Level Set Formulation Research Articles

ACTIVE CONTOURS DRIVEN BY LOCAL INTENSITY AND LOCAL GRADIENT FITTING ENERGIES

This paper presents a new local region-based and local gradient-based active contour model in a variational level set formulation for image segmentation. The model consists of three parts: the local region term, the local gradient term and the regularization term. The local region term is insensitive to noise, while the local gradient term has better capability of localization than the local region. The energy minimization is achieved by the level set evolution in an iterative strategy. In each iteration, the local intensity and local gradient are updated and fed into the level set evolution. Comparative experiments show that our model achieves the better performance on the ultrasound images with noise and low signal-to-noise ratio than the local binary fitting (LBF) energy model.

A Local Statistical Information Active Contour Model for Image Segmentation

This paper presents a local statistical information (LSI) active contour model. Assuming that the distribution of intensity belonging to each region is a Gaussian distribution with spatially varying statistical information, and defining an energy function, the authors integrate the entire image domain. Then, this energy is incorporated into a variational level set formulation. Finally, by minimizing the energy functional, a curve evolution equation can be obtained. Because the image local information is considered, the proposed model can effectively deal with the image with intensity inhomogeneity. Experimental results on synthetic and real images demonstrate that the proposed model can effectively segment the image with intensity inhomogeneity.

A conditioned level-set method with block-division strategy to flame front extraction based on OH-PLIF measurements

A novel approach to extract flame fronts, which is called the conditioned level-set method with block division (CLSB), has been developed. Based on a two-phase level-set formulation, the conditioned initialization and region-lock optimization appear to be beneficial to improve the efficiency and accuracy of the flame contour identification. The original block-division strategy enables the approach to be unsupervised by calculating local self-adaptive threshold values autonomously before binarization. The CLSB approach has been applied to deal with a large set of experimental data involving swirl-stabilized premixed combustion in diluted regimes operating at atmospheric pressures. The OH-PLIF measurements have been carried out in this framework. The resulting images are, thus, featured by lower signal-to-noise ratios (SNRs) than the ideal image; relatively complex flame structures lead to significant non-uniformity in the OH signal intensity; and, the magnitude of the maximum OH gradient observed along the flame front can also vary depending on flow or local stoichiometry. Compared with other conventional edge detection operators, the CLSB method demonstrates a good ability to deal with the OH-PLIF images at low SNR and with the presence of a multiple scales of both OH intensity and OH gradient. The robustness to noise sensitivity and intensity inhomogeneity has been evaluated throughout a range of experimental images of diluted flames, as well as against a circle test as Ground Truth (GT).

A variational level set segmentation formulation based on signal model for images in the presence of intensity inhomogeneity

ABSTRACTBiological images with significant intensity inhomogeneity are considerably difficult for the tissue segmentation. To overcome the difficulties caused by the intensity inhomogeneity, this study presents a variational level set method to simultaneous bias field estimation and tissue segmentation for images in the presence of intensity inhomogeneity. An energy function is defined in terms of two data fitting terms which incorporate the local clustering properties into the global region information. First, depended on the observed image mode, the local cluster property based on the observed signal is simplified to a criterion function which is similar to the Mumford‐Shah model. The local criterion energy is then integrated with a global region measure, which is based on intensity difference of the true signal. The energy is minimized in a variational level set formulation with a regularity term, thus avoiding the expensive computation of the level set reinitialization and keeping the curve close to the signal distance function. Experiment results on biological images show desirable performance and demonstrate the effectiveness of the proposed algorithm.

Transient filling modelling at meso-level for RTM process using a single phase LSM

A single phase level set method for capturing the propagating front within textile unit-cells is presented. The computational domain is embedded within an artificial Cartesian grid in order to avoid the difficulties associated with the meshing step. The flow is modelled by using the Stokes/Brinkman equations. Hence, the necessity of describing the yarn boundaries is avoided. The propagating front is represented by using a level set formulation where the extra step of reinitialization is avoided. The flow equations are solved only in the filled part of the domain. The velocity in the unfilled part is obtained through a harmonic continuation. Numerical tests on simple configurations in 2D and 3D space are presented in order to demonstrate the proposed method.

