# Skewed alpha-stable distribution for natural texture modeling and segmentation in contourlet domain

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## Abstract

Texture modeling is a very useful tool in image analysis. This model can be used in texture segmentation, denoising or texture synthesis. In this work, alpha-stable distribution has been proposed to model and segment textured images in contourlet domain. Contourlet transform’s ability to extract texture features in different scales and directions combined with alpha-stable distribution’s modeling capabilities prove to be an effective method for texture feature extraction. Kolmogorov–Smirnov distance has been used to evaluate how well the proposed distribution fits to the image in the contourlet domain. The performance of the proposed features on image segmentation has been also compared with that of features extracted using different texture analysis methods in the presence of noise. Experimental results have demonstrated the superior performance of the proposed features and their robust performance in the presence of noise.

## Keywords

Alpha-stable distribution Contourlet transform Image segmentation Texture modeling## Abbreviations

- PolSAR
Polarimetric synthetic aperture radar

- SAR
Synthetic aperture radar

## 1 Introduction

Over the recent years, texture analysis and modeling have been a hot topic in the field of image processing. Texture synthesis, texture retrieval, denoising of textured images, edge detection in textured images, and texture segmentation are some of the most important challenges in this domain. Having an accurate statistical model for textured images can be very useful to deal with these challenges, especially image segmentation.

Texture segmentation is dividing the image into regions with similar texture. Segmenting a textured image is a challenging task because using traditional segmentation methods in textured images often results in an over segmentation [1]. This is due to the fact that a textured region usually consists of many small homogeneous regions (texture elements). Traditional segmentation methods rely on the pixel value for segmentation while an efficient texture analysis method should consider its neighboring pixel values too.

In this paper, different distributions have been used for modeling and segmentation of textured natural images in contourlet domain. Skewed alpha-stable distribution which is a generalized form of Rayleigh distribution presents the best results. The proposed method’s performance is also compared to other algorithms. Simulation results show superiority of the proposed algorithm related to the other methods.

## 2 Literature review

Feature extraction is the most important step in texture segmentation. Having some texture discriminative descriptors can make the segmentation process straightforward. Therefore, various works in recent years have been carried out in texture feature extraction [1, 2, 3, 4, 5, 6, 7, 8, 9]. Texture analysis methods can be divided into four categories: statistical methods (e.g., co-occurrence matrices [10, 11, 12], autocorrelation features [13, 14]), geometrical methods (e.g., Voronoi tessellations [15, 16]), model base methods (e.g., random fields [5, 17, 18, 19], fractals [12, 20]) and signal processing methods (e.g., spatial domain filtering [21, 22], Fourier domain filtering [2], and wavelet-based methods [16, 23]).

Statistical modeling is one of the most popular texture analysis methods. In this approach, the pixel values are assumed to be random variables drawn from a specific distribution, then the distribution’s parameters are estimated and can be used as features in different tasks such as image segmentation [24, 25, 26] or denoising [27]. Various distributions have been used for texture modeling and segmentation. Gaussian distribution is one of the most commonly used models for texture analysis [24, 28, 29]. Rayleigh, Weibull [30, 31], and Wishart [25] distributions have been also used for texture feature extraction. However, due to its shape and properties, each of these distributions is suitable for a specific type of data. For example, Rayleigh and Gaussian distributions are unable to fit heavy-tailed or nonsymmetrical histograms. To solve this problem, we propose using alpha-stable distribution for texture modeling. Alpha-stable or Levy distributions firstly introduced by Levy [25] are a family of distributions which are able to model a wide range of data; this is because in its general form, alpha-stable distribution can be asymmetric having various shapes.

