Variational image registration by a total fractional-order variation model

In this paper, a new framework of nonlocal deformation in non-rigid image registration is presented. It is well known that many non-rigid image registration techniques may lead to unsteady deformation (e.g. not one to one) if the dissimilarity between the reference and template images is too large. We present a novel variational framework of the total fractional-order variation to derive the underlying fractional Euler-Lagrange equations and a numerical implementation combining the semi-implicit update and conjugate gradients (CG) solution to solve the nonlinear systems. Numerical experiments show that the new registration not only produces accurate and smooth solutions but also allows for a large rigid alignment, the evaluations of the new model demonstrate substantial improvements in accuracy and robustness over the conventional image registration approaches

A Total Fractional-Order Variation Model for Image Restoration with Non-homogeneous Boundary Conditions and its Numerical Solution

To overcome the weakness of a total variation based model for image restoration, various high order (typically second order) regularization models have been proposed and studied recently. In this paper we analyze and test a fractional-order derivative based total $\alpha$-order variation model, which can outperform the currently popular high order regularization models. There exist several previous works using total $\alpha$-order variations for image restoration; however first no analysis is done yet and second all tested formulations, differing from each other, utilize the zero Dirichlet boundary conditions which are not realistic (while non-zero boundary conditions violate definitions of fractional-order derivatives). This paper first reviews some results of fractional-order derivatives and then analyzes the theoretical properties of the proposed total $\alpha$-order variational model rigorously. It then develops four algorithms for solving the variational problem, one based on the variational Split-Bregman idea and three based on direct solution of the discretise-optimization problem. Numerical experiments show that, in terms of restoration quality and solution efficiency, the proposed model can produce highly competitive results, for smooth images, to two established high order models: the mean curvature and the total generalized variation.Comment: 26 page

Diffeomorphic Metric Mapping and Probabilistic Atlas Generation of Hybrid Diffusion Imaging based on BFOR Signal Basis

We propose a large deformation diffeomorphic metric mapping algorithm to align multiple b-value diffusion weighted imaging (mDWI) data, specifically acquired via hybrid diffusion imaging (HYDI), denoted as LDDMM-HYDI. We then propose a Bayesian model for estimating the white matter atlas from HYDIs. We adopt the work given in Hosseinbor et al. (2012) and represent the q-space diffusion signal with the Bessel Fourier orientation reconstruction (BFOR) signal basis. The BFOR framework provides the representation of mDWI in the q-space and thus reduces memory requirement. In addition, since the BFOR signal basis is orthonormal, the L2 norm that quantifies the differences in the q-space signals of any two mDWI datasets can be easily computed as the sum of the squared differences in the BFOR expansion coefficients. In this work, we show that the reorientation of the $q$-space signal due to spatial transformation can be easily defined on the BFOR signal basis. We incorporate the BFOR signal basis into the LDDMM framework and derive the gradient descent algorithm for LDDMM-HYDI with explicit orientation optimization. Additionally, we extend the previous Bayesian atlas estimation framework for scalar-valued images to HYDIs and derive the expectation-maximization algorithm for solving the HYDI atlas estimation problem. Using real HYDI datasets, we show the Bayesian model generates the white matter atlas with anatomical details. Moreover, we show that it is important to consider the variation of mDWI reorientation due to a small change in diffeomorphic transformation in the LDDMM-HYDI optimization and to incorporate the full information of HYDI for aligning mDWI

p-Laplace Variational Image Inpainting Model Using Riesz Fractional Differential Filter

In this paper, p-Laplace variational image inpainting model with symmetric Riesz fractional differential filter is proposed. Variational inpainting models are very useful to restore many smaller damaged regions of an image. Integer order variational image inpainting models (especially second and fourth order) work well to complete the unknown regions. However, in the process of inpainting with these models, any of the unindented visual effects such as staircasing, speckle noise, edge blurring, or loss in contrast are introduced. Recently, fractional derivative operators were applied by researchers to restore the damaged regions of the image. Experimentation with these operators for variational image inpainting led to the conclusion that second order symmetric Riesz fractional differential operator not only completes the damaged regions effectively, but also reducing unintended effects. In this article, The filling process of damaged regions is based on the fractional central curvature term. The proposed model is compared with integer order variational models and also GrunwaldLetnikov fractional derivative based variational inpainting in terms of peak signal to noise ratio, structural similarity and mutual information

Bayesian Estimation of White Matter Atlas from High Angular Resolution Diffusion Imaging

We present a Bayesian probabilistic model to estimate the brain white matter atlas from high angular resolution diffusion imaging (HARDI) data. This model incorporates a shape prior of the white matter anatomy and the likelihood of individual observed HARDI datasets. We first assume that the atlas is generated from a known hyperatlas through a flow of diffeomorphisms and its shape prior can be constructed based on the framework of large deformation diffeomorphic metric mapping (LDDMM). LDDMM characterizes a nonlinear diffeomorphic shape space in a linear space of initial momentum uniquely determining diffeomorphic geodesic flows from the hyperatlas. Therefore, the shape prior of the HARDI atlas can be modeled using a centered Gaussian random field (GRF) model of the initial momentum. In order to construct the likelihood of observed HARDI datasets, it is necessary to study the diffeomorphic transformation of individual observations relative to the atlas and the probabilistic distribution of orientation distribution functions (ODFs). To this end, we construct the likelihood related to the transformation using the same construction as discussed for the shape prior of the atlas. The probabilistic distribution of ODFs is then constructed based on the ODF Riemannian manifold. We assume that the observed ODFs are generated by an exponential map of random tangent vectors at the deformed atlas ODF. Hence, the likelihood of the ODFs can be modeled using a GRF of their tangent vectors in the ODF Riemannian manifold. We solve for the maximum a posteriori using the Expectation-Maximization algorithm and derive the corresponding update equations. Finally, we illustrate the HARDI atlas constructed based on a Chinese aging cohort of 94 adults and compare it with that generated by averaging the coefficients of spherical harmonics of the ODF across subjects

A Novel Diffeomorphic Model for Image Registration and Its Algorithm

In this work, we investigate image registration by mapping one image to another in a variational framework and focus on both model robustness and solver efficiency. We first propose a new variational model with a special regularizer, based on the quasi-conformal theory, which can guarantee that the registration map is diffeomorphic. It is well known that when the deformation is large, many variational models including the popular diffusion model cannot ensure diffeomorphism. One common observation is that the fidelity error appears small while the obtained transform is incorrect by way of mesh folding. However, direct reformulation from the Beltrami framework does not lead to effective models; our new regularizer is constructed based on this framework and added to the diffusion model to get a new model, which can achieve diffeomorphism. However, the idea is applicable to a wide class of models. We then propose an iterative method to solve the resulting nonlinear optimization problem and prove the convergence of the method. Numerical experiments can demonstrate that the new model can not only get a diffeomorphic registration even when the deformation is large, but also possess the accuracy in comparing with the currently best models