Mathematical Problems in Engineering

Volume 2016, Article ID 3909645, 9 pages

http://dx.doi.org/10.1155/2016/3909645

## Unsupervised Joint Image Denoising and Active Contour Segmentation in Multidimensional Feature Space

^{1}College of Telecommunications and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, China^{2}College of Automation, Nanjing University of Posts and Telecommunications, Nanjing 210003, China^{3}KTH Royal Institute of Technology, 10044 Stockholm, Sweden

Received 28 April 2016; Accepted 26 June 2016

Academic Editor: Giuseppina Colicchio

Copyright © 2016 Qi Ge et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1] and fuzzy C-means clustering (FCM) algorithm [2], are applied in classifying the pixels based on some particular image features segmentation criteria. In general, the statistical methods achieve the classification based on only one segmentation criterion. However, there is various kinds of features in an image and the features may vary spatially. Therefore it will be not precise to use one kind of these methods. How to extract the features of an image and how to utilize these features as the segmentation criterion are significant for segmentation.

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 [3] classify the pixels by only one image feature, that is, the image intensity based on uniform distribution [4–6]. Nevertheless, the image intensity varies spatially; thus the image intensity is not necessarily described by one kind of specific distribution. For improving the precision, the works of [7, 8] extract the multifeature to deal with more complex information content. Simultaneously, the additional artificial parameters are introduced; thus it needs the experience to set the parameters.

The Polyakov action was introduced in image processing by Sochen et al. in [9]. This segmentation model is different from the other segmentation methods in two ways. First, images are represented as Riemannian manifolds embedded in a higher dimensional spatial-feature manifold. Second, the Polyakov action provides an efficient mathematical framework to embed the multifeature of images in higher-dimensional Riemannian manifolds by harmonic maps. Bresson et al. [10] propose active contour models based on the Polyakov action. These models map several kinds of features, for example, color and texture, into higher dimensional space. Because these models choose a metric with artificial parameters on the feature space, it requires careful manual parameter-tuning.

In this paper, the proposed active contour model is formulated in the framework of the Polyakov action [9]. Unlike the other related works [7–9], a metric on the feature space manifold is defined by the invariant geometry of images. Consequently, the proposed method is purely based on the geometrical features of images without any artificial parameters. We implement the segmentation through two steps. First, an approximated image, removing the noise while preserving the main structures, is found in the feature space built on geometrical features of the original image. Second, the active contour is embedded into the feature space built on both the statistical and geometrical features of the approximated image. For efficiency, we solve the proposed model via the improved Chambolle dual formulation [10] of the minimization problem.

The paper is organized as follows. In Section 2, we introduce the mathematical framework based on the Polyakov action. In Section 3, we introduce the proposed model and the numerical algorithm of the proposed method is also summarized. In Section 4, we validate our model by some experiments on medical images. In Section 5, we end the paper by a brief conclusion.

#### 2. Geometrical Framework Based on Weighted Polyakov Action

Sochen et al. introduce a general geometrical framework for low-level vision, based on the Polyakov action [9]. In this framework, images are represented as the surfaces on a Riemannian manifold. The Polyakov action is a functional that measures the weight of a mapping between an -dimensional embedded manifold (e.g., the image manifold) with coordinates and the -dimensional manifold with the coordinates , . A Riemannian structure metric can be introduced to measure the local distances on the embedded manifold , whereas we use the metric to measure the distance on the manifold . To measure the weight of the mapping , the Polyakov action is used as a generalization of the -norm on the embedded image to space feature manifold :where is the determinant of the image metric tensor and is its inverse. The metric is chosen as the induced metric, obtained by the pullback relation: ; the Polyakov energy is shortened to

In the relevant works [7, 8], the authors get the denoised image and the segmentation results by minimizing the energy functional (2) with respect to denoising and segmentation, respectively. In seminal work [9], they embed grey images in the feature , where is the grey intensity value for pixel . They choose a metric ; is a constant. Based on this metric on feature space and the Polyakov energy, the regularization term on the intensity values is given by . Although it allows setting the scale of the feature dimension independently of the spatial dimensions, the accuracy of the scale is subject to the artificial parameter .

