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Concept: Level set method


In recent studies, both tumor morphology and vascularity played an important role in differentiating breast tumors. In this article, a computer-aided diagnosis (CAD) system was proposed to quantify the tumor morphology of vascularity on three-dimensional (3-D) power Doppler breast ultrasound (PDUS) images. We segmented the tumor margin by the level set method and skeletonized vessels by the 3-D thinning algorithm from 3-D PDUS data to capture the B-mode and vascularity features. The B-mode features including texture, shape and ellipsoid fitting and the vascularity features containing volume, complexity, length, radius and tortuosity were used to differentiate breast tumors. In the experiment, 82 biopsy-verified lesions including 41 benign and 41 malignant lesions were used to test the performance of the proposed system. The proposed method performed well, achieving accuracy, sensitivity, specificity and Az values of 85.37% (70/82), 85.37% (35/41), 85.37% (35/41) and 0.9104, respectively.

Concepts: Cancer, Breast cancer, Oncology, Brain tumor, Benign tumor, Tumor, Neoplasm, Level set method


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.

Concepts: Mathematics, Level set, Partial differential equation, Numerical analysis, Level set method, Finite difference, Finite difference method, Finite differences


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.

Concepts: Mathematics, Function, Normal distribution, Level set, Set, Ring, Image processing, Level set method


The level set method has been used for 20 years in a wide range of physical applications to track moving interfaces instead of an explicit description of the geometry. This paper studies in detail the shape of the level set function, delimiting a sub-domain in solid mechanics, with an innovative update method based on the computation of a displacement field obtained with the values of the level set function. A criterion based on the values of the level set function is proposed in order to assign the material properties. With the help of this criterion, an optimal approach is proposed, which predicts an accurate evolution of the sub-domain boundary. To validate this method, it was first applied in two dimensions to a through-thickness hole plate case, and then to the cases of brain tumour expansion and grasping to demonstrate the applicability of the method.

Concepts: Mathematics, Brain tumor, Continuum mechanics, Materials science, Manifold, Level set, Object-oriented programming, Level set method


The level set method is a popular technique for tracking moving interfaces in several disciplines, including computer vision and fluid dynamics. However, despite its high flexibility, the original level set method is limited by two important numerical issues. First, the level set method does not implicitly preserve the level set function as a distance function, which is necessary to estimate accurately geometric features, s.a. the curvature or the contour normal. Second, the level set algorithm is slow because the time step is limited by the standard Courant-Friedrichs-Lewy (CFL) condition, which is also essential to the numerical stability of the iterative scheme. Recent advances with graph cut methods and continuous convex relaxation methods provide powerful alternatives to the level set method for image processing problems because they are fast, accurate, and guaranteed to find the global minimizer independently to the initialization. These recent techniques use binary functions to represent the contour rather than distance functions, which are usually considered for the level set method. However, the binary function cannot provide the distance information, which can be essential for some applications, s.a. the surface reconstruction problem from scattered points and the cortex segmentation problem in medical imaging. In this paper, we propose a fast algorithm to preserve distance functions in level set methods. Our algorithm is inspired by recent efficient l(1) optimization techniques, which will provide an efficient and easy to implement algorithm. It is interesting to note that our algorithm is not limited by the CFL condition and it naturally preserves the level set function as a distance function during the evolution, which avoids the classical re-distancing problem in level set methods. We apply the proposed algorithm to carry out image segmentation, where our methods prove to be 5-6 times faster than standard distance preserving level set techniques. We also present two applications where preserving a distance function is essential. Nonetheless, our method stays generic and can be applied to any level set methods that require the distance information.

Concepts: Mathematics, Function, Set theory, Graph theory, Level set, Numerical analysis, Image processing, Level set method


BACKGROUND: X-ray micro-Computed Tomography (muCT) offers the ability to visualise the three-dimensional structure of plant roots growing in their natural environment – soil. Recovery of root architecture descriptions from X-ray CT data is, however, challenging. The X-ray attenuation values of roots and soil overlap, and the attenuation values of root material vary. Any successful root identification method must both explicitly target root material and be able to adapt to local changes in root properties.RooTrak meets these requirements by combining the level set method with a visual tracking framework and has been shown to be capable of segmenting a variety of plant roots from soil in X-ray muCT images. The approach provides high quality root descriptions, but tracks root systems top to bottom and so omits upward-growing (plagiotropic) branches. RESULTS: We present an extension to RooTrak which allows it to extract plagiotropic roots. An additional backward-looking step revisits the previous image, marking possible upward-growing roots. These are then tracked, leading to efficient and more complete recovery of the root system. Results show clear improvement in root extraction, without which key architectural traits would be underestimated. CONCLUSIONS: The visual tracking framework adopted in RooTrak provides the focus and flexibility needed to separate roots from soil in X-ray CT imagery and can be extended to detect plagiotropic roots. The extended software tool produces more complete descriptions of plant root structure and supports more accurate computation of architectural traits.