A robust level set method based on local statistical information for noisy image segmentation

The paper presents an improved local region-based active contour model for image segmentation, which is robust to noise. A data fitting energy functional is defined in terms of curves and the energy terms which are based on the differences between the local average intensity and the global intensity means. Then the energy is incorporated into a level set variational formulation, from which a curve evolution equation is derived for energy minimization. And then the level set function is regularized by Gaussian filter to keep smooth and eliminate the re-initialization. By using the local statistical information, the proposed model can handle with noisy images. The proposed model is first presented as a two-phase level set formulation and then extended to a multi-phase one. Experimental results show desirable performances of the proposed model for both noisy synthetic and real images, especially with high level noise.

Computing the dynamics of biomembranes by combining conservative level set and adaptive finite element methods

The numerical simulation of the deformation of vesicle membranes under simple shear external fluid flow is considered in this paper. A saddle-point approach is proposed for the imposition of the fluid incompressibility and the membrane inextensibility constraints, through Lagrange multipliers defined in the fluid and on the membrane respectively. Using a level set formulation, the problem is approximated by mixed finite elements combined with an automatic adaptive mesh procedure at the vicinity of the membrane boundary. Numerical experiments show that this combination of the saddle-point and adaptive mesh method enhances the robustness of the method. The effect of inertia on the stability of the vesicle in a shear flow is also investigated.

Active Contour Driven by Local Region Statistics and Maximum A Posteriori Probability for Medical Image Segmentation

This paper presents a novel active contour model in a variational level set formulation for simultaneous segmentation and bias field estimation of medical images. An energy function is formulated based on improved Kullback-Leibler distance (KLD) with likelihood ratio. According to the additive model of images with intensity inhomogeneity, we characterize the statistics of image intensities belonging to each different object in local regions as Gaussian distributions with different means and variances. Then, we use the Gaussian distribution with bias field as a local region descriptor in level set formulation for segmentation and bias field correction of the images with inhomogeneous intensities. Therefore, image segmentation and bias field estimation are simultaneously achieved by minimizing the level set formulation. Experimental results demonstrate desirable performance of the proposed method for different medical images with weak boundaries and noise.

Learning Distance Transform for Boundary Detection and Deformable Segmentation in CT Prostate Images.

Segmenting the prostate from CT images is a critical step in the radio-therapy planning for prostate cancer. The segmentation accuracy could largely affect the efficacy of radiation treatment. However, due to the touching boundaries with the bladder and the rectum, the prostate boundary is often ambiguous and hard to recognize, which leads to inconsistent manual delineations across different clinicians. In this paper, we propose a learning-based approach for boundary detection and deformable segmentation of the prostate. Our proposed method aims to learn a boundary distance transform, which maps an intensity image into a boundary distance map. To enforce the spatial consistency on the learned distance transform, we combine our approach with the auto-context model for iteratively refining the estimated distance map. After the refinement, the prostate boundaries can be readily detected by finding the valley in the distance map. In addition, the estimated distance map can also be used as a new external force for guiding the deformable segmentation. Specifically, to automatically segment the prostate, we integrate the estimated boundary distance map into a level set formulation. Experimental results on 73 CT planning images show that the proposed distance transform is more effective than the traditional classification-based method for driving the deformable segmentation. Also, our method can achieve more consistent segmentations than human raters, and more accurate results than the existing methods under comparison.

Medical image segmentation based on a hybrid region-based active contour model.

A novel hybrid region-based active contour model is presented to segment medical images with intensity inhomogeneity. The energy functional for the proposed model consists of three weighted terms: global term, local term, and regularization term. The total energy is incorporated into a level set formulation with a level set regularization term, from which a curve evolution equation is derived for energy minimization. Experiments on some synthetic and real images demonstrate that our model is more efficient compared with the localizing region-based active contours (LRBAC) method, proposed by Lankton, and more robust compared with the Chan-Vese (C-V) active contour model.

New region-scalable discriminant and fitting energy functional for driving geometric active contours in medical image segmentation.

We propose a novel region-based geometric active contour model that uses region-scalable discriminant and fitting energy functional for handling the intensity inhomogeneity and weak boundary problems in medical image segmentation. The region-scalable discriminant and fitting energy functional is defined to capture the image intensity characteristics in local and global regions for driving the evolution of active contour. The discriminant term in the model aims at separating background and foreground in scalable regions while the fitting term tends to fit the intensity in these regions. This model is then transformed into a variational level set formulation with a level set regularization term for accurate computation. The new model utilizes intensity information in the local and global regions as much as possible; so it not only handles better intensity inhomogeneity, but also allows more robustness to noise and more flexible initialization in comparison to the original global region and regional-scalable based models. Experimental results for synthetic and real medical image segmentation show the advantages of the proposed method in terms of accuracy and robustness.

