# Entropy and Contrast Enhancement of Infrared Thermal Images Using the Multiscale Top-Hat Transform

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## Abstract

**:**

## 1. Introduction

## 2. Entropy and Contrast in Digital Images

## 3. Enhancement of Thermal Infrared Images

#### 3.1. Classic Top-Hat Transform

#### 3.2. Modified Top-Hat Transform

#### 3.3. How Entropy is Changed by Top-Hat Transform

- The old value g was unique in the region, with a count of 1, hence it disappears from the region and is replaced by value h. No change in entropy occurs because in the old g bin of the histogram the count of 1 becomes 0, and in the new h bin the count of 0 becomes 1; or
- The old value g existed in k > 1 pixels in the region. In this case the count in the g bin decreases to $k-1$, and the count in the h bin increases to 1. The following Lemma shows that this change in the histogram increases the region’s entropy.

**Lemma**

**1.**

**Proof.**

#### 3.4. Proposed Method Using Multiscale Top-Hat Transform

Algorithm 1 Proposed method for TII Enhancement |

Input:I, G, ${G}^{\prime}$, n, $\omega $Output:${I}_{E}$(Enhanced image)Initialization: G, ${G}^{\prime}$ 1: for$i=1$ to n do2: Calculation of top-hat transform. $M{B}_{i}=I-((I\ominus {G}_{i})\oplus {G}_{i}^{\prime})$ (Equation (11)) $M{D}_{i}=((I\oplus {G}_{i})\ominus {G}_{i}^{\prime})-I$ (Equation (12)) 3: Calculation of subtractions from neighboring scales, obtained through the top-hat transform. The top-hat is subtracted with the previous difference, from the first subtraction of the first neighboring top-hat.
$$SNB{S}_{i-1}=\left\{\begin{array}{c}M{B}_{i}-M{B}_{i-1},\phantom{\rule{4.pt}{0ex}}\mathrm{to}\phantom{\rule{4.pt}{0ex}}i=2\\ M{B}_{i}-SNB{S}_{i-2},\phantom{\rule{4.pt}{0ex}}\mathrm{to}\phantom{\rule{4.pt}{0ex}}i>2\end{array}\right.\phantom{\rule{4.pt}{0ex}}(\mathrm{Equation}\phantom{\rule{4.pt}{0ex}}(13))$$
$$SND{S}_{i-1}=\left\{\begin{array}{c}M{D}_{i}-M{D}_{i-1},\phantom{\rule{4.pt}{0ex}}\mathrm{to}\phantom{\rule{4.pt}{0ex}}i=2\\ M{D}_{i}-SND{S}_{i-2},\phantom{\rule{4.pt}{0ex}}\mathrm{to}\phantom{\rule{4.pt}{0ex}}i>2\end{array}\right.\phantom{\rule{4.pt}{0ex}}(\mathrm{Equation}\phantom{\rule{4.pt}{0ex}}(14))$$
4: end for5: Calculation of the maximum values of all the multiple scales obtained. $SMB={\sum}_{i=1}^{n}M{B}_{i}$ (Equation (15)) $SMD={\sum}_{i=1}^{n}M{D}_{i}$ (Equation (16)) $SSNBS={\sum}_{i=1}^{n-1}SNB{S}_{i-1}$ (Equation (17)) $SSNDS={\sum}_{i=1}^{n-1}SND{S}_{i-1}$ (Equation (18)) 6: TII enhancement calculation.The contrast enhancement calculation consists of adding the results of the multiple bright scales to the original image and subtracting the results of the multiple dark scales. ${I}_{E}=I+\omega \times (SMB+SSNBS)-\omega \times (SND+SSNDS)$ (Equation (19)) 7: return ${I}_{E}$ |

## 4. Results and Discussion

- In the first part (Section 4.1) we perform a parameter adjustment to find good parameter values that maximize the entropy of the output image after applying the proposed method.
- Then, in the second part (Section 4.2) we analyze the proposed method per iteration and compare its performance with Multiscale Morphological Infrared Image Enhancement (MMIIE) (mathematical morphology-based multiscale approach) [4].
- Finally, in the last part (Section 4.3), we apply the proposed method and compare the results achieved with the proposed techniques with the following competitive methods from the literature: HE, Contrast Limited Adaptive Histogram Equalization (CLAHE) [51], the method of Kun Liang et al. [6] called IRHE2PL for infrared images, and the MMIIE method for infrared images.