Concepts: Plant, Data analysis, Tree, Vascular plant, Root, Level set method, Architecture


Underwater optical environments are seriously affected by various optical inputs, such as artificial light, sky light, and ambient scattered light. The latter two can block underwater object segmentation tasks, since they inhibit the emergence of objects of interest and distort image information, while artificial light can contribute to segmentation. Artificial light often focuses on the object of interest, and, therefore, we can initially identify the region of target objects if the collimation of artificial light is recognized. Based on this concept, we propose an optical feature extraction, calculation, and decision method to identify the collimated region of artificial light as a candidate object region. Then, the second phase employs a level set method to segment the objects of interest within the candidate region. This two-phase structure largely removes background noise and highlights the outline of underwater objects. We test the performance of the method with diverse underwater datasets, demonstrating that it outperforms previous methods.

Concepts: Optics, Light, Laser, Object, Polarization, Ambient noise level, Level set method, Beam divergence


Developmental biology has made great strides in recent years towards the quantification of cellular properties during development. This requires tissues to be imaged and segmented to generate computerised versions that can be easily analysed. In this context, one of the principal technical challenges remains the faithful detection of cellular contours, principally due to variations in image intensity throughout the tissue. Watershed segmentation methods are especially vulnerable to these variations, generating multiple errors due notably to the incorrect detection of the outer surface of the tissue.

Concepts: Developmental biology, Cellular differentiation, English-language films, Computer graphics, IMAGE, Level set method, Watershed, Segmentation


Segmenting the optic disc (OD) is an important and essential step in creating a frame of reference for diagnosing optic nerve head pathologies such as glaucoma. Therefore, a reliable OD segmentation technique is necessary for automatic screening of optic nerve head abnormalities. The main contribution of this paper is in presenting a novel OD segmentation algorithm based on applying a level set method on a localized OD image. To prevent the blood vessels from interfering with the level set process, an inpainting technique was applied. As well an important contribution was to involve the variations in opinions among the ophthalmologists in detecting the disc boundaries and diagnosing the glaucoma. Most of the previous studies were trained and tested based on only one opinion, which can be assumed to be biased for the ophthalmologist. In addition, the accuracy was calculated based on the number of images that coincided with the ophthalmologists' agreed-upon images, and not only on the overlapping images as in previous studies. The ultimate goal of this project is to develop an automated image processing system for glaucoma screening. The disc algorithm is evaluated using a new retinal fundus image dataset called RIGA (retinal images for glaucoma analysis). In the case of low-quality images, a double level set was applied, in which the first level set was considered to be localization for the OD. Five hundred and fifty images are used to test the algorithm accuracy as well as the agreement among the manual markings of six ophthalmologists. The accuracy of the algorithm in marking the optic disc area and centroid was 83.9%, and the best agreement was observed between the results of the algorithm and manual markings in 379 images.

Concepts: Scientific method, Optics, Glaucoma, Ophthalmology, Eye examination, Optic nerve, Image processing, Level set method


This paper presents an image reconstruction method based on parametric level set (PLS) method using electrical impedance tomography. The conductivity to be reconstructed was assumed to be piecewise constant and the geometry of the anomaly was represented by a shape-based PLS function, which we represent using Gaussian radial basis functions (GRBF). The representation of the PLS function significantly reduces the number of unknowns, and circumvents many difficulties that are associated with traditional level set (TLS) methods, such as regularization, re-initialization and use of signed distance function. PLS reconstruction results shown in this article are some of the first ones using experimental EIT data. The performance of the PLS method was tested with water tank data for twophase visualization and with simulations which demonstrate the most popular biomedical application of EIT: lung imaging. In addition, robustness studies of the PLS method w.r.t width of the Gaussian function and GRBF centers were performed on simulated lung imaging data. The experimental and simulation results show that PLS method has significant improvement in image quality compared with the TLS reconstruction.

Concepts: Simulation, Group, Level set, Complex number, Distance, Level set method, Radial basis function network, Radial basis function