Marc Droske

An Adaptive Level Set Method for Medical Image Segmentation

An efficient adaptive multigrid level set method for front propagation purposes in three dimensional medical image segmentation is presented. It is able to deal with non sharp segment boundaries. A flexible, interactive modulation of the front speed depending on various boundary and regularization criteria ensure this goal. Efficiency is due to a graded underlying mesh implicitly defined via error or feature indicators. A suitable saturation condition ensures an important regularity condition on the resulting adaptive grid. As a casy study the segmentation of glioma is considered. The clinician interactively selects a few parameters describing the speed function and a few seed points. The automatic process of front propagation then generates a family of segments corresponding to the evolution of the front in time, from which the clinician finally selects an appropriate segment covered by the gliom. Thus, the overall glioma segmentation turns into an efficient, nearly real time process with intuitive and usefully restricted user interaction.

A Variational Approach to Nonrigid Morphological Image Registration

A variational method for nonrigid registration of multimodal image data is presented. A suitable deformation will be determined via the minimization of a morphological, i.e., contrast invariant, matching functional along with an appropriate regularization energy. The aim is to correlate the morphologies of a template and a reference image under the deformation. Mathematically, the morphology of images can be described by the entity of level sets of the image and hence by its Gauss map. A class of morphological matching functionals is presented which measure the defect of the template Gauss map in the deformed state with respect to the deformed Gauss map of the reference image. The problem is regularized by considering a nonlinear elastic regularization energy. Existence of homeomorphic, minimizing deformation is proved under assumptions on the class of admissible deformations. With respect to actual medical applications, suitable generalizations of the matching energies and the boundary conditions are pre...

Image Registration by a Regularized Gradient Flow. A Streaming Implementation in DX9 Graphics Hardware

Abstract.The presented image registration method uses a regularized gradient flow to correlate the intensities in two images. Thereby, an energy functional is successively minimized by descending along its regularized gradient. The gradient flow formulation makes use of a robust multi-scale regularization, an efficient multi-grid solver and an effective time-step control. The data processing is arranged in streams and mapped onto the functionality of a stream processor. This arrangement automatically exploits the high data parallelism of the problem, and local data access helps to maximize throughput and hide memory latency. Although dedicated stream processors exist, we use a DX9 compatible graphics card as a stream architecture because of its ideal price-performance ratio. The new floating point number formats guarantee a sufficient accuracy of the algorithm and eliminate previously present concerns about the use of graphics hardware for medical computing. Therefore, the implementation achieves reliable results at very high performance, registering two 2572 images in approximately 3sec, such that it could be used as an interactive tool in medical image analysis.

An image processing approach to surface matching

Establishing a correspondence between two surfaces is a basic ingredient in many geometry processing applications. Existing approaches, which attempt to match two meshes directly in 3D, can be cumbersome to implement and it is often hard to produce accurate results in a reasonable amount of time. In this paper, we present a new variational method for matching surfaces that addresses these issues. Instead of matching two surfaces directly in 3D, we apply well-established matching methods from image processing in the parameter domains of the surfaces. A matching energy is introduced that can depend on curvature, feature demarcations or surface textures, and a regularization energy controls length and area changes in the induced non-rigid deformation between the two surfaces. The metric on both surfaces is properly incorporated into the formulation of the energy. This approach reduces all computations to the 2D setting while accounting for the original geometries. Consequently a fast multiresolution numerical algorithm for regular image grids can be used to solve the global optimization problem. The final algorithm is robust, generically much simpler than direct matching methods, and very fast for highly resolved triangle meshes.

Multiscale Joint Segmentation and Registration of Image Morphology

Multimodal image registration significantly benefits from previous denoising and structure segmentation and vice versa. In particular, combined information of different image modalities makes segmentation significantly more robust. Indeed, fundamental tasks in image processing are highly interdependent. A variational approach is presented, which combines the detection of corresponding edges, an edge preserving denoising, and the morphological registration via a nonrigid deformation for a pair of images with structural correspondence. The morphology of an image function is split into a singular part consisting of the edge set and a regular part represented by the field of normals on the ensemble of level sets. A Mumford-Shah type free discontinuity problem is applied to treat the singular morphology and the matching of corresponding edges under the deformation. The matching of the regular morphology is quantified by a second contribution, which compares deformed normals and normals at deformed positions. Finally, a nonlinear elastic energy controls the deformation itself and ensures smoothness and injectivity. A multiscale approach that is based on a phase field approximation leads to an effective and efficient algorithm. Numerical experiments underline the robustness of the presented approach and show applications on medical images.

