Variational Level Set Method Research Articles

A Registration Method for Multimodal Medical Images Using Contours Mutual Information

In recent years, mutual information has developed as a popular image registration measure especially in multimodality image registration. For different modality medical images, the contour of tissues or organs is similarity. In this paper, an effective new registration method of the multimodal medical images based on the contour mutual information is proposed. Firstly, get the contour through variational level set method. Secondly, within the contour pixels are assigned the same grayscale value, obtain two contour images. Finally, two contour images using mutual information as similarity measure for image registration. The experiment results show that the registration algorithm proposed in this paper can do more effectively and more accurately work than normalized mutual information and gradient mutual information.

Variational Level Set Method for Two-Stage Image Segmentation Based on Morphological Gradients

We use variational level set method and transition region extraction techniques to achieve image segmentation task. The proposed scheme is done by two steps. We first develop a novel algorithm to extract transition region based on the morphological gradient. After this, we integrate the transition region into a variational level set framework and develop a novel geometric active contour model, which include an external energy based on transition region and fractional order edge indicator function. The external energy is used to drive the zero level set toward the desired image features, such as object boundaries. Due to this external energy, the proposed model allows for more flexible initialization. The fractional order edge indicator function is incorporated into the length regularization term to diminish the influence of noise. Moreover, internal energy is added into the proposed model to penalize the deviation of the level set function from a signed distance function. The results evolution of the level set function is the gradient flow that minimizes the overall energy functional. The proposed model has been applied to both synthetic and real images with promising results.

Application of radial basis functions to evolution equations arising in image segmentation

In this paper, radial basis functions are used to obtain the solution of evolution equations which appear in variational level set method based image segmentation. In this method, radial basis functions are used to interpolate the implicit level set function of the evolution equation with a high level of accuracy and smoothness. Then, the original initial value problem is discretized into an interpolation problem. Accordingly, the evolution equation is converted into a set of coupled ordinary differential equations, and a smooth evolution can be retained. Compared with finite difference scheme based level set approaches, the complex and costly re-initialization procedure is unnecessary. Numerical examples are also given to show the efficiency of the method.

An adaptive level set evolution equation for contour extraction

Level set evolution without re-initialization is a novel variational level set method for edge contour extraction, which has many advantages over the traditional level set formulations. The resulting evolution equation is derived from the minimization of an energy functional which is a linear combination of a weighted length, weighted area and deviation penalization energy. However, the zero level set always keeps evolving in only one direction during evolution, depending on the sign of the scale parameter associated with the weighted area. This inconvenient makes the evolution highly sensitive to the contour initializations. In this paper, we propose an adaptive level set evolution equation following this method, wherein the scale parameter associated with the weighted area is modified as an adaptive variable sign function and one of two terms associated with the weighted length is removed to reduce computational cost. The proposed equation avoids completely the intrinsic limitation mentioned above and offers many advantages over the original equation, as illustrated by several examples of contour extraction, such as robustness to noise and detection of objects with discontinuous boundaries.

A Variational Binary Level Set Method for Structural Topology Optimization

AbstractThis paper proposes a variational binary level set method for shape and topology optimization of structural. First, a topology optimization problem is pre-sented based on the level set method and an algorithm based on binary level set method is proposed to solve such problem. Considering the difficulties of coordination between the various parameters and efficient implementation of the proposed method, we present a fast algorithm by reducing several parameters to only one parameter, which would substantially reduce the complexity of computation and make it easily and quickly to get the optimal solution. The algorithm we constructed does not need to re-initialize and can produce many new holes automatically. Furthermore, the fast algorithm allows us to avoid the update of Lagrange multiplier and easily deal with constraints, such as piecewise constant, volume and length of the interfaces. Finally, we show several optimum design examples to confirm the validity and efficiency of our method.

Level set segmentation of medical images based on local region statistics and maximum a posteriori probability.

This paper presents a variational level set method for simultaneous segmentation and bias field estimation of medical images with intensity inhomogeneity. In our model, the statistics of image intensities belonging to each different tissue in local regions are characterized by Gaussian distributions with different means and variances. According to maximum a posteriori probability (MAP) and Bayes' rule, we first derive a local objective function for image intensities in a neighborhood around each pixel. Then this local objective function is integrated with respect to the neighborhood center over the entire image domain to give a global criterion. In level set framework, this global criterion defines an energy in terms of the level set functions that represent a partition of the image domain and a bias field that accounts for the intensity inhomogeneity of the image. Therefore, image segmentation and bias field estimation are simultaneously achieved via a level set evolution process. Experimental results for synthetic and real images show desirable performances of our method.

