Journal of Healthcare Engineering

Volume 2017 (2017), Article ID 6506049, 11 pages

https://doi.org/10.1155/2017/6506049

## An Improved Random Walker with Bayes Model for Volumetric Medical Image Segmentation

^{1}Department of Mathematics and Computer Science, Fort Valley State University, Fort Valley, GA, USA^{2}College of Computer Science and Technology, Zhejiang University, Hangzhou, China^{3}Radiology Department, Sir Run Run Shaw Hospital, Medical School of Zhejiang University, Hangzhou, China^{4}Graduate School of Information Science and Engineering, Ritsumeikan University, Kyoto, Japan

Correspondence should be addressed to Yen-Wei Chen

Received 24 February 2017; Accepted 23 April 2017; Published 23 October 2017

Academic Editor: Pan Lin

Copyright © 2017 Chunhua Dong 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

Random walk (RW) method has been widely used to segment the organ in the volumetric medical image. However, it leads to a very large-scale graph due to a number of nodes equal to a voxel number and inaccurate segmentation because of the unavailability of appropriate initial seed point setting. In addition, the classical RW algorithm was designed for a user to mark a few pixels with an arbitrary number of labels, regardless of the intensity and shape information of the organ. Hence, we propose a prior knowledge-based Bayes random walk framework to segment the volumetric medical image in a slice-by-slice manner. Our strategy is to employ the previous segmented slice to obtain the shape and intensity knowledge of the target organ for the adjacent slice. According to the prior knowledge, the object/background seed points can be dynamically updated for the adjacent slice by combining the narrow band threshold (NBT) method and the organ model with a Gaussian process. Finally, a high-quality image segmentation result can be automatically achieved using Bayes RW algorithm. Comparing our method with conventional RW and state-of-the-art interactive segmentation methods, our results show an improvement in the accuracy for liver segmentation ().

#### 1. Introduction

Segmentation of organ from CT volume is an important prerequisite for computer-aided surgery, computer-assisted intervention, and image-guided surgery. The accurate segmentation of the organ from clinical CT images is considered a challenging task: Large variations in shape make an accurate segmentation difficult, and existing lesions (e.g., tumors) exhibit considerable variation for the organ anatomical structure. To accurately segment an organ, various approaches have been proposed in literatures [1–8], such as intensity-based [9–11], classification-based [12, 13], clustering-based [14–18], statistical shape model- (SSM-) based [19, 20], probabilistic atlas- (PA-) based [21–25], active contour- (AC-) based [26, 27], and watershed-based [28, 29] segmentation methods. However, the main challenge of the abovementioned methods is the fast and efficient segmentation of large image data. This can be observed particularly in medical applications where a resolution of three-dimensional CT and MRI body scans constantly increases.

Recently, a growing interest is attracted by an interactive graph-based image segmentation algorithms such as graph cut (GC) [30–36] and random walker (RW) [37–41] algorithms. The random walker algorithm represents a recent noteworthy development in the weighted graph-based interactive segmentation methods. This technique with user interaction is more suitable for volumetric medical images to guarantee the reliability, accuracy, and fast speed demands.

However, due to the classical RW algorithm definitions on the weighted graphs, for a high-resolution volumetric medical image, RW method needs to construct the corresponding large-scale graph to solve the resulting sparse linear system, which leads to high computation cost: the long computation time and the high memory usage. Hence, over the past years, a large amount of research has been conducted to extend and enhance the random walker algorithm. Grady et al. [40] extended the classical RW segmentation approach by combining the regional intensity priors. The sparse linear equations can be addressed by the preconditioned conjugate gradient to achieve an acceptable memory consumption and easy parallelization. In [41], the computational demands with RW are alleviated by introducing an “offline” precomputation before user interaction with RW in real-time “online.” Using a similar principle, an offline precomputation was used to further speed up the online segmentation in [42]. Both methods used the “offline” and “online” strategies to minimize the time spent waiting. In addition, Goclawski et al. [43] proposed a superpixel-based random walker method to reduce the graph size, while the computation time increases linearly with the number of superpixels. The accuracy of superpixels plays an immediate decisive role in the process of organ segmentation.

To resolve these limitations, in our previous research [44], we proposed a knowledge-based segmentation framework for the volumetric medical image in a slice-by-slice manner based on the classical random walker. This algorithm employs the previous segmented slice as the prior knowledge for automatically setting the object/background seed points for the adjacent slices. It can reduce the graph scale and significantly speed up the optimization procedure of the graph. However, the classical RW algorithm was designed to be a general purpose interactive segmentation method, such that a user could mark a few pixels with an arbitrary number of labels and expect a quality result, regardless of the data set or the segmentation goal. Segmentation of a medical image ignores itself absolute intensity and shape information. If a consistent intensity and shape profile characterize an object of interest, then this information should be incorporated into the RW segmentation process.

Taking these into consideration, in our study, we extended a classical random walker algorithm by incorporating the prior (shape and intensity) knowledge in the optimization of sparse linear system. The objective of our work is to combine the prior knowledge with the spatial cohesion of the random walker algorithm in a principled way that produces the correct result. Based on the extended random walker, we applied a knowledge-based segmentation framework for the volumetric medical image in a slice-by-slice manner. Our strategy is to employ the previous segmented slice to obtain the prior (shape and intensity) knowledge of the target organ for the adjacent slice. With a small number of user-defined seed points, we can obtain the segmentation results of the start slice in the volume which can be used as the prior knowledge of the target organ. According to this prior knowledge, the object/background seed points are automatically defined and the corresponding Bayes model can be generated. Integrating this Bayes model into the RW sparse system, the organ is automatically segmented for the adjacent slice.

