# Texture Segmentation Using Laplace Distribution-Based Wavelet-Domain Hidden Markov Tree Models

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## Abstract

**:**

## 1. Introduction

## 2. Previous Works

#### 2.1. Laplacian Distribution

#### 2.2. HMT Models

#### 2.3. EM Algorithm

- E step: Computes the conditional expectation of the complete log-likelihood, given the observed data $\mathcal{X}$ and the current estimate ${\widehat{\mathrm{\Theta}}}^{t}$ as follows:$$\mathrm{Q}(\mathrm{\Theta}|{\widehat{\mathrm{\Theta}}}^{t})=\mathrm{E}[ln(p(\mathcal{X},\mathcal{Y}|\mathrm{\Theta}))|\mathcal{X},{\widehat{\mathrm{\Theta}}}^{t}]$$
- M step: Update the parameters by maximizing the function:$${\widehat{\mathrm{\Theta}}}^{t+1}=\underset{\mathrm{\Theta}}{\text{argmax}}\mathrm{Q}(\mathrm{\Theta}|{\widehat{\mathrm{\Theta}}}^{t})$$

#### 2.4. HMT Based Texture Segmentation

#### 2.4.1. Raw Maximum Likelihood Segmentation

#### 2.4.2. Context-Based Multiscale Fusion

#### 2.4.3. Pixel-Level Segmentation

## 3. LMM-HMT Based Description of Texture

#### 3.1. LMM Based HMT Model

- The probability of the state $m$ at the root node in the coarsest scale ${p}_{{s}_{1}}\left(m\right);$
- The state transition probability is:$${\mathsf{\epsilon}}_{i,\rho \left(i\right)}^{m,n}={p}_{{S}_{i}|{S}_{{\rho}_{\left(i\right)}}}\left({S}_{i}=m|{S}_{{\rho}_{\left(i\right)}}=n\right).$$
- The scale parameter ${b}_{i,m}$, given ${S}_{i}=m$.

#### 3.2. Parameter Estimation

#### 3.3. Pixel-Level Texture Description

## 4. Texture Segmentation

- (1)
- Model training. For each texture class, we train the wavelet domain LMM-HMT model with the homogeneous texture samples by using EM algorithm as the Section 3.2, and obtain the model parameters ${\mathcal{M}}_{c}$, in which c denotes the cth texture class. Meanwhile, the pixel-level multivariate Laplace mixture model parameters are gotten with Equations (19)–(24).
- (2)
- Raw maximum likelihood segmentation. For a heterogeneous texture to be segmented, the likelihood of each subtree at different scale can be computed by using the HMT likelihood computation method and the Equation (7). The raw segmentation ${c}^{J}$ at the coarest scale is accomplished by using Equation (5) with the trained LMM-HMT model parameters ${\mathcal{M}}_{c}$.
- (3)
- Context-based multiscale fusion. At the scale j, the context vectors ${v}_{i}^{j}$s are constructed from the segmentation label ${c}^{j+1}$ at scale j + 1. The segmentation result ${c}^{j}$ is obtained by using EM algorithm and maximizing the contextual posterior distribution as the work [13].
- (4)
- Pixel-level segmentation. Compute the likelihood of each pixel with the trained pixel-level multivariate Laplace mixture models. Perform the context-based fusion scheme from the scale j = 1 to the pixel-level as the step (3). The output is the final segmentation result.

#### 4.1. Image Texture Segmentation

#### 4.2. Dynamic Texture Segmentation

## 5. Experimental Results

#### 5.1. Image Texture Segmentation

#### 5.2. Dynamic Texture Segmentation

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) “Floor” texture and (

**b**) its wavelet coefficient histogram fitted with two-state, zero mean Gaussian mixture model (GMM) and Laplace mixture model (LMM).

**Figure 4.**Neighbor system. (

**a**) first-order 4-neighbor; (

**b**) second-order 8-neighbor; (

**c**) third-order 12-neighbor.

**Figure 5.**The flow diagram of texture segmentation based on LMM-HMT model. As GMM-HMT base segmentation method, texture segmentation based on LMM-HMT model consists of the raw segmentation, the multiscale fusion and the pixel-level segmentation.

