Initial Vector Field Research Articles

Vector Field-Based Volume Peeling for Multi-Axis Machining

Abstract This paper presents an easy-to-control volume peeling method for multi-axis machining based on the computation taken on vector fields. The current scalar field-based methods are not flexible and the vector field-based methods do not guarantee the satisfaction of the constraints in the final results. We first conduct an optimization formulation to compute an initial vector field that is well aligned with those anchor vectors specified by users according to different manufacturing requirements. The vector field is further optimized to be an irrotational field so that it can be completely realized by a scalar field’s gradients. Iso-surfaces of the scalar field will be employed as the layers of working surfaces for multi-axis volume peeling in the rough machining. Algorithms are also developed to remove and process singularities of the fields. Our method has been tested on a variety of models and verified by physical experimental machining.

Intelligent Interactive Deformable Image Registration for Online Adaptive Radiotherapy.

The goal of this study is to streamline the time-consuming contouring process in online adaptive radiotherapy (ART) by utilizing a deep learning-based interactive deformable image registration (DIR) algorithm. The objective is to minimize manual review and editing of automatically generated initial contours of organs-at-risk (OARs) and targets, thereby improving the efficiency and effectiveness of the treatment process. Our proposed method reforms the current DIR-based contour propagation method in clinical practice through the implementation of a deep learning-based interactive approach. The steps include: 1) generation of an initial deformable vector field (DVF) using a DL model, based on fixed and moving image pairs, resulting in the initial contours of OARs and targets; 2) clinician review/edit one the OAR/target contours as needed; 3) updated contour is sent to DL model to update the DVF and the remaining OARs/targets contours. Repeat this process until satisfactory contour qualities are achieved. We used the Open Access Series of Imaging Studies (OASIS) as the testbed, including 394 (train) and 20 (test) brain T1-weighted MRI scans, each containing 35 annotated organs. The U-Net architecture was employed to update the DVF from fixed/moving images, initial contours, and updated contours. We compared our approach to traditional manual editing without interaction and quantified the effort reduction using the added path length (APL) metric which is supposed to be proportional to the absolute time spent on the contour editing. We conducted paired t-test to show the significance. For comparison purpose, we assumed the clinicians edit the contours with the largest APL, i.e., the contours that require the most editing efforts. The editing effort, as measured by APL, was reduced by 18.5% to 25.4% with a mean of 23.3%, median of 23.6%, and standard deviation of 1.9%. The significance of the results was confirmed with a p-value of 1.47e-24. Our study demonstrates a significant reduction in editing effort, as measured by APL, compared to traditional manual contour editing. These results demonstrate the potential of our deep learning-based interactive approach to improve the efficiency and accuracy of the contouring process in clinical practice.

A class of exact solutions of the Navier–Stokes equations in three and four dimensions

A few basic, intuitive, properties of the Navier–Stokes system of equations for incompressible fluid flows are discussed in this paper. We present a rephrased interpretation of the Navier–Stokes equation in a space having an arbitrary number of dimensions. We then derive spatially periodic solutions for the velocity and pressure fields that span an unbounded domain in three and four dimensions, given a smooth solenoidal initial velocity vector field. In these solutions all velocity components depend non-trivially on all coordinate directions.

Accuracy and clinical feasibility of 3D-myocardial thickness map measured by cardiac computed tomogram

BackgroundAlthough myocardial thickness is an important variable for therapeutic catheter ablation of cardiac arrhythmias, quantification of wall thickness has been overlooked. We developed a software (AMBER) that measures 3D-myocardial thickness using a cardiac computed tomogram (CT) image, verified its accuracy, and tested its clinical feasibility.MethodsWe generated 3D-thickness maps by calculating wall thickness (WT) from the CT images of 120 patients’ hearts and a 3D-phantom model (PhM). The initial vector field of the Laplace equation was oriented to calculate WT with the field lines derived from the 3D mesh. We demonstrate the robustness of the Laplace WT algorithm by comparing with the real thickness of 3D-PhM, echocardiographically measured left ventricular (LV) WT, and regional left atrial (LA) WT reported from previous studies. We conducted a pilot case of catheter ablation for atrial fibrillation (AF) utilizing real-time LAWT map-guided radiofrequency (RF) energy titration.ResultsAMBER 3D-WT had excellent correlations with the real thickness of the PhM (R = 0.968, p < 0.001) and echocardiographically measured LVWT in 10 patients (R = 0.656, p = 0.007). AMBER 3D-LAWT (n = 120) showed a relatively good match with 12 previously reported regional LAWT. We successfully conducted pilot AF ablation utilizing AMBER 3D-LAWT map-guided real-time RF energy titration.ConclusionWe developed and verified an AMBER 3D-cardiac thickness map measured by cardiac CT images for LAWT and LVWT, and tested its feasibility for RF energy titration during clinical catheter ablation.

