Variational Level Set Formulation Research Articles

Level-set evolution for medical image segmentation with alternating direction method of multipliers

Segmentation via level-set methods plays a significant role in natural and medical image analysis. Recently, the distance regularized level-set evolution (DRLSE) model, which is based on variational level-set formulation has been proven very useful. However, the DRLSE model suffers from difficulties related to its computational cost due to the sign distance function (SDF) and Courant-Friedrichs-Lewy (CFL) condition applied by the gradient descent method. This paper analyzes and tests the well-known fast computing algorithm known as alternating direction method of multipliers (ADMM) for level-set based image segmentation. That is, we compute the DRLSE model by using the ADMM algorithm. The comparison between the classical gradient descent method and the proposed ADMM for two-phase and multi-phase level-set formulation is provided. The numerical experiments on standard public data sets, particularly in the case of medical image segmentation, demonstrate the effectiveness and efficiency of the proposed method. The average segmentation coefficients for Dice and Jaccard against ground truth are 0.97 and 0.92, respectively, and the average running time is 1.70 seconds. 0.0932, 0.993, 0.981, and 0.964 are average estimated values for mean absolute distance (MAD), accuracy, sensitivity, and specificity, respectively. The source code can be found at link https://github.com/WaliSamad/DRLSE-ADMM-Medical-Image-Segmentation.

Local reverse entropy weighted LBF model solving by Split Bregman for image segmentation

In this paper, an efficient level set model is proposed for image segmentation. Firstly, the original local binary fitting (LBF) model is redefined as a weighted energy integral, whose weight coefficient is the fast local reverse entropy of the image, and the total energy functional is then incorporated into a variational level set formulation. Secondly, the global convex segmentation method is used to construct a simplified convex segmentation model, at the same time, the edge information obtained by an edge indicator function is embedded into the total variation norm to further enhance the model’s target capture capability. Thirdly, the Split Bregman method is introduced to solve the generated convex optimization problem. Experimental results on synthetic and real images demonstrate that the proposed model has considerable improvements in terms of quantitative evaluation (being verified on the complete PASCAL VOC 2012 dataset), convergence rate, sensitivity to initial contour and robustness to noise interference compared with the state-of-the-art models. We also compare the proposed model with the famous FCN and Mask R-CNN, and make a special analysis on the adaptability of our method to occluded targets.

An efficient level set method based on global statistical information for image segmentation

In this paper, an improved region-based active contour model in a novel variational level set formulation employing global information constraints is proposed for image segmentation. Using the image global characteristics, the proposed model can accurately segment objects with weak or blurred boundaries in the presence of noise or intensity inhomogeneity and achieve robustness against different kinds of images. We present a global intensity fitting energy functional based on the differences between the original image intensity and the global intensity means. This energy is then integrated into a geodesic variational level set formulation, from which a curve evolution equation is derived for energy minimization. The level set function is regularized by the Gaussian filter which keeps it smooth and eliminates the reinitialization. Experimental results on different images demonstrate the performance of our method in terms of accuracy and robustness. Compared with the well-known active contour models, our method is less sensitive to the initial contour and more computationally efficient.

MSPSF: A Multi-Scale Local Intensity Measurement Function for Segmentation of Breast Thermogram

Accurate segmentation of region(s) of abnormality (RoAs) in a grayscale breast thermogram is an essential step in the automated identification of cancer in the breast. However, low contrast, intensity nonuniformity, noise, and complex background pose challenges to this segmentation task. To overcome those challenges, a novel segmentation method that amalgamates the breast blood perfusion (BBP) model, an adaptive triangular histogram-based thresholding (ATHT) method, and a new energy functional-based level set method (LSM) is proposed here. The BBP model improves the contrast between the RoA and normal tissue region in grayscale breast thermogram. The ATHT method is used for robust initialization of the proposed LSM. To deal with the complexity caused by intensity nonuniformity and noise, we have formulated a new multiscale local intensity measurement function, called multiscale spatially weighted pixel-contribution and shape-feature-embedded force (MSPSF), which is then incorporated into a variational level set formulation and minimized using interleaved operations of the level set function to segment the RoAs accurately. Extensive experimentation on two different databases justifies the superiority of our segmentation method over the other widely used state-of-the-art segmentation methods on the basis of different qualitative and quantitative measures. We have also proposed a novel two-stage breast abnormality prediction framework based on the segmented RoA.