#### 4.1. Parameter Tuning

#### 4.2. Performance of Proposed Method per Iteration

- The Standard Deviation (SD), which quantifies the global contrast of the infrared images, is defined as [16]:$$SD\left(I\right)=\sqrt{\sum _{k=0}^{L-1}{(k-A\left(I\right))}^{2}\times p\left(k\right)},$$
- The metric adopted to measure the signal-to-noise ratio of an image is the PSNR.Given the original infrared image I and the infrared image with enhancement ${I}_{EN}$ where the size of the images is $M\times N$, the PSNR between I and ${I}_{EN}$ is given by [30]:$$PSNR(I,{I}_{E})=10\times lo{g}_{10}{\displaystyle \frac{{(L-1)}^{2}}{MSE(I,{I}_{E})}}.$$The Mean Squared Error (MSE) is defined as:$$MSE(I,{I}_{E})={\displaystyle \frac{1}{M\times N}}\sum _{u=0}^{M-1}\sum _{v=0}^{N-1}{(I(u,v)-{I}_{EN}(u,v))}^{2}.$$
- The Absolute Mean Brightness Error (AMBE) [11], which quantifies the conservation of the mean brightness of the processed image, is given by:$$AMBE(I,{I}_{E})=|A\left(I\right)-A\left({I}_{E}\right)|,$$
- The linear blur index $\gamma $ [4] is used to measure the performance of the infrared image enhancement. It is defined as follows:$$\gamma \left(I\right)={\displaystyle \frac{2}{M\times N}}\sum _{u=1}^{M}\sum _{v=1}^{N}min\{{p}_{uv},(1-{p}_{uv})\},$$$${p}_{uv}=sin[{\displaystyle \frac{\pi}{2}}\times (1-{\displaystyle \frac{I(u,v)}{L-1}})].$$

#### 4.3. Comparison of the Performance of the Proposed Method with State of the Art Methods

#### 4.3.1. Analysis of Methods by Scenes

- E metric: The CLAHE method and the proposed method are the methods that have the best performance in terms of entropy for scenes 1 to 8. However, in scene 9 the CLAHE and MMIIE methods have the best results.
- SD metric: The HE, CLAHE, IRHE2PL methods and the proposed method enhance the contrast of the TII in the 9 scenes. The MMIIE method did not enhance the contrast of scenes 2, 4, 7, 8, and 9. The HE method is the best performing method for all scenes and the proposed method is in second place.
- PSNR metric: The methods that produce the less distortion to TII are the IRHE2PL, the proposed method, and CLAHE.
- AMBE metric: For all scenes, the best method in regards to maintaining the average brightness is the proposed method.
- $\gamma $ metric: The MMIIE method and the proposed method present the best results in terms of blurring.

#### 4.3.2. General Analysis of Methods

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Loss of information with enhanced contrast. (

**a**) TII enhanced with HE; (

**b**) Histogram of the TII enhanced with HE.

**Figure 5.**Visual results obtained with the proposed method and the configuration of $\omega $ and n. (

**a**) Original TII with $E=6.7893$; (

**b**) TII enhancement with proposed method with $\omega =0.35$, $n=8$ and $E=7.4696$; (

**c**) TII enhancement with proposed method with $\omega =0.45$, $n=7$ and $E=7.4467$.

**Figure 6.**Example of a thermal infrared imaging (TII) to compare the proposed method and Multiscale Morphological Infrared Image Enhancement (MMIIE) method. (

