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<p>Cryo-electron microscopy (cryo-EM) single-particle reconstruction stands out as a robust technique for elucidating the three-dimensional (3D) structure of particle samples from their 2D projection images with unknown orientations. A crucial aspect of cryo-EM 3D reconstruction involves determining the orientations of particle samples. The orientation of a particle is characterized by a rotation matrix. However, the non-convexity of the orthogonal constraint poses numerical challenges during the minimization process. The conventional strategy involves transforming the orthogonal constrained minimization problem into a semi-definite programming problem, which can be computationally expensive, especially when dealing with a large amount of cryo-EM images. In this study, we propose a novel solution to the orientation estimation problem by leveraging the Riemannian gradient method. All rotation matrices are concatenated into a matrix value vector, and then, the corresponding manifold is defined. By deducing the Riemannian gradient of the objective function, we preserve the structure of the orthogonality constraints through the retraction mapping on the Riemannian manifold. This unique approach enables the decoupling of the minimization problem with non-convex constraints, resulting in improved computational efficiency. We demonstrate that every limit point of the sequence generated by our proposed method constitutes a stationary point of the constrained minimization problem. Additionally, we provide the iteration complexity required to obtain an <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varepsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-stationary solution. Experimental results underscore the efficacy of our proposed algorithm, showcasing significantly reduced running times compared to existing algorithms.</p>

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Orientation Estimation of Cryo-EM Images Using the Riemannian Gradient Method

  • 期刊论文

  • 发表日期：

    2025年12月2日

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  • Huan Pan,
  • You-Wei Wen,
  • Jian Lu,
  • Chen Xu,
  • Tie-Yong Zeng,
  • Li-Xin Shen

摘要

Cryo-electron microscopy (cryo-EM) single-particle reconstruction stands out as a robust technique for elucidating the three-dimensional (3D) structure of particle samples from their 2D projection images with unknown orientations. A crucial aspect of cryo-EM 3D reconstruction involves determining the orientations of particle samples. The orientation of a particle is characterized by a rotation matrix. However, the non-convexity of the orthogonal constraint poses numerical challenges during the minimization process. The conventional strategy involves transforming the orthogonal constrained minimization problem into a semi-definite programming problem, which can be computationally expensive, especially when dealing with a large amount of cryo-EM images. In this study, we propose a novel solution to the orientation estimation problem by leveraging the Riemannian gradient method. All rotation matrices are concatenated into a matrix value vector, and then, the corresponding manifold is defined. By deducing the Riemannian gradient of the objective function, we preserve the structure of the orthogonality constraints through the retraction mapping on the Riemannian manifold. This unique approach enables the decoupling of the minimization problem with non-convex constraints, resulting in improved computational efficiency. We demonstrate that every limit point of the sequence generated by our proposed method constitutes a stationary point of the constrained minimization problem. Additionally, we provide the iteration complexity required to obtain an \(\varepsilon \) ε -stationary solution. Experimental results underscore the efficacy of our proposed algorithm, showcasing significantly reduced running times compared to existing algorithms.