Geodesic Active Fields:A Geometric Framework for Image Registration

Image registration is the concept of mapping homologous points in a pair of images. In other words, one is looking for an underlying deformation field that matches one image to a target image. The spectrum of applications of image registration is extremely large: It ranges from bio-medical imaging and computer vision, to remote sensing or geographic information systems, and even involves consumer electronics. Mathematically, image registration is an inverse problem that is ill-posed, which means that the exact solution might not exist or not be unique. In order to render the problem tractable, it is usual to write the problem as an energy minimization, and to introduce additional regularity constraints on the unknown data. In the case of image registration, one often minimizes an image mismatch energy, and adds an additive penalty on the deformation field regularity as smoothness prior. Here, we focus on the registration of the human cerebral cortex. Precise cortical registration is required, for example, in statistical group studies in functional MR imaging, or in the analysis of brain connectivity. In particular, we work with spherical inflations of the extracted hemispherical surface and associated features, such as cortical mean curvature. Spatial mapping between cortical surfaces can then be achieved by registering the respective spherical feature maps. Despite the simplified spherical geometry, inter-subject registration remains a challenging task, mainly due to the complexity and inter-subject variability of the involved brain structures. In this thesis, we therefore present a registration scheme, which takes the peculiarities of the spherical feature maps into particular consideration. First, we realize that we need an appropriate hierarchical representation, so as to coarsely align based on the important structures with greater inter-subject stability, before taking smaller and more variable details into account. Based on arguments from brain morphogenesis, we propose an anisotropic scale-space of mean-curvature maps, built around the Beltrami framework. Second, inspired by concepts from vision-related elements of psycho-physical Gestalt theory, we hypothesize that anisotropic Beltrami regularization better suits the requirements of image registration regularization, compared to traditional Gaussian filtering. Different objects in an image should be allowed to move separately, and regularization should be limited to within the individual Gestalts. We render the regularization feature-preserving by limiting diffusion across edges in the deformation field, which is in clear contrast to the indifferent linear smoothing. We do so by embedding the deformation field as a manifold in higher-dimensional space, and minimize the associated Beltrami energy which represents the hyperarea of this embedded manifold as measure of deformation field regularity. Further, instead of simply adding this regularity penalty to the image mismatch in lieu of the standard penalty, we propose to incorporate the local image mismatch as weighting function into the Beltrami energy. The image registration problem is thus reformulated as a weighted minimal surface problem. This approach has several appealing aspects, including (1) invariance to re-parametrization and ability to work with images defined on non-flat, Riemannian domains (e.g., curved surfaces, scalespaces), and (2) intrinsic modulation of the local regularization strength as a function of the local image mismatch and/or noise level. On a side note, we show that the proposed scheme can easily keep up with recent trends in image registration towards using diffeomorphic and inverse consistent deformation models. The proposed registration scheme, called Geodesic Active Fields (GAF), is non-linear and non-convex. Therefore we propose an efficient optimization scheme, based on splitting. Data-mismatch and deformation field regularity are optimized over two different deformation fields, which are constrained to be equal. The constraint is addressed using an augmented Lagrangian scheme, and the resulting optimization problem is solved efficiently using alternate minimization of simpler sub-problems. In particular, we show that the proposed method can easily compete with state-of-the-art registration methods, such as Demons. Finally, we provide an implementation of the fast GAF method on the sphere, so as to register the triangulated cortical feature maps. We build an automatic parcellation algorithm for the human cerebral cortex, which combines the delineations available on a set of atlas brains in a Bayesian approach, so as to automatically delineate the corresponding regions on a subject brain given its feature map. In a leave-one-out cross-validation study on 39 brain surfaces with 35 manually delineated gyral regions, we show that the pairwise subject-atlas registration with the proposed spherical registration scheme significantly improves the individual alignment of cortical labels between subject and atlas brains, and, consequently, that the estimated automatic parcellations after label fusion are of better quality

Geodesic Active Fields on the Sphere

In this paper, we propose a novel method to register images defined on spherical meshes. Instances of such spherical images include inflated cortical feature maps in brain medical imaging or images from omnidirectional cameras. We apply the Geodesic Active Fields (GAF) framework locally at each vertex of the mesh. Therefore we define a dense deformation field, which is embedded in a higher dimensional manifold, and minimize the weighted Polyakov energy. While the Polyakov energy itself measures the hyperarea of the embedded deformation field, its weighting allows to account for the quality of the current image alignment. Iteratively minimizing the energy drives the deformation field towards a smooth solution of the registration problem. Although the proposed approach does not necessarily outperform state-of-the-art methods that are tightly tailored to specific applications, it is of methodological interest due to its high degree of flexibility and versatility

