Level Set Method

The key idea which underpins our work is this. Both the theory and computational practice of the level set method for geometric inverse problems is potentially hampered by the fact that the mapping from the space of the level set function to the physical parameter space is discontinuous. This discontinuity occurs when the level set function is flat at the critical levels, and in particular where the desired level set has non-zero Lebesgue measure. This is dealt with in various ad hoc ways in the applied literature. The beauty of the Bayesian approach is that, with the right choice of prior in level set space, these discontinuities have probability zero. As a result a well-posedness theory (the posterior is Lipschitz in the data) follows automatically, and computational algorithms such as MCMC may be formulated in level set space. We thus have practical algorithms which are simultaneously founded on a sound theoretical bedrock.

The key idea which underpins our work is this. Both the theory and computational practice of the level set method for geometric inverse problems is potentially hampered by the fact that the mapping from the space of the level set function to the physical parameter space is discontinuous. This discontinuity occurs when the level set function is flat at the critical levels, and in particular where the desired level set has non-zero Lebesgue measure. This is dealt with in various ad hoc ways in the applied literature. The beauty of the Bayesian approach is that, with the right choice of prior in level set space, these discontinuities have probability zero. As a result a well-posedness theory (the posterior is Lipschitz in the data) follows automatically, and computational algorithms such as MCMC may be formulated in level set space. We thus have practical algorithms which are simultaneously founded on a sound theoretical bedrock.

Abstract. Level Set Method is a popular method for image segmentation. One of the problems in Level Set Method is finding the right initial surface parameter, which implicitly affects the curve evolution and ultimately the segmentation result. By setting the initial curve too far away from the target object, Level Set Method could potentially miss the target altogether, whereas by setting the initial curve as general as possible – i.e. capturing the whole image – makes Level Set Method susceptible to noise. Recently, deep-learning methods, especially Convolutional Neural Network (CNN), have been proven to achieve state-of-the- art performance in many computer vision tasks such as image classification and detection. In this paper, a new method is proposed, called Deep Convolutional Level Set Method (DCLSM). The idea is to use the CNN object detector as a prior for Level Set Method segmentation. Using DCLSM it is possible to significantly improve the segmentation accuracy and precision of the classic Level Set Method. It was also found that the prior used in the proposed method is the lower and upper bound for DCLSM’s precision and recall, respectively.

We applied our local level-set method to this prac- tical data set to compare the results of the local and the global method. Both methods were applied with the same initial signed-distance function, the same iso- value, and the same smoothness parameter. Renderings of the obtained zero level sets after convergence of both methods are shown in Figure 7. The presented local level-set method required only 6 minutes of compu- tation time (including all computations) until reaching steady state. In comparison, the global level-set process using actually asynchronous time integration required 68 minutes. In terms of quality, there are no significant differences visible between the resulting surfaces. We have measured the distance between the points of the two extracted zero level sets. The maximum deviation between the surfaces was always of the same order as the stopping criterion for the level-set process. This re- sults from the linear interpolation of surface points from the quasi-linear level-set function.

2 Abstract This article presents a fixed-mesh approach to model convective-diffusive particle deposition onto surfaces. The deposition occurring at the depositing front is modeled as a first order reaction. The evolving depositing front is captured implicitly using the level-set method. Within the level-set formulation, the particle consumed during the deposition process is accounted for via a volumetric sink term in the species conservation equation for the particles. Fluid flow is modeled using the incompressible Navier-Stokes equations. The presented approach is implemented within the framework of a finite volume method. Validations are made against solutions of the total concentration approach for one- and two-dimensional depositions with and without convective effect. The presented approach is then employed to investigate deposition on single- and multi-tube arrays in a cross-flow configuration.

