Conjugate-Gradient Preconditioning Methods for Shift-Variant PET Image Reconstruction

Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e., for those with approximately block-Toeplitz or block-circulant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging, the Hessian of the weighted least-squares objective function is quite shift-variant, and circulant preconditioners perform poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shift-variant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. We also propose a new efficient method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85979/1/Fessler85.pd

Preconditioning Methods for Shift-Variant Image Reconstruction

Preconditioning methods can accelerate the convergence of gradient-based iterative methods for tomographic image reconstruction and image restoration. Circulant preconditioners have been used extensively for shift-invariant problems. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. For inverse problems that are approximately shift-invariant (i.e. approximately block-Toeplitz or block-circulant Hessians), circulant or Fourier-based preconditioners can provide remarkable acceleration. However, in applications with nonuniform noise variance (such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging), the Hessian of the (penalized) weighted least-squares objective function is quite shift-variant, and the Fourier preconditioner performs poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that more accurately approximate the Hessian matrices of shift-variant imaging problems. Compared to diagonal or Fourier preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. Applications to position emission tomography (PET) illustrate the method.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85856/1/Fessler147.pd

Structural Variability from Noisy Tomographic Projections

In cryo-electron microscopy, the 3D electric potentials of an ensemble of molecules are projected along arbitrary viewing directions to yield noisy 2D images. The volume maps representing these potentials typically exhibit a great deal of structural variability, which is described by their 3D covariance matrix. Typically, this covariance matrix is approximately low-rank and can be used to cluster the volumes or estimate the intrinsic geometry of the conformation space. We formulate the estimation of this covariance matrix as a linear inverse problem, yielding a consistent least-squares estimator. For $n$ images of size $N$-by-$N$ pixels, we propose an algorithm for calculating this covariance estimator with computational complexity $\mathcal{O}(nN^4+\sqrt{\kappa}N^6 \log N)$, where the condition number $\kappa$ is empirically in the range $10$--$200$. Its efficiency relies on the observation that the normal equations are equivalent to a deconvolution problem in 6D. This is then solved by the conjugate gradient method with an appropriate circulant preconditioner. The result is the first computationally efficient algorithm for consistent estimation of 3D covariance from noisy projections. It also compares favorably in runtime with respect to previously proposed non-consistent estimators. Motivated by the recent success of eigenvalue shrinkage procedures for high-dimensional covariance matrices, we introduce a shrinkage procedure that improves accuracy at lower signal-to-noise ratios. We evaluate our methods on simulated datasets and achieve classification results comparable to state-of-the-art methods in shorter running time. We also present results on clustering volumes in an experimental dataset, illustrating the power of the proposed algorithm for practical determination of structural variability.Comment: 52 pages, 11 figure

Fast Predictions of Variance Images for Fan-Beam Transmission Tomography With Quadratic Regularization

Accurate predictions of image variances can be useful for reconstruction algorithm analysis and for the design of regularization methods. Computing the predicted variance at every pixel using matrix-based approximations is impractical. Even most recently adopted methods that are based on local discrete Fourier approximations are impractical since they would require a forward and backprojection and two fast Fourier transform (FFT) calculations for every pixel, particularly for shift-variant systems like fan-beam tomography. This paper describes new "analytical" approaches to predicting the approximate variance maps of 2-D images that are reconstructed by penalized-likelihood estimation with quadratic regularization in fan-beam geometries. The simplest of the proposed analytical approaches requires computation equivalent to one backprojection and some summations, so it is computationally practical even for the data sizes in X-ray computed tomography (CT). Simulation results show that it gives accurate predictions of the variance maps. The parallel-beam geometry is a simple special case of the fan-beam analysis. The analysis is also applicable to 2-D positron emission tomography (PET).Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86007/1/Fessler37.pd

Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page