An improved variational level set method for MR image segmentation and bias field correction

In this paper, we propose an improved variational level set approach to correct the bias and to segment the magnetic resonance (MR) images with inhomogeneous intensity. First, we use a Gaussian distribution with bias field as a local region descriptor in two-phase level set formulation for segmentation and bias field correction of the images with inhomogeneous intensities. By using the information of the local variance in this descriptor, our method is able to obtain accurate segmentation results. Furthermore, we extend this method to three-phase level set formulation for brain MR image segmentation and bias field correction. By using this three-phase level set function to replace the four-phase level set function, we can reduce the number of convolution operations in each iteration and improve the efficiency. Compared with other approaches, this algorithm demonstrates a superior performance.

22 Semi-Automatic OEDEMA Quantification from Direct T2 Map Cardiac MRI

The jeopardised ischaemic area-at-risk (AAR) is a key prognostic determinant in acute myocardial infarction. Myocardial oedema imaging with T2-weighted cardiac magnetic resonance CMR is validated for imaging the AAR and T2 ‘mapping’ is a new method for AAR imaging with clinical and research potential. We aimed to develop an automated post-processing method that would enable straightforward volumetric quantification of AAR with T2 maps. Our approach retains user input (i.e. clinical judgement) to confirm the presence of oedema on an image which is then subjected to an automated analysis. The new method was tested on 12 acute MI patients who had a CMR within 48 hours of hospital admission. Manual segmentation of the left ventricular wall and oedema were available for comparison. Left ventricular wall boundaries were delineated automatically by variational level set methods followed by automated detection of myocardial oedema by fitting a Gaussian-Gaussian mixture statistical model. The mean perpendicular distances between automatically detected left ventricular boundaries and corresponding manual delineated boundaries were 1.8±0.2mm for endocardial boundaries and 2.3±0.3mm for endocardial boundaries. Dice similarity coefficients for agreement (0=no agreement, 1=perfect agreement) between manual delineation and automated segmentation of the left ventricular wall boundaries and oedema regions were 0.85±0.02 and 0.74±0.05, respectively. Compared to standard manual approaches, the new semi-automated method for estimating myocardial oedema is straightforward and accurate.

LV wall segmentation using the variational level set method (LSM) with additional shape constraint for oedema quantification

In this paper an automatic algorithm for the left ventricle (LV) wall segmentation and oedema quantification from T2-weighted cardiac magnetic resonance (CMR) images is presented. The extent of myocardial oedema delineates the ischaemic area-at-risk (AAR) after myocardial infarction (MI). Since AAR can be used to estimate the amount of salvageable myocardial post-MI, oedema imaging has potential clinical utility in the management of acute MI patients. This paper presents a new scheme based on the variational level set method (LSM) with additional shape constraint for the segmentation of T2-weighted CMR image. In our approach, shape information of the myocardial wall is utilized to introduce a shape feature of the myocardial wall into the variational level set formulation. The performance of the method is tested using real CMR images (12 patients) and the results of the automatic system are compared to manual segmentation. The mean perpendicular distances between the automatic and manual LV wall boundaries are in the range of 1–2 mm. Bland–Altman analysis on LV wall area indicates there is no consistent bias as a function of LV wall area, with a mean bias of −121 mm2 between individual investigator one (IV1) and LSM, and −122 mm2 between individual investigator two (IV2) and LSM when compared to two investigators. Furthermore, the oedema quantification demonstrates good correlation when compared to an expert with an average error of 9.3% for 69 slices of short axis CMR image from 12 patients.

Fuzzy-Based Classification of Breast Lesions Using Ultrasound Echography and Elastography

Common breast lesions have different elasticity properties. Segmentation of contours of breast lesions from elastography and B mode images by incorporating variational level set method is involved in the proposed work. After segmentation, strain and shape features, such as differences in area, perimeter, and contour and width to height difference and solidity, as well as texture features like contrast, entropy, standard deviation, dissimilarity, homogeneity and energy, are estimated. A nonlinear fuzzy inference system is applied for classifying the breast lesions as benign cyst, benign solid mass, or malignant solid mass. Detection of malignant solid masses is our primary objective. A classification accuracy of 83% is obtained. One hundred percent sensitivity is reported. It can be concluded that the proposed fuzzy-based classification technique can be used as an aid for the automated detection of breast lesions.