**Figure 6.**Spatial-temporal wavelet transform. There are one approximate sub-band and seven detail sub-bands after one-level wavelet decomposition of the dynamic texture.

**Figure 8.**The results of texture image segmentation using GMM-HMT and LMM-HMT. (

**a**) Textures to be segmented; (

**b**) Ground truths; (

**c**) The results using GMM-HMT model and pixel-level GM model; (

**d**) The results using LMM-HMT model and pixel-level GM model; (

**e**) Factorization [8] (the parameters are the same as that of the Texture Mosaics in Section IV of [8]. The better results can be obtained by adjusting the parameters for different textures); (

**f**) The results using LMM-HMT and pixel-level LM model.

**Figure 10.**Results of multiscale fusion. (

**a**) Three adjacent frames selected from a synthesized video; (

**b**) Results with the context labeling tree based on the ordinary spatial second-order neighbors; (

**c**) Result with the spatial-temporal parent neighbors based context labeling tree.

**Figure 11.**Segmentation experiments. (

**a**) Textures to be segmented; (

**b**) The results using GMM-HMT model and pixel-level GM model; (

**c**) The result using LMM-HMT model and pixel-level GM model; (

**d**) The result using LMM-HMT and pixel-level LM model.

**Figure 12.**Segmentation experiments. (

**a**) Textures to be segmented; (

**b**) The results using GMM-HMT model and pixel-level GM model; (

**c**) The result using LMM-HMT model and pixel-level GM model; (

**d**) The result using LMM-HMT and pixel-level LM model.

**Figure 13.**Segmentation experiments. (

**a**) Textures to be segmented; (

**b**) The results using GMM-HMT model and pixel-level GM model; (

**c**) The result using LMM-HMT model and pixel-level GM model; (

**d**) The result using LMM-HMT and pixel-level LM model.

Texture | GMM-HMT | LMM-HMT | Factorization [8] | LMM-HMT with LM-Pixel |
---|---|---|---|---|

IT1 | 95.17 | 95.34 | 97.52 | 96.21 |

IT2 | 96.56 | 97.29 | 96.80 | 97.42 |

IT3 | 94.32 | 94.65 | 92.43 | 95.04 |

IT4 | 96.01 | 96.23 | 95.23 | 96.37 |

Texture | GMM-HMT | LMM-HMT with LM-Pixel |
---|---|---|

IT5 | 93.55 | 94.74 |

IT6 | 93.64 | 93.82 |

IT7 | 93.37 | 93.56 |

IT8 | 95.44 | 94.79 |

IT9 | 90.28 | 89.14 |

IT10 | 65.29 | 66.73 |

Texture | GMM-HMT | LMM-HMT | |||||||
---|---|---|---|---|---|---|---|---|---|

GM-Pixel | LM-Pixel | ||||||||

Max | Min | Avg | Max | Min | Avg | Max | Min | Avg | |

DT1 | 97.41 | 92.97 | 94.92 | 97.29 | 91.52 | 96.11 | 97.45 | 93.56 | 96.43 |

DT2 | 97.33 | 90.31 | 95.23 | 96.57 | 95.43 | 95.96 | 96.85 | 96.24 | 96.31 |

DT3 | 98.14 | 94.98 | 97.19 | 98.92 | 94.85 | 97.06 | 98.78 | 95.80 | 97.63 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Qiao, Y.; Zhao, G.
Texture Segmentation Using Laplace Distribution-Based Wavelet-Domain Hidden Markov Tree Models. *Entropy* **2016**, *18*, 384.
https://doi.org/10.3390/e18110384

**AMA Style**

Qiao Y, Zhao G.
Texture Segmentation Using Laplace Distribution-Based Wavelet-Domain Hidden Markov Tree Models. *Entropy*. 2016; 18(11):384.
https://doi.org/10.3390/e18110384

**Chicago/Turabian Style**

Qiao, Yulong, and Ganchao Zhao.
2016. "Texture Segmentation Using Laplace Distribution-Based Wavelet-Domain Hidden Markov Tree Models" *Entropy* 18, no. 11: 384.
https://doi.org/10.3390/e18110384