Evolution Of Level Set Function Research Articles

A parameterized level set method for structural topology optimization based on reaction diffusion equation and fuzzy PID control algorithm

<abstract> <p>We propose a parameterized level set method (PLSM) for structural topology optimization based on reaction diffusion equation (RDE) and fuzzy PID control algorithm. By using the proposed method, the structural compliance minimization problem under volume constraints is studied. In this work, the RDE is used as the evolution equation of level set function, and the topological derivative of the material domain is used as the reaction term of the RDE to drive the evolution of level set function, which has little dependence on the initial design domain, and can generate holes in the material domain; the compactly supported radial basis function (CS-RBF) is used to interpolate the level set function and modify the RDE, which can improve the computational efficiency, and keep the boundary smooth in the optimization process. Meanwhile, the fuzzy PID control algorithm is used to deal with the volume constraints, so that the convergence process of the structure volume is relatively stable. Furthermore, the proposed method is applied to 3D structural topology optimization. Several typical numerical examples are provided to demonstrate the feasibility and effectiveness of this method.</p> </abstract>

Magnetic Resonance Imaging Segmentation via Weighted Level Set Model Based on Local Kernel Metric and Spatial Constraint.

Magnetic resonance imaging (MRI) segmentation is a fundamental and significant task since it can guide subsequent clinic diagnosis and treatment. However, images are often corrupted by defects such as low-contrast, noise, intensity inhomogeneity, and so on. Therefore, a weighted level set model (WLSM) is proposed in this study to segment inhomogeneous intensity MRI destroyed by noise and weak boundaries. First, in order to segment the intertwined regions of brain tissue accurately, a weighted neighborhood information measure scheme based on local multi information and kernel function is designed. Then, the membership function of fuzzy c-means clustering is used as the spatial constraint of level set model to overcome the sensitivity of level set to initialization, and the evolution of level set function can be adaptively changed according to different tissue information. Finally, the distance regularization term in level set function is replaced by a double potential function to ensure the stability of the energy function in the evolution process. Both real and synthetic MRI images can show the effectiveness and performance of WLSM. In addition, compared with several state-of-the-art models, segmentation accuracy and Jaccard similarity coefficient obtained by WLSM are increased by 0.0586, 0.0362 and 0.1087, 0.0703, respectively.

A Novel Evolution Strategy of Level Set Method for the Segmentation of Overlapping Cervical Cells

Development of an accurate and automated algorithm to completely segment cervical cells in Pap images is still one of the most challenging tasks. The main reasons are the presence of overlapping cells and the lack of guiding mechanism for the convergence of ill-defined contours to the actual cytoplasm boundaries. In this paper, we propose a novel method to address these problems based on level set method (LSM). Firstly, we proposed a morphological scaling-based topology filter (MSTF) and derived a new mathematical toolbox about vector calculus for evolution of level set function (LSF). Secondly, we combine MSTF and the mathematical toolbox into a multifunctional filtering algorithm 2D codimension two-object level set method (DCTLSM) to split touching cells. The DCTLSM can morphologically scale up and down the contour while keeping part of the contour points fixed. Thirdly, we design a contour scanning strategy as the evolution method of LSF to segment overlapping cells. In this strategy, a cutting line can be detected by morphologically scaling the union LSF of the pairs of cells. Then, we used this cutting line to construct a velocity field with an effective guiding mechanism for attracting and repelling LSF. The performance of the proposed algorithm was evaluated quantitatively and qualitatively on the ISBI-2014 dataset. The experimental results demonstrated that the proposed method is capable of fully segmenting cervical cells with superior segmentation accuracy compared with recent peer works.

Structural topology optimization using a level set method with finite difference updating scheme

In conventional level set methods for topology optimization, the evolution of level set function is solved by upwind scheme and needs to be reinitialized frequently throughout the optimization. The upwind scheme is complicated in numerical implementation and the reinitialization is time-consuming and slows down the optimization efficiency. This paper presents a level set method updated with finite difference scheme for structural topology optimization. The velocity field used to update the level set function is interpolated by piecewise basis function in order to incorporate the universal mathematical programming algorithm into the optimization. The diffusion is introduced into the Hamilton-Jacobi equation and the three-step splitting method is adopted to solve the equation. By applying the proposed method, the evolution of the level set can be updated by simple central difference scheme and the procedure of reinitialization is avoided without sacrificing the numerical stability. The classical mean compliance minimization problem with several numerical examples is used to test the validity of the proposed method.

