A new active contour model for shape extraction

We propose a new active contour model for shape extraction of objects in grey‐valued two‐dimensional images based on an energy‐minimization formulation. The energy functional that we consider takes into account the two requirements of object isolation and smoothness of the contour. After deriving the Euler–Lagrange equations corresponding to the energy functional, we bring out some important geometric properties of a solution to these equations. The discussion on our solution method—with the help of which we try to minimize the energy functional by evolving an initial curve—also focuses on how to prescribe the initial curve fully automatically. The effectiveness of our algorithms is demonstrated with the help of experimental results. Copyright © 2000 John Wiley & Sons, Ltd.     

A fractional order derivative based active contour model for inhomogeneous image segmentation

Segmenting intensity inhomogeneous images is a challenging task for both local and global methods. Some hybrid methods have great advantages over the traditional methods in inhomogeneous image segmentation. In this paper, a new hybrid method is presented, which incorporates image gradient, local environment and global information into a framework, called adaptive-weighting active contour model. The energy or level set functions in the framework mainly include two parts: a global term and local term. The global term aims to enhance the image contrast, and it can also accelerate the convergence rate when minimizing the energy function. The local term integrates fractional order differentiation, fractional order gradient magnitude, and difference image information into the well-known local Chan–Vese model, which has been shown to be effective and efficient in modeling the local information. The local term can also enhance low frequency information and improve the inhomogeneous image segmentation. An adaptive weighting strategy is proposed to balance the actions of the global and local terms automatically. When minimizing the level set functions, regularization can be imposed by applying Gaussian filtering to ensure smoothness in the evolution process. In addition, a corresponding stopping criterion is proposed to ensure the evolving curve automatically stops on true boundaries of objects. Dice similarity coefficient is employed as the comparative quantitative measures for the segmented results. Experiments on synthetic images as well as real images are performed to demonstrate the segmentation accuracy and computational efficiency of the presented hybrid method.     

A fast level set model of multi‐atlas labels fusion for 3D MRI tissues segmentation with application of split Bregman method

In this paper, we propose a new fast level set model of multi‐atlas labels fusion for 3D magnetic resonance imaging (MRI) tissues segmentation. The proposed model is aimed at segmenting regions of interest in MR images, especially the tissues such as the amygdala, the caudate, the hippocampus, the pallidum, the putamen, and the thalamus. We first define a new energy functional by taking full advantage of an image data term, a length term, and a label fusion term. Different from using the region‐scalable fitting image data term and length term directly, we define a new image data term and a new length term, which is also incorporated with an edge detect function. By introducing a spatially weight function into the label fusion term, segmentation sensitivity to warped images can be largely improved. Furthermore, the special structure of the new energy functional ensures the application of the split Bregman method, which is a significant highlight and can improve segmentation efficiency of the proposed model. Because of these promotions, several good characters, such as accuracy, efficiency, and robustness have been exhibited in experimental results. Quantitative and qualitative comparisons with other methods have demonstrated the superior advantages of the proposed model.     

On the blowing up of solutions to quantum hydrodynamic models on bounded domains

The blow-up in finite time for the solutions to the initial-boundary value problem associated to the multi-dimensional quantum hydrodynamic model in a bounded domain is proved. The model consists on conservation of mass equation and a momentum balance equation equivalent to a compressible Euler equations corrected by a dispersion term of the third order in the momentum balance. The proof is based on a priori estimates for the energy functional for a new observable constructed with an auxiliary function, and it is shown that, under suitable boundary conditions and assumptions on the initial data, the solution blows up after a finite time.     

Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations

The existence of global-in-time weak solutions to the one-dimensional viscous quantum hydrodynamic equations is proved. The model consists of the conservation laws for the particle density and particle current density, including quantum corrections from the Bohm potential and viscous stabilizations arising from quantum Fokker-Planck interaction terms in the Wigner equation. The model equations are coupled self-consistently to the Poisson equation for the electric potential and are supplemented with periodic boundary and initial conditions. When a diffusion term linearly proportional to the velocity is introduced in the momentum equation, the positivity of the particle density is proved. This term, which introduces a strong regularizing effect, may be viewed as a classical conservative friction term due to particle interactions with the background temperature. Without this regularizing viscous term, only the nonnegativity of the density can be shown. The existence proof relies on the Faedo-Galerkin method together with a priori estimates from the energy functional.     

