the context of non-minimal solvers, where one is offered sufficiently many polynomials, the tightness obtained using only the Shor's relaxation may not be an issue.…”