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The travel time computation of microseismic waves in different directions (particularly, the diagonal direction) in three-dimensional space has been found to be inaccurate, which seriously affects the localization accuracy of three-dimensional microseismic sources. In order to solve this problem, this research study developed a method of calculating the P-wave travel time based on a 3D high-order fast marching method (3D_H_FMM). This study focused on designing a high-order finite-difference operator in order to realize the accurate calculation of the P-wave travel time in three-dimensional space. The method was validated using homogeneous velocity models and inhomogeneous layered media velocity models of different scales. The results showed that the overall mean absolute error (MAE) of the two homogenous models using 3D_H_FMM had been reduced by 88.335%, and 90.593% compared with the traditional 3D_FMM. On that basis, the three-dimensional localization of microseismic sources was carried out using a particle swarm optimization algorithm. The developed 3D_H_FMM was used to calculate the travel time, then to conduct the localization of the microseismic source in inhomogeneous models. The mean error of the localization results of the different positions in the three-dimensional space was determined to be 1.901 m, and the localization accuracy was found to be superior to that of the traditional 3D_FMM method (mean absolute localization error: 3.447 m) with the small-scaled inhomogeneous model.

When dealing with computed tomography volume data, the accurate segmentation of lung nodules is of great importance to lung cancer analysis and diagnosis, being a vital part of computer-aided diagnosis systems. However, due to the variety of lung nodules and the similarity of visual characteristics for nodules and their surroundings, robust segmentation of nodules becomes a challenging problem. A segmentation algorithm based on the fast marching method is proposed that separates the image into regions with similar features, which are then merged by combining regions growing with k-means. An evaluation was performed with two distinct methods (objective and subjective) that were applied on two different datasets, containing simulation data generated for this study and real patient data, respectively. The objective experimental results show that the proposed technique can accurately segment nodules, especially in solid cases, given the mean Dice scores of 0.933 and 0.901 for round and irregular nodules. For non-solid and cavitary nodules the performance dropped—0.799 and 0.614 mean Dice scores, respectively. The proposed method was compared to active contour models and to two modern deep learning networks. It reached better overall accuracy than active contour models, having comparable results to DBResNet but lesser accuracy than 3D-UNet. The results show promise for the proposed method in computer-aided diagnosis applications.

We address the numerical computation of distance maps with respect to Riemannian metrics of strong anisotropy. For that purpose we solve generalized eikonal equations, discretized using adaptive upwind finite differences on a Cartesian grid, in a single pass over the domain using a variant of the fast-marching algorithm. The key ingredient of our PDE numerical scheme is Voronoi's first reduction, a tool from discrete geometry which characterizes the interaction of a quadratic form with an additive lattice. This technique, never used in this context, which is simple and cheap to implement, allows us to efficiently handle Riemannian metrics of eigenvalue ratio 10² and more. Two variants of the introduced scheme are also presented, adapted to sub-Riemannian and to Ränder metrics, which can be regarded as degenerate Riemannian metrics and as Riemannian metrics perturbed with a drift term, respectively. We establish the convergence of the proposed scheme and of its variants, with convergence rates. Numerical experiments illustrate the effectiveness of our approach in various contexts, in dimension up to five, including an original sub-Riemannian model related to the penalization of path torsion.

Accurate determination of earthquake parameters is vital task for seismologists due to their potential hazards and the importance of risk mitigation. Hypocenter is one of the paramaters which is critical for tomography and inversion processing. This study introduces a novel approach for hypocenter localization based on the Reciprocal Method of Fast Marching Wavefront Modeling (RFMW). This method models seismic wavefronts by solving the eikonal equation through the Fast Marching Method (FMM). We evaluate the effectiveness of RFMW in locating hypocenters in highly heterogeneous subsurface media and in addressing the nonlinear aspects of wave propagation. Additionally, we investigate how hypocenter accuracy is affected by the spatial configuration and distribution of seismograph stations. The RFMW approach was applied to determine several hypocenters beneath Lake Toba in the North Sumatra. The results reveal the strong correlation between the seismograph network configuration—particularly station spacing and distribution—and the accuracy of hypocenter localization. Interestingly, increasing the number of seismographs did not significantly enhance the accuracy of hypocenter determination, highlighting the importance of optimal station placement position.

The geodesic model based on the eikonal partial differential equation (PDE) has served as a fundamental tool for the applications of image segmentation and boundary detection in the past two decades. However, the existing approaches commonly only exploit the image edge-based features for computing minimal geodesic paths, potentially limiting their performance in complicated segmentation situations. In this paper, we introduce a new variational image segmentation model based on the minimal geodesic path framework and the eikonal PDE, where the region-based appearance term that defines then regional homogeneity features can be taken into account for estimating the associated minimal geodesic paths. This is done by constructing a Randers geodesic metric interpretation of the region-based active contour energy functional. As a result, the minimization of the active contour energy functional is transformed into finding the solution to the Randers eikonal PDE. We also suggest a practical interactive image segmentation strategy, where the target boundary can be delineated by the concatenation of several piecewise geodesic paths. We invoke the Finsler variant of the fast marching method to estimate the geodesic distance map, yielding an efficient implementation of the proposed region-based Randers geodesic model for image segmentation. Experimental results on both synthetic and real images exhibit that our model indeed achieves encouraging segmentation performance.

