Unbiasing Procedures for Scale-invariant Multi-reference Alignment

This article discusses a generalization of the 1dimensional multi-reference alignment problem. The goal is to recover a hidden signal from many noisy observations, where each noisy observation includes a random translation and random dilation of the hidden signal, as well as high additive noise. We propose a method that recovers the power spectrum of the hidden signal by applying a data-driven, nonlinear unbiasing procedure, and thus the hidden signal is obtained up to an unknown phase. An unbiased estimator of the power spectrum is defined, whose error depends on the sample size and noise levels, and we precisely quantify the convergence rate of the proposed estimator. The unbiasing procedure relies on knowledge of the dilation distribution, and we implement an optimization procedure to learn the dilation variance when this parameter is unknown. Our theoretical work is supported by extensive numerical experiments on a wide range of signals.

[1]  A. Singer.  Angular Synchronization by Eigenvectors and Semidefinite Programming. , 2009, Applied and computational harmonic analysis.

[2]  Gregory S. Chirikjian,et al.  A stochastic kinematic model of class averaging in single-particle electron microscopy , 2011, Int. J. Robotics Res..

[3]  Roberto Marabini,et al.  Maximum-likelihood multi-reference refinement for electron microscopy images. , 2005, Journal of molecular biology.

[4]  Lisa M. Brown,et al.  A survey of image registration techniques , 1992, CSUR.

[5]  Roberto Cusani,et al.  A correlation based technique for shift, scale, and rotation independent object identification , 1987, ICASSP '87. IEEE International Conference on Acoustics, Speech, and Signal Processing.

[6]  Manuel Rosa-Zurera,et al.  Using Multilayer Perceptrons to Align High Range Resolution Radar Signals , 2005, ICANN.

[7]  Yuxin Chen,et al.  The Projected Power Method: An Efficient Algorithm for Joint Alignment from Pairwise Differences , 2016, Communications on Pure and Applied Mathematics.

[8]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[9]  Amit Singer,et al.  Multireference alignment using semidefinite programming , 2013, ITCS.

[10]  P. White,et al.  HIGHER-ORDER SPECTRA: THE BISPECTRUM AND TRISPECTRUM , 1998 .

[11]  Amit Singer,et al.  Tightness of the maximum likelihood semidefinite relaxation for angular synchronization , 2014, Math. Program..

[12]  Z. Kam.  The reconstruction of structure from electron micrographs of randomly oriented particles. , 1980, Journal of theoretical biology.

[13]  Nicolas Boumal,et al.  On the low-rank approach for semidefinite programs arising in synchronization and community detection , 2016, COLT.

[14]  Ankur Moitra,et al.  Message‐Passing Algorithms for Synchronization Problems over Compact Groups , 2016, ArXiv.

[15]  Matthew Hirn,et al.  Wavelet invariants for statistically robust multi-reference alignment. , 2021, Information and inference : a journal of the IMA.

[16]  F. Groen,et al.  Fast Translation Invariant Classification of (HRR) Range Profiles in a Zero Phase Representation , 2003 .

[17]  Douglas L. Theobald,et al.  Optimal simultaneous superpositioning of multiple structures with missing data , 2012, Bioinform..

[18]  Zhizhen Zhao,et al.  Bispectrum Inversion With Application to Multireference Alignment , 2017, IEEE Transactions on Signal Processing.

[19]  Hassan Foroosh,et al.  Extension of phase correlation to subpixel registration , 2002, IEEE Trans. Image Process..

[20]  Yutong Chen,et al.  NON-UNIQUE GAMES OVER COMPACT GROUPS AND ORIENTATION ESTIMATION IN CRYO-EM , 2015, Inverse problems.

[21]  L. Hansen.  Large Sample Properties of Generalized Method of Moments Estimators , 1982 .

[22]  Tomás Pajdla,et al.  Robust Rotation and Translation Estimation in Multiview Reconstruction , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[23]  Leonidas J. Guibas,et al.  Near-Optimal Joint Object Matching via Convex Relaxation , 2014, ICML.

[24]  Gregory S. Chirikjian,et al.  An Assembly Automation Approach to Alignment of Noncircular Projections in Electron Microscopy , 2014, IEEE Transactions on Automation Science and Engineering.

[25]  Federico Forneris,et al.  Identifying and Visualizing Macromolecular Flexibility in Structural Biology , 2016, Front. Mol. Biosci..

[26]  Peyman Milanfar,et al.  Optimal Registration Of Aliased Images Using Variable Projection With Applications To Super-Resolution , 2008, Comput. J..

[27]  Robert P. Davey,et al.  NanoOK: multi-reference alignment analysis of nanopore sequencing data, quality and error profiles , 2015, Bioinform..

[28]  Vinod Chandran,et al.  Position, rotation, and scale invariant recognition of images using higher-order spectra , 1992, [Proceedings] ICASSP-92: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[29]  Georgios B. Giannakis,et al.  Translation, rotation and scaling invariant object and texture classification using polyspectra , 1990 .

[30]  Brian M. Sadler,et al.  Shift- and rotation-invariant object reconstruction using the bispectrum , 1992 .

[31]  Afonso S. Bandeira,et al.  Optimal rates of estimation for multi-reference alignment , 2017, Mathematical Statistics and Learning.

[32]  Nicolas Boumal,et al.  Nonconvex Phase Synchronization , 2016, SIAM J. Optim..

[33]  Amit Singer,et al.  Multireference Alignment Is Easier With an Aperiodic Translation Distribution , 2017, IEEE Transactions on Information Theory.

[34]  Takio Kurita,et al.  Scale Invariant Face Detection and Classification Method Using Shift Invariant Features Extracted from Log-Polar Image , 2001 .

[35]  Amit Singer,et al.  Method of moments for 3D single particle ab initio modeling with non-uniform distribution of viewing angles , 2019, Inverse Problems.

[36]  Nicolas Boumal,et al.  Near-Optimal Bounds for Phase Synchronization , 2017, SIAM J. Optim..