The fuzzy C means clustering algorithm with spatial constraint (FCMS) is effective for image segmentation. However, it lacks essential smoothing constraints to the cluster boundaries and enough robustness to the noise. Samson et al. proposed a variational level set model for image clustering segmentation, which can get the smooth cluster boundaries and closed cluster regions due to the use of level set scheme. However it is very sensitive to the noise since it is actually a hard C means clustering model. In this paper, based on Samson’s work, we propose a new variational level set model combined with FCMS for image clustering segmentation. Compared with FCMS clustering, the proposed model can get smooth cluster boundaries and closed cluster regions due to the use of level set scheme. In addition, a block-based energy is incorporated into the energy functional, which enables the proposed model to be more robust to the noise than FCMS clustering and Samson’s model. Some experiments on the synthetic and real images are performed to assess the performance of the proposed model. Compared with some classical image segmentation models, the proposed model has a better performance for the images contaminated by different noise levels.
Image segmentation is separating the image domain into dissimilar homogeneous regions, which is the precondition and foundation of further image analysis and understanding. The quality of segmentation affects the result of the following analysis and processing directly. So, it is an important technique in image processing and has drawn much research attention at the theory and application. In recent years, variational level set method and clustering technology have been widely exploited in image segmentation because of their good experimental performance and sound theoretical foundation.
The variational level set model used in image segmentation is formulated as follows [
Clustering is to partition a given input dataset or image pixels into
The above mentioned clustering algorithms are all based on the discrete data. They utilize the intensity, statistics properties, and spatial features of image pixels to perform pixels clustering. But they cannot obtain the smooth cluster boundaries and closed cluster regions due to the lack of the essential smoothing constraint to the cluster boundaries, while the variational level set method just can deal with the above problems. So, the image segmentation quality will be improved if the clustering algorithms are appropriately combined with the variational level set method. However, most of the current clustering algorithms are based on the discrete dataset, while variation method is based on continuous function. So, these two techniques cannot be combined together easily. Samson et al. [
The remainder of this paper is organized as follows: in Section
In [
In [
Although FCMS clustering and its variants (e.g., FCMS1 and FCMS2) have the benefits that it is simple and easy to manipulate, it cannot obtain the smooth cluster boundaries and the closed cluster regions for the lack of the essential smoothing constraints for the cluster boundaries. In addition, although the spatial constraint is incorporated into the objective function, FCMS cannot achieve good clustering result when the dataset is contaminated by strong noise.
The mentioned above clustering algorithms are all based on the discrete data, so they cannot be solved by the variational method directly. It was the first time for Samson et al. to employ the variational method for data clustering by using level set method. In [
As the first attempt of data clustering manipulated by the use of variational method, Samson’s model has an advantage over the traditional clustering algorithms in obtaining the smooth cluster boundaries and the closed cluster regions. However, there are still some drawbacks as follows. Samson’s model is actually a hard C means clustering which lacks the ability to retain abundant information from the original image and is also very sensitive to noise. The external clustering energy (the first term of It is a supervised image classification model. The cluster number The evolving level set function
In this paper, following Samson’s work, we propose a variational level set model combined with FCMS clustering (VFCMS) for image clustering segmentation. Four schemes are introduced to resolve the above mentioned drawbacks of Samson’s model ( A block-based clustering energy and a spatial constraint are introduced into the energy functional. In addition, the fuzziness of belongingness of each pixel to the cluster centers is introduced. These improvements enable the new model to be more robust to noises than Samson’s model and FCMS clustering. A variational formulation is proposed for updating membership functions and cluster centers, which makes the new model more robust to the initial cluster centers and achieves a semisupervised clustering. A regularization term based on a new edge stopping function is proposed, which enables the active contours to move quickly through the noise regions and reach the right boundaries of image. A regularization term is introduced to eliminate the need of the costly reinitialization procedure.
