An inexact Newton-Krylov algorithm for constrained diffeomorphic image registration

We propose numerical algorithms for solving large deformation diffeomorphic image registration problems. We formulate the nonrigid image registration problem as a problem of optimal control. This leads to an infinite-dimensional partial differential equation (PDE) constrained optimization problem. The PDE constraint consists, in its simplest form, of a hyperbolic transport equation for the evolution of the image intensity. The control variable is the velocity field. Tikhonov regularization on the control ensures well-posedness. We consider standard smoothness regularization based on $H^1$- or $H^2$-seminorms. We augment this regularization scheme with a constraint on the divergence of the velocity field rendering the deformation incompressible and thus ensuring that the determinant of the deformation gradient is equal to one, up to the numerical error. We use a Fourier pseudospectral discretization in space and a Chebyshev pseudospectral discretization in time. We use a preconditioned, globalized, matrix-free, inexact Newton-Krylov method for numerical optimization. A parameter continuation is designed to estimate an optimal regularization parameter. Regularity is ensured by controlling the geometric properties of the deformation field. Overall, we arrive at a black-box solver. We study spectral properties of the Hessian, grid convergence, numerical accuracy, computational efficiency, and deformation regularity of our scheme. We compare the designed Newton-Krylov methods with a globalized preconditioned gradient descent. We study the influence of a varying number of unknowns in time. The reported results demonstrate excellent numerical accuracy, guaranteed local deformation regularity, and computational efficiency with an optional control on local mass conservation. The Newton-Krylov methods clearly outperform the Picard method if high accuracy of the inversion is required.Comment: 32 pages; 10 figures; 9 table

Real-time diffuse optical tomography using reduced-order light propagation models based on a priori anatomical and functional information

This paper proposes a new fast 3D image reconstruction algorithm for Diffuse Optical Tomography using reduced order polynomial mappings from the space of optical tissue parameters into the space of flux measurements at the detector locations. The polynomial mappings are constructed through an iterative estimation process involving structure detection, parameter estimation and cross-validation using data generated by simulating a diffusion approximation of the radiative transfer equation incorporating a priori anatomical and functional information provided by MR scans and prior psychological evidence. Numerical simulation studies demonstrate that reconstructed images are remarkably similar in quality as those obtained using the standard approach, but obtained at a fraction of the time

A Levinson-Galerkin algorithm for regularized trigonometric approximation

Trigonometric polynomials are widely used for the approximation of a smooth function $f$ from a set of nonuniformly spaced samples $\{f(x_j)\}_{j=0}^{N-1}$. If the samples are perturbed by noise, controlling the smoothness of the trigonometric approximation becomes an essential issue to avoid overfitting and underfitting of the data. Using the polynomial degree as regularization parameter we derive a multi-level algorithm that iteratively adapts to the least squares solution of optimal smoothness. The proposed algorithm computes the solution in at most $\cal{O}(NM + M^2)$ operations ($M$ being the polynomial degree of the approximation) by solving a family of nested Toeplitz systems. It is shown how the presented method can be extended to multivariate trigonometric approximation. We demonstrate the performance of the algorithm by applying it in echocardiography to the recovery of the boundary of the Left Ventricle

Magnetic resonance multitasking for motion-resolved quantitative cardiovascular imaging.

Quantitative cardiovascular magnetic resonance (CMR) imaging can be used to characterize fibrosis, oedema, ischaemia, inflammation and other disease conditions. However, the need to reduce artefacts arising from body motion through a combination of electrocardiography (ECG) control, respiration control, and contrast-weighting selection makes CMR exams lengthy. Here, we show that physiological motions and other dynamic processes can be conceptualized as multiple time dimensions that can be resolved via low-rank tensor imaging, allowing for motion-resolved quantitative imaging with up to four time dimensions. This continuous-acquisition approach, which we name cardiovascular MR multitasking, captures - rather than avoids - motion, relaxation and other dynamics to efficiently perform quantitative CMR without the use of ECG triggering or breath holds. We demonstrate that CMR multitasking allows for T1 mapping, T1-T2 mapping and time-resolved T1 mapping of myocardial perfusion without ECG information and/or in free-breathing conditions. CMR multitasking may provide a foundation for the development of setup-free CMR imaging for the quantitative evaluation of cardiovascular health

Four-dimensional Cone Beam CT Reconstruction and Enhancement using a Temporal Non-Local Means Method

Four-dimensional Cone Beam Computed Tomography (4D-CBCT) has been developed to provide respiratory phase resolved volumetric imaging in image guided radiation therapy (IGRT). Inadequate number of projections in each phase bin results in low quality 4D-CBCT images with obvious streaking artifacts. In this work, we propose two novel 4D-CBCT algorithms: an iterative reconstruction algorithm and an enhancement algorithm, utilizing a temporal nonlocal means (TNLM) method. We define a TNLM energy term for a given set of 4D-CBCT images. Minimization of this term favors those 4D-CBCT images such that any anatomical features at one spatial point at one phase can be found in a nearby spatial point at neighboring phases. 4D-CBCT reconstruction is achieved by minimizing a total energy containing a data fidelity term and the TNLM energy term. As for the image enhancement, 4D-CBCT images generated by the FDK algorithm are enhanced by minimizing the TNLM function while keeping the enhanced images close to the FDK results. A forward-backward splitting algorithm and a Gauss-Jacobi iteration method are employed to solve the problems. The algorithms are implemented on GPU to achieve a high computational efficiency. The reconstruction algorithm and the enhancement algorithm generate visually similar 4D-CBCT images, both better than the FDK results. Quantitative evaluations indicate that, compared with the FDK results, our reconstruction method improves contrast-to-noise-ratio (CNR) by a factor of 2.56~3.13 and our enhancement method increases the CNR by 2.75~3.33 times. The enhancement method also removes over 80% of the streak artifacts from the FDK results. The total computation time is ~460 sec for the reconstruction algorithm and ~610 sec for the enhancement algorithm on an NVIDIA Tesla C1060 GPU card.Comment: 20 pages, 3 figures, 2 table