Adaptive Segmentation Model for Images with Intensity Inhomogeneity based on Local Neighborhood Contrast

Segmentation of images with intensity inhomogeneity is a significant task in the field of image processing, especially in medical image processing and analysis. Some local region-based models work well on handling intensity inhomogeneity, but they are always sensitive to contour initialization and high noise. In this paper, we present an adaptive segmentation model for images with intensity inhomogeneity in the form of partial differential equation. Firstly, a global intensity fitting term and a local intensity fitting term are constructed by employing the global and local image information, respectively. Secondly, a tradeoff function is defined to adjust adaptively the weight between two fitting terms, which is based on the neighborhood contrast of image pixel. Finally, a weighted regularization term related to local entropy is used to ensure the smoothness of evolution curve. Meanwhile, a distance regularization term is added for stable level set evolution. Experimental results show that the proposed model without initial contour can segment inhomogeneous images stably and effectively, which thereby avoiding the influence of contour initialization on segmentation results. Besides, the proposed model works better on noise images comparing with two relevant segmentation models.     

New normalized nonlocal hybrid level set method for image segmentation

This article introduces a new normalized nonlocal hybrid level set method for image segmentation. Due to intensity overlapping, blurred edges with complex backgrounds, simple intensity and texture information, such kind of image segmentation is still a challenging task. The proposed method uses both the region and boundary information to achieve accurate segmentation results. The region information can help to identify rough region of interest and prevent the boundary leakage problem. It makes use of normalized nonlocal comparisons between pairs of patches in each region, and a heuristic intensity model is proposed to suppress irrelevant strong edges and constrain the segmentation. The boundary information can help to detect the precise location of the target object, it makes use of the geodesic active contour model to obtain the target boundary. The corresponding variational segmentation problem is implemented by a level set formulation. We use an internal energy term for geometric active contours to penalize the deviation of the level set function from a signed distance function. At last, experimental results on synthetic images and real images are shown in the paper with promising results.     

A Retinex modulated piecewise constant variational model for image segmentation and bias correction

In this paper, we propose a novel Retinex induced piecewise constant variational model for simultaneous segmentation of images with intensity inhomogeneity and bias correction. Firstly, we obtain an additive model by decomposing the original image into a smooth bias component and a structure part based on the Retinex theory. Secondly, the structure part can be modeled by the piecewise constant variational model and thus deduced a new data fidelity term. Finally, we formulate a new energy functional by incorporating the data fidelity term into the level set framework and introducing a GL-regularizer to the level set function and a smooth regularizer to model the bias component. Based on the alternating minimization algorithm and the operator splitting method, we present a numerical scheme to solve the minimization problem efficiently. Experimental results on images from diverse modalities demonstrate the competitive performances of the proposed model and algorithm over other representative methods in term of efficiency and robustness.     

A variational formulation for physical noised image segmentation

Image segmentation is a hot topic in image science. In this paper we present a new variational segmentation model based on the theory of Mumford-Shah model. The aim of our model is to divide noised image, according to a certain criterion, into homogeneous and smooth regions that should correspond to structural units in the scene or objects of interest. The proposed region-based model uses total variation as a regularization term, and different fidelity term can be used for image segmentation in the cases of physical noise, such as Gaussian, Poisson and multiplicative speckle noise. Our model consists of five weighted terms, two of them are responsible for image denoising based on fidelity term and total variation term, the others assure that the three conditions of adherence to the data, smoothing, and discontinuity detection are met at once. We also develop a primal-dual hybrid gradient algorithm for our model. Numerical results on various synthetic and real images are provided to compare our method with others,these results show that our proposed model and algorithms are effective.     

Minimal energy configurations of strained multi-layers

We derive an effective plate theory for internally stressed thin elastic layers as are used, e.g., in the fabrication of nano- and microscrolls. The shape of the energy minimizers of the effective energy functional is investigated without a priori assumptions on the geometry. For configurations in two dimensions (corresponding to Euler-Bernoulli theory) we also take into account a non-interpenetration condition for films of small but non-vanishing thickness.      

Perturbed integral equation method on determining unknown geometry in fluid flow

We consider an inverse problem arising in fluid flow. An algorithm to find the shape of a body in uniform flow is proposed when the tangential velocity on its boundary is given a priori. The fluid flow is assumed to be inviscid, incompressible and irrotational.The essential idea to develop our algorithm is the boundary modification process toward the solution shape with the help of the perturbed integral equations. The perturbed integral equations are derived from the boundary perturbation. We also give examples exhibiting the reliability for our proposed algorithm.     

