We study the problem of 3D shape reconstruction from 2D landmarks extracted
in a single image. We adopt the 3D deformable shape model and formulate the
reconstruction as a joint optimization of the camera pose and the linear shape
parameters. Our first contribution is to apply Lasserre's hierarchy of convex
Sums-of-Squares (SOS) relaxations to solve the shape reconstruction problem and
show that the SOS relaxation of minimum order 2 empirically solves the original
non-convex problem exactly. Our second contribution is to exploit the structure
of the polynomial in the objective function and find a reduced set of basis
monomials for the SOS relaxation that significantly decreases the size of the
resulting semidefinite program (SDP) without compromising its accuracy. These
two contributions, to the best of our knowledge, lead to the first certifiably
optimal solver for 3D shape reconstruction, that we name Shape*. Our third
contribution is to add an outlier rejection layer to Shape* using a truncated
least squares (TLS) robust cost function and leveraging graduated non-convexity
to solve TLS without initialization. The result is a robust reconstruction
algorithm, named Shape#, that tolerates a large amount of outlier measurements.
We evaluate the performance of Shape* and Shape# in both simulated and real
experiments, showing that Shape* outperforms local optimization and previous
convex relaxation techniques, while Shape# achieves state-of-the-art
performance and is robust against 70% outliers in the FG3DCar dataset.

该研究通过采用三维可变形形状模型，将三维形状重建问题转化为相机姿态和线性形状参数的联合优化，使用 Lasserre 的 SOS 松弛层次来解决问题，并通过添加一个异常值剔除层和使用新的 TLS robust 代价函数来解决 TLS 问题，研发出一个鲁棒的三维形状重建算法 Shape#，可以容忍 70% 的异常值，达到了最先进的性能。

完美形状：从 2D 标志点中可证明的最优 3D 形状重建

In Perfect Shape: Certifiably Optimal 3D Shape Reconstruction from 2D  Landmarks

Cascaded regression has been recently applied to reconstructing 3D faces from
single 2D images directly in shape space, and achieved state-of-the-art
performance. This paper investigates thoroughly such cascaded regression based
3D face reconstruction approaches from four perspectives that are not well
studied yet: (i) The impact of the number of 2D landmarks; (ii) the impact of
the number of 3D vertices; (iii) the way of using standalone automated landmark
detection methods; and (iv) the convergence property. To answer these
questions, a simplified cascaded regression based 3D face reconstruction method
is devised, which can be integrated with standalone automated landmark
detection methods and reconstruct 3D face shapes that have the same pose and
expression as the input face images, rather than normalized pose and
expression. Moreover, an effective training method is proposed by disturbing
the automatically detected landmarks. Comprehensive evaluation experiments have
been done with comparison to other 3D face reconstruction methods. The results
not only deepen the understanding of cascaded regression based 3D face
reconstruction approaches, but also prove the effectiveness of proposed method.

本研究探讨基于级联回归的 3D 人脸重建方法在 2D 标记点数、3D 顶点数、标记点检测方法及收敛性等四个方面的影响，并提出了一种简化的级联回归算法以及有效的训练方法，实验证明了该方法的有效性。

基于级联回归在形状空间中进行三维面部重建

On 3D Face Reconstruction via Cascaded Regression in Shape Space

We investigate the problem of estimating the 3D shape of an object, given a
set of 2D landmarks in a single image. To alleviate the reconstruction
ambiguity, a widely-used approach is to confine the unknown 3D shape within a
shape space built upon existing shapes. While this approach has proven to be
successful in various applications, a challenging issue remains, i.e., the
joint estimation of shape parameters and camera-pose parameters requires to
solve a nonconvex optimization problem. The existing methods often adopt an
alternating minimization scheme to locally update the parameters, and
consequently the solution is sensitive to initialization. In this paper, we
propose a convex formulation to address this problem and develop an efficient
algorithm to solve the proposed convex program. We demonstrate the exact
recovery property of the proposed method, its merits compared to alternative
methods, and the applicability in human pose and car shape estimation.

本文提出了一种基于凸形式的方法，用于解决在单个图像中给定一组二维路标的情况下估计物体三维形状的问题，解决了现有方法中对于相机姿态和形状参数的联合估计需要解决非凸优化问题导致的初始化敏感性问题，本文方法能够恢复精确的三维形状且应用在人体姿势和汽车形状估计中。