3D cerebrovascular segmentation combining fuzzy vessel enhancement and level-sets with anisotropic energy weights

The aim of this work is to present and evaluate a level-set segmentation approach with vesselness-dependent anisotropic energy weights, which focuses on the exact segmentation of malformed as well as small vessels from time-of-flight (TOF) magnetic resonance angiography (MRA) datasets.In a first step, a vesselness filter is used to calculate the vesselness dataset, which quantifies the likeliness of each voxel to belong to a bright tubular-shaped structure and estimate the corresponding vessel directions from a given TOF dataset. The vesselness and TOF datasets are then combined using fuzzy-logic and used for initialization of a variational level-set method. The proposed level-set model has been extended in a way that the weight of the internal energy is locally adapted based on the vessel direction information. Here, the main idea is to weight the internal energy lower if the gradient direction of the level-set is similar to the direction of the eigenvector extracted by the vesselness filter. Furthermore, an additional vesselness force has been integrated in the level-set formulation.The proposed method was evaluated based on ten TOF MRA datasets from patients with an arteriovenous malformation. Manual segmentations from two observers were available for each dataset and used for quantitative comparison. The evaluation revealed that the proposed method yields significantly better segmentation results than four other state-of-the-art segmentation methods tested. Furthermore, the segmentation results are within the range of the inter-observer variation.In conclusion, the proposed method allows an improved delineation of small vessels, especially of those represented by low intensities and high surface curvatures.

Reinitialization-Free Level Set Evolution via Reaction Diffusion

This paper presents a novel reaction-diffusion (RD) method for implicit active contours that is completely free of the costly reinitialization procedure in level set evolution (LSE). A diffusion term is introduced into LSE, resulting in an RD-LSE equation, from which a piecewise constant solution can be derived. In order to obtain a stable numerical solution from the RD-based LSE, we propose a two-step splitting method to iteratively solve the RD-LSE equation, where we first iterate the LSE equation, then solve the diffusion equation. The second step regularizes the level set function obtained in the first step to ensure stability, and thus the complex and costly reinitialization procedure is completely eliminated from LSE. By successfully applying diffusion to LSE, the RD-LSE model is stable by means of the simple finite difference method, which is very easy to implement. The proposed RD method can be generalized to solve the LSE for both variational level set method and partial differential equation-based level set method. The RD-LSE method shows very good performance on boundary antileakage. The extensive and promising experimental results on synthetic and real images validate the effectiveness of the proposed RD-LSE approach.

Optimization of unsteady incompressible Navier–Stokes flows using variational level set method

SUMMARYThis paper presents the optimization of unsteady Navier–Stokes flows using the variational level set method. The solid–liquid interface is expressed by the level set function implicitly, and the fluid velocity is constrained to be zero in the solid domain. An optimization problem, which is constrained by the Navier–Stokes equations and a fluid volume constraint, is analyzed by the Lagrangian multiplier based adjoint approach. The corresponding continuous adjoint equations and the shape sensitivity are derived. The level set function is evolved by solving the Hamilton–Jacobian equation with the upwind finite difference method. The optimization method can be used to design channels for flows with or without body forces. The numerical examples demonstrate the feasibility and robustness of this optimization method for unsteady Navier–Stokes flows.Copyright © 2012 John Wiley & Sons, Ltd.

Active Curve Recovery of Region Boundary Patterns

This study investigates the recovery of region boundary patterns in an image by a variational level set method which drives an active curve to coincide with boundaries on which a feature distribution matches a reference distribution. We formulate the scheme for both the Kullback-Leibler and the Bhattacharyya similarities, and apply it in two conditions: the simultaneous recovery of all region boundaries consistent with a given outline pattern, and segmentation in the presence of faded boundary segments. The first task uses an image-based geometric feature, and the second a photometric feature. In each case, the corresponding curve evolution equation can be viewed as a geodesic active contour (GAC) flow having a variable stopping function which depends on the feature distribution on the active curve. This affords a potent global representation of the target boundaries, which can effectively drive active curve segmentation in a variety of otherwise adverse conditions. Detailed experimentation shows that the scheme can significantly improve on current region and edge-based formulations.

