# A Novel Geometric Dictionary Construction Approach for Sparse Representation Based Image Fusion

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## Abstract

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## 1. Introduction

- A geometric-information based classification method is proposed and applied to a sub-dictionary learning of image patches. The proposed classification method can accurately split source image patches into different groups for sub-dictionary learning based on the corresponding geometry features. Sub-dictionary bases extracted from each image-patch group contain the key geometry features of source images. These extracted sub-dictionary bases are trained to form informative and compact sub-dictionaries for image fusion.
- A dictionary combination method is developed to construct an informative and compact sub-dictionary. Each image patch of a fused image is composed of corresponding source image patches using a constructed-sub-dictionary (CSD). According to the classification of geometry features, each source image patch is trained and categorized into a group of sub-dictionaries. Corresponding image patches, that appear at the same place of the two source images, at most have two groups of sub-dictionary. Redundant geometric information of source image patches is eliminated.

## 2. Geometry-Based Image Fusion Framework

#### 2.1. Dictionary Learning

Algorithm 1 SCC Algorithm. |

Input:Image patches of Wth cluster ${P}^{w}=({p}_{1}^{w},{p}_{2}^{w},...,{p}_{n}^{w})\in {\mathbb{R}}^{64\times n}$ Output:Sub-dictionary $S={S}_{n}^{\partial}\in {\mathbb{S}}^{64\times n}$, and ${z}_{i}={z}_{i}^{\partial}$ for $i=1,2,...,n$ Initialize ${S}_{1}^{1}$, $H=0$, and ${z}_{i}^{0}=0$ for $i=1,2,...,n$ for $k=1$ to ∂ dofor $i=1$ to n doGet image patch ${p}_{i}^{n}$ Update ${z}_{i}^{k}$ via one or a few steps of coordinate descent: ${z}_{i}^{k}\leftarrow CD({D}_{i}^{k},{z}_{i}^{k-1},{x}_{i})$ Update the Hessian matrix and the learning rate: $H\leftarrow H+{z}_{i}^{k}{\left({z}_{i}^{k}\right)}^{T},{\eta}_{i,j}^{k}=1/{h}_{jj}$ Update the support of the dictionary via SGD (Stochastic Gradient Descent): ${d}_{i+1}^{k}\leftarrow {d}_{i,j}^{k}-{\eta}_{i,j}^{k}{z}_{i,j}({S}_{i}^{k}{z}_{i}^{k}-{x}_{i})$ if $i=n$ thenSet ${S}_{1}^{k+1}={S}_{n+1}^{k}$ end ifend forend for |

Algorithm 2 SOMP Algorithm. |

Input:Dictionary D, image patches ${\left\{{x}_{k}\right\}}_{k=1}^{K}$, ${x}_{k}\in {R}^{n}$, threshold $\epsilon $, an empty matrix $\Phi $ Output:Sparse coefficients ${\left\{{z}_{k}\right\}}_{k=1}^{K},{z}_{j}\in {R}^{\mathrm{w}}$ Initialize the residuals ${r}_{k}^{\left(0\right)}={x}_{k}$, for $k=1,2,...,K$, set iteration counter $l=1$. Select the index ${\widehat{t}}_{l}$ which indicates the next best coefficient atom to simultaneously provide good reconstruction for all signals by solving: ${\widehat{t}}_{l}=arg\underset{t=1,2,...,T}{max}{\displaystyle \sum _{k=1}^{K}}\left|\left(\right),{r}_{k}^{l-1},{d}_{t}\right|$. Update sets $\Phi =\left(\right)open="["\; close="]">{\Phi}_{l-1},{d}_{\widehat{{t}_{l}}}$. Compute new coefficients (sparse representations), approximations, and residuals as: ${\alpha}_{k}^{\left(l\right)}=\underset{\alpha}{argmin}{\left(\right)}_{{x}_{k}-{\Phi}_{l}\alpha}2{{\Phi}_{l}}^{T}{x}_{k}$, for $k=1,2,...,K$, ${\widehat{x}}_{k}^{\left(l\right)}={\Phi}_{t}{\alpha}_{k}^{\left(l\right)}$, for $k=1,2,...,K$, ${r}_{k}^{\left(l\right)}={x}_{k}-{{\widehat{x}}_{k}}^{\left(l\right)}$, for $k=1,2,...,K$. Increase the iteration counter $l=l+1$, if $\sum _{k=1}^{K}}{\left(\right)}_{{r}_{k}^{\left(l\right)}}^{}22$, go back to step 2. |