**a**) Original TII with $E=6.8387$ and $SD=31.8204$; (

**b**) Histogram of the original image; (

**c**) TII enhanced with the MMIIE method with $E=6.8619$, $SD=42.5731$, and $AMBE=24.1018$; (

**d**) Histogram of the image enhanced with MMIIE; (

**e**) TII enhanced with the proposed method with $E=7.3629$, $SD=58.8730$, and $AMBE=2.1848$; (

**f**) Histogram of the image enhanced with the proposed method.

**Figure 7.**(

**a**) Original TII 449.png, $E=7.2210$, $SD=53.1827$; (

**b**) TII enhanced with IRHE2PL method, $E=7.2058$, $SD=53.3164$, and $PSNR=48.0940$ and (

**c**) TII enhanced with the proposed method, $E=7.2411$, $SD=62.8887$, and $PSNR=20.6578$.

**Figure 8.**An example of comparison of a TII with a dark background. (

**a**) Original TII, (

**b**) TII enhanced with HE method, (

**c**) TII enhanced with Contrast Limited Adaptive Histogram Equalization (CLAHE) method, (

**d**) TII enhanced with IRHE2PL method, (

**e**) TII enhanced with MMIIE method, and (

**f**) TII enhanced with the proposed method.

**Figure 9.**This example shows a TII with a dark background and semi-bright objectives. (

**a**) original TII, (

**b**) TII enhanced with HE, (

**c**) TII enhanced with CLAHE, (

**d**) TII enhanced with IRHE2PL, (

**e**) TII enhanced with MMIIE, and (

**f**) TII enhanced with the proposed method.

**Figure 10.**TII with improved contrast and detail, (

**a**) the Original TII with $E=6.9334$ and $SD=58.24$ and (

**b**) the TII enhanced with the proposed method with $E=7.5783$ and $SD=70.7615$

Parameter | Value(s) |
---|---|

n | $[2,10]$ |

$\omega $ | $[0,1]$ |

G | $3\times 3$ |

${G}^{\prime}$ | $15\times 15$ |

**Table 2.**Entropy values of the enhanced thermal infrared imaging (TII) obtained by the proposed method with parameters $\omega $ and n.

$\mathit{\omega}$ | n | ||||||||
---|---|---|---|---|---|---|---|---|---|

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

0.05 | 6.5931 | 6.5962 | 6.6041 | 6.6188 | 6.6396 | 6.6656 | 6.6950 | 6.7271 | 6.7607 |

0.10 | 6.5971 | 6.6120 | 6.6394 | 6.6777 | 6.7224 | 6.7696 | 6.8174 | 6.8658 | 6.9129 |

0.15 | 6.6037 | 6.6315 | 6.6756 | 6.7326 | 6.7900 | 6.8502 | 6.9100 | 6.9653 | 7.0099 |

0.20 | 6.6145 | 6.6577 | 6.7190 | 6.7861 | 6.8548 | 6.9228 | 6.9848 | 7.0326 | 7.0593 |

0.25 | 6.6242 | 6.6821 | 6.7540 | 6.8316 | 6.9081 | 6.9788 | 7.0349 | 7.0661 | 7.0648 |

0.30 | 6.6293 | 6.6970 | 6.7790 | 6.8678 | 6.9498 | 7.0185 | 7.0633 | 7.0702 | 7.0348 |

0.35 | 6.6430 | 6.7217 | 6.8089 | 6.9025 | 6.9851 | 7.0475 | 7.0740 | 7.0519 | 6.9828 |

0.40 | 6.6518 | 6.7420 | 6.8398 | 6.9380 | 7.0181 | 7.0673 | 7.0688 | 7.0161 | 6.9169 |

0.45 | 6.6568 | 6.7545 | 6.8607 | 6.9637 | 7.0394 | 7.0735 | 7.0505 | 6.9706 | 6.8448 |

0.50 | 6.6806 | 6.7872 | 6.8957 | 6.9957 | 7.0596 | 7.0726 | 7.0225 | 6.9165 | 6.7668 |

0.55 | 6.6824 | 6.7928 | 6.9068 | 7.0085 | 7.0647 | 7.0610 | 6.9900 | 6.8619 | 6.6915 |

0.60 | 6.6914 | 6.8113 | 6.9314 | 7.0295 | 7.0702 | 7.0450 | 6.9516 | 6.8011 | 6.6111 |

0.65 | 6.6945 | 6.8201 | 6.9451 | 7.0404 | 7.0682 | 7.0249 | 6.9112 | 6.7406 | 6.5330 |

0.70 | 6.7066 | 6.8408 | 6.9655 | 7.0516 | 7.0631 | 7.0010 | 6.8676 | 6.6786 | 6.4556 |

0.75 | 6.7161 | 6.8588 | 6.9835 | 7.0602 | 7.0550 | 6.9745 | 6.8216 | 6.6152 | 6.3778 |

0.80 | 6.7221 | 6.8707 | 6.9979 | 7.0639 | 7.0433 | 6.9462 | 6.7747 | 6.5531 | 6.3032 |

0.85 | 6.7259 | 6.8791 | 7.0077 | 7.0650 | 7.0303 | 6.9167 | 6.7289 | 6.4929 | 6.2319 |

0.90 | 6.7309 | 6.8906 | 7.0183 | 7.0641 | 7.0140 | 6.8843 | 6.6802 | 6.4311 | 6.1612 |

0.95 | 6.7326 | 6.8959 | 7.0235 | 7.0604 | 6.9973 | 6.8515 | 6.6320 | 6.3702 | 6.0927 |