Structured graph regularized shape prior and cross-entropy induced active contour model for myocardium segmentation in CTA images

Accurate segmentation of myocardium tissue from computed tomography angiography (CTA) images is a crucial step in development of cardiac clinical applications. Automatic methods are highly desirable to release the burden of manual delineation, however, it is challenging due to the presence of images with intensity non-uniformity and coarse, weak and even pseudo boundary. To this end, a new geometric active contour (GAC) model that integrates high-level shape prior and low-level local intensity statistics is proposed. First, the cardiac-type specific shape prior is learned by structured graph-regularized principal component analysis, so that allows to be faithful to the shape of the desired myocardium. Second, local intensity distribution with inhomogeneity is modeled by cross-entropy energy functional and segmentation-oriented image decomposition. Third, the proposed model is solved as variational level set formulation and a distance regularized energy functional is also incorporated for the stability of numerical computation. The model was evaluated on a set of cardiac CTA images with comparison to related shape prior and local region-based methods and multi-atlas joint label fusion methods, and experimental results show it achieves competitive accuracies of segmenting myocardial epicardium and endocardium parts. It also presents advantage of computational burden in comparison to some popular methods such as multi-atlas joint label fusion. Since the framework of proposed method is general and adaptive, it could be potentially extended to segment similar objects in CT images.

A local mean and variance active contour model for biomedical image segmentation

To effectively segment the biomedical images with intensity inhomogeneity, a local mean and variance (LMV) active contour model is presented in this paper. In the model, the distribution of intensity belonging to each region is assumed as a Gaussian distribution with spatially varying mean and variance. At the same time, an energy function is defined and incorporated into a variational level set formulation. The curve evolution equation is then obtained from energy minimization. Because both image local mean and variance are considered, the proposed model can more effectively deal with the images with intensity inhomogeneity than the existing active contour model. Experimental results on synthetic and real images show the advantages of our method in terms of both effectiveness and robustness. Compared with the local binary fitting (LBF) model and the local hybrid image fitting (LHIF) model, our model is much most computationally efficient and much least sensitive to the initial contour.

Accurate tooth segmentation with improved hybrid active contour model

In orthodontic diagnosis and oral treatment planning, 3D tooth model constructed by dental computed tomography (CT) images is an essential and useful assisted tool. In virtue of the higher spatial resolution and lower radiation of x-ray, cone beam computed tomography (CBCT) has been widely used in dental application. However, due to lower signal to noise ratio, vague and weak edge between tooth root and sockets as well as intensity inhomogeneity, the tooth root is easy to be under-segmented and appears false boundary. This paper presents a new hybrid active contour model in a variational level set formulation to segment the tooth root accurately. Initial shape and intensity information from the upper layer is used for next layer’s enhancement and shape constraint. The hybrid level set model is constituted by multi-scale local likelihood image fitting (LLIF) energy term, prior shape constraint energy term with adaptive weight and reaction–diffusion (RD) regularization energy term. For detailed interpretation of this hybrid energy model, the intensity information in a narrowband region outside the contour was used to enhance the contrast between tooth dentine and sockets. The LLIF energy term was incorporated into the level set function to overcome the edge fuzziness and intensity inhomogeneity. The shape prior energy term with adaptive weight was used to differentiate the constraint of the contour evolution inside and outside the level set function to improve the capability of curve topology changes. The RD energy term was introduced to effectively regularize the level set evolution. A new measurement for tooth segmentation evaluation was proposed for quantitative validation. The experimental result of the proposed method was compared with two other typical approaches, and was demonstrated to achieve a higher segmentation accuracy.

A fast computational approach for illusory contour reconstruction

Illusory contour reconstruction can be modeled as a minimization problem with a tractable variational level set formulation, utilizing Euler’s elastica to reconstruct the illusory boundaries. However, this kind of formulation is very difficult to solve numerically as it is hard to implement such optimization algorithms efficiently in practice. In this paper, we propose an equivalently reduced variational level set formulation by taking the level set functions as signed distance functions. Technically, an alternating direction method of multipliers (ADMM) is developed by introducing some auxiliary variables, Lagrange multipliers and applying an alternating optimization strategy. With the proposed ADMM method, the minimization problem can be transformed into a series of sub-problems, which can be solved easily via using the Gauss-Seidel iterations and Fast Fourier Transform (FFT). The corresponding level set functions are regarded as signed distance functions during computation process using a simple algebraic projection method, which avoids the traditional re-initialization process for conventional level set functions. Extensive experiments have been conducted on both synthetic and real images, which validated the proposed approach, and demonstrated the advantages of the proposed ADMM-Projection (ADMM-P) method over the existing algorithms based on traditional gradient descent method (GDM) in terms of computational efficiency.

