An Improved Riemannian Metric Approximation for Graph Cuts

The graph cut framework presents a popular energy minimization tool. In order to be able to minimize contour length dependent energy terms an appropriate metric approximation has to be embedded into the graph such that the cost of every cut approximates the length of a corresponding contour under a given metric. Formulas giving a good approximation have been introduced by Boykov and Kolmogorov for both Euclidean and Riemannian metrics. In this paper, we improve their method and obtain a better approximation in case of the Euclidean metric. In our approach, we combine the well-known Cauchy-Crofton formulas with Voronoi diagrams theory to devise a general method with straightforward extension from 2D to 3D space. Our edge weight formulas are invariant to mirroring and directly applicable to grids with anisotropic node spacing. Finally, we show that our approach yields smaller metrication errors in both the isotropic and anisotropic case and demonstrate an application of the derived formulas to biomedical image segmentation.

Graph cut is a popular technique for interactive image segmentation. However, it has certain shortcomings. In particular, graph cut has problems with segmenting thin elongated objects due to the ldquoshrinking biasrdquo. To overcome this problem, we propose to impose an additional connectivity prior, which is a very natural assumption about objects. We formulate several versions of the connectivity constraint and show that the corresponding optimization problems are all NP-hard. For some of these versions we propose two optimization algorithms: (i) a practical heuristic technique which we call DijkstraGC, and (ii) a slow method based on problem decomposition which provides a lower bound on the problem. We use the second technique to verify that for some practical examples DijkstraGC is able to find the global minimum.

Developing weak annotations is one of the effective methods to reduce the workload of labeling the training data and extract the valuable priori information. In this paper, we propose a novel weakly supervised variational segmentation model that requires only a single randomly selected point within the target region. First, We construct an anisotropic Riemannian metric that incorporates not only the texture patterns, the local directional information at the boundaries, but also the intensity heterogeneity through a conformal factor. This step is the cornerstone of our method, as it enables the effective integration of multi-scale image features. Second, we calculate two types of distances: geodesic distances based on the new Riemannian metric and Euclidean distances, from all the points in the image to the taken point. These pointwise distances are then integrated into the Chan-Vese model, enabling the simultaneous exclusion of regions with similar intensities both near and far from the target region. The distances serve as adaptive weighting parameters in the variational model, further enhancing segmentation accuracy. We employ the Douglas-Rachford algorithm for efficient numerical implementation. Experimental results show that our method achieves similar or better segmentation performance than other weakly annotated methods and the single-point annotation has better robustness.

This paper is devoted to the study of the evolution of positively curved metrics on the Wallach spaces $SU(3)/T_{\max}$, $Sp(3)/Sp(1)\times Sp(1)\times Sp(1)$, and $F_4/Spin(8)$. We prove that for all Wallach spaces, the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature. Moreover, we prove that for the spaces $Sp(3)/Sp(1)\times Sp(1)\times Sp(1)$ and $F_4/Spin(8)$, the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive Ricci curvature into metrics with mixed Ricci curvature. We also get similar results for some more general homogeneous spaces.

The key and difficult issue in the research of binocular vision-based 3D measurement is the accurate calibration of internal and external parameters of the camera and stereo matching. Matlab calibration is more efficient and accurate compared with manual or OpenCV calibration. In this paper, binocular camera is calibrated by Matlab calibration toolbox, and calibration parameters imported in OpenCV for follow-up image correction and stereo matching. By studying and comparing Block Matching (BM) and Graph Cut (GC) stereo matching algorithms, a disparity image of the object is obtained, thus laying foundation for follow-up 3D data information acquisition and reconstruction.

Let $M$ be a closed connected spin manifold of dimension $2$ or $3$ with a fixed orientation and a fixed spin structure. We prove that for a generic Riemannian metric on $M$ the non-harmonic eigenspinors of the Dirac operator are nowhere zero. The proof is based on a transversality theorem and the unique continuation property of the Dirac operator.

We introduce a class of Falconer distance problems, which we call of restricted type, lying between the classical version and its pinned variant. Prototypical restricted distance sets are the diagonal distance sets, k-point configuration sets given by $$ \begin{align*}\Delta^{\mathrm{diag}}(E)= \{ \,|(x,x,\dots,x)-(y_1,y_2,\dots,y_{k-1})| : x, y_1, \dots,y_{k-1} \in E\, \}\end{align*} $$ for a compact $E\subset \mathbb {R}^d$ and $k\ge 3$ . We show that $\Delta ^{\mathrm{diag}}(E)$ has non-empty interior if the Hausdorff dimension of E satisfies (0.1) $$ \begin{align} \dim(E)> \begin{cases} \frac{2d+1}3, & k=3, \\ \frac{(k-1)d}k,& k\ge 4. \end{cases} \end{align} $$ We prove an extension of this to $C^\omega $ Riemannian metrics g close to the product of Euclidean metrics. For product metrics, this follows from known results on pinned distance sets, but to obtain a result for general perturbations g, we present a sequence of proofs of partial results, leading up to the proof of the full result, which is based on estimates for multilinear Fourier integral operators.