An Adaptive Fuzzy Level Set Model With Local Spatial Information for Medical Image Segmentation and Bias Correction

Medical image segmentation is still a challenging task due to noise and intensity inhomogeneity. An adaptive fuzzy level set model (AFLSM) with local spatial information is presented in this paper for accurately segmenting medical images and correcting bias field. A weighting scheme that can ensure each pixel in the neighborhood to have anisotropic weight is first introduced to remove noisy pixels and hence improve the robustness to noise. Then, a linear combination of orthogonal basis functions is used to represent bias field to ensure its smoothly and slowly varying property. Besides, to improve the robustness to initialization, this adaptive fuzzy level set model fuses a level set model with the membership function of fuzzy clustering, which can adaptively adjust the evolution of level set function. Finally, the distance regularization term in energy formulation is redefined with a novel double-well potential function to inherently maintain the accuracy and stability of the AFLSM. The AFLSM is first represented in the two-phase case and subsequently extended to the multi-phase formulation. The numerous visual segmentation results and quantitative evaluation can demonstrate the performance of the AFLSM on synthetic and real medical images. Comparison with the state-of-the-art models shows that the AFLSM can achieve better segmentation results with an improvement of 0.2286 ± 0.1477 in Dice coefficient and 0.1350 ± 0.0661 in Jaccard similarity coefficient in terms of robustness and the capability to correct bias field, respectively.

Saliency driven region-edge-based top down level set evolution reveals the asynchronous focus in image segmentation

Level set method (LSM) is popular in image segmentation due to its intrinsic features for handling complex shapes and topological changes. Existing LSM-based segmentation models can be generally grouped into region- and edge-based models. The former often have problems to deal with images whose objects have similar color intensity to that of the background when the region descriptor is insufficient. The latter usually suffer to boundary leakage problem when the images’ edges are weak. To overcome these problems, we present a novel hierarchical level set evolution protocol (SDREL), wherein we propose to use both saliency map and color intensity as region external energy to motivate an initial evolution of level set function (LSF), followed by the LSF and further smoothed by an internal energy (regulation term) to recognize a more precise boundary positioning. Our results show that the newly introduced saliency map term improves extracting objects from complex background and the asynchronous evolution of a single LSF results in a better segmentation. The new hierarchical SDREL model has been evaluated extensively and the results indicate that it has the merits of flexible initialization, robust evolution, and fast convergence. SDREL is available at: www.csbio.sjtu.edu.cn/bioinf/SDREL/.

Ochotona Curzoniae Image Segmentation Based on Improved CV Model

Ochotona curzoniae image possessed the characteristics of low contrast, intensity inhomogeneity and complex background. This paper proposed an improved Chan-Vese (CV) model. This model is combined image mean filtering information with object fitting term. As CV model has a problem with easily falling to segment image with object’s intensity inhomogeneity. In addition, to suppress the effects of background interference by using rectangular dirac delta function, to reduce the calculation of initialization of the evolving level set function by using Quadtree, to automatically obtain the initial contour by using Otsu. Tests and comparisons show that the new model has a better effect for Ochotona curzoniae image segmentation in autumn and winter season.

Globally supported radial basis function based collocation method for evolution of level set in mass segmentation using mammograms

Computer-aided detection systems play an important role for the detection of breast abnormalities using mammograms. Global segmentation of mass in mammograms is a complex process due to low contrast mammogram images, irregular shape of mass, speculated margins, and the presence of intensity variations of pixels. This work presents a new approach for mass detection in mammograms, which is based on the variational level set function. Mesh-free based radial basis function (RBF) collocation approach is employed for the evolution of level set function for segmentation of breast as well as suspicious mass region. The mesh-based finite difference method (FDM) is used in literature for evolution of level set function. This work also showcases a comparative study of mesh-free and mesh-based approaches. An anisotropic diffusion filter is employed for enhancement of mammograms. The performance of mass segmentation is analyzed by computing statistical measures. Binarized statistical image features (BSIF) and variants of local binary pattern (LBP) are computed from the segmented suspicious mass regions. These features are given as input to the supervised support vector machine (SVM) classifier to classify suspicious mass region as mass (abnormal) or non-mass (normal) region. Validation of the proposed algorithm is done on sample mammograms taken from publicly available Mini-mammographic image analysis society (MIAS) and Digital Database for Screening Mammography (DDSM) datasets. Combined BSIF features perform better as compared to LBP variants with the performance reported as 97.12% sensitivity, 92.43% specificity, and 98% AUC with 5.12 FP/I on DDSM dataset; and 95.12% sensitivity, 92.41% specificity, and 95% AUC with 4.01FP/I on MIAS dataset.