Combining learning-based intensity distributions with nonparametric shape priors for image segmentation

Integration of shape prior information into level set formulations has led to great improvements in image segmentation in the presence of missing information, occlusion, and noise. However, most shape-based segmentation techniques incorporate image intensity through simplistic data terms. A common underlying assumption of such data terms is that the foreground and the background regions in the image are homogeneous, i.e., intensities are piecewise constant or piecewise smooth. This situation makes integration of shape priors inefficient in the presence of intensity inhomogeneities. In this paper, we propose a new approach for combining information from shape priors with that from image intensities. More specifically, our approach uses shape priors learned by nonparametric density estimation and incorporates image intensity distributions learned in a supervised manner. Such a combination has not been used in previous work. Sample image patches are used to learn the intensity distributions, and segmented training shapes are used to learn the shape priors. We present an active contour algorithm that takes these learned densities into account for image segmentation. Our experiments on synthetic and real images demonstrate the robustness of the proposed approach to complicated intensity distributions, and occlusions, as well as the improvements it provides over existing methods.

An adaptive spectral/DG method for a reduced phase-space based level set approach to geometrical optics on curved elements

We present an adaptive spectral/discontinuous Galerkin (DG) method on curved elements to simulate high-frequency wavefronts within a reduced phase-space formulation of geometrical optics. Following recent work, the approach is based on the use of level sets defined by functions satisfying the Liouville equations in reduced phase-space and, in particular, it relies on the smoothness of these functions to represent them by rapidly convergent spectral expansions in the phase variables. The resulting (hyperbolic) system of equations for the coefficients in these expansions are then amenable to a high-order accurate treatment via DG approximations. In the present work, we significantly expand on the applicability and efficiency of the approach by incorporating mechanisms that allow for its use in scattering simulations and for a reduced overall computational cost. With regards to the former we demonstrate that the incorporation of curved elements is necessary to attain any kind of accuracy in calculations that involve scattering off non-flat interfaces. With regards to efficiency, on the other hand, we also show that the level-set formulation allows for a space p-adaptive scheme that under-resolves the level-set functions away from the wavefront without incurring in a loss of accuracy in the approximation of its location. As we show, these improvements enable simulations that are beyond the capabilities of previous implementations of these numerical procedures.

Multi-branched cerebrovascular segmentation based on phase-field and likelihood model

Angiograms have been extensively used by neurosurgeons for vascular and non-vascular pathology. Indeed, examining the cerebral vessel network is helpful in revealing arteriosclerosis, diabetes, hypertension, cerebrovascular diseases and strokes. Thus, accurate segmentation of blood vessels in the brain is of major importance to radiologists. Many algorithms have been proposed for blood vessel segmentation. Although they work well for segmenting major parts of vessels, these techniques cannot handle challenging problems including (a) segmentation of thinner blood vessels due to low contrast around thin blood vessels; (b) inhomogeneous intensities, which lead to inaccurate segmentation. In order to tackle these challenges, we developed a new Allen Cahn (AC) equation and likelihood model to segment blood vessels in angiograms. Its level set formulation combines length, region-based and regularization terms. The length term is represented by the AC equation with a double well potential. The region-based term combines both local and global statistical information, where the local part deals with the intensity inhomogeneity, and the global part solves the low contrast problem. Finally, the regularization term ensures the stability of contour evolution. Experimental results show that the proposed method is both efficient and robust, and is able to segment inhomogeneous images with an arbitrary initial contour. It outperforms other methods in detecting finer detail.

Computational and experimental studies of rapid free-surface granular flows around obstacles

The study of the rapid free-surface granular flows that are driven under gravitational acceleration has attracted much attention in recent years. This is not only because such flows occur in many industrial and natural examples, but also because their transport mechanisms can be observed through small-scale lab experiments, modeled with continuum theories, and simulated by computers. When granular particles rapidly propagate around obstacles, the resulting phenomena – shock waves, particle free regions or granular vacua, expansion fans and stagnation zones – are of particular theoretical and practical importance. In this paper we develop a computational method for a hydraulic-type avalanche model that is able to simulate such phenomena. It numerically solves the avalanche model over a structured grid by including the topography of the obstacles. Different finite-difference TVD methods based on the non-oscillatory central (NOC) scheme are tested in computation to compare the resolution of the shock waves when granular flows propagate against an oblique wedge. This is also involved in the choice of the limiters, but which are shown rather insensitive in such computations. A level set formulation is coupled into the governing equations to follow the evolution of the boundaries of the granular vacua. It is tested in a situation when granular particles flow around a circular cylinder, where bow shock waves, granular vacua, expansion fans and stationary zones are all captured in the computation. These results agree well and consistently with the laboratory experiments.