The external clustering energy proposed by Samson et al. in [
Based on the points discussed above, we introduce the following fuzzy and block-based clustering energy:
In what follows, we analyze this energy in the theory. Denote
In order to further increase the robustness to noises, we introduce the following spatial constraint into the energy functional. Note that here we adopt a modified form proposed by Chen and Zhang in [
Combining (
Fixing level set functions
Fixing level set functions
In this section, we introduce two internal energies
In the traditional variational level set method for image processing, in order to maintain the stability of the level set function during the evolution, the evolving level set function needs periodical reinitialization to keep it close to a signed distance function [
The constraint
Samson et al. [
In this paper, we use a new stopping function in regularization energy (
Combined with external energy
Formally minimizing the energy (
Using updating formulas (
Given the number of the classes
Fixed level set functions
Fixed cluster centers
Fixed membership functions
If
Output the result of image clustering segmentation
In this section, we show the experimental results of image segmentation on several synthetic and real images. There are a total of twelve models used in this section, that is, (1) clustering models (FCM, FCMS1, FCMS2, and GFCMS), (2) variational level set models (CV, IVC, LIF, MCV, and MLCV), and (3) the integration of variational level set and clustering (Samson’s model, VFCMS1, and VFCMS2).
For the choice of the initial level set function, under the combined effects of the clustering energy and internal energy, the choice of the initial level set function is very flexible. In our experiments, the initial level set functions are all chosen as
The parameter
Figure
We take a set of values for
Figure
Figure
Figure
Table
SA % of twelve models on noisy synthetic image.
Figure |
Figure |
Figure |
Figure |
|
---|---|---|---|---|
FCM | 90.34 | 86.45 | 87.32 | 85.47 |
|
99.01 | 97.87 | 98.84 | 98.74 |
|
99.32 | 99.24 | 99.14 | 99.07 |
GFCMS | 99.48 | 99.43 | 99.23 | 99.15 |
CV | 89.75 | 87.27 | 68.14 | 68.11 |
LIF | 72.34 | 71.45 | 62.32 | 61.46 |
RBACM | 96.57 | 96.53 | 70.34 | 70.28 |
MCV | 99.34 | 99.32 | ||
MLCV | 99.47 | 99.44 | ||
Samson’s model | 85.89 | 83.42 | 96.42 | 95.47 |
VFCMS1 | 99.53 | 99.47 | 99.45 | 99.43 |
VFCMS2 | 99.82 | 99.78 | 99.64 | 99.62 |
Comparison of segmentation results on a synthetic image with mixed 1% Salt and Pepper, Gaussian, and Speckle noise (cluster number
Noisy data
FCM
FCMS1
FCMS2
GFCMS
CV (ZLSs)
CV
LIF (ZLSs)
LIF
RBACM (ZLSs)
RBACM
Samson’s model (ZLSs)
Samson’s model
VFCMS1 (ZLSs)
VFCMS1
VFCMS2 (ZLSs)
VFCMS2
The surface plot of
The surface plot of
Comparison of classification errors on two-phase synthetic image with different mixed noise levels under different values of alpha.
Comparison of classification errors on synthetic image with 1% mixed noise under different values of alpha
Comparison of classification errors on synthetic image with 2% mixed noise under different values of alpha
Comparison of segmentation results on a synthetic image with mixed 2% Salt and Pepper, Gaussian, and Speckle noise (cluster number
Noisy image
FCM
FCMS1
FCMS2
GFCMS
CV (ZLSs)
CV
LIF (ZLSs)
LIF
RBACM (ZLSs)
RBACM
Samson’s model (ZLSs)
Samson’s model
VFCMS1 (ZLSs)
VFCMS1
VFCMS2 (ZLSs)
VFCMS2
Comparison of segmentation results on a synthetic image with mixed 0.5% Salt and Pepper, Gaussian, and Speckle noise (cluster number
Noisy data
FCM
FCMS1
FCMS2
GFCMS
CV (ZLSs)
CV
LIF (ZLSs)
LIF
RBACM (ZLSs)
RBACM
Samson’s model (ZLSs)
Samson’s model
MCV (ZLSs)
MCV
MLCV (ZLSs)
MLCV
VFCMS1 (ZLSs)
VFCMS1
VFCMS2 (ZLSs)
VFCMS2
The surface plot of
The surface plot of
The surface plot of