Fractional Differentiation-Based Active Contour Model Driven by Local Intensity Fitting Energy

A novel active contour model is proposed for segmentation images with inhomogeneity. Firstly, fractional order filter is defined by eight convolution masks corresponding to the image orientation in the eight compass directions. Then, the fractional order differentiation image is obtained and applied to the level set method. Secondly, we defined a new energy functional based on local image information and fractional order differentiation image; the proposed model not only can describe the input image more accurately but also can deal with intensity inhomogeneity. Local fitting term can enhance the ability of the model to deal with intensity inhomogeneity. The defined penalty term is used to reduce the occurrence of false boundaries. Finally, in order to eliminate the time-consuming step of reinitialization and ensure stable evolution of level set function, the Gaussian filtering method is used. Experiments on synthetic and real images show that the proposed model is efficient for images with intensity inhomogeneity and flexible to initial contour.

A New Nonlinear Diffusion Equation Model for Noisy Image Segmentation

Image segmentation and image denoising are two important and fundamental topics in the field of image processing. Geometric active contour model based on level set method can deal with the problem of image segmentation, but it does not consider the problem of image denoising. In this paper, a new diffusion equation model for noisy image segmentation is proposed by incorporating some classical diffusion equation denoising models into the segmental process. An assumption about the connection between the image intensity and level set function is given firstly. Some classical denoising models are employed to describe the evolution of level set function secondly. The final nonlinear diffusion equation model for noisy image segmentation is built thirdly. Then image segmentation and image denoising are combined in a united framework. The segmental results can be presented by level set function. Experimental results show that the new model has the advantage of noise resistance and is superior to traditional segmentation model.

Region-based active contours with cosine fitting energy for image segmentation.

In this paper, by employing the cosine function to express the so-called data fitting term in traditional active contour models, we propose an active contour model with the global cosine fitting energy for segmenting synthetic and real-world images. After that, in order to segment the image with intensity inhomogeneity, we extend the proposed global model to the local cosine fitting energy. In addition, we introduce level set regularization terms into the proposed models to avoid the expensive computational cost which is usually caused by the reinitialization of the evolving level set function. Experimental results indicate that the proposed models are accurate and effective when applied to segment different types of images. Moreover, our models are more efficient and robust for segmenting the images with strong noise and clutter than the Chan-Vese model and the local binary fitting model.

Hybrid geodesic region-based active contours for image segmentation

In this paper, we propose novel hybrid edge and region based active contour models. First, we consider geodesic curve and region-based model, and evolve contours based on global information to segment images with intensity homogeneity. Second, we extend the global model to the local intensity fitting energy for segmenting the images with intensity inhomogeneity. Moreover, the level set regularization term is added to the energy functional to ensure accurate computation and avoid expensive re-initialization of the evolving level set function. Experimental results indicate the proposed method has advantage over the geodesic active contour (GAC) model, the Chan–Vese (C–V) model, the Lankton’s method and the local binary fitting (LBF) model in terms of efficiency and robustness.

An Enhanced Segmentation Method by Combining Super Resolution and Level Set

High accuracy in image segmentation is highly demanded in today’s technological world. In this paper combining super resolution image reconstruction with level set image is proposed as a way to improve image segmentation. The term Super resolution, resolution enhancement, is a process to increase the resolution of an image. This improvement quality is due to sub-pixel shift of low resolution (LR) images from each other between images. In fact, each LR image has new information of the image and the main aim of super resolution is to combining these LR images to enhance the image resolution. Following this method, allows users that without any demand for additional hardware, overcoming the limitations of the imaging system. Moreover, the main goal of segmentation is to distinguish an object from background. Segmentation can do that by dividing pixels of an image into prominent image regions. In fact, a specific region is corresponding to individual objects or natural parts of objects. Segmentation can be used in different fields such as image compression and image editing. Vandewalle algorithm and level set segmentation are used in the super resolution and segmentation part respectively. Additionally the regularity of the level set function is conserved via level set regularization term to evade expensive evolving level set function re-initialization. Experiment results for real and magnetic resolution (MR) images indicate the performance of our method. Using level set segmentation technique with super resolution, improves segmentation results in both normal and MR images.

