SIAM J. Imaging Sci.最新文献

{"title":"Compactification of the Rigid Motions Group in Image Processing","authors":"Tamir Bendory, Ido Hadi, N. Sharon","doi":"10.1137/21m1429448","DOIUrl":"https://doi.org/10.1137/21m1429448","url":null,"abstract":"Image processing problems in general, and in particular in the field of single-particle cryo-electron microscopy, often require considering images up to their rotations and translations. Such problems were tackled successfully when considering images up to rotations only, using quantities which are invariant to the action of rotations on images. Extending these methods to cases where translations are involved is more complicated. Here we present a computationally feasible and theoretically sound approximate invariant to the action of rotations and translations on images. It allows one to approximately reduce image processing problems to similar problems over the sphere, a compact domain acted on by the group of 3D rotations, a compact group. We show that this invariant is induced by a family of mappings deforming, and thereby compactifying, the group structure of rotations and translations of the plane, i.e., the group of rigid motions, into the group of 3D rotations. Furthermore, we demonstrate its viability in two image processing tasks: multi-reference alignment and classification. To our knowledge, this is the first instance of a quantity that is either exactly or approximately invariant to rotations and translations of images that both rests on a sound theoretical foundation and also applicable in practice.","PeriodicalId":185319,"journal":{"name":"SIAM J. Imaging Sci.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128784915","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An Image Registration Model in Electron Backscatter Diffraction","authors":"M. Graf, Sebastian Neumayer, R. Hielscher, G. Steidl, M. Liesegang, T. Beck","doi":"10.1137/21m1426353","DOIUrl":"https://doi.org/10.1137/21m1426353","url":null,"abstract":"Variational methods were successfully applied for registration of gray and RGB-valued image sequences. A common assumption in these models is that pixel-values do not change under transformations. Nowadays, modern image acquisition techniques such as electron backscatter tomography (EBSD), which is used in material sciences, can capture images with values in nonlinear spaces. Here, the image values belong to the quotient space SO(3)/S of the special orthogonal group modulo the discrete symmetry group of the crystal. For such data, the assumption that pixel-values remain unchanged under transformations appears to be no longer valid. Hence, we propose a variational model for the registration of SO(3)/S-valued image sequences, taking the dependence of pixel-values on the transformation into account. More precisely, the data is transformed according to the rotation part in the polar decomposition of the Jacobian of the transformation. To model non-smooth transformations without obtaining so-called staircasing effects, we propose to use a total generalized variation like prior. Then, we prove existence of a minimizer for our model and explain how it can be discretized and minimized by a primal-dual algorithm. Numerical examples illustrate the performance of our method.","PeriodicalId":185319,"journal":{"name":"SIAM J. Imaging Sci.","volume":"140 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127326597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Limits of Accuracy for Parameter Estimation and Localization in Single-Molecule Microscopy via Sequential Monte Carlo Methods","authors":"A. M. d'Avigneau, S. S. Singh, R. Ober","doi":"10.1137/21m1422823","DOIUrl":"https://doi.org/10.1137/21m1422823","url":null,"abstract":"Assessing the quality of parameter estimates for models describing the motion of single molecules in cellular environments is an important problem in fluorescence microscopy. In this work, we consider the fundamental data model, where molecules emit photons at random time instances and these photons arrive at random locations on the detector according to complex point spread functions (PSFs). The randomness and non-Gaussian PSF of the detection process, and the random trajectory of the molecule, makes inference challenging. Moreover, the presence of other closely spaced molecules causes further uncertainty in the origin of the measurements, which impacts the statistical precision of the estimates. We quantify the limits of accuracy of model parameter estimates and separation distance between closely spaced molecules (known as the resolution problem) by computing the Cramér-Rao lower bound (CRLB), or equivalently the inverse of the Fisher information matrix (FIM), for the variance of estimates. Results on the CRLB obtained from the fundamental model are crucial, in that they provide a lower bound for more practical scenarios. While analytic expressions for the FIM can be derived for static and deterministically moving molecules, the analytical tools to evaluate the FIM for molecules whose trajectories follow stochastic differential equations (SDEs) are still for the most part missing. We address this by presenting a general sequential Monte Carlo (SMC) based methodology for both parameter inference and computing the desired accuracy limits for non-static molecules and a non-Gaussian fundamental detection model. For the first time, we are able to estimate the FIM for stochastically moving molecules observed through the Airy and Born and Wolf detection models. This is achieved by estimating the score and observed information matrix via SMC. We summarise the outcome of our numerical work by delineating the qualitative behaviours for the accuracy limits as functions of various experimental settings like collected photon count, molecule diffusion, etc. We also verify that we can recover known results from the static molecule case.","PeriodicalId":185319,"journal":{"name":"SIAM J. Imaging Sci.","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125415589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Variational Inequality Model for the Construction of