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# Active contours with neighborhood-extending and noise-smoothing gradient vector flow external force

- Lixiong Liu
^{1}Email author and - Alan C Bovik
^{2}

**2012**:9

https://doi.org/10.1186/1687-5281-2012-9

© Liu and Bovik; licensee Springer. 2012

**Received:**15 January 2012**Accepted:**10 May 2012**Published:**10 May 2012

## Abstract

We propose a novel external force for active contours, which we call *neighborhood-extending and noise-smoothing gradient vector flow* (NNGVF). The proposed NNGVF snake expresses the gradient vector flow (GVF) as a convolution with a neighborhood-extending Laplacian operator augmented by a noise-smoothing mask. We find that the NNGVF snake provides better segmentation than the GVF snake in terms of noise resistance, weak edge preservation, and an enlarged capture range. The NNGVF snake accomplishes this with a reduced computational cost while maintaining other desirable properties of the GVF snake, such as initialization insensitivity and good convergences at concavities. We demonstrate the advantages of NNGVF on synthetic and real images.

## Keywords

- image segmentation
- active contour
- gradient vector flow
- Laplacian operator
- neighborhood-extending and noise-smoothing gradient vector flow

## 1. Introduction

D uring the last two decades, variational and PDE-based methods for image segmentation and analysis have become standard tools [1]. Active contours or snakes which have deeply influenced variational approaches to image segmentation since their introduction [2] are curves that can conform to object boundaries or other image features under the influence of internal and external forces [2]. Generally, active contours can be categorized as parametric snakes [2] or as geometric snakes [3–5] according to their representation. Parametric snakes require an explicit representation while geometric snakes are defined implicitly. Here, we show how to construct an effective external force for parametric active contour models that can also be integrated into geometric active contours using a level set formulation [4].

Since the external force defines the evolution of an active contour, many external force models have been proposed [5–14]. Among these, the gradient vector flow (GVF) [10] has been most successful, as it provides a large capture range and the ability to capture concavities by diffusing the gradient vectors of an edge map generated from the image. Although the GVF model has proved effective and has widely been used in image segmentation, it has some disadvantages, such as a high computational cost, substantial noise sensitivity, and an inability to capture and preserve weak edges. Various improved models based on the GVF have been developed. For example, generalized gradient vector flow [11], harmonic gradient vector flow [12], motion gradient vector flow [13] and generalized dynamic directional gradient vector flow [14], but none of these are able to resolve all of the problems mentioned above.

We propose a novel external force for active models, called neighborhood-extending and noise-smoothing gradient vector flow (NNGVF), which incorporates a neighborhood-extended Laplacian operator mask and modifies the mask by adding a noise-smoothing mask. The proposed NNGVF snake outperforms the GVF snake in terms of computation, capture range, noise resistance, and weak edge preserving ability, while maintaining other desirable properties of the GVF snake such as initialization insensitivity and good convergence at concavities.

## 2. Background

### 2.1. Snakes: active contours

*c*(

*s*) = [

*x*(

*s*),

*y*(

*s*)], s∊ [0, 1] which deforms to minimize the energy functional [2]:

**c**

_{s}(

*s*) and

**c**

_{ss}(

*s*) are the first and second derivative of

**c**(

*s*) with respect to

*s*weighed by positive

*α*and

*β*, respectively. A typical external force for a gray-scale image I is

*E*

_{ext}= -|∇

*G*

_{ σ }*

*I*|, where

*G*

_{ σ }is the Gaussian kernel with standard deviation

*σ*and where * denotes convolution. Using standard variational methods, the Euler equation to minimize

*E*

_{snake}is expressed

where *F*_{int} = *αc*_{ss}(*s*)-*βc*_{ssss}(*s*) and *F*_{ext} = -∇*E*_{ext}. The internal force *F*_{int} forces the snake contour to be smooth while the external force *F*_{ext} attracts the snake to the desired image features.

### 2.2. GVF: gradient vector flow external force

**v**(

*x, y*) = (

*u*(

*x, y*),

*v*(

*x, y*)) obtained by minimizing the following energy functional [15]:

*f*is the edge map of an image, and variable

*μ*is a regularization parameter. The functions

*u*(

*x, y*) and

*v*(

*x, y*) are at least

*C*

^{2}when

*E*

_{GVF}is minimized. The Euler equation to minimize

*E*

_{GVF}is:

where ∇^{2} is the Laplacian operator.

## 3. NNGVF snakes

### 3.1. Extended neighborhood

### 3.2. Decomposition of the Laplacian operator

In this model, the AP filter is the 2D linear identity (do-nothing) filter, while the RM filter is a 2D low-pass filter. The difference yields high-frequency components over a large area. Since the purpose of the AP filter is to estimate the image at the center pixel, but is highly sensitive to noise [17], it is advisable to replace the AP filter with a better designed filter that can augment edge-preservation and noise robustness.

### 3.3. The proposed NNGVF external force

In (11), NS_{24} and RM_{24} are 5 × 5 masks, which make use of larger areas of image information. Since the convolution is used in (11), the computational cost of NNGVF is greatly reduced relative to GVF.

## 4. Experimental results

Next, we demonstrate some desirable properties of the NNGVF snake and compare the performances of the NNGVF and GVF snakes. Since NNGVF is an improvement over GVF, we focus primarily on some common concerns encountered in snake-based image segmentation, which include (1) capture range enlargement and U-shape convergence, (2) weak edge preservation, (3) noise robustness, and (4) real images. The parameters for all snakes in our experiments are *α* = 0.1, *β* = 0 and time step *τ* = 1. The weight *μ* for the GVF and NNGVF snakes is set to 0.15 in all experiments unless otherwise stated.

### 4.1. Capture range enlargement and U-shape convergence

### 4.2. Weak edge preserving

### 4.3. Noise robustness

### 4.4. Real images

## 5. Conclusion

We proposed a novel external force called NNGVF for active contours. The NNGVF snake deploys the GVF as a convolution operation using a neighborhood-extending Laplacian mask, modifying the mask to improve noise-smoothing, yields a good performance in terms of capture range, weak edge preservation, and noise robustness while maintaining the other desirable properties of GVF, such as initialization insensitivity and good convergence at concavities. The experimental results showed that the NNGVF snake outperforms the GVF snake in terms of computation as well.

## Declarations

### Acknowledgements

This work is supported by the NSFC (60805004).

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.