Variational Level Set Method Research Articles

Image Segmentation with Depth Information via Simplified Variational Level Set Formulation

Image segmentation with depth information can be modeled as a minimization problem with Nitzberg–Mumford–Shiota functional, which can be transformed into a tractable variational level set formulation. However, such formulation leads to a series of complicated high-order nonlinear partial differential equations which are difficult to solve efficiently. In this paper, we first propose an equivalently reduced variational level set formulation without using curvatures by taking level set functions as signed distance functions. Then, an alternating direction method of multipliers (ADMM) based on this simplified variational level set formulation is designed by introducing some auxiliary variables, Lagrange multipliers via using alternating optimization strategy. With the proposed ADMM method, the minimization problem for this simplified variational level set formulation is transformed into a series of sub-problems, which can be solved easily via using the Gauss–Seidel iterations, fast Fourier transform and soft thresholding formulas. The level set functions are treated as signed distance functions during computation process via implementing a simple algebraic projection method, which avoids the traditional re-initialization process for conventional variational level set methods. Extensive experiments have been conducted on both synthetic and real images, which validate the proposed approach, and show advantages of the proposed ADMM projection over algorithms based on traditional gradient descent method in terms of computational efficiency.

Parametric structural shape & topology optimization with a variational distance-regularized level set method

The signed distance function (SDF) gives the shortest distance from a given point to the boundary, and the sign indicates whether this point is inside or outside the closed boundary or enclosed region. The SDF property is highly preferred in classical level set methods to maintain the numerical stability during the topology optimization process and provide a metric for the distance-based interpolation of different material properties. In conventional level set methods, a common way of achieving a level set function with the signed distance property is to periodically implement the so-called reinitialization scheme by solving an additional Hamilton–Jacobi partial differential equation. However, such reinitialization scheme is implemented outside the optimization loop with the optimization process suspended, which may shift the optimization result and bring convergence issues. In this paper, a double-well potential functional is employed for distance regularization inside the topology optimization loop, which can enforce the signed distance property of the level set function in a narrow band along the design boundaries while keeping the level set function flat in the rest area of the computational domain. The radial basis function (RBF) based parameterization technique is combined with mathematical programming to improve the performance of the proposed distance-regularized topology optimization method in handling problems with non-convex objective functions and multiple constraints. The flatness of the level set function in the material region also enables easy creation of new holes to the design in the topology optimization process. Both 2D and 3D benchmark examples are employed to demonstrate the validity of the proposed method.

Level set framework of multi-atlas label fusion with applications to magnetic resonance imaging segmentation of brain region of interests and cardiac left ventricles

ABSTRACT Background and Objectives: This paper evaluates the performance of a variational level set method for performing label fusion through the use of a penalty term, label fusion term, and length regularization term, which automatically labels objects of interest in biomedical images. This paper is an extension of our preliminary work in the conference paper. We mainly focus on the validation of the variational level set method. Subjects and Methods: Label fusion is achieved by combining the three terms: label fusion term, image data term, and regularization term. The curve evolution derived from the energy minimization is impacted by the three terms simultaneously to achieve optimal label fusion. Each label obtained from the nonlinear registration method is represented by a level set function whose zero level contour encloses the labeled region. In Lu et al.'s paper, they employ the level set formulation only for hippocampus segmentation. Results: Our method is compared with majority voting (MV), local weighted voting (LWV), and Simultaneous Truth and Performance Level Estimation (STAPLE). The method is evaluated on MICCAI 2012 Multi-Atlas Labeling challenge and MICCAI 2012 ventricle segmentation challenge. The mean Dice metric is computed using different atlases and produces results with 0.85 for the hippocampus, 0.77 for the amygdala, 0.87 for the caudate, 0.78 for the pallidum, 0.89 for the putamen, 0.91 for the thalamus, and 0.78 for cardiac left ventricles. Conclusions: Experimental results demonstrate that our method is robust to parameter setting and outperforms MV, LWV, and STAPLE. The image data term plays a key role in improving the segmentation accuracy. Our method can obtain satisfactory results with fewer atlases.

Normal Vector Projection Method used for Convex Optimization of Chan-Vese Model for Image Segmentation

The variational level set method is one of the main methods of image segmentation. Due to signed distance functions as level sets have to keep the nature of the functions through numerical remedy or additional technology in an evolutionary process, it is not very efficient. In this paper, a normal vector projection method for image segmentation using Chan-Vese model is proposed. An equivalent formulation of Chan-Vese model is used by taking advantage of property of binary level set functions and combining with the concept of convex relaxation. Threshold method and projection formula are applied in the implementation. It can avoid the above problems and obtain a global optimal solution. Experimental results on both synthetic and real images validate the effects of the proposed normal vector projection method, and show advantages over traditional algorithms in terms of computational efficiency.