Many researchers have used texture modeling and feature extraction in multiresolution domains like wavelet transform [26, 27]. Most of these methods use transform coefficients’ energy [28], fractal dimension [20], or statistical modeling parameters [29, 30, 31] as texture features. Different multiresolution transforms have been introduced in recent years. Wavelet transform is the simplest and most popular one used for texture feature extraction. Wavelet’s advantages are simplicity and low redundancy but decomposing in these two directions is its serious disadvantage that limits the number of informative features extracted from the image. Contourlet transform has been introduced to overcome this problem. Contourlet [32] is a two-dimensional transform that uses a pyramidal directional filter bank to extract image details at different scale and directions. Since there is no limitation on the number of directions in contourlet, various texture features can be extracted.

Statistical modeling in transform domain is one of the most effective texture feature extraction and classification methods. For example, Euclidean or Kullback-Liebler distance between the distributions can be used as a similarity measurement for texture classification [29, 33], or the distribution’s parameters can be used as texture features [29]. Different distributions have been used to model texture in the multiresolution domain. Generalized-Gaussian distribution has been used to model wavelet [29, 34, 35] and contourlet [31] coefficients. Kwitt and Uhl [36] have used gamma and Weibull distributions to model texture in complex DWT domain. Generalized gamma distribution has been also used for texture modeling in transform domain [37, 38].

In our work, alpha-stable distribution is proposed to model and segment textured images in contourlet domain. We have used the alpha-stable parameters of these detail images for texture segmentation. Contourlet transform’s ability to extract texture features in different scales and directions combined with alpha-stable distribution’s modeling capabilities prove to be an effective method for texture analysis and feature extraction. Simulation results show that a significant improvement has been achieved by this method compared with other texture segmentation methods.

## 3 Contourlet transform

Recently, wavelet transform has been widely used in signal processing. This is because wavelet transform provides a good nonlinear approximation for piecewise smooth functions in one dimension [39]. However, since two-dimensional wavelet is the tensor product of one-dimensional wavelet, it can only model zero-dimension discontinuities (points). On the other hand, in an image, edges are one-dimension discontinuities and thus cannot be modeled using wavelets. Contourlet transform is a “true two-dimensional representation of images which can capture the intrinsic geometrical structure inherent in visual information” [32].

In our work, we use Contourlet transform’s ability to extract image details in various scales and directions to build a strong texture descriptor.

## 4 Skewed alpha-stable distribution

*ℝ*

^{+}(dispersion parameter), and μ ∈

*ℝ*(location parameter). Gaussian distribution is a special case of alpha-stable distribution with α = 2. Figures 2, 3, and 4 show the alpha-stable probability density function for different values of α, β, and γ.

Due to its flexibility in representing heavy-tailed and non-symmetric data, alpha stable distribution can be used to model a wide range of texture data. Combining this with contourlet’s ability to extract texture features in different scales and directions gives us a full texture descriptor.

## 5 Materials and method

In simulation process, different texture mosaics from Brodatz album [40], Vistex database, and the KTH-TIPS2 database [41] which contain multiple natural textures, in addition to two polarimetric SAR images, have been used. These polarimetric images consist of a RADARSAT-2 image of Vancouver area and an AIRSAR image of San Francisco Bay.

### 5.1 Statistical modeling

In our work, Gaussian, Rayleigh, Log-Normal, Inverse-Gaussian, Weibull, and alpha-stable distributions have been used for modeling. These distributions have been chosen because they have been used in many state of the art texture analysis publications [24, 42, 43, 44, 45].

The estimation has been done for all the Brodatz and Vistex images and for each of the Vancouver image polarizations.

### 5.2 Image segmentation

Used feature sets in our work

Method | Number of features | |
---|---|---|

Feature set 1 | 2 | |

Feature set 2 | Rayleigh distribution [42] | 1 |

Feature set 3 | Alpha-stable distribution | 4 |

Feature set 4 | Gaussian distribution of contourlet subbands | 66 |

Feature set 5 | Rayleigh distribution of contourlet subbands | 33 |

Feature set 6 | Alpha-stable distribution of contourlet subbands | 132 |

Feature set 7 | Fractal dimension [20] | 1 |

Feature set 8 | Fractal dimension of contourlet subbands [20] | 33 |

Feature set 9 | Alpha-stable distribution of wavelet subbands | 16 |

For comparison, fractal analysis which is another texture analysis method has also been used for feature extraction [20].