1.00 | 6.7791 | 6.9368 | 7.0460 | 7.0610 | 6.9780 | 6.8134 | 6.5783 | 6.3055 | 6.0210 |

**Table 3.**Results achieved with the proposed method and Multiscale Morphological Infrared Image Enhancement (MMIIE) method.

n | Proposed Method | MMIIE | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

E | SD | PSNR | AMBE | $\mathit{\gamma}$ | Time (ms) | E | SD | PSNR | AMBE | $\mathit{\gamma}$ | Time (ms) | |

2 | 6.643 | 40.835 | 40.417 | 0.129 | 0.332 | 2328 | 6.324 | 30.247 | 16.236 | 38.072 | 0.194 | 454 |

3 | 6.722 | 41.969 | 33.564 | 0.286 | 0.320 | 3752 | 6.447 | 32.069 | 16.176 | 37.775 | 0.193 | 902 |

4 | 6.809 | 43.632 | 29.256 | 0.466 | 0.311 | 6719 | 6.441 | 33.016 | 16.211 | 37.528 | 0.192 | 1605 |

5 | 6.902 | 45.854 | 26.023 | 0.714 | 0.301 | 10,629 | 6.519 | 34.438 | 16.204 | 37.193 | 0.193 | 2759 |

6 | 6.985 | 48.556 | 23.509 | 1.107 | 0.293 | 11,947 | 6.515 | 35.486 | 16.216 | 36.902 | 0.194 | 4770 |

7 | 7.047 | 51.749 | 21.434 | 1.636 | 0.286 | 16,429 | 6.570 | 36.798 | 16.177 | 36.580 | 0.195 | 7187 |

8 | 7.074 | 55.353 | 19.693 | 2.299 | 0.281 | 19,979 | 6.565 | 37.558 | 16.194 | 36.318 | 0.196 | 9255 |

9 | 7.052 | 59.216 | 18.219 | 3.034 | 0.275 | 20,176 | 6.612 | 38.453 | 16.178 | 36.093 | 0.197 | 18,318 |

10 | 6.983 | 63.111 | 16.980 | 3.830 | 0.268 | 21,527 | 6.604 | 39.017 | 16.201 | 35.884 | 0.197 | 20,003 |

Methods | Percentage of Images Improved (%) |
---|---|

HE | 98.89% |

CLAHE | 82.67% |

IRHE2PL | 90.22% |

MMIIE | 47.56% |

Proposed method | 100% |

Methods | E | SD | PSNR | AMBE | $\mathit{\gamma}$ | |
---|---|---|---|---|---|---|

