A fast edge detection algorithm using binary labels
Yuying Shi Ying Gu Li-Lian Wang Xue-Cheng Tai
Inverse Problems & Imaging 2015, 9(2): 551-578 doi: 10.3934/ipi.2015.9.551
Edge detection (for both open and closed edges) from real images is a challenging problem. Developing fast algorithms with good accuracy and stability for noisy images is difficult yet and in demand. In this work, we present a variational model which is related to the well-known Mumford-Shah functional and design fast numerical methods to solve this new model through a binary labeling processing. A pre-smoothing step is implemented for the model, which enhances the accuracy of detection. Ample numerical experiments on grey-scale as well as color images are provided. The efficiency and accuracy of the model and the proposed minimization algorithms are demonstrated through comparing it with some existing methodologies.
keywords: edge detection. open/closed curves binary level set method Mumford-Shah model offset curves
Four color theorem and convex relaxation for image segmentation with any number of regions
Ruiliang Zhang Xavier Bresson Tony F. Chan Xue-Cheng Tai
Inverse Problems & Imaging 2013, 7(3): 1099-1113 doi: 10.3934/ipi.2013.7.1099
Image segmentation is an essential problem in imaging science. One of the most successful segmentation models is the piecewise constant Mumford-Shah minimization model. This minimization problem is however difficult to carry out, mainly due to the non-convexity of the energy. Recent advances based on convex relaxation methods are capable of estimating almost perfectly the geometry of the regions to be segmented when the mean intensity and the number of segmented regions are known a priori. The next important challenge is to provide a tight approximation of the optimal geometry, mean intensity and the number of regions simultaneously while keeping the computational time and memory usage reasonable. In this work, we propose a new algorithm that combines convex relaxation methods with the four color theorem to deal with the unsupervised segmentation problem. More precisely, the proposed algorithm can segment any a priori unknown number of regions with only four intensity functions and four indicator (``labeling") functions. The number of regions in our segmentation model is decided by one parameter that controls the regularization strength of the geometry, i.e., the total length of the boundary of all the regions. The segmented image function can take as many constant values as needed.
keywords: four color theorem Mumford-Shah model Unsupervised segmentation convex relaxation method.
High-order total variation regularization approach for axially symmetric object tomography from a single radiograph
Raymond H. Chan Haixia Liang Suhua Wei Mila Nikolova Xue-Cheng Tai
Inverse Problems & Imaging 2015, 9(1): 55-77 doi: 10.3934/ipi.2015.9.55
In this paper, we consider tomographic reconstruction for axially symmetric objects from a single radiograph formed by fan-beam X-rays. All contemporary methods are based on the assumption that the density is piecewise constant or linear. From a practical viewpoint, this is quite a restrictive approximation. The method we propose is based on high-order total variation regularization. Its main advantage is to reduce the staircase effect while keeping sharp edges and enable the recovery of smoothly varying regions. The optimization problem is solved using the augmented Lagrangian method which has been recently applied in image processing. Furthermore, we use a one-dimensional (1D) technique for fan-beam X-rays to approximate 2D tomographic reconstruction for cone-beam X-rays. For the 2D problem, we treat the cone beam as fan beam located at parallel planes perpendicular to the symmetric axis. Then the density of the whole object is recovered layer by layer. Numerical results in 1D show that the proposed method has improved the preservation of edge location and the accuracy of the density level when compared with several other contemporary methods. The 2D numerical tests show that cylindrical symmetric objects can be recovered rather accurately by our high-order regularization model.
keywords: high-order total variation augmented Lagrangian method. radiograph Tomography Abel inversion
A dual EM algorithm for TV regularized Gaussian mixture model in image segmentation
Shi Yan Jun Liu Haiyang Huang Xue-Cheng Tai
Inverse Problems & Imaging 2019, 13(3): 653-677 doi: 10.3934/ipi.2019030

A dual expectation-maximization (EM) algorithm for total variation (TV) regularized Gaussian mixture model (GMM) is proposed in this paper. The algorithm is built upon the EM algorithm with TV regularization (EM-TV) model which combines the statistical and variational methods together for image segmentation. Inspired by the projection algorithm proposed by Chambolle, we give a dual algorithm for the EM-TV model. The related dual problem is smooth and can be easily solved by a projection gradient method, which is stable and fast. Given the parameters of GMM, the proposed algorithm can be seen as a forward-backward splitting method which converges. This method can be easily extended to many other applications. Numerical results show that our algorithm can provide high quality segmentation results with fast computation speed. Compared with the well-known statistics based methods such as hidden Markov random field with EM method (HMRF-EM), the proposed algorithm has a better performance. The proposed method could also be applied to MRI segmentation such as SPM8 software and improve the segmentation results.