The remainder of this paper is organized as follows. Section 2 presents a brief recapitulation of the random walker algorithm and then extends to incorporate the prior (shape and intensity) knowledge. Section 3 elaborates our proposed knowledge-based framework using the extended RW with the Bayes model. Section 4 contains experimental work, and Section 5 discusses the implementation of our method, followed by the conclusion (Section 6).

#### 2. Development

The random walk algorithm treats image segmentation as an optimization problem on a weighted graph, where each node represents a pixel or voxel. Therefore, we firstly define the graph that we are working on. We use the following notations for the rest of the paper. Given an image, , a graph consists of with vertices (nodes) and edges . Each node in uniquely identifies an image pixel . An edge, , spanning two vertices and , is denoted by . A weighted graph assigns a weight to each edge. The weight of an edge, , is denoted by . It represents the similarity between two neighboring nodes and . The degree of a vertex is for all edges incident on .

##### 2.1. Review of Random Walker Method

The random walker segmentation algorithm of [37] computes the probability, for each pixel, that a random walker leaving that pixel will first arrive at a foreground seed before arriving at a background seed. It was shown in [37] that these probabilities may be calculated analytically by solving a linear system of equations with the graph Laplacian matrix. The Laplacian matrix is defined as where is indexed by vertices and . is the edge weight, and and indicate the image intensity at vertices and , respectively. represents a tuning constant that depends on the user.

Given a weighted graph, a set of marked (labeled) nodes, , and a set of unmarked nodes, , such that and , we would like to label each node with a label . stands for the foreground, and stands for the background. Assuming that each node has also been assigned with a label , we can compute the probabilities, , that a random walker leaving node arrives at a marked node by solving the minimization of

All nodes are divided into two sets: the marked (prelabeled) nodes and unlabeled (i.e., free) nodes . Therefore, the above function can be reformulated as follows:

Minimization of (3) with respect to , the random walker problem can be solved by the following system of equations:

The variable represents the set of probabilities corresponding to unmarked nodes; is the set of probabilities corresponding to marked nodes (i.e., “1” for foreground nodes and “0” for background nodes). By virtue of being a probability,

The random walk algorithm is explained in detail elsewhere [37]. Next, we will now present how the incorporation of the Bayes model into the above framework yields a segmentation algorithm.

##### 2.2. Random Walker with Bayes Model

According to the above priori knowledge, we can calculate a posterior probability at the node which belongs to the label . Assuming that each label is equally likely, Bayes theorem gives the probability that a node belongs to label as where is the likelihood map for an organ and is the shape map for the targeted organ. can be obtained by dilating the targeted organ region in the previous segmented slice. can be estimated by the previous segmented slice of the organ. is the foreground, and is the background.

Equation (6) can be also written in vector notation: where is a diagonal matrix with the values of on the diagonal.

According to (6), the minimum energy distribution for the external function is

To incorporate the posteriori probability function (external term) into the RW algorithm (internal term), we may optimize the following energy:

The first term is the driving force behind the spatial cohesion of the random walker algorithm. The second term is a Bayes penalty term with the weight used to guarantee robustness to small disconnected pieces. The used Bayes model is generated according to the prior knowledge of an organ: shape and intensity. In this work, we set the weight to .

The minimum energy of the above equation is obtained when satisfies the solution to

Optimizing this energy leads to the system of linear equations:

The usage of the proposed Bayes-based RW algorithm is strongly limited by the enormous size of the graph represented in 3D volumetric medical image and the necessity of solving a huge sparse linear system. It results in the relative increase of the unlabeled seed points relative to a 2D image. Hence, in order to estimate the probability of each unlabeled seed point, the extended RW algorithm needs to calculate the larger inverse matrix , which leads to high computation costs: long computation time and high memory usage. We integrated our extended RW algorithm into a knowledge-based framework to make it more suitable and workable for our application. The following details our knowledge-based framework and results.

#### 3. Knowledge-Based Framework

Our knowledge-based strategy employs the previous segmented slice as the prior (shape and intensity) knowledge of the target organ for automatic segmentation of the adjacent slice. Using a small number of user-defined seed points, we can obtain the segmentation results of the start slice of the volume for use as the prior knowledge of the target organ. According to the prior knowledge, the object/background seed points can be dynamically updated for the adjacent slice by combining the narrow band threshold (NBT) method and the organ model with a Gaussian process. Meanwhile, the corresponding Bayes model can be generated. Finally, an extended Bayes-based random walker algorithm is applied to automatically segment the whole volume in a slice-by-slice manner. In our work, “object” means the target organ to be segmented and “background” means the other tissues except the target organ. The whole procedure of the proposed approach is shown in Figure 1. In this method, there is a three-step pipeline consisting of the following: (1)Selecting and segmenting the start slice, as shown in the middle-part of Figure 1: (a) Manually defining the object/background seed points. (b) Generating a Gaussian model (GM) using the seed points. (c) Segmenting the organ (“Candicate Pixels” for the liver) using the classical RW method.(2)Segmenting the adjacent slice, as shown in the upper-part and bottom-part of Figure 1: (a) Generating a Gaussian model (GM) according to the previous segmented organ (intensity knowledge). (b) Automatic setting the object/background seeds based on the restricted region by morphological operation of the previous segmented organ (shape knowledge). (c) Refining the seed points based on NBT. (d) Segmenting the organ using our proposed Bayes-based RW methods. Thus, it automatically segments the whole organ in the remaining slices based on the updated prior knowledge of the organ.(3)Smoothing the boundary of the whole volume: Finally, the boundary of the output volume is smoothed by “Fourier transform” that forms the final organ surface.