Signals from Inconsistent Nonlinear Equations","authors":"P. L. Combettes, Zev Woodstock","doi":"10.1137/21m1420368","DOIUrl":"https://doi.org/10.1137/21m1420368","url":null,"abstract":"Building up on classical linear formulations, we posit that a broad class of problems in signal synthesis and in signal recovery are reducible to the basic task of finding a point in a closed convex subset of a Hilbert space that satisfies a number of nonlinear equations involving firmly nonexpansive operators. We investigate this formalism in the case when, due to inaccurate modeling or perturbations, the nonlinear equations are inconsistent. A relaxed formulation of the original problem is proposed in the form of a variational inequality. The properties of the relaxed problem are investigated and a provenly convergent block-iterative algorithm, whereby only blocks of the underlying firmly nonexpansive operators are activated at a given iteration, is devised to solve it. Numerical experiments illustrate robust recoveries in several signal and image processing applications.","PeriodicalId":185319,"journal":{"name":"SIAM J. Imaging Sci.","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123917617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Modulo Radon Transform: Theory, Algorithms and Applications","authors":"Matthias Beckmann, A. Bhandari, F. Krahmer","doi":"10.1137/21m1424615","DOIUrl":"https://doi.org/10.1137/21m1424615","url":null,"abstract":"Recently, experiments have been reported where researchers were able to perform high dynamic range (HDR) tomography in a heuristic fashion, by fusing multiple tomographic projections. This approach to HDR tomography has been inspired by HDR photography and inherits the same disadvantages. Taking a computational imaging approach to the HDR tomography problem, we here suggest a new model based on the Modulo Radon Transform (MRT), which we rigorously introduce and analyze. By harnessing a joint design between hardware and algorithms, we present a single-shot HDR tomography approach, which to our knowledge, is the only approach that is backed by mathematical guarantees. On the hardware front, instead of recording the Radon Transform projections that my potentially saturate, we propose to measure modulo values of the same. This ensures that the HDR measurements are folded into a lower dynamic range. On the algorithmic front, our recovery algorithms reconstruct the HDR images from folded measurements. Beyond mathematical aspects such as injectivity and inversion of the MRT for different scenarios including band-limited and approximately compactly supported images, we also provide a first proof-of-concept demonstration. To do so, we implement MRT by experimentally folding tomographic measurements available as an open source data set using our custom designed modulo hardware. Our reconstruction clearly shows the advantages of our approach for experimental data. In this way, our MRT based solution paves a path for HDR acquisition in a number of related imaging problems.","PeriodicalId":185319,"journal":{"name":"SIAM J. Imaging Sci.","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125484813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A New Variational Model for Shape Graph Registration with Partial Matching Constraints","authors":"Yashil Sukurdeep, Martin Bauer, N. Charon","doi":"10.1137/21m1418587","DOIUrl":"https://doi.org/10.1137/21m1418587","url":null,"abstract":"This paper introduces a new extension of Riemannian elastic curve matching to a general class of geometric structures, which we call (weighted) shape graphs, that allows for shape registration with partial matching constraints and topological inconsistencies. Weighted shape graphs are the union of an arbitrary number of component curves in Euclidean space with potential connectivity constraints between some of their boundary points, together with a weight function defined on each component curve. The framework of higher order invariant Sobolev metrics is particularly well suited for constructing notions of distances and geodesics between unparametrized curves. The main difficulty in adapting this framework to the setting of shape graphs is the absence of topological consistency, which typically results in an inadequate search for an exact matching between two shape graphs. We overcome this hurdle by defining an inexact variational formulation of the matching problem between (weighted) shape graphs of any underlying topology, relying on the convenient measure representation given by varifolds to relax the exact matching constraint. We then prove the existence of minimizers to this variational problem when we choose Sobolev metrics of sufficient regularity and a total variation (TV) regularization on the weight function. We propose a numerical optimization approach which adapts the smoothed fast iterative shrinkage-thresholding (SFISTA) algorithm to deal with TV norm minimization and allows us to reduce the matching problem to solving a sequence of smooth unconstrained minimization problems. We finally illustrate the capabilities of our new model through several examples showcasing its ability to tackle partially observed and topologically varying data.","PeriodicalId":185319,"journal":{"name":"SIAM J. Imaging Sci.","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129631130","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Harmonic Beltrami Signature: A Novel 2D Shape Representation for Object Classification","authors":"Chen-Hsuan Lin, L. Lui","doi":"10.1137/22m1470852","DOIUrl":"https://doi.org/10.1137/22m1470852","url":null,"abstract":"There is a growing interest in shape analysis in recent years. We present a novel shape signature for 2D bounded simply-connected domains, named the Harmonic Beltrami signature (HBS). The proposed signature is based on the harmonic extension of the conformal welding map of a unit circle and its Beltrami coefficient. We show that there is a one-to-one correspondence between the quotient space of HBS and the space of 2D simply-connected shapes up to a translation, rotation and scaling. With a suitable