perform procrustes alignment

This paper studies the perturbed Procrustes point-set alignment problem, which seeks an orthogonal transformation minimizing the matrix norm discrepancy between two observed point sets. Motivated by robust alignment requirements under random matrix models, we systematically analyze and compare the performance of spectral, Frobenius, and robust norms within a nonsmooth Riemannian optimization framework. Theoretical analysis and empirical experiments demonstrate that, in canonical settings—including low-dimensional alignment and random network hypothesis testing—the Frobenius norm achieves statistical accuracy comparable to the computationally more expensive spectral or robust norms, while substantially reducing optimization complexity and computational cost. This finding establishes a principled, efficient norm selection criterion for Procrustes-type problems, offering both theoretical guarantees and practical scalability.

This study addresses the susceptibility of alternating minimization to local optima and its computational inefficiency in correspondence-free point set alignment. We propose a global optimization method based on support vectors derived from the convex hull vertices of permuted polygons. By proving a tight bound of $n(n-1)$ vertices, we resolve an open problem posed by Rote. Integrating the Procrustes-Wasserstein framework with a branch-and-bound algorithm, our approach achieves exact solutions in 2D and extends naturally to 3D. Evaluated on the MPEG-7 benchmark, the method requires only 12ms on average, achieving a 50-fold speedup over grid search while delivering superior accuracy. These improvements substantially enhance shape retrieval performance, demonstrating both theoretical rigor and practical efficiency for robust point set registration.

This paper addresses the robust alignment of two rotation sets in SO(3) under challenging conditions: no point-wise correspondences, temporal asynchrony, high outlier ratios (up to 90%), and inconsistent axis conventions. We propose the Permutation- and Sign-Invariant (PASI) framework, which decomposes rotations into spherical basis vectors and achieves axis-level decoupled matching via exhaustive enumeration of the 24 valid sign permutations—bypassing conventional correspondence search. PASI integrates weighted correlation scoring, spherical point-set matching (SPMC/FRS), and projection-based or Karcher mean estimation to ensure globally consistent alignment. The algorithm exhibits linear time complexity, achieving 6–60× speedup over state-of-the-art methods. It requires neither initial correspondences nor temporal synchronization. Extensive experiments on synthetic and real-world data demonstrate significant accuracy improvements over baseline approaches.

This paper addresses the robust rotation estimation problem between spherical point clouds without correspondences. We propose three novel linear-time algorithms—SPMC, FRS, and a hybrid method—with computational complexity $O(n)$. By modeling spherical patterns as discrete point sets on the unit sphere, our methods directly solve the Wahba problem, thereby avoiding the $O(n^3)$ discretization overhead and explicit correspondence requirements inherent in conventional spherical cross-correlation and spherical convolution-based approaches. The algorithms exhibit exceptional accuracy and robustness even under extreme outlier ratios (>90%). Evaluated on a newly constructed “Robust Vector Alignment Dataset,” they achieve over 10× speedup and more than 10× reduction in rotation error compared to state-of-the-art methods. Our approach is successfully deployed in two practical applications: point cloud registration and spherical image rotation estimation.

This work addresses the problem of establishing dense intrinsic correspondences between non-rigid manifolds. We propose a matrix completion framework that jointly incorporates geometric priors—specifically, manifold Laplacian-guided constraints—and sparse functional landmark localization. Methodologically, we are the first to integrate Laplacian-based geometric regularization with ℓ₁-norm sparsity promotion into a unified matrix completion model, effectively mitigating overfitting under limited supervision and enabling precise identification of functionally consistent regions. Optimization is performed via an efficient numerical algorithm. Extensive evaluation on standard non-rigid matching benchmarks (e.g., FAUST, TOSCA) demonstrates state-of-the-art performance: our method achieves the highest overall accuracy and, under highly sparse supervision (fewer than 10 seed points), reduces average correspondence error by 18.7% compared to the best prior approach—substantially improving robustness and generalization capability.

This work addresses the computationally demanding yet critical nonlinear least-squares problem in pose estimation for real-time computer vision. By introducing a suitable parametrization of rotations, the problem is reformulated as a system of polynomial equations. The authors propose a novel class of resultant solvers based on Sylvester matrices that enable efficient closed-form solutions. This approach substantially reduces computational complexity while preserving high numerical accuracy. Experimental results demonstrate that the method outperforms state-of-the-art techniques in terms of runtime on both 3D–3D and 3D–2D pose estimation tasks, offering a practical solution for time-sensitive applications.

This work addresses the problem of estimating the pose of a known 3D shape from an unoccluded orthographic silhouette without relying on feature point correspondences. The method leverages the continuity of silhouette area along rotational trajectories to construct a precomputed silhouette signature response surface and introduces the aspect ratio of a fitted ellipse as a global shape signature. This enables an efficient, resolution-guided branch-and-bound search over the rotation space. To the best of our knowledge, this is the first approach capable of achieving globally optimal pose estimation using only silhouettes for arbitrary shapes, including non-convex and high-genus geometries. Experiments on both synthetic and real-world data demonstrate that the proposed method significantly outperforms existing techniques in both accuracy and computational efficiency.

This work addresses the problem of exactly recovering an unknown permutation that aligns two sets of Gaussian vectors in high dimensions (where \( d \gg \log n \)), given that the vectors are related by this permutation, an unknown rotation, and a fixed correlation coefficient \( \rho \). The authors propose a polynomial-time algorithm based on weighted “wide-tree” counting, which succeeds with high probability when \( d \geq \mathrm{polylog}(n) \) and \( \rho^2 > \sqrt{\alpha} \) with \( \alpha \approx 0.338 \). This is the first efficient method achieving exact recovery under constant correlation, significantly relaxing the previously required condition that \( \rho \) be close to 1. The tightness and necessity of this condition are corroborated through information-theoretic lower bounds—namely \( \rho^2 \gtrsim \max\{\log n / d, \sqrt{\log n / n}\} \)—and low-degree polynomial analysis.

This study addresses the challenge of constructing rotation-invariant vector representations for planar shapes by proposing a method that strictly encodes star-shaped normalized contours into Euclidean vectors. The resulting representation guarantees that Euclidean distances between vectors faithfully reflect shape dissimilarities while enabling efficient shape analysis. The approach is the first to simultaneously achieve strict invariance under rotation (and controllable reflection), injectivity, and robustness to small perturbations. By discretizing functions defined on the unit circle and employing an offset-based parameterization, the method constructs an ε-approximate vector in O((1/ε) log(1/ε)) time, yielding an O(1/ε)-dimensional embedding amenable to efficient nearest-neighbor search and clustering. Experimental results confirm that the representation maintains high accuracy and computational efficiency without compromising invariance properties.

本文提出一种基于几何驱动的数据优化方法，通过主轴对齐解决物体姿态估计中的噪声敏感、对称性混淆问题，且无需修改现有网络架构。