2 Centrum Wiskunde & Informatica, Amsterdam, The Netherlands Abstract. This paper introduces a parametric level-set method for tomographic reconstruction of partially discrete images. Such images consist of a continuously varying background and an anomaly with a constant (known) grey-value. We express the geometry of the anomaly using a level-set function, which we represent using radial basis func- tions. We pose the reconstruction problem as a bi-level optimization problem in terms of the background and coefficients for the level-set func- tion. To constrain the background reconstruction, we impose smoothness through Tikhonov regularization. The bi-level optimization problem is solved in an alternating fashion; in each iteration we first reconstruct the background and consequently update the level-set function. We test our method on numerical phantoms and show that we can successfully reconstruct the geometry of the anomaly, even from limited data. On these phantoms, our method outperforms Total Variation reconstruc- tion, DART and P-DART.

Breast thermography is a physiological test that provides information based on the tempera- ture changes in breast. It records the temperature distribution of a body using the infrared ra- diation emitted by the surface of that body. Precancerous tissue and the area around a cancer- ous tumor have higher temperature due to angiogenesis, and higher chemical and blood vessel activity than a normal breast; hence breast thermography has potential to detect early abnor- mal changes in breast tissues. It can detect the first sign of forming up cancer before mam- mography can detect . The thermal information can be shown in a pseudo colored image where each color represents a specific range of temperature. Various methods can be applied to extract hot regions for detecting suspected regions of interests in the breast infrared images and potentially suspicious tissues. Image segmentation techniques can play an important role to segment and extract these regions in the breast infrared images. Shape, size and borders of the hottest regions of the images can help to determine features which are used to detect ab- normalities. In this paper, three image segmentation methods: k-means, fuzzy c-means and level set are discussed and compared. These three methods are tested for different cases such as fibrocystic, inflammatory cancer cases. The hottest regions of thermal breast images in all cases are extracted and compared to the original images. According to the results, level set method is a more accurate approach and has potential to extract almost exact shape of tumors. Keywords: breast, thermography, level set, k-means fuzzy c-means

This paper proposes a level set method to solve minimum stress and stress-constrained shape and topology optimization problems. The method solves a sub-optimization problem every iteration to obtain optimal boundary velocities. A p-norm stress functional is used to aggregate stresses in a single constraint. The shape sensitivity function is derived and a computational procedure based on a least squares interpolation approach is devised in order to compute sensitivities at the boundaries. Adaptive constraint scaling is used to enforce exact control of stress limits. Numerical results show that the method is able to solve the problem efficiently for single and multiple load cases obtaining solutions with smooth boundaries.

3 Renault DREAM-DELT’A Guyancourt, France. 4 EADS Innovation Works, Suresnes, France. Abstract. We consider the optimal distribution of several elastic materials in a fixed working domain. In order to optimize both the geometry and topology of the mixture we rely on the level set method for the de- scription of the interfaces between the different phases. We discuss various approaches, based on Hadamard method of boundary variations, for computing shape derivatives which are the key ingredients for a steep- est descent algorithm. The shape gradient obtained for a sharp interface involves jump of discontinuous quantities at the interface which are difficult to numerically evaluate. Therefore we suggest an alternative smoothed interface approach which yields more convenient shape derivatives. We rely on the signed distance function and we enforce a fixed width of the transition layer around the interface (a crucial property in order to avoid increasing ”grey” regions of fictitious materials). It turns out that the optimization of a diffuse interface has its own interest in material science, for example to optimize functionally graded materials.

The basic task of this existing system is to detect the edges of the image and separating the foreground and background. In this existing system, two types of models will be used which are named to be edge-based models and the region- based models. In this existing system, the segmentation will be independent of the neighbor pixels which in turn obtain many noises in it. The edge based regions depends on the edges of the images like the detection of edge in airplane. The region based models will be dependent on the regions like homogenous regions in which it will be more dependent on the intensity homogenous regions. The simulation result of this existing system is done. The algorithm used in this method is level set algorithm. It is very difficult to apply this level set method and the time consuming will be high in this method. It will limit the length in each and every iteration where it will waste so much of time in limiting. In this existing method, the filters are used to reduce the noises and the filter which is used in existing system is median filter S. Osher et al 4 , M. Abdulghafour et al 19 . This will almost reduce all the noises but the noises will not be removed 100% accurately. The disadvantages which are contained in the existing system will be overcome by the proposed method.