normalization, each equivalence class in the quotient space of HBS is associated to a unique representative. It gets rid of the conformal ambiguity. As such, each shape is associated to a unique HBS. Conversely, the associated shape of a HBS can be reconstructed based on quasiconformal Teichmuller theories, which is uniquely determined up to a translation, rotation and scaling. The HBS is thus an effective fingerprint to represent a 2D shape. The robustness of HBS is studied both theoretically and experimentally. With the HBS, simple metric, such as L2, can be used to measure geometric dissimilarity between shapes. Experiments have been carried out to classify shapes in different classes using HBS. Results show good classification performance, which demonstrate the efficacy of our proposed shape signature.","PeriodicalId":185319,"journal":{"name":"SIAM J. Imaging Sci.","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132599748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A PDE-based Method for Shape Registration","authors":"Esten Nicolai Wøien, M. Grasmair","doi":"10.1137/21m1408932","DOIUrl":"https://doi.org/10.1137/21m1408932","url":null,"abstract":"In the square root velocity framework, the computation of shape space distances and the registration of curves requires solution of a non-convex variational problem. In this paper, we present a new PDE-based method for solving this problem numerically. The method is constructed from numerical approximation of the Hamilton-Jacobi-Bellman equation for the variational problem, and has quadratic complexity and global convergence for the distance estimate. In conjunction, we propose a backtracking scheme for approximating solutions of the registration problem, which additionally can be used to compute shape space geodesics. The methods have linear numerical convergence, and improved efficiency compared previous global solvers.","PeriodicalId":185319,"journal":{"name":"SIAM J. Imaging Sci.","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126814462","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bayesian Imaging With Data-Driven Priors Encoded by Neural Networks: Theory, Methods, and Algorithms","authors":"M. Holden, M. Pereyra, K. Zygalakis","doi":"10.1137/21m1406313","DOIUrl":"https://doi.org/10.1137/21m1406313","url":null,"abstract":"This paper proposes a new methodology for performing Bayesian inference in imaging inverse problems where the prior knowledge is available in the form of training data. Following the manifold hypothesis and adopting a generative modelling approach, we construct a data-driven prior that is supported on a sub-manifold of the ambient space, which we can learn from the training data by using a variational autoencoder or a generative adversarial network. We establish the existence and well-posedness of the associated posterior distribution and posterior moments under easily verifiable conditions, providing a rigorous underpinning for Bayesian estimators and uncertainty quantification analyses. Bayesian computation is performed by using a parallel tempered version of the preconditioned Crank-Nicolson algorithm on the manifold, which is shown to be ergodic and robust to the non-convex nature of these data-driven models. In addition to point estimators and uncertainty quantification analyses, we derive a model misspecification test to automatically detect situations where the data-driven prior is unreliable, and explain how to identify the dimension of the latent space directly from the training data. The proposed approach is illustrated with a range of experiments with the MNIST dataset, where it outperforms alternative image reconstruction approaches from the state of the art. A model accuracy analysis suggests that the Bayesian probabilities reported by the data-driven models are also remarkably accurate under a frequentist definition of probability.","PeriodicalId":185319,"journal":{"name":"SIAM J. Imaging Sci.","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124499492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Solving Inverse Problems by Joint Posterior Maximization with Autoencoding Prior","authors":"Mario Gonz'alez, Andrés Almansa, Pauline Tan","doi":"10.1137/21M140225X","DOIUrl":"https://doi.org/10.1137/21M140225X","url":null,"abstract":"In this work we address the problem of solving ill-posed inverse problems in imaging where the prior is a variational autoencoder (VAE). Specifically we consider the decoupled case where the prior is trained once and can be reused for many different log-concave degradation models without retraining. Whereas previous MAP-based approaches to this problem lead to highly non-convex optimization algorithms, our approach computes the joint (space-latent) MAP that naturally leads to alternate optimization algorithms and to the use of a stochastic encoder to accelerate computations. The resulting technique (JPMAP) performs Joint Posterior Maximization using an Autoencoding Prior. We show theoretical and experimental evidence that the proposed objective function is quite close to bi-convex. Indeed it satisfies a weak bi-convexity property which is sufficient to guarantee that our optimization scheme converges to a stationary point. We also highlight the importance of correctly training the VAE using a denoising criterion, in order to ensure that the encoder generalizes well to out-of-distribution images, without affecting the quality of the generative model. This simple modification is key to providing robustness to the whole procedure. Finally we show how our joint MAP methodology relates to more common MAP approaches, and we propose a continuation scheme that makes use of our JPMAP algorithm to provide more robust MAP estimates. Experimental results also show the higher quality of the solutions obtained by our JPMAP approach with respect to other non-convex MAP approaches which more often get stuck in spurious local optima.","PeriodicalId":185319,"journal":{"name":"SIAM J. Imaging Sci.","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121239144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}