Spatio-temporal Saliency-based Motion Vector Refinement for Frame Rate Up-conversion

A spatio-temporal saliency-based frame rate up-conversion (FRUC) approach is proposed, which achieves better quality of interpolated frames and invalidates existing texture variation-based FRUC detectors. A spatio-temporal saliency model is designed to select salient frames. After obtaining initial motion vector field by texture- and color-based bilateral motion estimation, two motion vector refining (MVR) schemes are adopted for high and low saliency frames to hierarchically refine the motion vectors, respectively. To produce high-quality interpolated frames, image enhancement are performed for salient frames after frame interpolation. Due to distinct MVR schemes, there are different degrees of texture information in interpolated frames. Some edge and texture information is supplemented into salient frames as post-processing, which can invalidate existing texture variation-based FRUC detectors. Experimental results show that the proposed approach outperforms state-of-the-art works in both objective and subjective qualities of interpolated frames, and achieves the purpose of FRUC anti-forensics.

Improved rainfall nowcasting using Burgers’ equation

Nowcasting of surface precipitation from radar data typically relies on algorithms that calculate advection, such as the McGill Algorithm for Precipitation nowcasting by Lagrangian Extrapolation (MAPLE). This method offers high spatial and temporal resolution but it cannot represent the growth-decay of precipitation and non-stationary advection vector fields.In this study, we propose some nowcasting rainfall models based on advection-diffusion equation with non-stationary motion vectors. The diffusion term of this equation gives to smoother rainfall predictions for lead times and increased skill scores. The motion vectors are updated in each time step by solving a system of two-dimensional (2D) Burgers’ equation. The proposed forecasting models use the following three steps. First, an initial motion vector field is approximated using the Variational Echo Tracking (VET) algorithm. Second, a forecast is obtained for each time step by solving a time-dependent advection or advection-diffusion equation. In this step, the motion vectors are updated by solving Burgers’ equation. Lastly, forecasts are evaluated with lead times from 2.5 min to 3 h, and forecasts are compared with rain rate observations for six events over a 250×250 km2 region in southeastern South Korea.To observe the effects of the diffusion term and Burgers’ equation, four variants of the proposed modeling methods are considered, depending on the equations: advection equation (Type 1), advection and Burgers’ equations (Type 2), advection-diffusion equation (Type 3), and combination of the advection–diffusion and Burgers’ equations (Type 4). The forecasts from the Type 1 method are very similar to those of MAPLE. The other models (Type 2–4) yielded clearly better skill scores and correlation on average, with up to 3 h’ lead time. Models that use Burgers’ equation (Type 2 and Type 4) give much better scores than other methods using fixed motion vectors when the temporal variation of the motion vectors is large.

A robust deformable image registration enhancement method based on radial basis function.