Segmentation of differential structures on computed tomography images for diagnosis lung-related diseases

Abstract Computer-aided diagnostics (CAD) systems for automatic detection of lung cancer or lung-related diseases have highly depended on the segmentation accuracy of differential structures from computed tomography (CT) scan images. By detection of differential structures such as right/left Lungs, lung nodules, human airways and pulmonary trees, the new segmentation algorithm (PropSeg) is proposed. The PropSeg method is developed based on four major phases such as pre-processing, detection of candidate regions, segmentation, and post-processing. The pre-processing step is performed to enhance by reconstruction of an input image into the 4 frequency subbands through discrete wavelet transform (DWT) and un-sharp energy mask (UEM). The 3 levels of fuzzy c-means (FCM) clustering is used to detect candidate regions by an integration of local energy constraints (LEC) and variational level set (VLS) method is then utilized to segment differential regions. Moreover, the post-processing step is performed by morphological edge detection to enhance the results of segmentation. The system is tested with manually draw radiologist contours on the 220 images by using statistical measures. The performance of PropSeg is also compared with other four state-of-the-art segmentation methods. The achieve results show that the PropSeg system is outperformed compared to other techniques and it is favorable for automatic diagnosis of lung cancer or to detect lung-related diseases.

Hepatic vessel segmentation using variational level set combined with non-local robust statistics

Hepatic vessel segmentation is a challenging step in therapy guided by magnetic resonance imaging (MRI). This paper presents an improved variational level set method, which uses non-local robust statistics to suppress the influence of noise in MR images. The non-local robust statistics, which represent vascular features, are learned adaptively from seeds provided by users. K-means clustering in neighborhoods of seeds is utilized to exclude inappropriate seeds, which are obviously corrupted by noise. The neighborhoods of appropriate seeds are placed in an array to calculate the non-local robust statistics, and the variational level set formulation can be constructed. Bias correction is utilized in the level set formulation to reduce the influence of intensity inhomogeneity of MRI. Experiments were conducted over real MR images, and showed that the proposed method performed better on small hepatic vessel segmentation compared with other segmentation methods.

Wood Surface Defect Image Segmentation Method based on Canny Operator and GGAC Model of Reaction Diffusion Equation

GGAC is an improvement based on geodesic active contour model (GAC). GGAC model is a widely used method for image segmentation. But, it will be difficult to achieve satisfactory segmentation results to the texture, uneven structure, edge particles, weak edge and other features of the wood surface image. Therefore, the author proposes a segmentation method that integrates the improved Canny edge detection result integrated into the improved GGAC model redrawing boundary stop function, and uses the improved variational level set method to achieve the numerical solution. The algorithm has reduced the choice sensitivity to the initial contour and enhanced the scalability, which can make the profile curve converge to defect edges more rapidly, avoid the local optimum, and improve segmentation effects of weak edges and uneven image. The results are clearer, more consistent and real-time. It has provided a more effective way to segment the wood surface defects, and broadened the application scope of Canny operator and an improved geodesic active contour model.

Robust Cell Segmentation for Histological Images of Glioblastoma.

Glioblastoma (GBM) is a malignant brain tumor with uniformly dismal prognosis. Quantitative analysis of GBM cells is an important avenue to extract latent histologic disease signatures to correlate with molecular underpinnings and clinical outcomes. As a prerequisite, a robust and accurate cell segmentation is required. In this paper, we present an automated cell segmentation method that can satisfactorily address segmentation of overlapped cells commonly seen in GBM histology specimens. This method first detects cells with seed connectivity, distance constraints, image edge map, and a shape-based voting image. Initialized by identified seeds, cell boundaries are deformed with an improved variational level set method that can handle clumped cells. We test our method on 40 histological images of GBM with human annotations. The validation results suggest that our cell segmentation method is promising and represents an advance in quantitative cancer research.

基于变分水平集方法的散乱数据曲面重构研究

工程学和医学等领域中的很多问题进行抽象后可转化为散乱数据的光滑曲面重构问题。在散乱数据附近设定初始轮廓，将初始轮廓间接地表示为水平集函数的零水平集形式，最终将初始活动轮廓的演化转化成水平集函数的演化。定义一个以水平集函数为自变量的能量泛函来度量它与散乱点的逼近程度，利用改进的变分水平集方法演化初始轮廓，达到重构目的。本文中，一是对Heaviside函数进行改进，使初始轮廓能更好的收敛于目标轮廓；二是给出了模型数值计算的具体过程，并以二维实例说明散乱数据重构的效果。 A lot of problems in the field of engineering and medicine after abstraction can be converted into scattered data reconfiguration problems. We set initial contour near the scattered data, expressed as the zero level set of the level set function, which will eventually turn the evolution of initial active contour into the evolution of the level set function. We define an energy functional who takes the level set function as its independent variable, to measure approximation degree with the scattered data, and use the improved variational level set method to evolve the initial contour and achieve reconstruction. In this paper, firstly, putting forward an improved Heaviside function to make the initial contour can better converge in the target contour; secondly, the specific process of numerical calculation model is given, and a two-dimensional example illustrates the effect of the scattered data reconstruction.

