A new binary level set model using L0 regularizer for image segmentation

A binary level set variational model with L1 data term for image segmentation

We present a modified binary level set method for two-phase image segmentation, which is based on the binary level set method originally proposed by X.-C. Tai et al. [Int. J. Comput. Vis. 73, 61–76 (2007)]. The modified binary level set method is superior to Tai et al.'s binary level set method in preserving the curve evolution property of gradually shrinking and expanding the partition interface represented by a binary level set function in the segmentation process. Some experiments conducted on real images show the good results of the modified binary level set method.

The Mumford-Shah model is an important variational image segmentation model. A popular multiphase level set approach, the Chan-Vese model, was developed for this model by representing the phases by several overlapping level set functions. Recently, exactly the same model was also formulated by using binary level set functions. In both approaches, the gradient descent equations had to be solved numerically, a procedure which is slow and has the potential of getting stuck in a local minima. In this work, we develop an efficient and global minimization method for the binary level set representation of the multiphase Chan-Vese model based on graph cuts. If the average intensity values of the different phases are sufficiently evenly distributed, the discretized energy function becomes submodular. Otherwise, a novel method for minimizing nonsubmodular functions is proposed with particular emphasis on this energy function.

In this paper, we propose a novel saliency-based active contour model for image segmentation and region detection (SAIR) to overcome the problems of noise and intensity inhomogeneity in image segmentation. The proposed new level set protocol evolves adaptively and eliminates the need for initialization. In the proposed energy function, we formulate an adaptive weight function that adaptively changes the intensity of the internal and external energy functions according to the image. Moreover, according to the inner and outer regions, modulating the signs in the proposed energy function influences the elimination of noise in the image. Finally, SAIR is tested on multiple images with different initial contour positions, intensity inhomogeneity, and noise to demonstrate the robustness of SAIR.

In this paper we present a novel geometric framework called geodesic active fields for general image registration. In image registration, one looks for the underlying deformation field that best maps one image onto another. This is a classic ill-posed inverse problem, which is usually solved by adding a regularization term. Here, we propose a multiplicative coupling between the registration term and the regularization term, which turns out to be equivalent to embed the deformation field in a weighted minimal surface problem. Then, the deformation field is driven by a minimization flow toward a harmonic map corresponding to the solution of the registration problem. This proposed approach for registration shares close similarities with the well-known geodesic active contours model in image segmentation, where the segmentation term (the edge detector function) is coupled with the regularization term (the length functional) via multiplication as well. As a matter of fact, our proposed geometric model is actually the exact mathematical generalization to vector fields of the weighted length problem for curves and surfaces introduced by Caselles-Kimmel-Sapiro. The energy of the deformation field is measured with the Polyakov energy weighted by a suitable image distance, borrowed from standard registration models. We investigate three different weighting functions, the squared error and the approximated absolute error for monomodal images, and the local joint entropy for multimodal images. As compared to specialized state-of-the-art methods tailored for specific applications, our geometric framework involves important contributions. Firstly, our general formulation for registration works on any parametrizable, smooth and differentiable surface, including nonflat and multiscale images. In the latter case, multiscale images are registered at all scales simultaneously, and the relations between space and scale are intrinsically being accounted for. Second, this method is, to the best of our knowledge, the first reparametrization invariant registration method introduced in the literature. Thirdly, the multiplicative coupling between the registration term, i.e. local image discrepancy, and the regularization term naturally results in a data-dependent tuning of the regularization strength. Finally, by choosing the metric on the deformation field one can freely interpolate between classic Gaussian and more interesting anisotropic, TV-like regularization.

Magnetic resonance images are often corrupted by intensity inhomogeneity (i.e., bias field effects), which manifests itself as slow intensity variations over the image domain. Such shading artifacts must be corrected before performing computerized analyses such as intensity-based segmentation and quantitative analysis. In this paper, we present a novel strategy in the fuzzy c-means (FCM) framework that simultaneously estimates the bias field while segmenting the image. An additive field term that models the bias field is incorporated into the FCM objective function. We propose a new term based on the spectral parameterization (i.e., wavelet coefficients) of the bias field that serves as a regularizer to enforce the smoothness of the estimated bias field. We also introduce a second regularization term that causes the labeling of each pixel to be influenced by its immediate neighborhood pixels. The latter regularization term renders the algorithm less sensitive to noise. We show that the novel objective functional could be optimized efficiently using an iterative process. The efficacy of the algorithm is demonstrated on synthesized images as well as on clinical breast MR images. With the synthesize images, segmentation accuracy using standard FCM is 89.07% while segmentation accuracy with the proposed algorithm is 99.95%.

This study proposes a novel local region-based active contour model (ACM) for image segmentation based on fractional calculus with consideration of the Riemann–Liouville operators. The proposed method aims to achieve accurate boundary segmentation in the presence of severe intensity inhomogeneity. The strength of fractional calculus is exploited with Gaussian Kernel namely, Fractional Gaussian Kernel (FGK) that provides an effective method of edge detection and has good noise immunity. An adaptive window mechanism with various sizes and orientations is employ to maintain and enhance image details especially at the object’s boundary and angle. The powerful combination of adaptive window and Fractional Gaussian Kernel (AFGK) provide an efficient way to utilize image information in local regions. Specifically, the fractional differential Heaviside function (FDH) extracts the image gradient and its various intensities for an accurate boundary segmentation outcome. Experiments on both synthetics and medical images demonstrate that the proposed local region-based ACM including fractional calculus realizes accurate boundary segmentation on images even under the most challenging situations, such as severe intensity inhomogeneity interface.