To develop and evaluate a robust deformable image registration (DIR) enhancement method based on radial basis function (RBF) expansion. To improve DIR accuracy using sparsely available measured displacements, it is crucial to estimate the motion correlation between the voxels. In the proposed method, we chose to derive this correlation from the initial displacement vector fields (DVFs), and represent it in the form of RBF expansion coefficients of the voxels. The method consists of three steps: (I) convert an initial DVF to a coefficient matrix comprising expansion coefficients of the Wendland's RBF; (II) modify the coefficient matrix under the guidance of sparely distributed landmarks to generate the post-enhancement coefficient matrix; and (III) convert the post-enhancement coefficient matrix to the post-enhancement DVF. The method was tested on five DIR algorithms using a digital phantom. 3D registration errors were calculated for comparisons between the pre-/post-enhancement DVFs and the ground-truth DVFs. Effects of the number and locations of landmarks on DIR enhancement were evaluated. After applying the DIR enhancement method, the 3D registration errors per voxel (unit: mm) were reduced from pre-enhancement to post-enhancement by 1.3 (2.4 to 1.1, 54.2%), 0.0 (0.9 to 0.9, 0.0%), 6.1 (8.2 to 2.1, 74.4%), 3.2 (4.7 to 1.5, 68.1%), and 1.7 (2.9 to 1.2, 58.6%) for the five tested DIR algorithms respectively. The average DIR error reduction was 2.5±2.3 mm (percentage error reduction: 51.1%±29.1%). 3D registration errors decreased inverse-exponentially as the number of landmarks increased, and were insensitive to the landmarks' locations in relation to the down-sampling DVF grids. We demonstrated the feasibility of a robust RBF-based method for enhancing DIR accuracy using sparsely distributed landmarks. This method has been shown robust and effective in reducing DVF errors using different numbers and distributions of landmarks for various DIR algorithms.

Weak Topologies for Carathéodory Differential Equations: Continuous Dependence, Exponential Dichotomy and Attractors

We introduce new weak topologies and spaces of Carath\'eodory functions where the solutions of the ordinary differential equations depend continuously on the initial data and vector fields. The induced local skew-product flow is proved to be continuous, and a notion of linearized skew-product flow is provided. Two applications are shown. First, the propagation of the exponential dichotomy over the trajectories of the linearized skew-product flow and the structure of the dichotomy or Sacker-Sell spectrum. Second, how particular bounded absorbing sets for the process defined by a Carath\'eodory vector field $f$ provide bounded pullback attractors for the processes with vector fields in the alpha-limit set, the omega-limit set or the whole hull of $f$. Conditions for the existence of a pullback or a global attractor for the skew-product semiflow, as well as application examples are also given.

Identification of the shear-type displacement vector field and evaluation of its characteristics

A problem of identification of a shear-type solid flow is considered. Characteristics of the shear flow are determined on the basis of the initial displacement vector field. An algorithm of formation of a finite-size model displacement field of the shear type is given. A functional is proposed to search for the amplitude and direction of the shear flow for each local area of the vector field. The influence of the size of this area and of the degree of its overlapping with the shear flow on the error of shear angle measurement is shown. The choice of the local area size is analyzed. Functional operation is demonstrated by an example of the displacement field obtained in the case of compressive loading of the Hadfield steel single crystal.

Multi-Channel Mixed-Pattern Based Frame Rate Up-Conversion Using Spatio-Temporal Motion Vector Refinement and Dual-Weighted Overlapped Block Motion Compensation

In this paper, a novel motion compensated frame rate up-conversion (MC-FRUC) algorithm is proposed to enhance the visual quality of video sequences. First of all, the multi-channel mixed pattern (MCMP) is proposed to design a block matching criterion which uses a few computations to reveal the variances of one luminance channel and two chrominance channels in a video frame. Second, in basis of the forward and backward initial motion vector fields (MVFs) estimated by a variant of 3DRS algorithm, the spatio-temporal motion vector refinement (ST-MVR) algorithm is proposed to obtain the more smoother MVF by implicitly adding the spatio-temporal smooth constraint into the process of motion vector refinement, and then the highly fault-tolerant motion vector smoothing (HFT-MVS) algorithm is proposed to prevent the emergence of outliers in MVF. Finally, in order to reduce the edge blurring and occlusions, the proposed dual-weighted overlapped block motion compensation (DW-OBMC) algorithm uses the forward and backward MVFs to jointly produce the interpolated frame. Experimental results show that the proposed algorithm can significantly improve both objective and subjective quality of the interpolated frame with a low computational complexity, and provide the better performance than the existing algorithms.

Identification criteria for vortex structures in a deformed solid

The paper considers algorithms of automatic identification of vortex structures arising in a deformed solid. The initial vector fields characterizing the deformation of a solid are obtained by the digital image correlation method. Circulation and a functional related to vector and radius vector orthogonality are proposed for use as vortex identification criteria. The application of these criteria is considered on the example of model vector fields with rotational and irrotational vortices. The operation of the algorithms is exemplified by displacement fields for material flow in a silicon-manganese steel weld. The role of finite sizes of a sliding window is demonstrated. Additional criteria to assist vortex structure identification are proposed.