#### 2.2. Image Sparse Coding and Fusion

- Step 1: Use the sliding window technique to divide each source image ${I}_{j}$, from left-top to right-bottom, into patches of size $8\times 8$, i.e., the size of the atom in the dictionary. These image patches are vectorized to image pixel vectors in the linewise direction. The obtained image pixel vectors only have one dimension.
- Step 2: For the ith image patch ${x}_{ji}$ of one source image ${I}_{j}$, it can be sparse coded using the trained dictionary D.
- Step 3: When all of the image patches are sparse coded, the corresponding image patches of each image use Equation (5) to do fusion:$$\begin{array}{c}\hfill \begin{array}{c}\hfill {{\mathbf{z}}^{f}}_{i}=\sum _{j=1}^{k}{\mathbf{z}}_{ji}\times {O}_{ji},where\left(\right)open="\{"\; close>\begin{array}{c}{O}_{ji}=1,ifmax({\left(\right)}_{{\mathbf{z}}_{j1}}1,{\left(\right)}_{{\mathbf{z}}_{j2}}1...,{\left(\right)}_{{\mathbf{z}}_{jk}}1& )\\ ={\left(\right)}_{{\mathbf{z}}_{ji}}1\end{array}\\ {O}_{ji}=0,otherwise\end{array}\end{array}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}},$$
- Step 4: Fused coefficients are inversely transformed to fused image pixel vectors, using Equation (6), and transform these vectors back to the fused image patches. Then, it reconstructs the fused image by using fused image patches. The dictionary D in Equation (6) is the same as dictionary D in Algorithm 3:$$\begin{array}{c}\hfill \begin{array}{c}\hfill {I}^{f}=D{z}^{f}\end{array}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}.\end{array}$$

Algorithm 3 CSD Construction Algorithm. |

Input:Sub-dictionaries $S=({S}_{1},{S}_{2}...{S}_{n})$, image patches for fusion ${i}_{1}$ and ${i}_{2}$ Output:CSD D Figure out sub-dictionaries ${S}_{j}\in S$, ${S}_{k}\in S$, that correspond to image patch groups of ${i}_{1}$ and ${i}_{2}$. if$j=k$thenSet $D={S}_{j}$ elseSet $D=[{S}_{j},{S}_{k}]$ end if |

## 3. Experiments and Analyses

- Twenty pairs of $520\times 520$ size multi-focus images are obtained from the Lytro-multi-focus data-set http://mansournejati.ece.iut.ac.ir.
- Thirty pairs of medical images are from www.med.harvard.edu/aanlib/home.html. All of them are $256\times 256$ size.
- Ten pairs of visible-infrared images are obtained from from www.quxiaobo.org consisting of four $320\times 240$ and six $256\times 256$ image pairs.