#### 3. The Active Contour Model in Multifeature Space

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.

##### 3.1. Approximating Image under an Improved Geometrical Framework

The original image is defined on the image manifold with coordinates . The approximated image is defined on the image manifold and denoted by . To preserve the main edges of the original image, we extract the geometrical features of edges, , derived from the anisotropic diffusion equation [2]. Considering the intensity value as another feature, we build the feature space, , denoted by for the sake of simplicity. To avoid the influence of the artificial parameter, we choose a metric tensor on the feature space , which is defined by the invariant geometry of the original image . Consider . The pullback relation yields the determinant of metric tensor on manifold :Analogizing based on the Polyakov energy (2), we get the approximated image by minimizing the energy functional as follows:where the weight coefficient of second term, corresponding to the third element of the metric , denotes the coefficients of first fundamental form in differential geometry. When this weight coefficient is larger, the edge structure is enhanced in the vicinity of the edges; otherwise, smoothing the image is strengthened. The weight coefficient of the third term, corresponding to the last element of the metric , denotes the coefficients of second fundamental form in differential geometry. Approximating the intensity is strengthened when this coefficient is larger, whereas smoothing the image is strengthened when the weight is smaller.

##### 3.2. Active Contour Evolution under the Improved Geometrical Framework

The active contour is represented as the zero level set function on the image manifold . For avoiding the effects of the intensity nonuniformity, we extract the statistical features on the local region of size , where , denote the mean intensity in the local region inside and outside the zero level set. The feature space is , denoted by . The metric tensor defined on this feature space is . The pullback relation yields the determinant of metric tensor on manifold :According to the Polyakov energy (2), we drive the curve evolution by minimizing the energy functional as follows:where the weight of first term is actually an edge detector. The curve evolution tends to stop when it decreases to zero, whereas the evolution goes on.

##### 3.3. Dual Algorithm

To apply the dual gradient algorithm, we introduce the dual variable, . The total variation term in (4) and (6) can be formulated as follows:The approximation formulation of the energy of our model can be rewritten aswhere . We then apply the split Chambolle dual algorithm [10] to solve the optimization problem.

Introducing the auxiliary variables , , solving the energy functional (8) is equivalent to minimizing the problem as follows:where the parameter is chosen to be small for avoiding smearing the edges (in this paper, we choose .), is an exact penalty function provided that the constant is chosen large enough compared to such as , , and . The minimization problem (9) can be divided into four subproblems as follows and can be solved alternatively.

(a) Given image , , update . we search for as the solution of The solution of (10) is given bywhere can be updated by fixed point method: initializing and updating In this paper, we choose to ensure convergence.

(b) Given , , we search for by solving the minimization problem as follows:The solution of (13) is given byEquation (14) is solved by a fixed point method:

(c) Given the solution of , we search by solvingThe solution of (16) is given by

(d) Given , , we search for as the solution ofThe solution of (18) is given byAfter is solved, it is utilized in (12) for next iteration.

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

*Step **1*. Initialize .

*Step **2*. Given the fixed threshold of iterations , if , then stop; else go to Step 3.

*Step **3*. Do the iteration for solving subproblem.

*Step **3.1*. While do

*Step **3.2*. Update the dual variable by (14) and then update by (13); go to Step 3.3.

*Step **3.3*. Given , update the dual variable according to (15) and then update by (16); go to Step 3.4.

*Step **3.4*. Given , , compute and update by (17). Go to Step 3.5.

*Step **3.5*. Given , , update by (19); then go to Step 3.1; otherwise, go to Step 4.

*Step **4*. End while.

#### 4. Experimental Results

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) [11], the structure-based level set method (SLM) [12], and the region-scale fitting model (RSF) [4]. Figure 1 shows the experiments on the synthesized noisy images. This image is of size with 10% white Gaussian noise.