Cauchy Problem for Dissipative Hölder Solutions to the Incompressible Euler Equations

We consider solutions to the Cauchy problem for the incompressible Euler equations on the 3-dimensional torus which are continuous or Holder continuous for any exponent \({\theta < \frac{1}{16}}\) . Using the techniques introduced in De Lellis and Szekelyhidi (Inventiones Mathematicae 9:377–407, 2013; Dissipative Euler flows and Onsager’s conjecture, 2012), we prove the existence of infinitely many (Holder) continuous initial vector fields starting from which there exist infinitely many (Holder) continuous solutions with preassigned total kinetic energy.

On the structure of singularities of holomorphic flows in dimension 3

We study singularities of semi-complete holomorphic vector fields in dimension 3. Under the very generic assumption that the singularities of the associated foliation can be resolved with a single blow-up, we establish Ghys's conjecture asserting that the second jet of the initial vector field cannot vanish at an isolated singularity. In fact, it results from our proof that, under the preceding assumption, it suffices to assume that the singular set of the mentioned vector field has codimension at least 2. Related to this question and, in fact, playing a central role in our proof, we also present a detailed picture of the semi-complete singularities of vector fields whose associated foliation has a single eigenvalue different from zero.

A LAGRANGIAN FORMULATION FOR STATISTICAL FLUID REGISTRATION.

We defined a new statistical fluid registration method with Lagrangian mechanics. Although several authors have suggested that empirical statistics on brain variation should be incorporated into the registration problem, few algorithms have included this information and instead use regularizers that guarantee diffeomorphic mappings. Here we combine the advantages of a large-deformation fluid matching approach with empirical statistics on population variability in anatomy. We reformulated the Riemannian fluid algorithm developed in [4], and used a Lagrangian framework to incorporate 0 th and 1 rst order statistics in the regularization process. 92 2D midline corpus callosum traces from a twin MRI database were fluidly registered using the non-statistical version of the algorithm (algorithm 0), giving initial vector fields and deformation tensors. Covariance matrices were computed for both distributions and incorporated either separately (algorithm 1 and algorithm 2) or together (algorithm 3) in the registration. We computed heritability maps and two vector and tensor-based distances to compare the power and the robustness of the algorithms.

Global solvability of the problem on the motion of two fluids without surface tension

Unsteady motion of viscous incompressible fluids is considered in a bounded domain. The liquids are separated by an unknown interface on which the surface tension is neglected. This motion is governed by an interface problem for the Navier-Stokes system. First, a local existence theorem is established for the problem in Holder classes of functions. The proof is based on the solvability of a model problem for the Stokes system with a plane interface, which was obtained earlier. Next, for a small initial velocity vector field and small mass forces, we prove the existence of a unique smooth solution to the problem on an infinite time interval. Bibliography: 7 titles.

Good rough path sequences and applications to anticipating stochastic calculus

We consider anticipative Stratonovich stochastic differential equations driven by some stochastic process lifted to a rough path. Neither adaptedness of initial point and vector fields nor commuting conditions between vector field is assumed. Under a simple condition on the stochastic process, we show that the unique solution of the above SDE understood in the rough path sense is actually a Stratonovich solution. We then show that this condition is satisfied by the Brownian motion. As application, we obtain rather flexible results such as support theorems, large deviation principles and Wong--Zakai approximations for SDEs driven by Brownian motion along anticipating vectorfields. In particular, this unifies many results on anticipative SDEs.