A novel ship detection method for SAR images based on nonlinear diffusion filtering and Gaussian curvature

ABSTRACTThis letter takes account of the radiation feature of ships in SAR images and proposes a novel automatic approach to achieve high ship–sea contrast for ship detection. The proposed method consists of two steps. Nonlinear diffusion filtering is used to produce a smooth image, a regular surface. After the filtering process, the Gaussian curvature of the regular surface is calculated using the coefficients of the fundamental form defined on the surface. The variational level set method is applied to locate reflective small regions in the revised Gaussian curvature matrix. In this manner, ships on the sea area are detected. Experimental results show that the proposed method can accurately detect targets from the surrounding sea clutter.

Brain Tumour Segmentation using Level Set Method and Affected Area Calculation

Medical image processing is the most important and challenging field now a days. MRI image processing is one of the parts of this field. Brain tumour segmentation in magnetic resonance imaging (MRI) has become an emergent research area in the field of medical imaging system. In this paper we proposed a variational level set method and some morphological operation to segment the brain tumour from MRI image by using MATLAB. Actually we describe variational formulation on geometric active contours that forces the level set function at zero level to be close to signed distance function and without re-initialization process. The variational formulation uses energy function and partial diferential equation to evolve the level set function. Tumourshape area is connected component in binary image and calculated this connected area using some properties of morphological operation .

Regional active contours based on variational level sets and machine learning for image segmentation

Image segmentation is the problem of partitioning an image into different subsets, where each subset may have a different characterization in terms of color, intensity, texture, and/or other features. Segmentation is a fundamental component of image processing, and plays a significant role in computer vision, object recognition, and object tracking. Active Contour Models (ACMs) constitute a powerful energy-based minimization framework for image segmentation, which relies on the concept of contour evolution. Starting from an initial guess, the contour is evolved with the aim of approximating better and better the actual object boundary. Handling complex images in an efficient, effective, and robust way is a real challenge, especially in the presence of intensity inhomogeneity, overlap between the foreground/background intensity distributions, objects characterized by many different intensities, and/or additive noise. In this thesis, to deal with these challenges, we propose a number of image segmentation models relying on variational level set methods and specific kinds of neural networks, to handle complex images in both supervised and unsupervised ways. Experimental results demonstrate the high accuracy of the segmentation results, obtained by the proposed models on various benchmark synthetic and real images compared with state-of-the-art active contour models.

An improved hybrid gradient variation level set method for image segmentation and bias correction

Minimizing an energy functional defined on the level set function with gradient descent method is called variational level set methods. The variational methods are characterized by deriving an energy functional from some priori mathematical models and minimizing this energy functional by calculating the gradient over all possible partitions. The Sobolev gradient is more effective for minimizing the curve length functional than the gradient by the gradient descent method. In this paper, we propose a novel hybrid gradient active contour model in partial differential equation formulation based on method of Li et al. to correct the bias for image segmentation. By using the Sobolev gradient for the internal energy (curve length), and using gradient for the external energy during the evolution of curve. The proposed model can effectively and efficiently segment images with intensity inhomogeneity. Experimental results obtained on synthetic and real images show the advantages of our method in terms of computational efficiency.

Minimum mutual information based level set clustering algorithm for fast MRI tissue segmentation.

Accurate and accelerated MRI tissue recognition is a crucial preprocessing for real-time 3d tissue modeling and medical diagnosis. This paper proposed an information de-correlated clustering algorithm implemented by variational level set method for fast tissue segmentation. The key idea is to design a local correlation term between original image and piecewise constant into the variational framework. The minimized correlation will then lead to de-correlated piecewise regions. Firstly, by introducing a continuous bounded variational domain describing the image, a probabilistic image restoration model is assumed to modify the distortion. Secondly, regional mutual information is introduced to measure the correlation between piecewise regions and original images. As a de-correlated description of the image, piecewise constants are finally solved by numerical approximation and level set evolution. The converged piecewise constants automatically clusters image domain into discriminative regions. The segmentation results show that our algorithm performs well in terms of time consuming, accuracy, convergence and clustering capability.

On the Relationship between Variational Level Set-Based and SOM-Based Active Contours.