A Mumford–Shah Level‐Set Approach for Geometric Image Registration

A new method for nonrigid registration of multimodal images is presented. Due to the large interdependence of segmentation and registration, the approach is based on simultaneous segmentation and edge alignment. The two processes are directly coupled and thus benefit from using complementary information of the entire underlying data set. The approach is formulated as a bivariate, variational, free discontinuity problem in the Mumford–Shah framework. A geometric variable describing the contour set and a functional variable which represents the underlying deformation are simultaneously identified. The contour set is represented by a level‐set function. We derive a regularized gradient flow and describe an efficient numerical implementation using finite element discretization and multigrid techniques. Finally, we illustrate the method in several applications, such as multimodal intrapatient registration and reconstruction by registration to a reference object.

Computational methods for nonlinear image registration

1 Institut fur Mathematik, Gerhard-Mercator Universitat Duisburg, Lotharstrase 63/65, 47048 Duisburg, Germany {clarenz|droske|rumpf}@math.uni-duisburg.de. 2 Lehrstuhl fur Mathematische Optimierung, Mathematisches Institut, Heinrich-Heine Universitat Dusseldorf, Universitatsstrase 1, D-40225 Dusseldorf, Germany. henn@am.uni-duesseldorf.de 3 Lehrstuhl fur Angewandte Mathematik, Mathematisches Institut, Heinrich-Heine Universitat Dusseldorf, Universitatsstrase 1, D-40225 Dusseldorf, Germany. witsch@am.uni-duesseldorf.de Summary. Image registration is the process of the alignment of two or more data sets recorded with the same or different imaging machineries. Especially nonlinear image registration techniques allow the alignment of data sets that are mismatched in a nonuniform manner. Mathematically, this yields a nonlinear ill–conditioned inverse problem. In this presentation, we introduce several computational methods based on variational PDE approaches to obtain an approximate solution of the nonlinear registration problem. In each approach we have to solve a sequence of subproblems. Each subproblem has to be well-posed and should be efficiently solvable.

Mumford–Shah Model for One-to-One Edge Matching

This paper presents a new algorithm based on the Mumford-Shah model for simultaneously detecting the edge features of two images and jointly estimating a consistent set of transformations to match them. Compared to the current asymmetric methods in the literature, this fully symmetric method allows one to determine one-to-one correspondences between the edge features of two images. The entire variational model is realized in a multiscale framework of the finite element approximation. The optimization process is guided by an estimation minimization-type algorithm and an adaptive generalized gradient flow to guarantee a fast and smooth relaxation. The algorithm is tested on T1 and T2 magnetic resonance image data to study the parameter setting. We also present promising results of four applications of the proposed algorithm: interobject monomodal registration, retinal image registration, matching digital photographs of neurosurgery with its volume data, and motion estimation for frame interpolation.

A PHASE FIELD METHOD FOR JOINT DENOISING, EDGE DETECTION, AND MOTION ESTIMATION IN IMAGE SEQUENCE PROCESSING ∗

The estimation of optical flow fields from image sequences is incorporated in a Mumford–Shah approach for image denoising and edge detection. Possibly noisy image sequences are considered as input and a piecewise smooth image intensity, a piecewise smooth motion field, and a joint discontinuity set are obtained as minimizers of the functional. The method simultaneously detects image edges and motion field discontinuities in a rigorous and robust way. It is able to handle information on motion that is concentrated on edges. Inherent to it is a natural multiscale approximation that is closely related to the phase field approximation for edge detection by Ambrosio and Tortorelli. We present an implementation for two-dimensional image sequences with finite elements in space and time. This leads to three linear systems of equations, which have to be solved in a suitable iterative minimization procedure. Numerical results and different applications underline the robustness of the approach presented.

A variational approach to joint denoising, edge detection and motion estimation

The estimation of optical flow fields from image sequences is incorporated in a Mumford–Shah approach for image denoising and edge detection. Possibly noisy image sequences are considered as input and a piecewise smooth image intensity, a piecewise smooth motion field, and a joint discontinuity set are obtained as minimizers of the functional. The method simultaneously detects image edges and motion field discontinuities in a rigorous and robust way. It comes along with a natural multi–scale approximation that is closely related to the phase field approximation for edge detection by Ambrosio and Tortorelli. We present an implementation for 2D image sequences with finite elements in space and time. It leads to three linear systems of equations, which have to be iteratively in the minimization procedure. Numerical results underline the robustness of the presented approach and different applications are shown.