Regularized Methods for Topology-Preserving Smooth Nonrigid Image Registration Using B-Spline Basis

B-splines are a convenient tool for nonrigid registration, but ensuring invertibility can be challenge. This paper describes a new penalty method that is devised to enforce a sufficient condition for local invertibility and smoothness of nth order B-spline based deformations. Traditional direct Jacobian penalty methods penalize negative Jacobian determinant values only at grid points. In contrast, our new penalty method enforces the sufficient condition for invertibility directly on the B-spline coefficients by using a modified quadratic penalty function so that it enforces invertibility globally over a 3D continuous domain. This approach also saves computation time and memory compared to using Jacobian determinant values. We apply this method to 3D CT images of a thorax at inhale and exhale.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85874/1/Fessler232.pd

A Simple Regularizer for B-spline Nonrigid Image Registration That Encourages Local Invertibility

Nonrigid image registration is an important task for many medical imaging applications. In particular, for radiation oncology it is desirable to track respiratory motion for thoracic cancer treatment. B-splines are convenient for modeling nonrigid deformations, but ensuring invertibility can be a challenge. This paper describes sufficient conditions for local invertibility of deformations based on B-spline bases. These sufficient conditions can be used with constrained optimization to enforce local invertibility. We also incorporate these conditions into nonrigid image registration methods based on a simple penalty approach that encourages diffeomorphic deformations. Traditional Jacobian penalty methods penalize negative Jacobian determinant values only at grid points. In contrast, our new method enforces a sufficient condition for invertibility directly on the deformation coefficients to encourage invertibility globally over a 3-D continuous domain. The proposed penalty approach requires substantially less compute time than Jacobian penalties per iteration.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85951/1/Fessler21.pd

Nonrigid Registration Using Regularization that Accomodates Local Tissue Rigidity

Regularized nonrigid medical image registration algorithms usually estimate the deformation by minimizing a cost function, consisting of a similarity measure and a penalty term that discourages “unreasonable” deformations. Conventional regularization methods enforce homogeneous smoothness properties of the deformation field; less work has been done to incorporate tissue-type-specific elasticity information. Yet ignoring the elasticity differences between tissue types can result in non-physical results, such as bone warping. Bone structures should move rigidly (locally), unlike the more elastic deformation of soft issues. Existing solutions for this problem either treat different regions of an image independently, which requires precise segmentation and incurs boundary issues; or use an empirical spatial varying “filter” to “correct” the deformation field, which requires the knowledge of a stiffness map and departs from the cost-function formulation. We propose a new approach to incorporate tissue rigidity information into the nonrigid registration problem, by developing a space variant regularization function that encourages the local Jacobian of the deformation to be a nearly orthogonal matrix in rigid image regions, while allowing more elastic deformations elsewhere. For the case of X-ray CT data, we use a simple monotonic increasing function of the CT numbers (in HU) as a “rigidity index” since bones typically have the highest CT numbers. Unlike segmentation-based methods, this approach is flexible enough to account for partial volume effects. Results using a B-spline deformation parameterization illustrate that the proposed approach improves registration accuracy in inhale-exhale CT scans with minimal computational penalty.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85935/1/Fessler216.pd

A Simple Penalty that Encourages Local Invertibility and Considers Sliding Effects for Respiratory Motion

Nonrigid image registration is a key tool in medical imaging. Because of high degrees of freedom in nonrigid transforms, there have been many efforts to regularize the deformation based on some reasonable assumptions. Especially, motion invertibility and local tissue rigidity have been investigated as reasonable priors in image registration. There have been several papers on exploiting each constraint separately. These constraints are reasonable in respiratory motion estimation because breathing motion is invertible and there are some rigid structures such as bones. Using both constraints seems very attractive in respiratory motion registration since using invertibility prior alone usually causes bone warping in ribs. Using rigidity prior seems natural and straightforward. However, the “sliding effect” near the interface between rib cage and diaphragm makes problem harder because it is not locally invertible. In this area, invertibility and rigidity priors have opposite forces. Recently, we proposed a simple piecewise quadratic penalty that encourages the local invertibility of motions. In this work we relax this penalty function by using a Geman-type function that allows the deformation to be piecewise smooth instead of globally smooth. This allows the deformation to be discontinuous in the area of the interface between rib cage and diaphragm. With some small sacrifice of regularity, we could achieve more realistic discontinuous motion near diaphragm, better data fitting error as well as less bone warping. We applied this Geman-type function penalty only to the x- and y-direction partial derivatives of the z-direction deformation to address the sliding effect. 192 × 128 × 128 3D CT inhale and exhale images of a real patient were used to show the benefits of this new penalty method.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85922/1/Fessler238.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

Robust Cardiac Motion Estimation using Ultrafast Ultrasound Data: A Low-Rank-Topology-Preserving Approach

Cardiac motion estimation is an important diagnostic tool to detect heart diseases and it has been explored with modalities such as MRI and conventional ultrasound (US) sequences. US cardiac motion estimation still presents challenges because of the complex motion patterns and the presence of noise. In this work, we propose a novel approach to estimate the cardiac motion using ultrafast ultrasound data. -- Our solution is based on a variational formulation characterized by the L2-regularized class. The displacement is represented by a lattice of b-splines and we ensure robustness by applying a maximum likelihood type estimator. While this is an important part of our solution, the main highlight of this paper is to combine a low-rank data representation with topology preservation. Low-rank data representation (achieved by finding the k-dominant singular values of a Casorati Matrix arranged from the data sequence) speeds up the global solution and achieves noise reduction. On the other hand, topology preservation (achieved by monitoring the Jacobian determinant) allows to radically rule out distortions while carefully controlling the size of allowed expansions and contractions. Our variational approach is carried out on a realistic dataset as well as on a simulated one. We demonstrate how our proposed variational solution deals with complex deformations through careful numerical experiments. While maintaining the accuracy of the solution, the low-rank preprocessing is shown to speed up the convergence of the variational problem. Beyond cardiac motion estimation, our approach is promising for the analysis of other organs that experience motion.Comment: 15 pages, 10 figures, Physics in Medicine and Biology, 201

Discontinuity Preserving Regularization for Modeling Sliding in Medical Image Registration

Sliding effects often occur along tissue/organ boundaries. However, most conventional registration techniques either use smooth parametric bases or apply homogeneous smoothness regularization, and fail to address the sliding issue. In this study, we propose a class of discontinuity-preserving regularizers that fit naturally into optimization-based registration. The proposed regularization encourages smooth deformations in most regions, but preserves large discontinuities supported by the data. Variational techniques are used to derive the descending flows. We discuss general conditions on such discontinuity-preserving regularizers, and their properties based on an anisotropic filtering interpretation. Preliminary tests with 2D CT data show promising results.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85986/1/Fessler234.pd

Robust Non-Rigid Registration with Reweighted Position and Transformation Sparsity

Non-rigid registration is challenging because it is ill-posed with high degrees of freedom and is thus sensitive to noise and outliers. We propose a robust non-rigid registration method using reweighted sparsities on position and transformation to estimate the deformations between 3-D shapes. We formulate the energy function with position and transformation sparsity on both the data term and the smoothness term, and define the smoothness constraint using local rigidity. The double sparsity based non-rigid registration model is enhanced with a reweighting scheme, and solved by transferring the model into four alternately-optimized subproblems which have exact solutions and guaranteed convergence. Experimental results on both public datasets and real scanned datasets show that our method outperforms the state-of-the-art methods and is more robust to noise and outliers than conventional non-rigid registration methods.Comment: IEEE Transactions on Visualization and Computer Graphic