## 6 Simulation results and discussion

We can see that for the Brodatz textures and both SAR polarizations alpha-stable distribution has the best performance. This is due to alpha-stable distribution’s ability to model both skewed and heavy tailed data.

Average segmentation accuracy of different methods (without noise)

Gaussian | Rayleigh | Alpha-Stable | Contourlet + Gaussian | Contourlet + Rayleigh | Contourlet + alpha-stable | Fractal | Contourlet + Fractal | Wavelet + alpha-stable | |
---|---|---|---|---|---|---|---|---|---|

Brodatz album | 36.19 | 35.26 | 35.15 | 98.92 | 99.39 | 99.50 | 40.92 | 65.52 | 98.71 |

Vistex Database | 40.65 | 35.56 | 35.40 | 99.01 | 99.43 | 99.53 | 44.69 | 64.71 | 98.85 |

KTH-TIPS2 | 34.00 | 32.48 | 32.60 | 99.04 | 99.40 | 99.48 | 37.47 | 56.53 | 98.91 |

Vancouver | 86.16 | 81.24 | 89.79 | 95.40 | 74.40 | 96.19 | 84.16 | 66.25 | 61.08 |

San Francisco Bay | 94.07 | 89.87 | 91.03 | 98.29 | 97.17 | 98.62 | 91.65 | 88.37 | 85.56 |

Standard deviation of segmentation accuracy for different methods (without noise)

Gaussian | Rayleigh | Alpha-stable | Contourlet + Gaussian | Contourlet + Rayleigh | Contourlet + alpha-stable | Fractal | Contourlet + Fractal | Wavelet + alpha-stable | |
---|---|---|---|---|---|---|---|---|---|

Brodatz album | 4.30E-02 | 2.11E-02 | 2.08E-02 | 3.87E-03 | 1.06E-03 | 1.07E-03 | 5.11E-02 | 4.32E-02 | 4.36E-03 |

Vistex Database | 8.47E-02 | 6.06E-02 | 4.80E-02 | 3.26E-03 | 1.57E-03 | 1.27E-03 | 7.25E-02 | 6.62E-02 | 3.71E-03 |

KTH-TIPS2 | 2.66E-02 | 2.38E-02 | 2.36E-02 | 2.80E-03 | 1.32E-03 | 8.13E-04 | 2.57E-02 | 5.81E-02 | 3.28E-03 |

## 7 Conclusion

In this paper, skewed alpha stable distribution has been proposed for modeling the contourlet subbands of textured images. Simulation results show that alpha stable distribution can model the textured data in contourlet domain better than other distributions. This is due to alpha-stable distribution’s ability to model a wide range of data combined of different histogram shapes. Since contourlet transform provides texture details in multiple directions, statistical modeling of these details can be used as an effective method for texture feature extraction and classification.

We have also proposed a new texture segmentation method based on alpha-stable distribution parameters in contourlet domain. The performance of the proposed features on image segmentation has been compared with that of features extracted using Gaussian and Rayleigh distributions and fractal features which have been used in earlier texture segmentation works. Results have demonstrated the superior performance of new features in segmentation of textured images.

Even though Gaussian and Rayleigh parameters of contourlet domain perform almost as well as the alpha-stable features for the noise-free case, as the noise density increases their segmentation accuracy decreases quickly. The proposed segmentation method has the best accuracy and is the most robust in the presence of noise.

## Notes

### Acknowledgements

The authors would like to thank Dr. Dehghani and Dr. Masnadi for their help.

### Funding

There has been no funding for this work.

### Availability of data and materials

Please contact author for data requests.

### Authors’ contributions

The authors have had equal contribution to the work. Both authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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