Level Set Partial Differential Equations Research Articles

Stochastic time-optimal path-planning in uncertain, strong, and dynamic flows

Accounting for uncertainty in optimal path planning is essential for many applications. We present and apply stochastic level-set partial differential equations that govern the stochastic time-optimal reachability fronts and time-optimal paths for vehicles navigating in uncertain, strong, and dynamic flow fields. To solve these equations efficiently, we obtain and employ their dynamically orthogonal reduced-order projections, maintaining accuracy while achieving several orders of magnitude in computational speed-up when compared to classic Monte Carlo methods. We utilize the new equations to complete stochastic reachability and time-optimal path planning in three test cases: (i) a canonical stochastic steady-front with uncertain flow strength, (ii) a stochastic barotropic quasi-geostrophic double-gyre circulation, and (iii) a stochastic flow past a circular island. For all the three test cases, we analyze the results with a focus on studying the effect of flow uncertainty on the reachability fronts and time-optimal paths, and their probabilistic properties. With the first test case, we demonstrate the approach and verify the accuracy of our solutions by comparing them with the Monte Carlo solutions. With the second, we show that different flow field realizations can result in paths with high spatial dissimilarity but with similar arrival times. With the third, we provide an example where time-optimal path variability can be very high and sensitive to uncertainty in eddy shedding direction downstream of the island.

Application of statistical representation of the microstructure to modeling of phase transformations in DP steels by solution of the diffusion equation

The approach based on solution of diffusion equation and multiple Level Set Method was proposed to model phase transformation phenomena occurring in Dual Phase (DP) steels during continuous annealing. Coupled solutions of the Second Fick Law and level set equations were used to create framework for dealing with a moving boundary problems. Diffusion and level set partial differential equations were solved using Finite Element (FE) method with local mesh refinement. Velocity function depended on the curvature of the interface and the carbon concentration gradient, was proposed to simulate grain growth. Statistically Similar Representative Volume Element with DP microstructure was generated. This element was subjected to plastic deformation in tension, compression and shear. Finally, homogenized properties of the DP steel were determine.

Interactive Medical Image Segmentation Using PDE Control of Active Contours

Segmentation of injured or unusual anatomic structures in medical imagery is a problem that has continued to elude fully automated solutions. In this paper, the goal of easy-to-use and consistent interactive segmentation is transformed into a control synthesis problem. A nominal level set PDE is assumed to be given; this open-loop system achieves correct segmentation under ideal conditions, but does not agree with a human expert's ideal boundary for real image data. Perturbing the state and dynamics of a level set PDE via the accumulated user input and an observer-like system leads to desirable closed-loop behavior. The input structure is designed such that a user can stabilize the boundary in some desired state without needing to understand any mathematical parameters. Effectiveness of the technique is illustrated with applications to the challenging segmentations of a patellar tendon in MR and a shattered femur in CT.

Real-time left ventricular speckle tracking in 3D echocardiography with parallel block matching

level set formulation and partial differential equations. The weighted endocardium surface and epicardium surface are then varying according to the local echo sampling density. To handle the noisy and non-uniform regions within the echo data, we used a regularization term to adaptively and continuously following the gradient descent of the topological change. Finally, the optimal the endocardium surface and epicardium surface are obtained by solving the minimal energy for the two weighted surfaces using a brief propagation algorithm. Results: Even for the typically low signal-to-noise and small field-ofview in echo images, our method can successfully segment the left ventricle by robustly extracting the endocardium surface and epicardium surface. The proposed method achieves a mean accuracy of 6.1±1.3% volume differences against the gold standard for MRI segmentation method and a mean accuracy of 14.5±2.1% for segmentation of all four chambers. Conclusion: Using a level set approach to segment left ventricle in 3D echocardiography provides a fully automatic tool for physicians to obtain the anatomical and diagnostic information for cardiac functional analysis, without requiring any user input or any other assistants. The ability of our method to automatically and accurately extract endocardium and epicardium from echo has promising clinical potential in further medical diagnosis and surgery planning.

