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

Spatial Resolution and Noise Properties of Regularized Motion-Compensated Image Reconstruction

Reducing motion artifacts is an important problem in medical image reconstruction. Using gating to partition data into separate frames can reduce motion artifacts but can increase noise in images reconstructed from individual frames. One can pool the frames to reduce noise by using motion-compensated image reconstruction (MCIR) methods. MCIR methods have been studied in many medical imaging modalities to reduce both noise and motion artifacts. However, there has been less analysis of the spatial resolution and noise properties of MCIR methods. This paper analyzes the spatial resolution and noise properties of MCIR methods based on a general parametric motion model. For simplicity we consider the motion to be given. We present a method to choose quadratic spatial regularization parameters to provide predictable resolution properties that are independent of the object and the motion. The noise analysis shows that the estimator variance depends on both the measurement covariance and the Jacobian determinant values of the motion. A 2D PET simulation demonstrates the theoretical results.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85889/1/Fessler240.pd

Joint Image Reconstruction and Nonrigid Motion Estimation with a Simple Penalty That Encourages Local Invertibility

Motion artifacts are a significant issue in medical image reconstruction. There are many methods for incorporating motion information into image reconstruction. However, there are fewer studies that focus on deformation regularization in motioncompensated image reconstruction. The usual choice for deformation regularization has been penalty functions based on the assumption that tissues are elastic. In the image registration field, there have been some methods proposed that impose deformation invertibility using constraints or regularization, assuming that organ motions are invertible transformations. However, most of these methods require very high memory or computation complexity, making them poorly suited for dealing with multiple images simultaneously in motion-compensated image reconstruction. Recently we proposed an image registration method that uses a simple penalty function based on a sufficient condition for the local invertibility of deformations.1 That approach encourages local invertibility in a fast and memory-efficient way. This paper investigates the use of that regularization method for the more challenging problem of joint image reconstruction and nonrigid motion estimation. A 2D PET simulation (based on realistic motion from real patient CT data) demonstrates the benefits of such motion regularization for joint image reconstruction/registration.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85929/1/Fessler237.pd

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

On Ambiguity in Linear Inverse Problems: Entrywise Bounds on Nearly Data-Consistent Solutions and Entrywise Condition Numbers

Ill-posed linear inverse problems appear frequently in various signal processing applications. It can be very useful to have theoretical characterizations that quantify the level of ill-posedness for a given inverse problem and the degree of ambiguity that may exist about its solution. Traditional measures of ill-posedness, such as the condition number of a matrix, provide characterizations that are global in nature. While such characterizations can be powerful, they can also fail to provide full insight into situations where certain entries of the solution vector are more or less ambiguous than others. In this work, we derive novel theoretical lower- and upper-bounds that apply to individual entries of the solution vector, and are valid for all potential solution vectors that are nearly data-consistent. These bounds are agnostic to the noise statistics and the specific method used to solve the inverse problem, and are also shown to be tight. In addition, our results also lead us to introduce an entrywise version of the traditional condition number, which provides a substantially more nuanced characterization of scenarios where certain elements of the solution vector are less sensitive to perturbations than others. Our results are illustrated in an application to magnetic resonance imaging reconstruction, and we include discussions of practical computation methods for large-scale inverse problems, connections between our new theory and the traditional Cram\'{e}r-Rao bound under statistical modeling assumptions, and potential extensions to cases involving constraints beyond just data-consistency

Benefits of Using a Spatially-variant Penalty Strength with Anatomical Priors in PET Reconstruction

In this study, we explore the use of a spatiallyvariant penalty strength in penalized image reconstruction using anatomical priors to reduce the dependence of lesion contrast on surrounding activity and lesion location. This work builds on a previous method to make the local perturbation response (LPR) approximately spatially invariant. While the dependence of lesion contrast on the local properties introduced by the anatomical penalty is intentional, the method aims to reduce the influence from surroundings lying along the lines of response (LORs) but not in the penalty neighborhood structure. The method is evaluated using simulated data, assuming that the anatomical information is absent or well-aligned with the corresponding activity images. Since the Parallel Level Sets (PLS) penalty is convex and has shown promising results in the literature, it is chosen as the representative anatomical penalty and incorporated into the previously proposed preconditioned algorithm (L-BFGSB-PC) for achieving both good image quality and fast convergence rate. A 2D disc phantom with a feature at the center and a 3D XCAT thorax phantom with lesions inserted in different slices are used to study how surrounding activity and lesion location affect the visual appearance and quantitative consistency. A bias and noise analysis is also performed with the 2D disc phantom. The consistency of the algorithm convergence rate with respect to different data noise and background levels is also investigated using the XCAT phantom. Finally, an example reconstruction for a patient dataset with inserted pseudo lesions is used as a demonstration in a clinical context. We show that applying the spatially-variant penalization with PLS can reduce the dependence of the lesion contrast on the surrounding activity and lesion location. It does not affect the bias and noise trade-off curves for matched local resolution. Moreover, when using the proposed penalization, significant improvement in algorithm convergence rate and convergence consistency is observed

X-ray CT Image Reconstruction on Highly-Parallel Architectures.

Model-based image reconstruction (MBIR) methods for X-ray CT use accurate models of the CT acquisition process, the statistics of the noisy measurements, and noise-reducing regularization to produce potentially higher quality images than conventional methods even at reduced X-ray doses. They do this by minimizing a statistically motivated high-dimensional cost function; the high computational cost of numerically minimizing this function has prevented MBIR methods from reaching ubiquity in the clinic. Modern highly-parallel hardware like graphics processing units (GPUs) may offer the computational resources to solve these reconstruction problems quickly, but simply "translating" existing algorithms designed for conventional processors to the GPU may not fully exploit the hardware's capabilities. This thesis proposes GPU-specialized image denoising and image reconstruction algorithms. The proposed image denoising algorithm uses group coordinate descent with carefully structured groups. The algorithm converges very rapidly: in one experiment, it denoises a 65 megapixel image in about 1.5 seconds, while the popular Chambolle-Pock primal-dual algorithm running on the same hardware takes over a minute to reach the same level of accuracy. For X-ray CT reconstruction, this thesis uses duality and group coordinate ascent to propose an alternative to the popular ordered subsets (OS) method. Similar to OS, the proposed method can use a subset of the data to update the image. Unlike OS, the proposed method is convergent. In one helical CT reconstruction experiment, an implementation of the proposed algorithm using one GPU converges more quickly than a state-of-the-art algorithm converges using four GPUs. Using four GPUs, the proposed algorithm reaches near convergence of a wide-cone axial reconstruction problem with over 220 million voxels in only 11 minutes.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113551/1/mcgaffin_1.pd