A modified level set approach to 2D modeling of dynamic recrystallization

The macroscopic properties of metallic materials depend on the state of the grain microstructure. Recrystallization acts as one of the most important mechanisms in the evolution of the microstructure and hence also of the macroscopic properties. This paper presents a mesoscale model of microstructure evolution due to recrystallization, based on a level set formulation employed in a finite element setting. The use of level sets to represent grains and grain boundaries in polycrystal microstructures is a relatively recent development in computational materials science and the present contribution suggests new methodologies such as interface reconstruction, allowing for example boundary conditions to be prescribed along grain boundary interfaces and distinct localization and representation of grain boundary junctions. Polycrystal plasticity is modeled by considering the evolution of dislocation density in the individual crystals. The influence of grain boundaries on dislocation accumulation is captured in the model, causing the formation of dislocation density gradients within the grains. The model is used in simulations of dynamic recrystallization, taking pure copper as example material. It is shown that the proposed model captures the salient features of dynamic recrystallization during thermomechanical materials processing.

Numerical Analysis of Film Boiling in Liquid Jet Impingement

A numerical approach is presented for computing film boiling in liquid jet impingement on a high-temperature plate. The conservation equations of mass, momentum and energy are numerically solved in the liquid, vapor, and air phases. The sharp-interface level-set formulation is employed to track the liquid-air interface, as well as the liquid-vapor interface with phase change. A simplified analytical model for a thin vapor film, whose thickness is several orders of magnitude smaller than the liquid layer, is incorporated into the level-set formulation. The multiscale approach is tested through the computations of film boiling in a circular water jet.

Gaussian beam methods for the Schrödinger equation with discontinuous potentials

We propose Eulerian and Lagrangian Gaussian beam methods for the Schrödinger equation with discontinuous potentials. At the quantum barriers where the potential is discontinuous, we derive suitable interface conditions to account for quantum scattering information. These scattering interface conditions are then built into the numerical fluxes in the Eulerian level set formulation of the Gaussian beam methods, and are also used in the Lagrangian formulation, including an interface condition for the Hessian matrix. We carry both 1D and 2D numerical examples to verify the accuracy of the method.

A Variation Level Set Formulation Local Based C-V Model for Medical Images Segmentation

Local image information is crucial for accurate segmentation of images with intensity inhomogeneity which usually occurs in medical images. However, image information in local region is not incorporated in popular region-based active contour models, such as piecewise constant models and piecewise smooth models. In this paper, a method which is able to use local information is proposed. The main point is the introduction of the local fitting information expressed by a kernel function which is crucial for segmentation. Our method is based on piecewise constant Chan-Vese model, and compare with different methods for several synthetic images and medical images.

Crystallization of supercooled water: A level-set-based modeling of the dendrite tip velocity

’It is well-known that solidification front of a supercooled liquid is unstable; consequently, this instability leads to the appearance of an array of dendrites of sub-micron diameter. The shape and the velocity of the dendrite propagation are determined by the thermodynamic properties of the liquid and solid phases, including interfacial energy as well as the initial temperatures of both. Accordingly, the numerical simulation of solidification process is a rather challenging problem which requires an accurate prediction of high temperature gradients near the moving solidification front. In this study a relevant level set formulation has been developed enabling correct determination of the position and the curvature of the liquid/solid interface. At this interface a Dirichlet boundary condition for the temperature field is imposed by applying a ghost-face method. For the purpose of updating the level set function and optimizing computing time a narrow-band around the interface is introduced. Within this band, whose width is temporally adjusted to the maximum curvature of the interface, the normal-to-interface velocity is appropriately expanded. The computational model is firstly validated along with the analytical solution of stable freezing. The tip velocity of dendritic patterns (pertinent to unstable freezing) is investigated by performing two-dimensional simulations. The computational results exhibit excellent qualitative and quantitative agreement with the marginal stability theory as well as with the available experiments in the heat-diffusion-dominated region.