Bundle Adjustment Based On L1 Norm For Image Sequence Alignment Of Cryo-EM

At present,cryo-electron microscopy has been widely used in the field of structural biology.In cryo-electron microscopy,cryo-electron tomography is an extremely important branch.It combines three-dimensional reconstruction technology,which can extract more accurate biological structure information and allow the study of biological materials at molecular resolution.In cryo-electron tomography,the alignment of image sequences plays an important role in the entire 3D reconstruction process,and has a direct impact on the accuracy of the final restored 3D structure.There are two main types of image sequence alignment,fiducial marker-based alignment and marker-free alignment.Although the fiducial marker-based alignment is currently the most accurate alignment method,there is still room for optimization.In the fiducial marker-based alignment for image sequences,the bundle adjustment method,as the core link,plays an indispensable role,which essentially optimizes the parameters by minimizing the error of reprojection.The traditional bundle adjustment solves the equation of the reprojection error based on the L2 norm,which is usually solved by nonlinear least squares method.When collecting data through cryo-electron microscopy,real data sets usually have acquisition errors,such as manual operation instrument errors,sample protrusion deformation caused by electron irradiation of cryo-electron microscope electron beams,etc.Therefore,it is very important to improve the robustness of the bundle adjustment algorithm in the image alignment process.Although methods based on the L2 norm have been developed and can quickly minmize the reprojection error,they are easily affected by outliers.For this reason,this paper considers replacing the L2 norm in the traditional binding adjustment algorithm with the L1 norm,and for the first time develops a binding adjustment algorithm based on the L1 norm that can be applied to the alignment of cryo-electron microscopy image sequences.Although the method based on L1 norm is more robust than the method based on L2 norm,the objective function based on L1 norm cannot be derived.In order to solve this problem,we use the interior point method to transform the original problem into an approximately equivalent problem that can be derived.We design a new update strategy based on the L1 norm,so that the step size obtained in each cycle can reduce the error of the reprojection of the L1 norm,so as to obtain the optimal parameter solution.In the cycle of the algorithm,the weight of the approximation problem is gradually increased by using the barrier method,making it more and more close to the original problem.In the cycle of the barrier method,we use the global Newton iteration method to solve the iterative step size in the descending direction,and ensure that each step is strictly limited within the feasible region.This bundle adjustment algorithm based on L1 norm reduces the weight of outliers,thereby reducing the adverse effect of outliers in the data on the results,it shows that the algorithm has good robustness.We found that the algorithm will perform a large number of multiplication operations of large sparse matrices in multiple cycles,and when the number of cycles is sufficient,the final result will not change much,therefore,we set appropriate termination parameters in the algorithm to effectively improve the efficiency of the algorithm.In the experiments and tests,compared with the traditional L2 norm bundle adjustment algorithm and the robust MADN algorithm,our algorithm has high accuracy and optimal robustness even on datasets containing a large number of outliers(30%).Taken together,the bundled adjustment algorithm based on the L1 norm we implemented can obtain highquality aligned image sequences,which can further improve the accuracy of structures obtained by cryo-electron microscopy 3D reconstruction.Finally,our code is open source so that everyone can discuss and improve it together.