Turbulent Flame Speeds of G-equation Models in Unsteady Cellular Flows

We perform a computationl study of front speeds of G-equation models in time dependent cellular flows. The G-equations arise in premixed turbulent combustion, and are Hamilton-Jacobi type level set partial differential equations (PDEs). The curvature-strain G-equations are also non-convex with degenerate diffusion. The computation is based on monotone finite difference discretization and weighted essentially nonoscillatory (WENO) methods. We found that the large time front speeds lock into the frequency of time periodic cellular flows in curvature-strain G-equations similar to what occurs in the basic inviscid G-equation. However, such frequency locking phenomenon disappears in viscous G-equation, and in the inviscid G-equation if time periodic oscillation of the cellular flow is replaced by time stochastic oscillation.

A numerical study of turbulent flame speeds of curvature and strain G-equations in cellular flows

We study front speeds of curvature and strain G-equations arising in turbulent combustion. These G-equations are Hamilton–Jacobi type level set partial differential equations (PDEs) with non-coercive Hamiltonians and degenerate nonlinear second order diffusion. The Hamiltonian of a strain G-equation is also non-convex. Numerical computation is performed based on monotone discretization and weighted essentially nonoscillatory (WENO) approximation of transformed G-equations on a fixed periodic domain. The advection field in the computation is a two dimensional Hamiltonian flow consisting of a periodic array of counter-rotating vortices, or cellular flows. Depending on whether the evolution is predominantly in the hyperbolic or parabolic regimes, suitable explicit and semi-implicit time stepping methods are chosen. The turbulent flame speeds are computed as the linear growth rates of large time solutions. A new nonlinear parabolic PDE is proposed for the reinitialization of level set functions to prevent piling up of multiple bundles of level sets on the periodic domain. We found that the turbulent flame speed sT of the curvature G-equation is enhanced as the intensity A of cellular flows increases, at a rate between those of the inviscid and viscous G-equations. The sT of the strain G-equation increases in small A, decreases in larger A, then drops down to zero at a large enough but finite value A∗. The flame front ceases to propagate at this critical intensity A∗, and is quenched by the cellular flow.

Interactive free-form level-set surface-editing operators

We present a set of interactive, free-form editing operators for direct manipulation of level-set models that supports the creation and removal of surface detail. The mathematics, data structures and algorithms needed to implement numerous level-set modeling capabilities have been developed. The first component of these capabilities identifies the region-of-influence (ROI) on the surface to be modified, and the specification of user handles, i.e. a point or a curve within the ROI, that are used to control the free-form surface edits. The editing operators include pulling the level-set surface by a handle with the surface changes occurring symmetrically around the handle or within the ROI, surface offsetting and carving, deformations towards a profile curve and localized smoothing. The editing operators are implemented with specialized speed functions, which are incorporated into the level set partial differential equation (PDE). The PDE is then evolved to produce the desired model modification. The specific form of each speed function is described in detail. The operators have been combined with an OpenGL interface and the VISPACK level-set library to create a preliminary interactive level-set modeling system. VISPACK's narrow-band data structures have been extended to localize all computations and updates to optimize running time and provide interactive performance. Additional sketch-based level-set editing operators have been implemented within the system, and are described elsewhere [1]. A variety of level-set models are presented to demonstrate the effectiveness of the editing operators.

Facial boundary detection with an active contour model

This paper presents an active contour model for extracting accurate facial regions in complex environments. In the model, a contour is represented by a zero level set of level function ϕ, and evolved via level set partial differential equations. Then, unlike general active contours, skin-color information that is represented by 2D-Gaussian model is used for evolving and stopping a curve, which allows the proposed method to be robust to noise and varying pose. To assess the effectiveness of the proposed method, it was tested with several natural scenes, and the results were compared with those of geodesic active contours. Experimental results demonstrate the superior performance of the proposed method.