Adaptive level set evolution starting with a constant function

In this paper, we propose a novel level set evolution model in a partial differential equation (PDE) formulation. According to the governing PDE, the evolution of level set function is controlled by two forces, an adaptive driving force and a total variation (TV)-based regularizing force that smoothes the level set function. Due to the adaptive driving force, the evolving level set function can adaptively move up or down in accordance with image information as the evolution proceeds forward in time. As a result, the level set function can be simply initialized to a constant function rather than the widely-used signed distance function or piecewise constant function in existing level set evolution models. Our model completely eliminates the needs of initial contours as well as re-initialization, and so avoids the problems resulted from contours initialization and re-initialization. In addition, the evolution PDE can be solved numerically via a simple explicit finite difference scheme with a significantly larger time step. The proposed model is fast enough for near real-time segmentation applications while still retaining enough accuracy; in general, only a few iterations are needed to obtain segmentation results accurately.

A novel active contour model for medical image segmentation

A novel segmentation method for medical image with intensity inhomogeneity is introduced. In the proposed active contour model, both region and gradient information are taken into consideration. The former, i.e., region-based fitting energy, draws upon the region information and guarantees the accurate extraction of inhomogeneous image’s local information. The latter, i.e., an edge indicator, weights the length penalizing term to consider the gradient constrain. Moreover, signed distance penalizing term is also added to ensure accurate computation and avoid the time-consuming re-initialization of evolving level set function. Experiments for synthetic and real images demonstrate the feasibility and superiority of the proposed model.

Region-based active contour driven by global intensity fitting energy

In this paper, we present a novel region-based active contour model based on global intensity fitting energy in a variational level set framework. Meanwhile, an internal energy term is introduced, and it forces the level set function to be close to a signed distance function. Image global information utilized efficiently makes the proposed model insensitive to noise, and the introduced penalty term can avoid the costly re-initialization for the evolving level set function, which not only speeds up the contour evolvement, but also improves accuracy of the final contour. Comparisons with other classical region-based models, such as Chan-Vese model and Region-Scalable Fitting (RSF) model, show the advantages of our model in terms of efficiency and accuracy. Moreover, the model is robust to noise.

A multi-phase level set framework for source reconstruction in bioluminescence tomography

We propose a novel multi-phase level set algorithm for solving the inverse problem of bioluminescence tomography. The distribution of unknown interior source is considered as piecewise constant and represented by using multiple level set functions. The localization of interior bioluminescence source is implemented by tracing the evolution of level set function. An alternate search scheme is incorporated to ensure the global optimal of reconstruction. Both numerical and physical experiments are performed to evaluate the developed level set reconstruction method. Reconstruction results show that the proposed method can stably resolve the interior source of bioluminescence tomography.

Minimization of region-scalable fitting energy for image segmentation.

Intensity inhomogeneities often occur in real-world images and may cause considerable difficulties in image segmentation. In order to overcome the difficulties caused by intensity inhomogeneities, we propose a region-based active contour model that draws upon intensity information in local regions at a controllable scale. A data fitting energy is defined in terms of a contour and two fitting functions that locally approximate the image intensities on the two sides of the contour. This energy is then incorporated into a variational level set formulation with a level set regularization term, from which a curve evolution equation is derived for energy minimization. Due to a kernel function in the data fitting term, intensity information in local regions is extracted to guide the motion of the contour, which thereby enables our model to cope with intensity inhomogeneity. In addition, the regularity of the level set function is intrinsically preserved by the level set regularization term to ensure accurate computation and avoids expensive reinitialization of the evolving level set function. Experimental results for synthetic and real images show desirable performances of our method.

Kernel Density Estimation and Intrinsic Alignment for Shape Priors in Level Set Segmentation

In this paper, we make two contributions to the field of level set based image segmentation. Firstly, we propose shape dissimilarity measures on the space of level set functions which are analytically invariant under the action of certain transformation groups. The invariance is obtained by an intrinsic registration of the evolving level set function. In contrast to existing approaches to invariance in the level set framework, this closed-form solution removes the need to iteratively optimize explicit pose parameters. The resulting shape gradient is more accurate in that it takes into account the effect of boundary variation on the object's pose. Secondly, based on these invariant shape dissimilarity measures, we propose a statistical shape prior which allows to accurately encode multiple fairly distinct training shapes. This prior constitutes an extension of kernel density estimators to the level set domain. In contrast to the commonly employed Gaussian distribution, such nonparametric density estimators are suited to model aribtrary distributions. We demonstrate the advantages of this multi-modal shape prior applied to the segmentation and tracking of a partially occluded walking person in a video sequence, and on the segmentation of the left ventricle in cardiac ultrasound images. We give quantitative results on segmentation accuracy and on the dependency of segmentation results on the number of training shapes.