This article is concerned with the problem of reconstructing the curl and divergence of two-dimensional (2D) vector fields from ultrasound time-of-flight measurements taking the refraction of the emitted ultrasound rays into account. Usually, in 2D vector field tomography the refraction of the ultrasound signal is neglected; i.e., one assumes that the signal propagates along straight lines. We address refractive effects, assuming that the ultrasound signal propagates along the geodesics of a Riemannian metric that is associated with the refractive index which is due to Fermat's principle. The investigated vector field, however, is still defined on a domain in $\mathbb{R}^2$ that is equipped with the Euclidean metric. We show a connection between the ray transforms and the Radon transform along geodesic curves which generalizes a well-known result from the Euclidean case. Relying on this relation, we develop an asymptotic reconstruction formula for computing the curl and divergence of the vector field using Fourier integral operators. This article concludes with detailed numerical tests proving on the one hand a good performance of our method and showing on the other hand an improvement compared with computations that assume a propagation along straight lines.

This paper proposes an iterative natural gradient algorithm to perform the optimization of switching probabilities in a space-varying hidden Markov model, in the context of human activity recognition in long-range surveillance. The proposed method is a version of the gradient method, developed under an information geometric viewpoint, where the usual Euclidean metric is replaced by a Riemannian metric on the space of transition probabilities. It is shown that the change in metric provides advantages over more traditional approaches, namely: 1) it turns the original constrained optimization into an unconstrained optimization problem; 2) the optimization behaves asymptotically as a Newton method and yields faster convergence than other methods for the same computational complexity; and 3) the natural gradient vector is an actual contravariant vector on the space of probability distributions for which an interpretation as the steepest descent direction is formally correct. Experiments on synthetic and real-world problems, focused on human activity recognition in long-range surveillance settings, show that the proposed methodology compares favorably with the state-of-the-art algorithms developed for the same purpose.

Dense stereo image matching in remotely sensed images is a challenging problem, though it has been studied for more than two decades, due to occlusions, discontinuities, geometric, and radiometric distortions. A novel multiobjective fitness function-based dense stereo matching approach using genetic algorithms (GAs) is proposed in this paper. The proposed method is useful for estimating dense disparity map with an improved number of inliers for a stereo image pair, despite the constraint of finding correct disparity at depth discontinuities. In this paper, the steps of GA, such as initialization of the population, fitness function, and crossover and mutation operation, are designed and implemented to effectively deal with the problem of dense stereo image matching. To initialize the population, a Scale Invariant Feature Transform (SIFT) descriptor is computed for each pixel and multiple-size window-based matching is performed, using the similarity measures: 1) Euclidean distance and 2) spectral angle mapper. The generated disparity maps are pruned to choose a suitable subset using the designed fitness functions, considering the constraints related to stereo image pair, such as epipolar constraint, which encodes the epipolar geometry and the similarity measure that is useful to decide accuracy of the correspondences. The two objective functions are the number of inliers computed using the fundamental matrix and an energy minimization function, considering discontinuities and occlusions. The usefulness of this approach for remotely sensed stereo image pairs is demonstrated by improving the number of inliers and favorably comparing with state-of-the-art dense stereo image matching methods.

For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a closed manifold $M^{n+1}$, $3\leq (n+1)\leq 7$, we prove that the union of all closed, smooth, embedded minimal hypersurfaces is dense. This implies there are infinitely many minimal hypersurfaces thus proving a conjecture of Yau (1982) for generic metrics.

Symmetric Positive Definite (SPD) matrices have been used in many fields of medical data analysis. Many Riemannian metrics have been defined on this manifold but the choice of the Riemannian structure lacks a set of principles that could lead one to choose properly the metric. This drives us to introduce the principle of balanced metrics that relate the affine-invariant metric with the Euclidean and inverse-Euclidean metric, or the Bogoliubov-Kubo-Mori metric with the Euclidean and log-Euclidean metrics. We introduce two new families of balanced metrics, the mixed-power-Euclidean and the mixed-power-affine metrics and we discuss the relation between this new principle of balanced metrics and the concept of dual connections in information geometry.

The Alpha Group is an abstract geometry group in R4. The way it was conceived allows a new interpretation of the structure of hypercomplex space with a new geometry and spatial topology, and a meaning for the geometric representation of R4 space to infinity. Therefore, it has been described as the tensorial metric formula in the Alpha Group. It will be shown that the Riemannian and Euclidean distance metrics between infinitesimal surfaces are represented as special cases of the metric of the Alpha group.

This paper presents an automatic method for brain tumour segmentation from magnetic resonance images. The method uses the feature vectors obtained by an efficient feature encoding approach which combines the advantages of the supervoxels and sparse coding techniques. Extremely Randomized Trees (ERT) algorithm is trained using these feature vectors to detect the whole tumour and for multi-label classification of abnormal tissues. A Conditional Random Field (CRF) algorithm is implemented to delimit the region where the brain tumour is located. The obtained predictions of the ERT are used to estimate probability maps. The probability maps of the images and the Euclidean distance between the feature vectors of neighbour supervoxels define the conditional random field energy function. The minimization of the energy function is performed via graph cuts. The proposed methods are evaluated on real patient data obtained from BraTS 2018 challenge. Results demonstrate that proposed method achieves a competitive performance on the validation dataset using Dice score is: 0.5719, 0.7992 and 0.6285 for enhancing tumuor, whole tumour and tumour core respectively. The achieved performance of this method on testing set using Dice score is: 0.5081, 0.7278 and 0.5778 for enhancing tumuor, whole tumour and tumour core respectively.

Mobile robot perception using vertical line stereo