A hybrid active contour model based on global and local information for medical image segmentation

For segmenting medical images with abundant noise, blurry boundaries, and intensity heterogeneities effectively, a hybrid active contour model that synthesizes the global information and the local information is proposed. A novel global energy functional is constructed, together with an adaptive weight by the statistical information of image pixels on the clustering idea. Minimizing this global energy functional in a variational level set formulation will drive the curve to desirable boundaries. The local energy functional contains the local threshold, which is used to correct the deviation of the level set function. Experiments demonstrate that the proposed method can segment synthetic and medical images effectively, and have a relatively higher performance compared to other representative methods.

Active contours driven by edge entropy fitting energy for image segmentation

Active contour models have been widely used for image segmentation purposes. However, they may fail to delineate objects of interest depicted on images with intensity inhomogeneity. To resolve this issue, a novel image feature, termed as local edge entropy, is proposed in this study to reduce the negative impact of inhomogeneity on image segmentation. An active contour model is developed on the basis of this feature, where an edge entropy fitting (EEF) energy is defined with the combination of a redesigned regularization term. Minimizing the energy in a variational level set formulation can successfully drive the motion of an initial contour curve towards optimal object boundaries. Experiments on a number of test images demonstrate that the proposed model has the capability of handling intensity inhomogeneity with reasonable segmentation accuracy.

Variational Level-set Formulation for Lithographic Source and Mask Optimization Generalized Euler'Constant Family

Variational Level-set Formulation for Lithographic Source and Mask Optimization Generalized Euler'Constant Family

Robust Active Contour Model Guided by Local Binary Pattern Stopping Function

Abstract Edge based active contour models are adequate to some extent in segmenting images with intensity inhomogeneity but often fail when applied to images with poorly defined or noisy boundaries. Instead of the classical and widely used gradient or edge stopping function which fails to stop contour evolution at such boundaries, we use local binary pattern stopping function to construct a robust and effective active contour model for image segmentation. In fact, comparing to edge stopping function, local binary pattern stopping function accurately distinguishes object’s boundaries and determines the local intensity variation dint to the local binary pattern textons used to classify the image regions. Moreover, the local binary pattern stopping function is applied using a variational level set formulation that forces the level set function to be close to a signed distance function to eliminate costly re-initialization and speed up the motion of the curve. Experiments on several gray level images confirm the advantages and the effectiveness the proposed model.

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.

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.

A Novel Brain Tumor Segmentation from Multi-Modality MRI via A Level-Set-Based Model

Segmentation of brain tumor from magnetic resonance imaging is a challenging and time-consuming task due to the unpredictable appearance of tumor tissue in practical applications. In this paper we propose a novel level-set-based model for tumor segmentation from multi-modality magnetic resonance imaging. We formulate the tumor segmentation on pixel-level with three classes: tumor, edema and healthy tissue. First, we detect abnormal regions by using a region-based active contour model on T2-weighted images with fluid-attenuated inversion recovery modality, and a variational level set formulation is applied locally to approximate the image intensities on two sides of the contour. In the second stage, we distinguish the edema and tumor tissues in the abnormal regions based on the contrast enhancement T1 modality. Compared with traditional one-modality methods, our model can better represent the specific tissue of tumor in a simple way. The validation experiments on synthetic and clinical brain magnetic resonance images demonstrate the effectiveness and simplicity of the proposed method for brain tumor segmentation.

Image segmentation and bias correction using local inhomogeneous iNtensity clustering (LINC): A region-based level set method

Image segmentation is still an open problem due to the existing of intensity inhomogeneity and noise. To accurately segment images with these biases, a local inhomogeneous intensity clustering (LINC) model is proposed. In LINC, a linear combination of a given set of smooth orthogonal basis functions is used to estimate the bias field. A local clustering criterion function is first defined to cluster the nearly homogeneous intensities in a relatively small neighborhood of each pixel. An energy functional is then defined by integrating the function with respect to the neighborhood center. This energy together with a regularization term and an arc length term are incorporated into a variational level set formulation in which de-nosing is implicitly included due to the implied convolution. Image segmentation and bias correction can be simultaneously achieved by updating variables of the final energy functional iteratively till it is stable or a predetermined iteration number is reached. The proposed model LINC has been extensively tested on both synthetic and real images. Experimental results and comparison with state-of-the-art methods demonstrate the advantages of the proposed model in terms of segmentation accuracy, bias field correction, dealing with noise, and robustness to initialization.

