# Exploiting Superpixels for Multi-Focus Image Fusion

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## Abstract

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## 1. Introduction

## 2. Background and Literature Review

## 3. Proposed Method

#### 3.1. Superpixels Computation

#### 3.2. Focus Map Estimation

#### 3.3. Fusion Rule

## 4. Experiments and Results

#### 4.1. Compared Methods

#### 4.2. Qualitative Performance Evaluation

#### 4.3. Quantitative Performance Evaluation

- Information theory based metrics analyze the quality of the fused images by checking the amount of information transferred from the source images to the fused image. These metrics include: normalized mutual information $\left(NMI\right)$ [79], visual information fidelity $\left(VIFF\right)$ [81], and the nonlinear correlation information entropy metric $\left({Q}_{NCIE}\right)$ [80].
- Image feature based metrics calculate the quality of fused image by analyzing the transferred features from the source images to the fused image. Gradient based fusion performance metrics ${Q}^{AB/F}$[82], $\left({Q}_{G}\right)$[83], image fusion metric based on multiscale scheme $\left({Q}_{M}\right)$[85], image fusion metric based on spatial frequency $\left({Q}_{SF}\right)$[84], and image fusion metric based on phase congruency $\left({Q}_{p}\right)$ [86,91] are examples of these metrics.
- Image structural similarity based metrics: The measurement of image similarity depends on the proof that the human visual system is profoundly adjusted to structural information, which is exploited in these metrics to assess the quality of a fused image. The loss of structural information is very important to estimate the image distortion. These metrics include Piella’s metric [87], Cvejie’s image fusion metric $\left({Q}_{C}\right)$[88], and Yang’s image fusion metric $\left({Q}_{Y}\right)$ [89].
- Human perception inspired fusion metrics, e.g., Chen Blum metric ${Q}_{CB}$ [90], measure the quality of fused images on the basis of the saliency map, which is generated by filtering and calculating the preservation of contrast.

#### 4.4. Computational Complexity

^{®}Core™ i7 $2.5$ GHz processor with 8 GB RAM and 64 bit operating system. The results show that the PCA algorithm is the fastest among all approaches; the proposed method takes an average $122.61$ seconds to generate a fused image. The main contributor to this time is the complex computation involved in estimating the superpixels and using the CIEDE2000 color distance model. An efficient implementation can help reduce its computational complexity.

