Traditional variational models often fail to segment images in the presence of inhomogeneity or weak boundaries, partly due to their reliance on unreliable region metrics that quantify inhomogeneity based on a single mean value or a smoothed image serving as a mean function. The former, such as variance-based methods, are highly sensitive to image inhomogeneity, whereas the latter, such as local convolution-based approaches, lack a global receptive field. To address these issues, we employ an optimal transport-based data fidelity term in our segmentation objective functional. This term accounts for global differences between regions, resolving problems arising from local convolutions. It can also adaptively seek an optimized match between two probability density functions, proving more robust than relying solely on their mean values. Our proposed functional is minimized by gradually performing region merging. Experimental results demonstrate that our model outperforms state-of-the-art variational and deep learning models.

Existing level set models employ regularization based only on gradient information, 1D curvature or 2D curvature. For 3D image segmentation, however, an appropriate curvature-based regularization should involve a well-defined 3D curvature energy. This is the first paper to introduce a regularization energy that incorporates 3D scalar curvature for 3D image segmentation, inspired by the Einstein-Hilbert functional. To derive its Euler-Lagrange equation, we employ a two-step gradient descent strategy, alternately updating the level set function and its gradient. The paper also establishes the existence and uniqueness of the viscosity solution for the proposed model. Experimental results demonstrate that our proposed model outperforms other state-of-the-art models in 3D image segmentation.

Abstract. The present paper studies mathematical models for representing, imaging, and analyzing polycrystalline materials. We introduce various techniques for converting grain maps into diagram or tessellation representations that rely on constrained clustering. In particular, we show how to significantly accelerate the computation of generalized balanced power diagrams and how to extend it to allow for optimization over all relevant parameters. A comparison of the accuracy of the proposed approaches is given based on a three-dimensional real-world data set of [Formula: see text] voxels.

Superresolution imaging aims at improving the resolution of an image by enhancing it with other images or data that might have been acquired using different imaging techniques or modalities. In this paper we consider the task of doubling, in each dimension, the resolution of grayscale images of binary objects by fusion with double-resolution tomographic data that have been acquired from two viewing angles. We show that this task is polynomial-time solvable if the gray levels have been reliably determined. The problem becomes ${N}{P}$-hard if the gray levels of some pixels come with an error of $\pm1$ or larger. The ${N}{P}$-hardness persists for any larger resolution enhancement factor. This means that noise does not only affect the quality of a reconstructed image but, less expectedly, also the algorithmic tractability of the inverse problem itself.

The present paper deals with the discrete inverse problem of reconstructing binary matrices from their row and column sums under additional constraints on the number and pattern of entries in specified minors. While the classical consistency and reconstruction problems for two directions in discret e tomography can be solved in polynomial time, it turns out that these window constraints cause various unexpected complexity jumps back and forth from polynomialtime solvability to ℕℙ-hardness.

The task of reconstructing binary images from the knowledge of their line sums (discrete X-rays) in a given finite number m of directions is ill-posed. Even some small noise in the physical measurements can lead to dramatically different yet still unique solutions. The present paper addresses in particular the following problems. Does discrete tomography have the power of error correction? Can noise be compensated by taking more X-ray images, and, if so, what is the quantitative effect of taking one more X-ray? Our main theorem gives the first nontrivial unconditioned (and best possible) stability result. In particular, we show that the Hamming distance between any two different sets of m X-ray images of the same cardinality is at least 2(m-1), and this is best possible. As a consequence, this result implies a Rényi-type theorem for denoising and shows that the noise compensating effect of X-rays is linear in their number. Our theoretical results are complemented by determining the computational complexity of some underlying algorithmic tasks. In particular, we show that while there always is a certain inherent stability, the possibility of making (worst-case) efficient use of it is rather limited.