keywords: Dual algorithm expectation-maximization Gaussian mixture model total variation projection gradient image segmentation
Augmented Lagrangian method for total variation restoration with non-quadratic fidelity
Chunlin Wu Juyong Zhang Xue-Cheng Tai
Inverse Problems & Imaging 2011, 5(1): 237-261 doi: 10.3934/ipi.2011.5.237
Recently augmented Lagrangian method has been successfully applied to image restoration. We extend the method to total variation (TV) restoration models with non-quadratic fidelities. We will first introduce the method and present an iterative algorithm for TV restoration with a quite general fidelity. In each iteration, three sub-problems need to be solved, two of which can be very efficiently solved via Fast Fourier Transform (FFT) implementation or closed form solution. In general the third sub-problem need iterative solvers. We then apply our method to TV restoration with $L^1$ and Kullback-Leibler (KL) fidelities, two common and important data terms for deblurring images corrupted by impulsive noise and Poisson noise, respectively. For these typical fidelities, we show that the third sub-problem also has closed form solution and thus can be efficiently solved. In addition, convergence analysis of these algorithms are given. Numerical experiments demonstrate the efficiency of our method.
keywords: impulsive noise Poisson noise Augmented Lagrangian method TV-KL convergence. TV-$L^1$ total variation
Augmented Lagrangian method for a mean curvature based image denoising model
Wei Zhu Xue-Cheng Tai Tony Chan
Inverse Problems & Imaging 2013, 7(4): 1409-1432 doi: 10.3934/ipi.2013.7.1409
High order derivative information has been widely used in developing variational models in image processing to accomplish more advanced tasks. However, it is a nontrivial issue to construct efficient numerical algorithms to deal with the minimization of these variational models due to the associated high order Euler-Lagrange equations. In this paper, we propose an efficient numerical method for a mean curvature based image denoising model using the augmented Lagrangian method. A special technique is introduced to handle the mean curvature model for the augmented Lagrangian scheme. We detail the procedures of finding the related saddle-points of the functional. We present numerical experiments to illustrate the effectiveness and efficiency of the proposed numerical method, and show a few important features of the image denoising model such as keeping corners and image contrast. Moreover, a comparison with the gradient descent method further demonstrates the efficiency of the proposed augmented Lagrangian method.
keywords: mean curvature Augmented lagrangian method image denoising.
Augmented Lagrangian method for an Euler's elastica based segmentation model that promotes convex contours
Egil Bae Xue-Cheng Tai Wei Zhu
Inverse Problems & Imaging 2017, 11(1): 1-23 doi: 10.3934/ipi.2017001

In this paper, we propose an image segmentation model where an $L^1$ variant of the Euler's elastica energy is used as boundary regularization. An interesting feature of this model lies in its preference for convex segmentation contours. However, due to the high order and non-differentiability of Euler's elastica energy, it is nontrivial to minimize the associated functional. As in recent work on the ordinary $L^2$-Euler's elastica model in imaging, we propose using an augmented Lagrangian method to tackle the minimization problem. Specifically, we design a novel augmented Lagrangian functional that deals with the mean curvature term differently than in previous works. The new treatment reduces the number of Lagrange multipliers employed, and more importantly, it helps represent the curvature more effectively and faithfully. Numerical experiments validate the efficiency of the proposed augmented Lagrangian method and also demonstrate new features of this particular segmentation model, such as shape driven and data driven properties.

keywords: Euler's elastica augmented Lagrangian method image segmentation convex contour variational model
A two-level domain decomposition method for image restoration
Jing Xu Xue-Cheng Tai Li-Lian Wang
Inverse Problems & Imaging 2010, 4(3): 523-545 doi: 10.3934/ipi.2010.4.523
Image restoration has drawn much attention in recent years and a surge of research has been done on variational models and their numerical studies. However, there remains an urgent need to develop fast and robust methods for solving the minimization problems and the underlying nonlinear PDEs to process images of moderate to large size. This paper aims to propose a two-level domain decomposition method, which consists of an overlapping domain decomposition technique and a coarse mesh correction, for directly solving the total variational minimization problems. The iterative algorithm leads to a system of small size and better conditioning in each subspace, and is accelerated with a piecewise linear coarse mesh correction. Various numerical experiments and comparisons demonstrate that the proposed method is fast and robust particularly for images of large size.
keywords: Coarse mesh correction Total variation minimization Image restoration. Overlapping domain decomposition

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