In this paper, we introduce multi-symplectic Lagrangian variational integrators for solving Chan-Vese active contour models in image segmentation. Energy functionals are discretized firstly, and numerical schemes are derived from discrete Euler-Lagrange equations based on discrete variational principle. Lagrangian variational integrators preserve native differential structure-multi-symplecticity, that makes the numerical methods have a satisfied behavior. Experiments are performed on the benchmark images from literature. We further evaluated the methods in a segmentation database containing 1023 images. It shows that the proposed numerical schemes attain relatively faster convergence rates and better segmentation accuracy. Comparisons with the standard explicit Euler method of the original Chan-Vese model and other fast numerical optimization methods show that the proposed methods have better stability, higher accuracy, and are more robust when dealing with a large number of pictures. This study provides an example for further research to improve the performance of other existing image segmentation methods based on active contour models.

Segmentation of images with intensity inhomogeneity is always challenging due to low resolution, blurred boundaries and poor illumination. Although existing image segmentation methods were widely used, there exists some shortcomings in segmenting intensity inhomogeneous images, such as not considering the spatial interrelation within the central pixel and its neighborhood. Therefore, this paper proposes an enhanced region-based active contour model for segmenting images with intensity inhomogeneity. In this model, a range-based adaptive bilateral filter is utilized to preserve edge structures and resist the noise of the image. Then an effective energy functional is constructed into the level set framework. With the permission of keeping the original shape of the image, a regularization term is utilized to refrain from the process of re-initialization and speed up the curve evolution. In the end, some experiments on synthetic and real images and contrast with the classic segmentation models are executed. The proposed model is more accuracy than other classic models.

An improved region-based model with local statistical features for image segmentation

Accurate segmentation of medical images plays a very important role in clinical diagnosis so that the segmentation technology for medical images attracts more and more attention. However, most medical images usually suffer from severe intensity inhomogeneity and make accurate segmentation difficult. In this study, the authors propose an efficient and robust active contour model for simultaneous image segmentation and correction. The proposed model not only can accurately segment images with severe intensity inhomogeneity and serious noise but also can eliminate the intensity varying information to get the homogeneous correction images. They first present the level set formulation of the two-phase model, which is then extended to the multi-phase formulation. The split Bregman method is applied to efficiently minimise the proposed energy functionals. The proposed model is tested with lots of synthetic images and medical images with promising results. Experimental results demonstrate that the proposed model can accurately segment and correct the inhomogeneous images with serious noise. Quantitative comparison results of the proposed model and other models illustrate the proposed model is more accurate and more efficient. What's more, the proposed model not only is insensitive to the initial contour, but also is robust to the noise.

Image segmentation and selective smoothing based on p-harmonic Mumford–Shah functional

Accurate brain tissue segmentation from magnetic resonance (MR) images is an essential step in quantitative brain image analysis. However, due to the existence of noise and intensity inhomogeneity in brain MR images, many segmentation algorithms suffer from limited accuracy. In this paper, it is assumed that the local image data within each voxel's neighborhood satisfy the Gaussian mixture model (GMM), and thus propose the fuzzy local GMM (FLGMM) algorithm for automated brain MR image segmentation. In this paper results obtained from the proposed algorithm is compared with those obtained by using Level set function in both synthetic and clinical data is analyzed. Thus concluding that the proposed algorithm can largely overcome the difficulties raised by noise, low contrast, and bias field, and improves the accuracy of brain MR image segmentation.

In the Bayesian image super resolution (SR), a regularisation term is minimised along with a data‐fidelity term to generate a high‐resolution (HR) image from input low‐resolution (LR) image. The regularisation term is incorporated into the SR to fulfil a prior knowledge over the HR image. For instance, smoothness in the background and foreground regions is a prior knowledge about document images. The bilateral total variation (BTV), as a known regularisation term, uniformly smooths the image in all directions while preserving the edges. In this study, the authors present a document image SR method by introducing a new regularisation term called the stroke width‐based directional total variation (SWDTV). It is a modified version of the BTV, and adaptively performs smoothing based on the local width and local direction of the character strokes. By analysing the input LR image and its interpolation, the local stroke width and local stroke direction in the output image are approximated. This information is encapsulated into the SWDTV regularisation term. By minimising the linear combination of the regularisation and data‐fidelity terms, the HR image is reconstructed. The authors’ method achieves significant improvements in visual image quality and optical character recognition accuracy compared to related non‐example‐based document image SR methods.

We investigated a parametric kernel-driven active contour (PKAC) model, which implicitly transfers kernel mapping and piecewise constant to modeling the image data via kernel function. The proposed model consists of curve evolution functional with three terms: global kernel-driven and local kernel-driven terms, which evaluate the deviation of the mapped image data within each region from the piecewise constant model, and a regularization term expressed as the length of the evolution curves. In the local kernel-driven term, the proposed model can effectively segment images with intensity inhomogeneity by incorporating the local image information. By balancing the weight between the global kernel-driven term and the local kernel-driven term, the proposed model can segment the images with either intensity homogeneity or intensity inhomogeneity. To ensure the smoothness of the level set function and reduce the computational cost, the distance regularizing term is applied to penalize the deviation of the level set function and eliminate the requirement of re-initialization. Compared with the local image fitting model and local binary fitting model, experimental results show the advantages of the proposed method in terms of computational efficiency and accuracy.