Vector field design on surfaces

Vector field design on surfaces is necessary for many graphics applications: example-based texture synthesis, nonphotorealistic rendering, and fluid simulation. For these applications, singularities contained in the input vector field often cause visual artifacts. In this article, we present a vector field design system that allows the user to create a wide variety of vector fields with control over vector field topology, such as the number and location of singularities. Our system combines basis vector fields to make an initial vector field that meets user specifications.The initial vector field often contains unwanted singularities. Such singularities cannot always be eliminated due to the Poincaré-Hopf index theorem. To reduce the visual artifacts caused by these singularities, our system allows the user to move a singularity to a more favorable location or to cancel a pair of singularities. These operations offer topological guarantees for the vector field in that they only affect user-specified singularities. We develop efficient implementations of these operations based on Conley index theory . Our system also provides other editing operations so that the user may change the topological and geometric characteristics of the vector field.To create continuous vector fields on curved surfaces represented as meshes, we make use of the ideas of geodesic polar maps and parallel transport to interpolate vector values defined at the vertices of the mesh. We also use geodesic polar maps and parallel transport to create basis vector fields on surfaces that meet the user specifications. These techniques enable our vector field design system to work for both planar domains and curved surfaces.We demonstrate our vector field design system for several applications: example-based texture synthesis, painterly rendering of images, and pencil sketch illustrations of smooth surfaces.

Formal Normal Forms for the Perturbations of a Quasi-Homogeneous Hamiltonian Vector Field

We classify up to a formal change of variable the vector fields of two complex variables together with the foliations that they de.ne which are a perturbation of a quasi-homogeneous Hamiltonian vector field X0 by terms of higher degree of quasi-homogeneity. We do not require anything about the degree δ0 of the initial vector field X0, but we assume that the perturbed vector field still keeps invariant the separatrices of X0. We obtain formal normal forms which extend those obtained in the case of an initial vector field with a semisimple or nilpotent linear part. We give an interpretation of the dual version of these normal forms through the relative cohomology with respect to the dual initial di.erential form ω0.

Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

We study degree $n$ polynomial perturbations of quadratic reversible Hamiltonian vector fields with one center and one saddle point. It was recently proved that if the first Poincaré–Pontryagin integral is not identically zero, then the exact upper bound for the number of limit cycles on the finite plane is $n-1$. In the present paper we prove that if the first Poincaré–Pontryagin function is identically zero, but the second is not, then the exact upper bound for the number of limit cycles on the finite plane is $2(n-1)$. In the case when the perturbation is quadratic ($n=2$) we obtain a complete result—there is a neighborhood of the initial Hamiltonian vector field in the space of all quadratic vector fields, in which any vector field has at most two limit cycles.

Singular relative integral invariants and adiabatic processes of thermodynamics

UDC 517.987.5; 517.958 The vector field is rectified in the neighborhood of a nonsingular point. By means of local foliation into nonsingular hypersurfaces containing phase curves of the vector field, we obtain a two-dimensional vector quotient field. A quotient field in the phase plane defines the rectified phase pattern and the module (over the ring of first integrals) of invariant measures. The pair that is composed of the phase pattern and the invariant measure of the vector field in the plane has a stationary subaigebra in the algebra of all vector fields. The triple consisting of the initial vector field, its invariant measure, and an element of the stationary subalgebra of the pairs, admits a relative integral invariant, namely, the integral of the measure over sets that are bounded by the phase curves of two vector fields. The relative integral invariant is generated (over the ring of first integrals) by the action of an element of the stationary subalgebra (the product of the Hamiltonian or the potential by the time of the element of the subalgebra). This explicitly asymmetric construction relative to the pair of vector fields becomes symmetric if we consider singular invariant measures and smooth vector fields. Suppose that an element of the subalgebra is a nonsingular vector field at the point being considered. Then we can take it as the initial element for the next triple. Obviously, the iteration of this procedure generates a chain of triples and, respectively, a chain of measures. We are interested in the convergence of the arising measures (the sense of convergence is defined in the article). We say that a chain of triples is subjected to the weight k > 0 (say, k is a positive integer) if, in each triple, the invariant measure generated by the action of the element of the subaigebra is generated by the first integral of the first element of the triple of degree k (The invariant measure of a triple makes it possible to determine two numbers, namely, the successive weight, i.e., the order of singularity of the measure generated by the action of the element of the subalgebra over the ring of the first integrals of the subalgebra and the preceding weight, which is the same as above, only with the replacement of the element of the subalgebra by the initial vector field.). The order of singularity of an invariant measure is the weight of the measure relative to the action for the element of the subalgebra. Theorem. The orders of singularity of the invariant measures of a chain of triples subjected to the weight k correspond to the orbits of the Feigenbaum mapping.