We generalize discrete variational models involving the infimal convolution (IC) of first and second order differences and the total generalized variation (TGV) to manifold-valued images. We propose both extrinsic and intrinsic approaches. The extrinsic models are based on embedding the manifold into an Euclidean space of higher dimension with manifold constraints. An alternating direction methods of multipliers can be employed for finding the minimizers. However, the components within the extrinsic IC or TGV decompositions live in the embedding space which makes their interpretation difficult. Therefore we investigate two intrinsic approaches: for Lie groups, we employ the group action within the models; for more general manifolds our IC model is based on recently developed absolute second order differences on manifolds, while our TGV approach uses an approximation of the parallel transport by the pole ladder. For computing the minimizers of the intrinsic models we apply gradient descent algorithms. Numerical examples demonstrate that our approaches work well for certain manifolds.

Focused ion beam (FIB) tomography provides high resolution volumetric images on a micro scale. However, due to the physical acquisition process the resulting images are often corrupted by a so-called curtaining or waterfall effect. In this paper, a new convex variational model for removing such effects is proposed. More precisely, an infimal convolution model is applied to split the corrupted 3D image into the clean image and two types of corruptions, namely a striped part and a laminar one. As regularizing terms different direction dependent first and second order differences are used to cope with the specific structure of the corruptions. This generalizes discrete unidirectional total variational (TV) approaches. A minimizer of the model is computed by well-known primal dual techniques. Numerical examples show the very good performance of our new method for artificial and real-world data. Besides FIB tomography, we have also successfully applied our technique for the removal of pure stripes in Moderate Resolution Imaging Spectroradiometer (MODIS) data.

This paper addresses the problems of disparity and optical flow partitioning based on the brightness invariance assumption. We investigate new variational approaches to these problems with Potts priors and possibly box constraints. For the optical flow partitioning, our model includes vector-valued data and an adapted Potts regularizer. Using the notion of asymptotically level stable (als) functions, we prove the existence of global minimizers of our functionals. We propose a modified alternating direction method of multipliers. This iterative algorithm requires the computation of global minimizers of classical univariate Potts problems which can be done efficiently by dynamic programming. We prove that the algorithm converges both for the constrained and unconstrained problems. Numerical examples demonstrate the very good performance of our partitioning method.

In the last two decades, focused ion beam (FIB) systems have been used for sample preparation. For example, the edges of a sample can be polished for analytical measurements or continuous cross-sections can be milled for three-dimensional (3D) tomography and reconstruction. One major challenge in both procedures is the so-called curtaining effect, i.e., increasing surface roughness in the direction of the milling depth. The roughness of the cut can influence the result of the measurement and the segmentation process. In the present study, the authors report on two different methods to reduce the curtaining effect, namely, a hardware- and a software-based solution. For instance, Tescan implemented the so-called “rocking stage” in its plasma FIB. However, this is not available for other FIB systems. Therefore, for our FEI gallium FIB, an inhouse-developed goniometer stage is installed, which can be adapted as necessary. With this relatively inexpensive solution, the sample can be rotated around an additional axis and tilted by ±8°. Different sample heights are adjustable, and the sample's edge can be polished and imaged without stage movement. However, for automated milling and imaging procedures such as 3D tomography, such a tilting stage is not feasible. Therefore, as a second option, an image processing method is proposed that can be applied after the milling procedure on a whole image stack. A novel variation of this method for mathematical image processing is developed to reduce milling artifacts. Besides the curtaining effect, additional artifacts such as discontinuities caused by redeposition of previously removed materials or charging effects can be removed. The method is applied to the entire 3D dataset, and distortions are reduced by using information of their particular structure and directional dependence. The resulting new image stack can then be used to compose a 3D volume reconstruction. As an example, the geometries of silicon carbide particles reinforcing an aluminum matrix can be measured with nearly no milling artifacts.

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