Scene 1 | I | 6.814 | 32.336 | - | - | 0.284 |

HE | 6.596 | 73.420 | 11.543 | 48.519 | 0.406 | |

CLAHE | 7.557 | 50.808 | 15.984 | 24.259 | 0.401 | |

IRHE2PL | 6.814 | 36.776 | 29.603 | 7.065 | 0.293 | |

MMIIE | 6.910 | 43.573 | 17.005 | 25.392 | 0.164 | |

Proposed method | 7.418 | 60.136 | 16.767 | 1.887 | 0.273 | |

Scene 2 | I | 7.039 | 55.330 | - | - | 0.454 |

HE | 6.844 | 73.364 | 20.479 | 3.101 | 0.400 | |

CLAHE | 7.500 | 53.721 | 21.237 | 1.657 | 0.488 | |

IRHE2PL | 7.036 | 58.604 | 33.310 | 6.802 | 0.453 | |

MMIIE | 7.038 | 45.486 | 13.085 | 48.345 | 0.273 | |

Proposed method | 7.601 | 69.425 | 18.215 | 1.534 | 0.419 | |

Scene 3 | I | 5.945 | 18.269 | - | - | 0.477 |

HE | 5.881 | 73.063 | 10.789 | 47.807 | 0.408 | |

CLAHE | 6.970 | 32.154 | 19.839 | 18.275 | 0.485 | |

IRHE2PL | 5.945 | 42.197 | 20.011 | 13.270 | 0.326 | |

MMIIE | 6.133 | 20.900 | 17.288 | 32.819 | 0.127 | |

Proposed method | 6.826 | 30.332 | 23.205 | 0.136 | 0.273 | |

Scene 4 | I | 6.808 | 41.521 | - | - | 0.342 |

HE | 6.642 | 73.148 | 12.977 | 41.972 | 0.407 | |

CLAHE | 7.482 | 48.135 | 18.560 | 19.816 | 0.422 | |

IRHE2PL | 6.808 | 56.144 | 22.617 | 11.306 | 0.313 | |

MMIIE | 6.848 | 35.941 | 16.106 | 33.095 | 0.149 | |

Proposed method | 7.566 | 56.723 | 19.019 | 1.328 | 0.312 | |

Scene 5 | I | 7.052 | 40.839 | - | - | 0.356 |

HE | 6.901 | 73.319 | 14.793 | 32.298 | 0.404 | |

CLAHE | 7.620 | 50.630 | 17.786 | 15.352 | 0.433 | |

IRHE2PL | 7.048 | 45.807 | 32.631 | 11.444 | 0.307 | |

MMIIE | 7.025 | 44.123 | 15.660 | 34.956 | 0.173 | |

Proposed method | 7.505 | 61.669 | 17.599 | 2.584 | 0.317 | |

Scene 6 | I | 6.272 | 24.626 | - | - | 0.152 |

HE | 6.158 | 72.882 | 8.091 | 86.636 | 0.408 | |

CLAHE | 7.200 | 42.379 | 16.477 | 31.714 | 0.263 | |

IRHE2PL | 6.272 | 37.308 | 21.316 | 17.733 | 0.179 | |

MMIIE | 6.035 | 28.802 | 21.602 | 15.389 | 0.048 | |

Proposed method | 6.702 | 40.545 | 20.893 | 2.284 | 0.114 | |

Scene 7 | I | 6.990 | 67.015 | - | - | 0.348 |

HE | 6.783 | 73.516 | 18.535 | 19.921 | 0.398 | |

CLAHE | 7.548 | 66.298 | 19.828 | 7.159 | 0.400 | |

IRHE2PL | 6.987 | 75.173 | 33.982 | 14.009 | 0.295 | |

MMIIE | 7.125 | 53.062 | 12.017 | 54.459 | 0.327 | |

Proposed method | 7.204 | 75.462 | 19.735 | 1.405 | 0.323 | |

Scene 8 | I | 6.219 | 28.522 | - | - | 0.237 |

HE | 6.131 | 72.805 | 8.042 | 89.033 | 0.409 | |

CLAHE | 7.134 | 41.543 | 16.828 | 31.090 | 0.309 | |

IRHE2PL | 6.219 | 62.031 | 11.319 | 60.173 | 0.334 | |

MMIIE | 5.952 | 23.883 | 21.792 | 14.567 | 0.077 | |

Proposed method | 6.589 | 37.486 | 23.014 | 2.426 | 0.118 | |

Scene 9 | I | 6.191 | 53.458 | - | - | 0.448 |

HE | 6.001 | 80.909 | 13.975 | 41.396 | 0.329 | |

CLAHE | 6.459 | 59.638 | 19.045 | 13.075 | 0.440 | |

IRHE2PL | 6.188 | 67.644 | 30.537 | 10.979 | 0.386 | |

MMIIE | 6.438 | 50.305 | 11.052 | 65.813 | 0.433 | |

Proposed method | 6.254 | 66.401 | 18.794 | 7.105 | 0.378 |

© 2019 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/).

## Share and Cite

**MDPI and ACS Style**

Mello Román, J.C.; Vázquez Noguera, J.L.; Legal-Ayala, H.; Pinto-Roa, D.P.; Gomez-Guerrero, S.; García Torres, M. Entropy and Contrast Enhancement of Infrared Thermal Images Using the Multiscale Top-Hat Transform. *Entropy* **2019**, *21*, 244.
https://doi.org/10.3390/e21030244

**AMA Style**

Mello Román JC, Vázquez Noguera JL, Legal-Ayala H, Pinto-Roa DP, Gomez-Guerrero S, García Torres M. Entropy and Contrast Enhancement of Infrared Thermal Images Using the Multiscale Top-Hat Transform. *Entropy*. 2019; 21(3):244.
https://doi.org/10.3390/e21030244

**Chicago/Turabian Style**

Mello Román, Julio César, José Luis Vázquez Noguera, Horacio Legal-Ayala, Diego P. Pinto-Roa, Santiago Gomez-Guerrero, and Miguel García Torres. 2019. "Entropy and Contrast Enhancement of Infrared Thermal Images Using the Multiscale Top-Hat Transform" *Entropy* 21, no. 3: 244.
https://doi.org/10.3390/e21030244