Automatic optic disc localization and segmentation in retinal images by a line operator and level sets

Existing methods may fail to locate and segment the optic disc (OD) due to imprecise boundaries, inconsistent image contrast and deceptive edge features in retinal images. To locate the OD and detect the OD boundary accurately. The method exploits a multi-stage strategy in the detection procedure. Firstly, OD location candidate regions are identified based on high-intensity feature and vessels convergence property. Secondly, a line operator filter for circular brightness feature detection is designed to locate the OD accurately on candidates. Thirdly, an initialized contour is obtained by iterative thresholding and ellipse fitting based on the detected OD position. Finally, a region-based active contour model in a variational level set formulation and ellipse fitting are employed to estimate the OD boundary. The proposed methodology achieves an accuracy of 98.67% for OD identification and a mean distance to the closest point of 2 pixels in detecting the OD boundary. The results illuminate that the proposed method is effective in the fast, automatic, and accurate localization and boundary detection of the OD. The present work contributes to the more effective evaluation of the OD and realizing automatic screening system for early eye diseases to a large extent.

Robust level set image segmentation algorithm using local correntropy-based fuzzy c-means clustering with spatial constraints

Accurate image segmentation is a challenge task in image analysis and understanding, while fuzzy c-means clustering (FCM) with spatial constraints (FCM_S) is an effective algorithm suitable for this challenge. However, FCM_S has high computational complexity and still lacks enough robustness to noise and outliers, which will limit its usefulness. To overcome these difficulties, a local correntropy-based fuzzy c-means clustering algorithm with spatial constraints (LCFCM_S) and its simplified model (LCFCM_S1) are proposed in this paper. By utilizing the correntropy criterion, the clustering algorithm can efficiently emphasize the weights of the samples that are close to their corresponding cluster centers. Then, the proposed clustering algorithms are incorporated into a variational level set formulation with a level set regularization term. Finally, the iteratively re-weighted algorithm is adopted to solve the LCFCM_S and LCFCM_S1 based level set method. Experimental results on synthetic and real images show the superiority of our methods in terms of accuracy and robustness for segmenting images with intensity inhomogeneity and noise, when compared with several state-of-the-art approaches.

Segmentation of cell nuclei in fluorescence microscopy images: An integrated framework using level set segmentation and touching-cell splitting

Accurate segmentation of cells in fluorescence microscopy images plays a key role in high-throughput applications such as quantification of protein expression and the study of cell function. In this paper, an integrated framework consisting of a new level sets based segmentation algorithm and a touching-cell splitting method is proposed. For cell nuclei segmentation, a new region-based active contour model in a variational level set formulation is developed where our new level set energy functional minimizes the Bayesian classification risk. For touching-cell splitting, the touching cells are first distinguished from non-touching cells, and then a strategy based on the splitting area identification is proposed to obtain splitting point-pairs. To form the appropriate splitting line, the image properties from different information channels are used to define the surface manifold of the image patch around the selected splitting point-pairs and geodesic distance is used to measure the length of the shortest path on the manifold connecting the two splitting points. The performance of the proposed framework is evaluated using a large number of fluorescence microscopy images from four datasets with different cell types. A quantitative comparison is also performed with several existing segmentation approaches.

Active Contour Driven by Local Gaussian Distribution Fitting and Local Signed Difference Based on Local Entropy

Intensity inhomogeneity often causes considerable difficulties in image segmentation. In order to tackle this problem, we propose a novel region-based active contour model in a variational level set formulation. We first define a data fitting energy with a local Gaussian distribution fitting (LGDF) term, which induces a local force to attract the contour and stops it at object boundaries, and a local signed difference (LSD) term based on local entropy, which possesses both local separability and global consistency. This energy is then incorporated into a level set formulation with a level set regularization term that is necessary for accurate computation in the corresponding level set method. Experimental results show that the proposed model can not only segment images with intensity inhomogeneities and weak boundaries but also be robust to the noise, initial contours.