Spatiotemporal Motion Boundary Detection and Motion Boundary Velocity Estimation for Tracking Moving Objects With a Moving Camera: A Level Sets PDEs Approach With Concurrent Camera Motion Compensation

The purpose of this study is to investigate a method of tracking moving objects with a moving camera. This method estimates simultaneously the motion induced by camera movement. The problem is formulated as a Bayesian motion-based partitioning problem in the spatiotemporal domain of the image quence. An energy functional is derived from the Bayesian formulation. The Euler-Lagrange descent equations determine imultaneously an estimate of the image motion field induced by camera motion and an estimate of the spatiotemporal motion undary surface. The Euler-Lagrange equation corresponding to the surface is expressed as a level-set partial differential equation for topology independence and numerically stable implementation. The method can be initialized simply and can track multiple objects with nonsimultaneous motions. Velocities on motion boundaries can be estimated from geometrical properties of the motion boundary. Several examples of experimental verification are given using synthetic and real-image sequences.

Region tracking via level set PDEs without motion computation

We propose an approach to region tracking that is derived from a Bayesian formulation. The novelty of the approach is twofold. First, no motion field or motion parameters need to be computed. This removes a major burden since accurate motion computation has been and remains a challenging problem and the quality of region tracking algorithms based on motion critically depends on the computed motion fields and parameters. The second novelty of this approach, is that very little a priori information about the region being tracked is used in the algorithm. In particular, unlike numerous tracking algorithms, no assumption is made on the strength of the intensity edges of the boundary of the region being tracked, nor is its shape assumed to be of a certain parametric form. The problem of region tracking is formulated as a Bayesian estimation problem and the resulting tracking algorithm is expressed as a level set partial differential equation. We present further extensions to this partial differential equation, allowing the possibility of including additional information in the tracking process, such as priors on the region's intensity boundaries and we present the details of the numerical implementation. Very promising experimental results are provided.

Level set surface editing operators

We present a level set framework for implementing editing operators for surfaces. Level set models are deformable implicit surfaces where the deformation of the surface is controlled by a speed function in the level set partial differential equation. In this paper we define a collection of speed functions that produce a set of surface editing operators. The speed functions describe the velocity at each point on the evolving surface in the direction of the surface normal. All of the information needed to deform a surface is encapsulated in the speed function, providing a simple, unified computational framework. The user combines pre-defined building blocks to create the desired speed function. The surface editing operators are quickly computed and may be applied both regionally and globally. The level set framework offers several advantages. 1) By construction, self-intersection cannot occur, which guarantees the generation of physically-realizable, simple, closed surfaces. 2) Level set models easily change topological genus, and 3) are free of the edge connectivity and mesh quality problems associated with mesh models. We present five examples of surface editing operators: blending, smoothing, sharpening, openings/closings and embossing. We demonstrate their effectiveness on several scanned objects and scan-converted models.

Left ventricular endocardial surface detection based on real-time 3D echocardiographic data

Objective: A new computerized semi-automatic method for left ventricular (LV) chamber segmentation is presented. Methods: The LV is imaged by real-time three-dimensional echocardiography (RT3DE). The surface detection model, based on level set techniques, is applied to RT3DE data for image analysis. The modified level set partial differential equation we use is solved by applying numerical methods for conservation laws. The initial conditions are manually established on some slices of the entire volume. The solution obtained for each slice is a contour line corresponding with the boundary between LV cavity and LV endocardium. Results: The mathematical model has been applied to sequences of frames of human hearts (volume range: 34–109 ml) imaged by 2D and reconstructed off-line and RT3DE data. Volume estimation obtained by this new semi-automatic method shows an excellent correlation with those obtained by manual tracing ( r=0.992). Dynamic change of LV volume during the cardiac cycle is also obtained. Conclusion: The volume estimation method is accurate; edge based segmentation, image completion and volume reconstruction can be accomplished. The visualization technique also allows to navigate into the reconstructed volume and to display any section of the volume.