Variational level set methods for image segmentation based on both [formula omitted] and Sobolev gradients

Variational level set methods for image segmentation involve minimizing an energy functional over a space of level set functions using a continuous gradient descent method. The functional includes the internal energy (curve length, usually) for regularization and the external energy that aligns the curves with object boundaries. Current practice is, in general, to minimize the energy functional by calculating the L 2 gradient of the total energy. However, the Sobolev gradient is particularly effective for minimizing the curve length functional by the gradient descent method in that it produces the solution in a single iteration. In this paper, we thus propose to use the Sobolev gradient for the internal energy (curve length), while still using the L 2 gradient for the external energy. The test results show that the “ L 2 plus Sobolev” gradient scheme is significantly more computationally efficient than the methods only based on the L 2 gradient.

A Novel Approach for Color Tongue Image Fast Segmentation Based on Level Sets

To make variational level set method applicable to the application of tongue segmentation, this article introduced a kind of color tongue image fast segmentation method based on level sets. The early variational level set method is improved from the following aspects: the boundary feature weight function is improved to make this method adaptable to color tongue image segmentation, and a kind of variable time step method is introduced to increase the efficiency and accuracy of segmentation. As the comparison experiment results show, our method is about 1 time faster than the early method, and its accuracy is higher than the early one.

A variational binary level-set method for elliptic shape optimization problems

We present a variational binary level-set method to solve a class of elliptic problems in shape optimization. By the ‘ersatz material’ approach, which amounts to fill the holes by a weak phase, the original shape optimization model is approximated by a two-phase optimization problem. Under the binary level-set framework, we need to optimize a smooth functional under a binary constraint. We propose an augmented Lagrangian method to solve the constrained optimization problem. Numerical results are presented and compared with those obtained by level-set methods, which demonstrate the robustness and efficiency of our method.

Approaches to the Feather Stain Detection and Segmentation

Extraction of feather defects is difficult because of their complicated and diversified characteristics. It is proposed that feather stain defects be extracted in HSI color space according to color characters of different types of feather defects. The improved variational level set method is adopted to locate and segment the feather stains on that basis. Experiment results show that the proposed methods can extract feather stains effectively and segment them accurately.

Shape and topology optimization for elliptic boundary value problems using a piecewise constant level set method

The aim of this paper is to propose a variational piecewise constant level set method for solving elliptic shape and topology optimization problems. The original model is approximated by a two-phase optimal shape design problem by the ersatz material approach. Under the piecewise constant level set framework, we first reformulate the two-phase design problem to be a new constrained optimization problem with respect to the piecewise constant level set function. Then we solve it by the projection Lagrangian method. A gradient-type iterative algorithm is presented. Comparisons between our numerical results and those obtained by level set approaches show the effectiveness, accuracy and efficiency of our algorithm.

Iris segmentation using variational level set method

Continuous efforts have been made to process degraded iris images for enhancement of the iris recognition performance in unconstrained situations. Recently, many researchers have focused on developing the iris segmentation techniques, which can deal with iris images in a non-cooperative environment where the probability of acquiring unideal iris images is very high due to gaze deviation, noise, blurring, and occlusion by eyelashes, eyelids, glasses, and hair. Although there have been many iris segmentation methods, most focus primarily on the accurate detection of iris images captured in a closely controlled environment. The novelty of this research effort is that we propose to apply a variational level set-based curve evolution scheme that uses a significantly larger time step to numerically solve the evolution partial differential equation (PDE) for segmentation of an unideal iris image accurately, and thereby, speeding up the curve evolution process drastically. The iris boundary represented by the variational level set may break and merge naturally during evolution, and thus, the topological changes are handled automatically. The proposed variational model is also robust against poor localization and weak iris/sclera boundaries. In order to solve the size irregularities occurring due to arbitrary shapes of the extracted iris/pupil regions, a simple method is applied based on connection of adjacent contour points. Furthermore, to reduce the noise effect, we apply a pixel-wise adaptive 2D Wiener filter. The verification and identification performance of the proposed scheme is validated on three challenging iris image datasets, namely, the ICE 2005, the WVU Unideal, and the UBIRIS Version 1.