#### 4.5. Summary

## 5. Performance Analysis Using Different Color Distance Models and the Impact of other Parameters

#### 5.1. Fusion Performance with Different Color Distance Models

- Euclidean color distance in the RGB color space: The simplest and the most well-known way to compute the distance between two colors is using the Euclidean distance in the RGB color space. The distance between two pixel colors ${C}_{1}({R}_{1},{G}_{1},{B}_{1})$ and ${C}_{2}({R}_{2},{G}_{2},{B}_{2})$ is computed as,$$d=\sqrt{{({R}_{2}-{R}_{1})}^{2}+{({G}_{2}-{G}_{1})}^{2}+{({B}_{2}-{B}_{1})}^{2}}$$
- Color approximation distance (CAD): The perception of brightness by the human eye is non-linear, and for each color, non-linearity is not the same, as proven by different experiments [92,93,94]. Therefore, there have been many attempts to weight the RGB values to better fit human perception so that the color approximation would be more appropriate. The CAD between two colors ${C}_{1}({R}_{1},{G}_{1},{B}_{1})$ and $C2(R2,G2,B2)$ is computed as:$$d=\sqrt{\left(2+\frac{\overline{r}}{256}\right)\Delta {R}^{2}+4\Delta {G}^{2}+\left(2+\frac{255-\overline{r}}{256}\right)\Delta {B}^{2}}$$
- CIEXYZ: The visualization of the RGB color space was not perfect because the color space of the human eye is greater than the experiment results of CIERGB, and an updated color difference model, CIEXYZ, was introduced by Commission Internationale de L’éclairage (CIE) [95]. It was mathematically designed to avoid the negative numbers known as tristimulus values. It consists of three parameters: non-negative cone response curve $\left(X\right)$, luminance $\left(Y\right)$, and partially-equal to blue color $\left(Z\right)$.$$\left[\begin{array}{c}X\\ Y\\ Z\end{array}\right]=\left[\begin{array}{ccc}0.431& 0.342& 0.178\\ 0.222& 0.707& 0.071\\ 0.020& 0.130& 0.939\end{array}\right]\times \left[\begin{array}{c}R\\ G\\ B\end{array}\right]$$The normalized tristimulus values x, y, and z were calculated from X, Y, and Z as: $\left[x\phantom{\rule{3.33333pt}{0ex}}y\phantom{\rule{3.33333pt}{0ex}}z\right]=\left[\frac{X}{S}\phantom{\rule{3.33333pt}{0ex}}\frac{Y}{S}\phantom{\rule{3.33333pt}{0ex}}\frac{Z}{S}\right],$ where $S=X+Y+Z$. The difference between the tristimulus values of two colors was then computed using the Euclidean distance formula.
- CIE76: In the CIE76 color distance model, the CIELAB color space was used as it is considered to be the most exact means of representing color [96]. The color difference between two colors ${C}_{1}({L}_{1},{a}_{1},{b}_{1})$ and ${C}_{2}({L}_{2},{a}_{2},{b}_{2})$ in the CIELAB color space is calculated by following formula:$$d=\sqrt{{({L}_{1}-{L}_{2})}^{2}+{({a}_{1}-{a}_{2})}^{2}+{({b}_{1}-{b}_{2})}^{2}}$$
- CMC l:c: The color difference CMC (Colour Measurement Committee of the Society of Dyes and Colourists of Great Britain) l:c is calculated in the LCh color space and has two parameters: lightness (l) and chroma (c). For the application, it allows the users based on the ratio of $l:c$ to weight the difference that is deemed appropriate.
- CIEDE2000: The CIE Delta E (CIEDE2000) color distance model was introduced by CIE and provides an improved method for the calculation of color differences. A detailed discussion is presented in Section 3.1.

#### 5.2. Impact of the Number of Superpixels on the Fusion Quality

#### 5.3. Analysis of the Relative Contribution of Spatial and Color Distances

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Results of the proposed algorithm on sample multi-focus images from the test dataset. (

**a**) Source images with foreground focused, (

**b**) source images with background focused, and (

**c**) fused images using the proposed method.

**Figure 3.**Superpixel computation using the proposed strategy. (

**a**) Superpixels of ${I}_{1}$ (foreground focused). (

**b**) Superpixels of ${I}_{2}$ (background focused).

**Figure 4.**Initial focus map generated using the proposed strategy for the images shown in Figure 3.

**Figure 5.**Initial focus map refinement: (

**a**) a sample multi-focus image with superpixels structures overlaid on it, (

**b**) the obtained focus map, and (

**c**) the refined focus map.

**Figure 6.**Fusion results using the refined focus map: (

**a**) first source image ${I}_{1}$, (

**b**) second source image ${I}_{2}$, and (

**c**) fused image ${I}_{f}$.

**Figure 8.**Visual quality assessment of the fusion results achieved by the proposed and the compared methods on the “Swimmer” multi-focus image pair from the test dataset. WSSM, wavelet based statistical sharpness measure; PCNN, pulse coupled neural network; DCTLP, discrete cosine transform Laplacian pyramid; IFGD, image fusion using luminance gradients; CSR, convolutional sparse representation.

**Figure 9.**Visual quality assessment of the fusion results achieved by the proposed and the compared methods on the “Cookie” multi-focus image pair from the test dataset.

**Figure 10.**Impact of the number of superpixels k on the fusion quality of the proposed method. In all graphs, the x-axis represents k and the y-axis the quality score. The x-axis labels are coded as ${10}^{3}$.

**Figure 12.**Impact of the weight factor m on the fusion quality of the proposed method. In all graphs, the x-axis represents m and the y-axis the quality score.