#### 3.1. Objective Evaluation Methods

#### 3.1.1. Mutual Information

#### 3.1.2. ${Q}^{AB/F}$

#### 3.1.3. Visual Information Fidelity

#### 3.1.4. ${Q}_{Y}$

#### 3.1.5. ${Q}_{CB}$

#### 3.2. Image Quality

#### 3.2.1. Multi-Focus Comparison

#### 3.2.2. Medical Comparison

#### 3.2.3. Visible-Infrared Comparison

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 3.**Selected sample pairs of multi-focus, medical, and visible-infrared images; (

**a**,

**b**) are sample multi-focus image pairs; (

**c**,

**d**) are sample medical image pairs; (

**e**,

**f**) are sample visible-infrared images.

**Figure 4.**Fusion results of multi-focus image of ’Love Card and Hong-Kong’; (

**a**,

**b**) are source images, (

**c**–

**e**) are fused image of K-means generalized singular value decomposition, joint clustering patches dictionary and the proposed method, (

**f**–

**h**) are difference images between (

**a**) and fused image (

**c**–

**e**), (

**i**–

**k**) are difference images between (

**b**) and fused image (

**c**–

**e**).

**Figure 5.**Fusion results of the medical image of the “Brain”; (

**a**,

**b**) are source images, (

**c**–

**e**) are fused image of K-SVD, JCPD and proposed method, (

**f**–

**k**) are enlarged details in red and green frame of fused image (

**c**–

**e**).

**Figure 6.**Fusion results of visible-infrared images of “Downtown Street Scenes”; (

**a**,

**b**) are source images, (

**c**–

**e**) are fused images of K-SVD, JCPD and the proposed method, (

**f**–

**k**) are enlarged details in red and blue frames of fused images (

**c**–

**e**).

${\mathit{Q}}^{\mathit{AB}/\mathit{F}}$ | $\mathit{MI}$ | $\mathit{VIF}$ | ${\mathit{Q}}_{\mathit{Y}}$ | ${\mathit{Q}}_{\mathit{CB}}$ | |
---|---|---|---|---|---|

K-SVD | 0.4753 | 4.5992 | 0.7705 | 0.6897 | 0.6408 |

JCPD | 0.5331 | 4.5586 | 0.7571 | 0.7403 | 0.6317 |

Proposed | 0.5374 | 4.9561 | 0.7778 | 0.7420 | 0.6613 |

${\mathit{Q}}^{\mathit{AB}/\mathit{F}}$ | $\mathit{MI}$ | $\mathit{VIF}$ | ${\mathit{Q}}_{\mathit{Y}}$ | ${\mathit{Q}}_{\mathit{CB}}$ | |
---|---|---|---|---|---|

K-SVD | 0.2886 | 1.8554 | 0.2831 | 0.3294 | 0.6700 |

JCPD | 0.2880 | 1.8575 | 0.2829 | 0.3290 | 0.6680 |

Proposed | 0.3088 | 1.8869 | 0.2878 | 0.3591 | 0.6854 |

${\mathit{Q}}^{\mathit{AB}/\mathit{F}}$ | $\mathit{MI}$ | $\mathit{VIF}$ | ${\mathit{Q}}_{\mathit{Y}}$ | ${\mathit{Q}}_{\mathit{CB}}$ | |
---|---|---|---|---|---|

K-SVD | 0.4784 | 1.7713 | 0.3585 | 0.5670 | 0.5370 |

JCPD | 0.5648 | 1.4563 | 0.3173 | 0.6562 | 0.4653 |

Proposed | 0.6449 | 1.8398 | 0.3597 | 0.7647 | 0.5437 |

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**MDPI and ACS Style**

Wang, K.; Qi, G.; Zhu, Z.; Chai, Y.
A Novel Geometric Dictionary Construction Approach for Sparse Representation Based Image Fusion. *Entropy* **2017**, *19*, 306.
https://doi.org/10.3390/e19070306

**AMA Style**

Wang K, Qi G, Zhu Z, Chai Y.
A Novel Geometric Dictionary Construction Approach for Sparse Representation Based Image Fusion. *Entropy*. 2017; 19(7):306.
https://doi.org/10.3390/e19070306

**Chicago/Turabian Style**

Wang, Kunpeng, Guanqiu Qi, Zhiqin Zhu, and Yi Chai.
2017. "A Novel Geometric Dictionary Construction Approach for Sparse Representation Based Image Fusion" *Entropy* 19, no. 7: 306.
https://doi.org/10.3390/e19070306