The level-set method, which has several advantages, was investigated by Osher and Sethian [22] for numerically tracking fronts and free boundaries, and re- cently introduced in the field of shape optimization [4,5,11,21,24,27]. First, its main feature is to enable an accurate description of the boundaries on a fixed mesh. Therefore it leads to fast numerical algorithms. Second, its range of application is very wide, since the front velocity can be derived from the clas- sical shape sensitivity. Finally, it can handle some kinds of topology changes, namely the merging or cancellation of holes. However, the usual choice of a Hamilton-Jacobi equation to control the evolution of the level-set function im- plies that this latter obeys a maximum principle. The immediate consequence of this property is that the nucleation of holes inside the domain is prohib- ited. Holes can only appear by pinching two boundaries, which is possible in 3D but not in 2D. It follows that in this case the obtained design is strongly dependent on the initial guess which decides of the maximum number of holes allowed.

We have described an edge- and region-based level set method for accurate seg- mentation of vertebra CT images. The ERBLS model can efficiently segment the images with intensity inhomogenity and blurry or discontinuous boundaries by employing the image gradient information and the local image information. Meanwhile, the level set function is automatically initialized by Otsu threshold, which segmentation result is taken as the initial contours of the EBRLS model. Experimental results on both synthetic and real images demonstrated that the proposed ERBLS model is very robust and efficient. Compared with the well- known local binary fitting (LBF) model, the ERBLS model is not only much more computationally efficient and but also much less sensitive to the initial contours. The proposed method has also applied to 56 patient data sets and has produced very promising results.

In the present paper we propose a novel method for handling three major manufacturing constraints of geometric nature using the level-set method for structural optimization: minimal and maximal local thickness and minimal members’ distance. They all three rely on a definition of the structure thickness based on the signed distance function of the shape. We then introduce specific integral criteria which exactly measure the satisfaction of each constraint. Our work was previously announced in [5], [29]. Our approach is similar but significantly different from that recently appeared in [24]. We shall compare it, as well as other methods, to our work in Section 9. Of course, it was already noticed by many other authors that the signed distance function is a convenient tool for geometric problems, most notably in image processing (see e.g. [1], [26], [27], [31]). Closely linked to the signed distance function are other important notions that we shall use in the sequel: the ”skeleton” or ridge of a shape (which is the location of discontinuities on the distance gradient [1], [26]) and the offset sets of the boundary [1], [26].

Keywords: Level set method, Minimum flexibility, Topology optimization. Abstract. This paper takes the minimum flexibility of the structure as the objective function and the volume ratio of the solid material as the constraint condition to realize the topology optimization of the level set method. The radial basis function (RBF) can be used to interpolate discrete points into smooth curves, and the level set function can be used to implicitly describe the structure boundary. By designing the initial structure of the hole, it is obtained that the optimization result will be better if the hole is as wide as possible in the whole area. Through many experiments, it is proved that the restraint condition is that the remaining material volume ratio of 50% is more reasonable than other ratios. The effectiveness of the proposed method is verified by two classical examples of cantilever beam and simply supported beam. Comparing with the variable density method, the checkerboard format of gray level unit is eliminated, the smooth boundary is obtained, and the efficiency of hole nucleation is improved.

5. CONCLUSION We have proposed an efficient object recognition technique using level set method and neural network. In this paper, object shape contour is extracted by level set method and convex hull set descriptors have been proposed which are invariant to the viewpoint, geometry of the object and illumination conditions, this set have been served as the pattern for the forthcoming processes involved in the proposed approach. Subsequently we have trained feed forward neural network using different class of data set and tested using all set of data. We have attained distinct results concerned with different class of the objects. We have evaluated our proposed approach on the ALOI collection, a large collection of object images consists of 1000 objects recorded under various imaging circumstances. The experiments clearly demonstrated that our proposed approach significantly outperforms the state-of-the- art methods with level set method for shape features. The proposed method is shown to be effective under a wide variety of imaging conditions.