Split Bregman method for minimization of improved active contour model combining local and global information dynamically

This paper presents an improved active contour model by combining the Chan–Vese model, the region-scalable fitting energy model, the globally convex segmentation method and the split Bregman method. A weight function that varies with the location of a given image is used to control the influence of the local and global information dynamically. We first present our model in a 2-phase level set formulation and then extend it to a multi-phase formulation. By taking the local and global information into consideration together, our model can segment more general images, especially images with intensity inhomogeneity. Our model has been applied to synthetic and real images with promising results. Numerical results show the advantages of our model compared with other models. The accuracy and efficiency are demonstrated by the numerical results. Besides, our model is robust in the presence of noise.     

On the blowing up of solutions to quantum hydrodynamic models on bounded domains

The blow-up in finite time for the solutions to the initial-boundary value problem associated to the multi-dimensional quantum hydrodynamic model in a bounded domain is proved. The model consists on conservation of mass equation and a momentum balance equation equivalent to a compressible Euler equations corrected by a dispersion term of the third order in the momentum balance. The proof is based on a priori estimates for the energy functional for a new observable constructed with an auxiliary function, and it is shown that, under suitable boundary conditions and assumptions on the initial data, the solution blows up after a finite time. I.M. Gamba is supported by NSF-DMS0507038. M.P. Gualdani acknowledges partial support from the Deutsche Forschungsgemeinschaft, grants JU359/5 and was partially supported under the Feodor Lynen Research fellowship. P. Zhang is partially supported by the NSF of China under Grant 10525101 and 10421101, and the innovation grant from the Chinese Academy of Sciences. Part of the work was done when P. Zhang visited the Department of Mathematics of Texas University at Austin, the author would like to thank the hospitality of the department. Support from the Institute for Computational Engineering and Sciences at the University of Texas at Austin is also gratefully acknowledged.     

Noniterative Reconstruction Method for an Inverse Potential Problem Modeled by a Modified Helmholtz Equation

This paper deals with an inverse potential problem posed in two dimensional space whose forward problem is governed by a modified Helmholtz equation. The inverse problem consists in the reconstruction of a set of anomalies embedded into a geometrical domain from partial measurements of the associated potential. Since the inverse problem, we are dealing with, is written in the form of an ill-posed boundary value problem, the idea is to rewrite it as a topology optimization problem. In particular, a shape functional is defined to measure the misfit of the solution obtained from the model and the data taken from the partial measurements. This shape functional is minimized with respect to a set of ball-shaped anomalies using the concept of topological derivatives. It means that the shape functional is expanded asymptotically and then truncated up to the desired order term. The resulting expression is trivially minimized with respect to the parameters under consideration which leads to a noniterative second-order reconstruction algorithm. As a result, the reconstruction process becomes very robust with respect to noisy data and independent of any initial guess. Finally, some numerical experiments are presented to show the effectiveness of the proposed reconstruction algorithm.     

Image reconstruction model for the exterior problem of computed tomography based on weighted directional total variation

The exterior problem of computed tomography (CT) is a special imaging modality and it is an important research issue in Non-Destructive Testing (NDT). In the exterior problem, the missing projection data vary with the position of the pixels. Furthermore, amounts of theoretical and experimental results showed that the edges that are tangent to the radial direction are much harder to be reconstructed than edges that are tangent to the angular direction. In this paper, a weighted directional total variation (WDTV) based regularization model was proposed to better deal with the exterior problem. By introducing the WDTV regularization term, our model can reconstruct high quality images. First, WDTV of an image also describes the sparsity of image gradient magnitude which can enforce the reconstructed result to be a nearly flat image. Second, our model can preserve edges and reduce artifacts near edges that are tangent to the radial direction. Third, a numerical implementation called SART+WDTV algorithm was developed to solve our model. Simulated and real experiments demonstrated that our model was more capable of suppressing artifacts and preserving edges.     