In this paper, we introduce novel methods for computing middle surfaces between various 3D data sets such as point clouds and/or discrete surfaces. Traditionally the middle surface is obtained by detecting singularities in computed distance function such as ridges, triple junctions, etc. It requires to compute second order differential characteristics, and also some kinds of heuristics must be applied. Opposite to that, we determine the middle surface just from computing the distance function itself which is a fast and simple approach. We present and compare the results of the fast sweeping method, the vector distance transform algorithm, the fast marching method, and the Dijkstra-Pythagoras method in finding the middle surface between 3D data sets.

In this paper, we propose a novel curvature penalized minimal path model via an orientation-lifted Finsler metric and the Euler elastica curve. The original minimal path model computes the globally minimal geodesic by solving an Eikonal partial differential equation (PDE). Essentially, this first-order model is unable to penalize curvature which is related to the path rigidity property in the classical active contour models. To solve this problem, we present an Eikonal PDE-based Finsler elastica minimal path approach to address the curvature-penalized geodesic energy minimization problem. We were successful at adding the curvature penalization to the classical geodesic energy (Caselles et al. in Int J Comput Vis 22(1):61–79, 1997 ; Cohen and Kimmel in Int J Comput Vis 24(1):57–78, 1997 ). The basic idea of this work is to interpret the Euler elastica bending energy via a novel Finsler elastica metric that embeds a curvature penalty. This metric is non-Riemannian, anisotropic and asymmetric, and is defined over an orientation-lifted space by adding to the image domain the orientation as an extra space dimension. Based on this orientation lifting, the proposed minimal path model can benefit from both the curvature and orientation of the paths. Thanks to the fast marching method, the global minimum of the curvature-penalized geodesic energy can be computed efficiently. We introduce two anisotropic image data-driven speed functions that are computed by steerable filters. Based on these orientation-dependent speed functions, we can apply the proposed Finsler elastica minimal path model to the applications of closed contour detection, perceptual grouping and tubular structure extraction. Numerical experiments on both synthetic and real images show that these applications of the proposed model indeed obtain promising results.

The velocity model is a key factor that affects the accuracy of microseismic event location around tunnels. In this paper, we consider the effect of the empty area on the microseismic event location and present a 3D heterogeneous velocity model for excavated tunnels. The grid-based heterogeneous velocity model can describe a 3D arbitrarily complex velocity model, where the microseismic monitoring areas are divided into many blocks. The residual between the theoretical arrival time calculated by the fast marching method (FMM) and the observed arrival time is used to identify the block with the smallest residual. Particle swarm optimization (PSO) is used to improve the location accuracy in this block. Synthetic tests show that the accuracy of the microseismic event location based on the heterogeneous velocity model was higher than that based on the single velocity model, independent of whether an arrival time error was considered. We used the heterogeneous velocity model to locate 7 blasting events and 44 microseismic events with a good waveform quality in the Qinling No. 4 tunnel of the Yinhanjiwei project from 6 June 2017 to 13 June 2017 and compared the location results of the heterogeneous-velocity model with those of the single-velocity model. The results of this case study show that the events located by the heterogeneous velocity model were concentrated around the working face, which matched the actual conditions of the project, while the events located by the single-velocity model were scattered and far from the working face.

The precise determination of earthquake location is the fundamental basis in seismological community, and is crucial for analyzing seismic activity and performing seismic tomography. First arrivals are generally used to practically determine earthquake locations. However, first-arrival traveltimes are not sensitive to focal depths. Moreover, they cannot accurately constrain focal depths. To improve the accuracy, researchers have analyzed the depth phases of earthquake locations. The traveltimes of depth phases are sensitive to focal depths, and the joint inversion of depth phases and direct phases can be implemented to potentially obtain accurate earthquake locations. Generally, researchers can determine earthquake locations in layered models. Because layered models can only represent the first-order feature of subsurface structures, the advantages of joint inversion are not fully explored if layered models are used. To resolve the issue of current joint inversions, we use the traveltimes of three seismic phases to determine earthquake locations in heterogeneous models. The three seismic phases used in this study are the first P-, sPg- and PmP-waves. We calculate the traveltimes of the three seismic phases by solving an eikonal equation with an upwind difference scheme and use the traveltimes to determine earthquake locations. To verify the accuracy of the earthquake location method by the inversion of three seismic phases, we take the 2021 M S 6.4 Yangbi, Yunnan earthquake as an example and locate this earthquake using synthetic and real seismic data. Numerical tests demonstrate that the eikonal equation-based earthquake location method, which involves the inversion of multiple phase arrivals, can effectively improve earthquake location accuracy.