^{1}

^{2}

^{2}

^{1}

^{1,3}

^{1}

^{2}

^{3}

We describe a new method for simultaneous image denoising and level set-based active contour segmentation using multidimensional features. We consider an image to be a surface embedded in a Riemannian manifold. By defining a metric in the embedded space, which in our case includes multidimensional image features as well as a level set-based active contour model, a minimization problem in the image space can be obtained through the Polyakov action framework. The resulting minimization problem is solved with a dual algorithm for efficiency. Benefits of this new method include the fact that it is independent of any artificial “running” parameters, and experiments using both synthetic and real images show that the method is robust with respect to noise and blurry object boundaries.

Unsupervised image segmentation is an important problem with many applications in science, including medical imaging. Image segmentation is a postprocessing problem in many computer vision tasks; its aim is to divide an image into finite number of subregions. The features of different subregions are utilized as the segmentation criteria. The statistical methods, such as expectation-maximization (EM) algorithm [

Many works utilize the difference between invariable pixel intensities, as well as their spatial connectivity, in assessing whether two pixels belong to the same object. These active contour models based on the level set method [

The Polyakov action was introduced in image processing by Sochen et al. in [

In this paper, the proposed active contour model is formulated in the framework of the Polyakov action [

The paper is organized as follows. In Section

Sochen et al. introduce a general geometrical framework for low-level vision, based on the Polyakov action [

In the relevant works [

In this work, we utilize an improved geometrical framework based on the weighted Polyakov action without any artificial parameter. First, we get an approximated image by embedding it into the feature space constituted by the features of the original image. Second, given the approximated image, active contour is driven by embedding the level set function into the higher dimensional feature space composed of the geometrical and statistical features of the approximated image.

The original image

The active contour is represented as the zero level set function

To apply the dual gradient algorithm, we introduce the dual variable,

Introducing the auxiliary variables

(a) Given image

(b) Given

(c) Given the solution of

(d) Given

The algorithm of minimizing our model is described in the following.

All the experiments are run with Matlab code on the PC of CPU 3.2 GHz, RAM 728 M. we show the experiments results for medical image segmentation of Chan-Vese model (CV) [

Segmentation for synthetic noisy image.

Original image

CV model 200th

SLM 50th

The proposed model 50th

The approximated image

As shown in Figures

Figure

Segmentation for brain MRI. (a) Original image and initial level set contour. (b) Segmentation result of the CV model with 200 iterations. (c) Segmentation result of the SLM with 100 iterations. (d) Segmentation result of the RSF model with 200 iterations. (e) Segmentation result of the proposed model with 30 iterations. (f) The approximated result of the proposed model.

Figure

Segmentation for brain MRI. (a) Original image and initial level set contour. (b) Segmentation result of the CV model with 200 iterations. (c) Segmentation result of the SLM with 100 iterations. (d) Segmentation result of the RSF model with 200 iterations. (e) Segmentation result of the proposed model with 30 iterations. (f) The approximated result of the proposed model.

Figure

Iteration numbers and processing time in each iteration.

CV model | SLM model | The proposed model | |
---|---|---|---|

Single level set function | 0.35 sec | 0.23 sec | 0.18 sec |

Multilayer functions | 1.47 sec | 0.62 sec | 0.39 sec |

Iteration number for convergence | 100 iterations | 50 iterations | 30 iterations |

Quantitative evaluation for brain MR-data in Figure

The CV model | The SLM model | The proposed model | |
---|---|---|---|

Overall accuracy | 65.2% | 34.6% | 91.5% |

GM (dice metric) | 78.3% | 82.4% | 97.2% |

WM (dice metric) | 69.0% | 13.5% | 94.7% |

Segmentation for brain MRIs by the active contour methods based on multilayer level set functions. (a) and (e) are original images and initial level set contours. (b) and (f) are segmentation results of the CV model with 100 iterations. (c) and (g) are segmentation results of the SLM with 50 iterations. (d) and (h) are the results of the proposed method with 30 iterations.

The data in Figure

Quantitative evaluation for brain MR-data in Figure

The CV model | The SLM model | The RSF model | The proposed model | |
---|---|---|---|---|

Overall accuracy | 15.4% | 15.7% | 82.6% | 88.7% |

GM (dice metric) | 13.5% | 14.1% | 79.6% | 90.4% |

WM (dice metric) | 16.2% | 16.8% | 83.7% | 87.9% |

Segmentation for 3D brain image: (a) original image; (b) segmentation result of the CV model with 200 iterations; (c) segmentation result of the SLM model with 100 iterations; (d) segmentation result of the RSF model with 200 iterations; (e) segmentation result of the proposed model with 30 iterations; (f) the denoising result of the proposed model; (g) the 3D original image; (h) the 3D segmentation result of the proposed method; (i) the approximated 3D image.

In this paper, we propose a new variational model for image segmentation and image denoising simultaneously. We obtain the approximated image by embedding the approximating criteria into a specific multifeature space. And then the segmentation result is obtained by embedding the active contour into another multifeature space which is composed by the segmentation criteria depending on the approximated image. The segmentation and the denoising problems are solved by the split Chambolle dual algorithm alternately. The comparisons of the other popular segmentation models demonstrate the accuracy and efficiency of the proposed model.

The following are the research highlights of this paper. The proposed variational model incorporates segmentation and denoising together. Segmentation and denoising processing are achieved alternately by the Polyakov action framework. An improved Polyakov action framework is purely based on the geometric features of the image without any manual parameters. Minimizing the variational model is achieved by the improved Chambolle algorithm.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported in part by the Natural Science Foundation Science Foundation of China under Grant nos. 61502244, 61402239, and 71301081, the Science Foundation of Jiangsu Province under Grant nos. BK20150859, BK20130868, and BK20130877, the Science Foundation of Jiangsu Province University (15KJB520028), NJUPT Talent Introduction Foundation (NY213007), NJUPT Advanced Institute Open Foundation (XJKY14012), China Postdoctoral Science Foundation (2015M580433, 2014M551637), and Postdoctoral Science Foundation of Jiangsu Province (1401046C).