Comparison of segmentation results on a synthetic image with mixed 1.5% Salt and Pepper, Gaussian, and Speckle noise (cluster number
Noisy data
FCM
FCMS1
FCMS2
GFCMS
CV (ZLSs)
CV
LIF (ZLSs)
LIF
RBACM (ZLSs)
RBACM
Samson’s model (ZLSs)
Samson’s model
MCV (ZLSs)
MCV
MLCV (ZLSs)
MLCV
VFCMS1 (ZLSs)
VFCMS1
VFCMS2 (ZLSs)
VFCMS2
Figure
Figure
Comparison of segmentation results on a plane image with mixed 2% Salt and Pepper, Gaussian, and Speckle noise (cluster number
Noisy data
FCM
FCMS1
FCMS2
GFCMS
CV (ZLSs)
CV
LIF (ZLSs)
LIF
RBACM (ZLSs)
RBACM
Samson’s model (ZLSs)
Samson’s model
VFCMS1 (ZLSs)
VFCMS1
VFCMS2 (ZLSs)
VFCMS2
The surface plot of
The surface plot of
Comparison of segmentation results on a plane image with mixed 2% Salt and Pepper, Gaussian, and Speckle noise (cluster number
Noisy image
FCM
FCMS1
FCMS2
GFCMS
MCV (ZLSs)
MCV
MLCV (ZLSs)
MLCV
Samson’s model (ZLSs)
Samson’s model
VFCMS1 (ZLSs)
VFCMS1
VFCMS2 (ZLSs)
VFCMS2
The surface plot of
The surface plot of
The surface plot of
Finally, to show the practicability and validity of the proposed model, different kinds of real images corrupted by mixed Salt and Pepper, Gaussian, and Speckle noise are tested. Compared with the synthetic images, these real images contain much more complex boundaries, weak boundaries, inhomogeneous regions, and low-contrast regions. Figures
Segmentation results on a sun image with mixed 2% Salt and Pepper, Gaussian, and Speckle noise (cluster number
Original image and its noisy version
VFCMS1
VFCMS2
Segmentation results on a palm image with mixed 2% Salt and Pepper, Gaussian, and Speckle noise (cluster number
Original image and its noisy version
VFCMS1
VFCMS2
Segmentation results on a light image with mixed 2% Salt and Pepper, Gaussian, and Speckle noise (cluster number
Original image and its noisy version
VFCMS1
VFCMS2
Segmentation results on satellite image with mixed 2% Salt and Pepper, Gaussian, and Speckle noise (cluster number
Original image and its noisy version
VFCMS1
VFCMS2
Segmentation results on a butterfly image with mixed 2% Salt and Pepper, Gaussian, and Speckle noise (cluster number
Original image and its noisy version
VFCMS1
VFCMS2
Segmentation results on a panda image with mixed 2% Salt and Pepper, Gaussian, and Speckle noise (cluster number
Original image and its noisy version
VFCMS1
VFCMS2
Segmentation results on a CT image with mixed 2% Salt and Pepper, Gaussian, and Speckle noise (cluster number
Original image and its noisy version
VFCMS1
VFCMS2
Segmentation results on a CT image with mixed 2% Salt and Pepper, Gaussian, and Speckle noise (cluster number
Original image and its noisy version
VFCMS1
VFCMS2
Segmentation results on a CT image with mixed 2% Salt and Pepper, Gaussian, and Speckle noise (cluster number
Original image and its noisy version
VFCMS1
VFCMS2
In this paper, based on the Samson’s work and FCMS clustering, we proposed a new variational level set model combined with FCMS for image clustering segmentation. In addition, a block-based energy was incorporated into the energy functional, which enables the proposed models robust to the noise. Some synthetic and real noisy images with different noise levels were employed to compare the performance of 12 models. Experimental results show that the proposed model has a superior performance among these methods. And different kinds of real noisy image were also used to show the practicability and validity of the proposed model.
The experimental results reported in this paper show that the proposed VFCMS model is very effective for noise image clustering segmentation. This model can also be improved by incorporating other FCMS-based clustering algorithms, for example, Kernel-induced FCMS proposed by Chen and Zhang [
The author declares that there is no conflict of interests regarding the publication of this paper.