Metric | Metric Description and Reference |
---|---|

Information theory based metrics | |

$NMI$ | Normalized mutual information [78,79] |

${Q}_{NCIE}$ | Non-linear correlation metric [80] |

$VIFF$ | Visual information fidelity metric [81] |

Feature based metrics | |

${Q}^{AB/F}$ | Gradient based metric [82] |

${Q}_{G}$ | Gradient based metric [83] |

${Q}_{SF}$ | Spatial frequency based metric [84] |

${Q}_{M}$ | Multi-scale scheme metric [85] |

${Q}_{P}$ | Phase congruency based metric [86] |

Structural similarity based metrics | |

${Q}_{S}$ | Piella’s metric [87] |

${Q}_{C}$ | Cvejie’s metric [88] |

${Q}_{Y}$ | Yang’s metric [89] |

Human perception based metrics | |

${Q}_{CB}$ | Chen Blum’s metric [90] |

**Table 2.**Objective performance evaluation of the proposed and the compared methods on the Lytro dataset. The best scores for each metric are highlighted in bold.

Metrics | DCHWT | WSSM | PCNN | DCTLP | IFGD | ICA | NSCT | PCA | CNN | CSR | Proposed |
---|---|---|---|---|---|---|---|---|---|---|---|

${Q}^{AB/F}$ | 0.7212 | 0.7305 | 0.7036 | 0.6526 | 0.7174 | 0.7445 | 0.2314 | 0.5992 | 0.7514 | 0.7366 | 0.7539 |

$VIFF$ | 0.9021 | 0.9333 | 0.8565 | 0.8388 | 1.0456 | 0.9386 | 0.5851 | 0.7809 | 0.9548 | 0.9329 | 0.9344 |

$NMI$ | 0.9176 | 0.9748 | 1.2068 | 0.8296 | 0.5223 | 0.9374 | 0.5645 | 0.8939 | 1.0725 | 0.9921 | 1.1972 |

${Q}_{G}$ | 0.6153 | 0.6758 | 0.6745 | 0.5390 | 0.6181 | 0.6787 | 0.2070 | 0.5340 | 0.7029 | 0.6447 | 0.7291 |

${Q}_{NCIE}$ | 0.8291 | 0.8321 | 0.8497 | 0.8252 | 0.8145 | 0.8301 | 0.8159 | 0.8277 | 0.8384 | 0.8341 | 0.8472 |

${Q}_{M}$ | 0.8821 | 0.9331 | 2.2555 | 0.6983 | 0.5633 | 1.0264 | 0.1796 | 0.4820 | 2.3431 | 1.3400 | 2.5074 |

${Q}_{SF}$ | −0.0900 | −0.0899 | −0.1050 | −0.0941 | −0.0744 | −0.0684 | −0.1522 | −0.3890 | −0.0321 | −0.0490 | −0.0314 |

${Q}_{P}$ | 0.7839 | 0.8201 | 0.7483 | 0.6892 | 0.7632 | 0.8197 | 0.0486 | 0.7500 | 0.8428 | 0.8283 | 0.8289 |

${Q}_{S}$ | 0.9477 | 0.9438 | 0.9133 | 0.9298 | 0.8901 | 0.9547 | 0.5802 | 0.9217 | 0.9485 | 0.9467 | 0.9463 |

${Q}_{C}$ | 0.8019 | 0.8097 | 0.8183 | 0.7698 | 0.7485 | 0.8334 | 0.3853 | 0.7978 | 0.8253 | 0.7980 | 0.8258 |

${Q}_{Y}$ | 0.9219 | 0.9567 | 0.9672 | 0.8747 | 0.8526 | 0.9515 | 0.3792 | 0.8490 | 0.9653 | 0.9340 | 0.9838 |

${Q}_{CB}$ | 0.6977 | 0.7887 | 0.7476 | 0.6185 | 0.6118 | 0.7130 | 0.4778 | 0.6325 | 0.7816 | 0.7638 | 0.8054 |

Method | DCHWT | WSSM | PCNN | DCTLP | IFGD | ICA | NSCT | PCA | CNN | CSR | Proposed |
---|---|---|---|---|---|---|---|---|---|---|---|

Time | 9.39 | 215.49 | 1.31 | 0.33 | 1.01 | 11.30 | 174.00 | 0.04 | 106.32 | 345.76 | 122.61 |

**Table 4.**Comparison of different color distance models for the proposed multi-focus image fusion algorithm. The best values are in bold. Time (in seconds) is the average execution time of the proposed method with each color distance model. CAD, color approximation distance.

