提出行列式比矩阵法，统一求解3D与2D正交投影姿态估计问题。
The Determinant Ratio Matrix Approach to Solving 3D Matching and 2D Orthographic Projection Alignment Tasks
- 基于行列式比矩阵（DRaM）构建3D-3D与3D-2D正交投影的闭式解。
- 在无噪声情况下精确求解EnP与OnP问题，有噪声时可通过旋转修正处理。
- 方法可追溯至高斯时代，且可推广至任意维度的姿态估计任务。
姿态估计是计算机视觉中的基础问题，广泛应用于三维物体相对方位的确定。当已知三维参考物体的旋转版本或其在二维平面的投影时，可求解姿态。本文聚焦于正交投影（OnP）问题和完整的3D姿态估计（EnP）问题。通过行列式比矩阵（DRaM）方法，我们对无误差的EnP与OnP问题给出了最小二乘解。对于含噪声数据的情况，采用简单的旋转校正方案即可应对。尽管奇异值分解（SVD）和最优四元数特征系统能精确求解3D-3D对齐的噪声问题，但3D-2D正交投影（OnP）尚无已知闭式解。本文提出的DRaM类方法填补了这一空白。我们指出，此前研究虽使用过QR分解与Moore-Penrose伪逆，但未意识到这些方法属于更广义的DRaM家族。通过对比分析，揭示了该家族在不同姿态估计问题中的行为特性。本工作不仅提供了3D与2D正交投影姿态估计的新解法，还为相关问题提供了深刻洞察。从后见之明看，我们的DRaM解法可追溯至高斯时代，并可推广至所有N维欧氏空间的姿态估计问题。
原文摘要 · Abstract (English)
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.
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