The Determinant Ratio Matrix Approach to Solving 3D Matching and 2D Orthographic Projection Alignment Tasks

TL;DR

This work addresses pose estimation by solving exact-data EnP ($3D$–$3D$) and OnP ($3D$–$2D$ orthographic) problems with closed-form determinant-ratio matrices (DRaM) expressed via cross-covariances. It develops three derivations of DRaM, generalizes the approach to ND spaces, and situates DRaM alongside QR and Moore–Penrose methods within a broader class that yields near-orthonormal solutions after rotation correction under noise. The paper also delineates the RMSD-class methods that exactly solve EnP with or without noise and discusses their relation to DRaM, noting OnP lacks a direct RMSD analogue. Experimental results show corrected DRaM solutions closely approach the gold-standard ArgMin results while offering faster solutions for OnP, suggesting practical speedups and broader applicability to PnP and high-dimensional pose estimation in vision and related fields.

Abstract

Pose estimation is a general problem in computer vision with wide applications. The relative orientation of a 3D reference object can be determined from a 3D rotated version of that object, or from a projection of the rotated object to a 2D planar image. This projection can be a perspective projection (the PnP problem) or an orthographic projection (the OnP problem). We restrict our attention here to the OnP problem and the full 3D pose estimation task (the EnP problem). Here we solve the least squares systems for both the error-free EnP and OnP problems in terms of the determinant ratio matrix (DRaM) approach. The noisy-data case can be addressed with a straightforward rotation correction scheme. While the SVD and optimal quaternion eigensystem methods solve the noisy EnP 3D-3D alignment exactly, the noisy 3D-2D orthographic (OnP) task has no known comparable closed form, and can be solved by DRaM-class methods. We note that while previous similar work has been presented in the literature exploiting both the QR decomposition and the Moore-Penrose pseudoinverse transformations, here we place these methods in a larger context that has not previously been fully recognized in the absence of the corresponding DRaM solution. We term this class of solutions as the DRaM family, and conduct comparisons of the behavior of the families of solutions for the EnP and OnP rotation estimation problems. Overall, this work presents both a new solution to the 3D and 2D orthographic pose estimation problems and provides valuable insight into these classes of problems. With hindsight, we are able to show that our DRaM solutions to the exact EnP and OnP problems possess derivations that could have been discovered in the time of Gauss, and in fact generalize to all analogous N-dimensional Euclidean pose estimation problems.

Figures (3)

• Figure 1: Results for $\sigma = 0.1$, cloud data width $\sim 2.0$, using the DRaM analytical solution to the EnP 3D-to-3D alignment problem. The corrected EnP DRaM is not identical to the known-to-be-optimal SVD/Quaternion "Horn" solution, but it is very close, with 20X scaled differences plotted at the bottom.
• Figure 2: Results for $\sigma = 0.1$, cloud data width $\sim 2.0$, using the DRaM analytical solution to the OnP 3D-to-2D orthographic projection alignment problem. The corrected OnP DRaM is not identical to the known-to-be-optimal ArgMin solution, but it is very close, with 20X scaled differences plotted at the bottom.
• Figure 3: (a) EnP with noise. Behavior of the EnP loss ( Eq. (\ref{['3D3DPoseLSQ.eq']}) ) as a function of standard deviation $\sigma$, data width $\sim 2.0$, for a selection of pose estimation methods. (b) OnP with noise. Behavior of the OnP loss ( Eq. (\ref{['3D2DPoseLSQ.eq']}) ) as a function of standard deviation $\sigma$ for a selection of pose estimation methods. Here applying the SVD method with planar target data is highly inaccurate at $\sigma=0$ (see Table \ref{['dataTable.tbl']}), but approaches the other methods for $sigma > 0.3$. For the EnP problem with or without noise, the quaternion-eigenvalue:SVD method is exact, and we see that it corresponds to machine accuracy with the ArgMin gold standard. Note that in both plots, the bare DRaM's loss is better than the gold standard ArgMin only because it is deformed and not actually a valid rotation matrix.