Most Active Contour Models (ACMs) deal with the image segmentation problem as a functional optimization problem, as they work on dividing an image into several regions by optimizing a suitable functional. Among ACMs, variational level set methods have been used to build an active contour with the aim of modeling arbitrarily complex shapes. Moreover, they can handle also topological changes of the contours. Self-Organizing Maps (SOMs) have attracted the attention of many computer vision scientists, particularly in modeling an active contour based on the idea of utilizing the prototypes (weights) of a SOM to control the evolution of the contour. SOM-based models have been proposed in general with the aim of exploiting the specific ability of SOMs to learn the edge-map information via their topology preservation property and overcoming some drawbacks of other ACMs, such as trapping into local minima of the image energy functional to be minimized in such models. In this survey, we illustrate the main concepts of variational level set-based ACMs, SOM-based ACMs, and their relationship and review in a comprehensive fashion the development of their state-of-the-art models from a machine learning perspective, with a focus on their strengths and weaknesses.

Lesion segmentation and identification of breast tumor on dynamic contrast-enhanced magnetic resonance imaging

Dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) can show subtle lesion morphology, improve the display of lesion definitions, and objectively reflect the blood supply of breast tumors; it can also reflect different strengthening patterns of normal tissues and lesion areas after medical tracer injection. DCE-MRI has become an important basis for the clinical diagnosis of breast cancer. To DCE-MRI data acquired from several hospitals across multiple provinces, a series of in-silico computational methods were applied for lesion segmentation and identification of breast tumor in this paper. The image segmentation methods include Otsu segmentation of subtraction images, signal-interference-ratio segmentation method and an improved variational level set method, each has its own application scope. After that, the distribution of benign and malignant in lesion region is identified based on three-time-point theory. From the experiment, the analysis of DCE-MRI data of breast tumor can show the distribution of benign and malignant in lesion region, provide a great help for clinicians to diagnose breast cancer more expediently and lay a basis for medical diagnosis and treatment planning.

An efficient Self-Organizing Active Contour model for image segmentation

Active Contour Models (ACMs) constitute a powerful energy-based minimization framework for image segmentation, based on the evolution of an active contour. Among ACMs, supervised ACMs are able to exploit the information extracted from supervised examples to guide the contour evolution. However, their applicability is limited by the accuracy of the probability models they use. As a consequence, effectiveness and efficiency of supervised ACMs are among their main real challenges, especially when handling images containing regions characterized by intensity inhomogeneity. In this paper, to deal with such kinds of images, we propose a new supervised ACM, named Self-Organizing Active Contour (SOAC) model, which combines a variational level set method (a specific kind of ACM) with the weights of the neurons of two Self-Organizing Maps (SOMs). Its main contribution is the development of a new ACM energy functional optimized in such a way that the topological structure of the underlying image intensity distribution is preserved - using the two SOMs - in a parallel-processing and local way. The model has a supervised component since training pixels associated with different regions are assigned to different SOMs. Experimental results show the superior efficiency and effectiveness of SOAC versus several existing ACMs.

Anisotropic diffusion filter based edge enhancement for the segmentation of carotid intima-media layer in ultrasound images using variational level set method without re-initialisation.

In this work an attempt has been made to enhance the edges and segment the boundary of intima-media layer of Common Carotid Artery (CCA) using anisotropic diffusion filter and level set method. Ultrasound B mode longitudinal images of normal and abnormal images of common carotid arteries are used in this study. The images are subjected to anisotropic diffusion filter to generate edge map. This edge map is used as a stopping boundary in variational level set method without re-initialisation to segment the intima-media layer. Geometric features are extracted from this layer and analyzed statistically. Results show that anisotropic diffusion filtering is able to extract the edges in both normal and abnormal images. The obtained edge maps are found to have high contrast and sharp edges. The edge based variational level set method is able to segment the intima-media layer precisely from common carotid artery. The extracted geometrical features such as major axis and extent are found to be statistically significant in differentiating normal and abnormal images. Thus this study seems to be clinically useful in diagnosis of cardiovascular disease.

Integrating Variational Level Set Method and Fuzzy c-Means to Automatically Segment the MRI Brain Images

The main aim of this paper is to provide a method that could easily delineate the exact location of tumor region from brain MRI images by integrating the fuzzy c-means and level set method. The proposed method smoothly exploits the spatial function during FCM clustering which in turn proves the automaticity of the method by dividing the original image into clusters and then using one cluster for automatic initialization. Since automating the process reduces the time utilization of processing thereby making the work less tedious, hence if considered it could be competent tool in future. Another problem associated when utilizing manual methods is that it may lead to different results when produced by different medical experts which can be completely erased while using this method. Secondly, the proposed method uses the level set method to find the contour of tumor region in the original image which helps in situations where image changes their topologies by merging or splitting in two. Thus, the proposed method is using the convenient variational level set method in place of traditional level set method thereby eliminating one more drawback of re-initializing the contour several times during image segmentation.

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.