Geodesic Active Fields - A Geometric Framework for Image Registration

In this paper we present a novel geometric framework called geodesic active fields for general image registration. In image registration, one looks for the underlying deformation field that best maps one image onto another. This is a classic ill-posed inverse problem, which is usually solved by adding a regularization term. Here, we propose a multiplicative coupling between the registration term and the regularization term, which turns out to be equivalent to embed the deformation field in a weighted minimal surface problem. Then, the deformation field is driven by a minimization flow toward a harmonic map corresponding to the solution of the registration problem. This proposed approach for registration shares close similarities with the well-known geodesic active contours model in image segmentation, where the segmentation term (the edge detector function) is coupled with the regularization term (the length functional) via multiplication as well. As a matter of fact, our proposed geometric model is actually the exact mathematical generalization to vector fields of the weighted length problem for curves and surfaces introduced by Caselles-Kimmel-Sapiro. The energy of the deformation field is measured with the Polyakov energy weighted by a suitable image distance, borrowed from standard registration models. We investigate three different weighting functions, the squared error and the approximated absolute error for monomodal images, and the local joint entropy for multimodal images. As compared to specialized state-of-the-art methods tailored for specific applications, our geometric framework involves important contributions. Firstly, our general formulation for registration works on any parametrizable, smooth and differentiable surface, including non-flat and multiscale images. In the latter case, multiscale images are registered at all scales simultaneously, and the relations between space and scale are intrinsically being accounted for. Secondly, this method is, to the best of our knowledge, the first re-parametrization invariant registration method introduced in the literature. Thirdly, the multiplicative coupling between the registration term, i.e. local image discrepancy, and the regularization term naturally results in a data-dependent tuning of the regularization strength. Finally, by choosing the metric on the deformation field one can freely interpolate between classic Gaussian and more interesting anisotropic, TV-like regularization

Direct Fourier Tomographic Reconstruction Image-To-Image Filter

We present an open-source ITK implementation of a direct Fourier method for tomographic reconstruction, applicable to parallel-beam x-ray images. Direct Fourier reconstruction makes use of the central-slice theorem to build a polar 2D Fourier space from the 1D transformed projections of the scanned object, that is resampled into a Cartesian grid. Inverse 2D Fourier transform eventually yields the reconstructed image. Additionally, we provide a complex wrapper to the BSplineInterpolateImageFunction to overcome ITK’s current lack for image interpolators dealing with complex data types. A sample application is presented and extensively illustrated on the Shepp-Logan head phantom. We show that appropriate input zeropadding and 2D-DFT oversampling rates together with radial cubic b-spline interpolation improve 2D-DFT interpolation quality and are efficient remedies to reduce reconstruction artifacts

Synergistic NGF/B27 Gradients Position Synapses Heterogeneously in 3D Micropatterned Neural Cultures

Native functional brain circuits show different numbers of synapses (synaptic densities) in the cerebral cortex. Until now, different synaptic densities could not be studied in vitro using current cell culture methods for primary neurons. Herein, we present a novel microfluidic based cell culture method that combines 3D micropatterning of hydrogel layers with linear chemical gradient formation. Micropatterned hydrogels were used to encapsulate dissociated cortical neurons in laminar cell layers and neurotrophic factors NGF and B27 were added to influence the formation of synapses. Neurotrophic gradients allowed for the positioning of distinguishable synaptic densities throughout a 3D micropatterned neural culture. NGF and B27 gradients were maintained in the microfluidic device for over two weeks without perfusion pumps by utilizing a refilling procedure. Spatial distribution of synapses was examined with a pre-synaptic marker to determine synaptic densities. From our experiments, we observed that (1) cortical neurons responded only to synergistic NGF/B27 gradients, (2) synaptic density increased proportionally to synergistic NGF/B27 gradients; (3) homogeneous distribution of B27 disturbed cortical neurons in sensing NGF gradients and (4) the cell layer position significantly impacted spatial distribution of synapses

An Approach to Multimodal Image Segmentation

Radiotherapy treatment planning requires exact delineation of tumors in acquired images. Further, motion and deformation of a tumor due to respiratory movements must be known in order to better target and dose the treatment radiation. Here we present and implement a pipeline for the registration of respiratory correlated multimodal RC-CT-PET image sequences. Further, a multimodal region-based non-parametric active-mesh segmentation framework is presented, that allows to correctly delineate objects in multimodal volumetric images. Applications to MRI brain image registration and extraction, as well as RC-CT-PET lung tumor delineation are shown. These tools allow for the delineation and the volumetric motion tracking and deformation analysis of lung tumors throughout the respiratory cycle, and therefore represent a major achievement for radiotherapy treatment planning