Numerous works concerning head pose estimation (HPE) offer algorithms or
proposed neural network-based approaches for extracting Euler angles from
either facial key points or directly from images of the head region. However,
many works failed to provide clear definitions of the coordinate systems and
Euler or Tait-Bryan angles orders in use. It is a well-known fact that rotation
matrices depend on coordinate systems, and yaw, roll, and pitch angles are
sensitive to their application order. Without precise definitions, it becomes
challenging to validate the correctness of the output head pose and drawing
routines employed in prior works. In this paper, we thoroughly examined the
Euler angles defined in the 300W-LP dataset, head pose estimation such as
3DDFA-v2, 6D-RepNet, WHENet, etc, and the validity of their drawing routines of
the Euler angles. When necessary, we infer their coordinate system and sequence
of yaw, roll, pitch from provided code. This paper presents (1) code and
algorithms for inferring coordinate system from provided source code, code for
Euler angle application order and extracting precise rotation matrices and the
Euler angles, (2) code and algorithms for converting poses from one rotation
system to another, (3) novel formulae for 2D augmentations of the rotation
matrices, and (4) derivations and code for the correct drawing routines for
rotation matrices and poses. This paper also addresses the feasibility of
defining rotations with right-handed coordinate system in Wikipedia and SciPy,
which makes the Euler angle extraction much easier for full-range head pose
research.

该论文通过对数据集、头部姿态估计算法和绘制例程的深入研究，提出了从提供的源代码推断坐标系、应用顺序和提取精确旋转矩阵和欧拉角的代码和算法，以及将姿态从一个旋转系统转换到另一个旋转系统的代码和算法，2D 旋转矩阵的新公式，以及旋转矩阵和姿态的正确绘制例程的推导和代码，并讨论了在维基百科和 SciPy 中定义右手坐标系旋转的可行性，从而使全方位头部姿态研究中的欧拉角提取更加容易。

全角度头部姿态估计的数学基础和改正方法

Mathematical Foundation and Corrections for Full Range Head Pose  Estimation

Symmetric orthogonalization via SVD, and closely related procedures, are
well-known techniques for projecting matrices onto $O(n)$ or $SO(n)$. These
tools have long been used for applications in computer vision, for example
optimal 3D alignment problems solved by orthogonal Procrustes, rotation
averaging, or Essential matrix decomposition. Despite its utility in different
settings, SVD orthogonalization as a procedure for producing rotation matrices
is typically overlooked in deep learning models, where the preferences tend
toward classic representations like unit quaternions, Euler angles, and
axis-angle, or more recently-introduced methods. Despite the importance of 3D
rotations in computer vision and robotics, a single universally effective
representation is still missing. Here, we explore the viability of SVD
orthogonalization for 3D rotations in neural networks. We present a theoretical
analysis that shows SVD is the natural choice for projecting onto the rotation
group. Our extensive quantitative analysis shows simply replacing existing
representations with the SVD orthogonalization procedure obtains state of the
art performance in many deep learning applications covering both supervised and
unsupervised training.

本研究探讨了将奇异值分解正交化应用于神经网络的三维旋转问题，并表明该方法在计算机视觉中取得了优于现有方法的效果。