Speckle noise removal in ultrasound images by first- and second-order total variation

Speckle noise contamination is a common issue in ultrasound imaging system. Due to the edge-preserving feature, total variation (TV) regularization-based techniques have been extensively utilized for speckle noise removal. However, TV regularization sometimes causes staircase artifacts as it favors solutions that are piecewise constant. In this paper, we propose a new model to overcome this deficiency. In this model, the regularization term is represented by a combination of total variation and high-order total variation, while the data fidelity term is depicted by a generalized Kullback-Leibler divergence. The proposed model can be efficiently solved by alternating direction method with multipliers (ADMM). Compared with some state-of-the-art methods, the proposed method achieves higher quality in terms of the peak signal to noise ratio (PSNR) and the structural similarity index (SSIM). Numerical experiments demonstrate that our method can remove speckle noise efficiently while suppress staircase effects on both synthetic images and real ultrasound images.     

Wavelet frame based blind image inpainting

Image inpainting has been widely used in practice to repair damaged/missing pixels of given images. Most of the existing inpainting techniques require knowing beforehand where those damaged pixels are, either given as a priori or detected by some pre-processing. However, in certain applications, such information neither is available nor can be reliably pre-detected, e.g. removing random-valued impulse noise from images or removing certain scratches from archived photographs. This paper introduces a blind inpainting model to solve this type of problems, i.e., a model of simultaneously identifying and recovering damaged pixels of the given image. A tight frame based regularization approach is developed in this paper for such blind inpainting problems, and the resulted minimization problem is solved by the split Bregman algorithm first proposed by Goldstein and Osher (2009) [1]. The proposed blind inpainting method is applied to various challenging image restoration tasks, including recovering images that are blurry and damaged by scratches and removing image noise mixed with both Gaussian and random-valued impulse noise. The experiments show that our method is compared favorably against many available two-staged methods in these applications.     

Arbitrary Initial Term Structure within the CIR Model: A Perturbative Solution

Single‐factor interest rate models with constant coefficients are not consistent with arbitrary initial term structures. An extension which allows both arbitrary initial term structure and analytical tractability has been provided only in the Gaussian case. In this paper, within the context of the HJM methodology, an extension of the CIR model is provided which admits arbitrary initial term structure. It is shown how to calculate bond prices via a perturbative approach, and closed formulas are provided at every order. Since the parameter selected for the expansion is typically estimated to be small, the perturbative approach turns out to be adequate to our purpose. Using results on affine models, the extended CIR model is estimated via maximum likelihood on a time series of daily interest rate yields. Results show that the CIR model has to be rejected with respect to the proposed extension, and it is pointed out that the extended CIR model provides a more flexible characterization of the link between risk neutral and natural probability.     

A noniterative reconstruction method for the inverse potential problem with partial boundary measurements

In this paper, a noniterative reconstruction method for solving the inverse potential problem is proposed. The forward problem is governed by a modified Helmholtz equation. The inverse problem consists in the reconstruction of a set of anomalies embedded into a geometrical domain from partial or total boundary measurements of the associated potential. Since the inverse problem is written in the form of an ill‐posed boundary value problem, the idea is to rewrite it as a topology optimization problem. In particular, a shape functional measuring the misfit between the solution obtained from the model and the data taken from the boundary measurements is minimized with respect to a set of ball‐shaped anomalies by using the concept of topological derivatives. It means that the shape functional is expanded asymptotically and then truncated up to the desired order term. The resulting truncated expansion is trivially minimized with respect to the parameters under consideration that leads to a noniterative second order reconstruction algorithm. As a result, the reconstruction process becomes very robust with respect to the noisy data and independent of any initial guess. Finally, some numerical experiments are presented showing the capability of the proposed method in reconstructing multiple anomalies of different sizes and shapes by taking into account complete or partial boundary measurements.     

Lower Dimensional Representation of Text Data Based on Centroids and Least Squares

Dimension reduction in today's vector space based information retrieval system is essential for improving computational efficiency in handling massive amounts of data. A mathematical framework for lower dimensional representation of text data in vector space based information retrieval is proposed using minimization and a matrix rank reduction formula. We illustrate how the commonly used Latent Semantic Indexing based on the Singular Value Decomposition (LSI/SVD) can be derived as a method for dimension reduction from our mathematical framework. Then two new methods for dimension reduction based on the centroids of data clusters are proposed and shown to be more efficient and effective than LSI/SVD when we have a priori information on the cluster structure of the data. Several advantages of the new methods in terms of computational efficiency and data representation in the reduced space, as well as their mathematical properties are discussed.Experimental results are presented to illustrate the effectiveness of our methods on certain classification problems in a reduced dimensional space. The results indicate that for a successful lower dimensional representation of the data, it is important to incorporate a priori knowledge in the dimension reduction algorithms.