Metric | Euclidean | CAD | CIE76 | CIE94 | CIEXYZ | CMC | CIEDE2000 |
---|---|---|---|---|---|---|---|

${Q}^{AB/F}$ | 0.7483 | 0.7477 | 0.7520 | 0.7510 | 0.7491 | 0.7533 | 0.7539 |

$VIFF$ | 0.9192 | 0.9182 | 0.9256 | 0.9240 | 0.9182 | 0.9287 | 0.9344 |

$NMI$ | 1.1919 | 1.1919 | 1.1945 | 1.1953 | 1.1953 | 1.1946 | 1.1972 |

${Q}_{G}$ | 0.7252 | 0.7242 | 0.7291 | 0.7287 | 0.7287 | 0.7295 | 0.7291 |

${Q}_{NCIE}$ | 0.8478 | 0.8479 | 0.8476 | 0.8478 | 0.8479 | 0.8472 | 0.8472 |

${Q}_{M}$ | 2.4243 | 2.4406 | 2.4445 | 2.4363 | 2.3985 | 2.4603 | 2.5074 |

${Q}_{SF}$ | −0.0403 | −0.0387 | −0.0393 | −0.0412 | −0.0457 | −0.0335 | −0.0314 |

${Q}_{P}$ | 0.8085 | 0.8077 | 0.8239 | 0.8239 | 0.8214 | 0.8224 | 0.8289 |

${Q}_{S}$ | 0.9449 | 0.9449 | 0.9465 | 0.9463 | 0.9456 | 0.9467 | 0.9463 |

${Q}_{C}$ | 0.8199 | 0.8190 | 0.8240 | 0.8238 | 0.8226 | 0.8250 | 0.8258 |

${Q}_{Y}$ | 0.9803 | 0.9796 | 0.9827 | 0.9825 | 0.9822 | 0.9826 | 0.9838 |

${Q}_{CB}$ | 0.7904 | 0.7881 | 0.8006 | 0.7994 | 0.7950 | 0.8005 | 0.8054 |

Time | 52.08 | 56.87 | 44.71 | 45.31 | 40.94 | 53.67 | 122.61 |

**Table 5.**Execution time of the proposed method with different numbers of superpixels k. Time (in seconds) is the average execution time. The best time is marked in bold.

k | 1500 | 2000 | 2500 | 3000 | 3500 | 4000 | 4500 |
---|---|---|---|---|---|---|---|

Time | 97.86 | 109.16 | 118.69 | 122.61 | 119.98 | 132.39 | 131.43 |

**Table 6.**Impact of weight factor m on execution time. Time (in seconds) is the average execution time of the proposed method with different values of m. The best values are in bold.

m | 10 | 15 | 20 | 25 | 30 | 35 | 40 |
---|---|---|---|---|---|---|---|

Time | 151.72 | 141.16 | 125.76 | 122.61 | 110.32 | 110.32 | 98.54 |

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## Share and Cite

**MDPI and ACS Style**

Ilyas, A.; Farid, M.S.; Khan, M.H.; Grzegorzek, M. Exploiting Superpixels for Multi-Focus Image Fusion. *Entropy* **2021**, *23*, 247.
https://doi.org/10.3390/e23020247

**AMA Style**

Ilyas A, Farid MS, Khan MH, Grzegorzek M. Exploiting Superpixels for Multi-Focus Image Fusion. *Entropy*. 2021; 23(2):247.
https://doi.org/10.3390/e23020247

**Chicago/Turabian Style**

Ilyas, Areeba, Muhammad Shahid Farid, Muhammad Hassan Khan, and Marcin Grzegorzek. 2021. "Exploiting Superpixels for Multi-Focus Image Fusion" *Entropy* 23, no. 2: 247.
https://doi.org/10.3390/e23020247