The number of noisy images required for molecular reconstruction in single-particle cryo-electron microscopy (cryo-EM) is governed by the autocorrelations of the observed, randomly-oriented, noisy projection images. In this work, we consider the effect of imposing sparsity priors on the molecule. We use techniques from signal processing, optimization, and applied algebraic geometry to obtain new theoretical and computational contributions for this challenging non-linear inverse problem with sparsity constraints. We prove that molecular structures modeled as sums of Gaussians are uniquely determined by the second-order autocorrelation of their projection images, implying that the sample complexity is proportional to the square of the variance of the noise. This theory improves upon the non-sparse case, where the third-order autocorrelation is required for uniformly-oriented particle images and the sample complexity scales with the cube of the noise variance. Furthermore, we build a computational framework to reconstruct molecular structures which are sparse in the wavelet basis. This method combines the sparse representation for the molecule with projection-based techniques used for phase retrieval in X-ray crystallography.

翻译：单粒子冷冻电子显微镜（cryo-EM）中分子重构所需的含噪图像数量，取决于所观测到的、随机取向的含噪投影图像的自相关性。本研究探讨了在分子上施加稀疏先验条件的影响。我们运用信号处理、优化及应用代数几何中的技术，针对这一具有稀疏约束的非线性逆问题，获得了新的理论成果与计算方法。我们证明，以高斯函数之和为模型的分子结构可由其投影图像的二阶自相关唯一确定，这意味着样本复杂度与噪声方差的平方成正比。这一理论优于非稀疏情况，后者对均匀取向的粒子图像需依赖三阶自相关，且样本复杂度随噪声方差的三次方增长。此外，我们构建了一个计算框架，用于重构在小波基下具有稀疏性的分子结构。该方法将分子的稀疏表示与X射线晶体学中用于相位恢复的基于投影的技术相结合。