Abstract The active contour method is one of the most successful image segmentation techniques. It has received a tremendous amount of attention in medical image processing. The segmentation operation can be carried out manually or automatically. A manual segmentation requires a skilled operator trained to use a digital tool to mark the contours of the desired structures. In this project the three-phase formulation of the level set evolution (LSE) and bias field estimation and a Level Set Method for Image Segmentation in the Presence of Intensity is done. The three-phase formulation is used to segment an image into three regions. Intensity inhomogeneity often occurs in real-world images, which presents a considerable challenge in image segmentation. The most widely used image segmentation algorithms are region-based and typically rely on the homogeneity of the image intensities in the regions of interest, which often fail to provide accurate segmentation results due to the intensity inhomogeneity. Matlab code is used for the three phase formation and bias field estimation.

Abstract Segmentation of vasculature in retinal fundus image by level set methods employing classical edge detection methodologies is a tedious task. In this study, a revised level set-based retinal vasculature segmentation approach is proposed. During preprocessing, intensity inhomogeneity on the green channel of input image is corrected by utilizing all image channels, generating more efficient results compared to methods utilizing only one (green) channel. A structure-based level set method employing a modified phase map is introduced to obtain accurate skeletonization and segmentation of the retinal vasculature. The seed points around vessels are selected and the level sets are initialized automatically. Furthermore, the proposed method introduces an improved zero-level contour regularization term which is more appropriate than the ones introduced by other methods for vasculature structures. We conducted the experiments on our own dataset, as well as two publicly available datasets. The results show that the proposed method segments retinal vessels accurately and its performance is comparable to state-of-the-art supervised/unsupervised segmentation techniques.

A diffusion term is introduced into LSE, resulting in a diffusion-augmented level set method with efficient two step implementation. First we iteratively solve the diffusion term and then iteratively solve the level set equation. By solving equation in two steps we can stabilize the level set function without re-initialization. This is also called two step splitting method for image segmentation.

or organ should have similar metabolic property, a shape-based reconstruction strategy is implemented in FMT to reduce the number of unknown parameters. In this strategy, the reconstructed object is divided into numbers of disjoint subregions and the shapes of these subregions are represented by a level set method [11] or spherical harmonics expansion [12]. Accordingly, the fluorescent yield in each subregion is described by only one parameter. It changes the reconstruction problem from reconstruction of fluores- cent yield at each voxel or node to reconstruction of level set function or expansion coefficients with several parameters of fluorescent yield. When a problem with dynamic fluorescent yield is considered, the number of unknown parameters will be reduced because the shapes of the distributions of fluorescent yield at different projection angles are described by the same set of unknown parameters by using the level set method or spherical harmonics expansion. In this paper, the level set method is chosen to represent the shapes of subregions for its ability of arbitrary boundary representation. Although spherical harmonics expansion can also describe arbitrary boundary theoretically, the maximum degree of spherical harmonics, which controls the shape representation, needs to be determined manually [13].

In the area of 3D vessel image segmentation, Lorigo et al. [17] proposed the ’CURVES’ algorithm to extract vessels by using a geodesic active contour based on the codimension two level set method [18]. Nain et al. [1] proposed a region-based geometric deformable model for MRA segmentation, which is driven by the combination of image statistics and shape information. Yan and Kassim [19] presented a segmentation scheme for the extraction of vasculature from MRA images by using capillary active contour. In [20], the vessel contours are segmented from the enhanced dataset by means of a level set evolution, which relies on the image intensity statistics to either expand or shrink the evolving contour.In [21], a shape functional is used to regularize the ‘CURVES’ and flux maximizing flow functional [22], which is claimed to be a promising tool to improve the efficiency of both techniques. Law and Chung [23] presented a vessel extraction approach based on a weighted local variance based edge detection scheme, which is effective for the segmentation of vessel image with low contrast edges. Recently, Wang and Jiang presented a nonparametric shape constrained algorithm for segmentation of coronary arteries in computed tomography images within the framework of active contours [24]. However, as most of these active contour models only take the edge information or region information of the image into account, they may have difficulty in correctly extracting tiny vessels from images, due to the low image contrast of the thin vessel branches.