# A POSHE-Based Optimum Clip-Limit Contrast Enhancement Method for Ultrasonic Logging Images

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. Histogram Equalization

_{k}is used to represent the kth gray level, then the L gray levels can be described as {r

_{0}, r

_{1}, ..., r

_{L−}

_{1}}; the probability density function (PDF) corresponding to the gray level distribution of the original image, p(r

_{k}), is given by

_{k}and N represent the number of pixels with gray level r

_{k}and the total number of pixels of the input image X, respectively. Then, the cumulative distribution function (CDF), c(r

_{k}), is calculated from the original PDF as follows:

_{0}, r

_{L−}

_{1}) using the CDF as a transformation function. The formula of the transformation function f(r

_{k}) is given as:

#### 2.2. POSHE

- Step 1:
- Let us define an image with size M × N.
- Step 2:
- Assign an m × n sub-block at the top left corner. For computational simplicity, the size of the sub-block is selected to be equal to the quotient of the input image size divided by a multiple of two.
- Step 3:
- Perform local histogram equalization for the current sub-block.
- Step 4:
- The sub-block moves from left to right and from top to bottom by the horizontal step size and the vertical step size. Repeat Step 3 until POSHE covers the entire input image plane.
- Step 5:
- After sub-block histogram equalization is completed, because each pixel is obviously histogram equalized more than once, accumulated equalization results on each pixel can be divided by its histogram equalization frequency and then produce each pixel value in the output image array.

## 3. The Proposed Contrast Enhancement Method

#### 3.1. Clipped Histogram Equalization

_{k}) (defined in Section 2.1). The slope of c(r

_{k}) is given by

_{k}). In order to limit the contrast to a desired level, a technique proposed in the CLAHE method [64,65] is restraining the slope of CDF for the original image, and it can be achieved by clipping the histogram with values over a predefined threshold, and the residual is redistributed uniformly to the histogram. The maximum slope is limited using a clip limit β to clip all histograms; β can be defined as follows [64,70]:

_{t}and L

_{t}are total numbers of pixels and gray-levels in each region, respectively, ${s}_{\mathrm{max}}$ is the maximum allowable slope, and a is the clip factor. If α = 0, the minimum clip limit β is equal to ($\frac{{N}_{t}}{{L}_{t}}$) which can reach the maximum value of (${s}_{\mathrm{max}}\cdot \frac{{N}_{t}}{{L}_{t}}$) for α = 100. It is clear that various ${s}_{\mathrm{max}}$ values will affect the processed results. Normally, ${s}_{\mathrm{max}}$ is set to four for the application of still X-ray images as reported in Ref. [64]. However, for other applications, it is recommended that a good selection for ${s}_{\mathrm{max}}$ is obtained by practical experiment.

_{t}− 1) and set the average number of pixels per grayscale as N

_{av}for any sub-block with size m × n. Then, ${N}_{t}=m\cdot n$, for an 8-bit grayscale image, L

_{t}= 256, and ${N}_{av}=\frac{{N}_{t}}{{L}_{t}}=\frac{m\times n}{256}$. The total number of pixels whose histogram values exceed the clip limit β is represented as Excess with the initial value being 0.

_{t}− 1, then, Excess = Excess – β + h(k), h(k) = β. This step modifies the histogram by preserving the histograms that are less than or equal to β, while clipping the ones that exceed β.

_{m}is used to represent the number of pixels that should be equally assigned to each gray-level, then N

_{m}= Excess/L

_{t}. The histogram h(k) after the redistribution is given by Equation (12), and Excess after the redistribution is generated as in Equation (13):

_{av}, respectively. The over-enhancement phenomenon is avoided to different degrees with different β values, as presented in Figure 3c,d. It is clear that the enhanced image quality highly depends on the selection of β value, whose value can be chosen empirically. The clipping and redistribution processes are depicted in Figure 4. Figure 4a is the corresponding histogram of Figure 3a, where the horizontal axis and vertical axis represent the intensity values and the probability of occurrence of intensity levels, respectively. Let us assume that the clip limit β be given by the horizontal dotted line. Figure 4b shows the histogram of the original input image in Figure 3a and the modified limited histogram after redistribution according to the clipping and redistribution procedure mentioned above. The comparison curves of the cumulative density function for the original and the modified limited histogram with β = 2.5 N

_{av}and β = 1.5 N

_{av}are illustrated in Figure 4c,d, respectively. It is obvious that the slope of the clip-limited CDF (dotted line) is less than the original CDF (solid line) in the range of the 20–100 gray-level, in which most pixels for the original image are concentrated. The input gray values in the range of 20–100 are extended to the full range by the original CDF and a narrower range of output values by the clip limited CDF, which is the reason for the excessive enhancement shown in Figure 3b but avoided in Figure 3c,d. Comparing CDF curves in Figure 4c,d, the smaller the clip limit value, the lower the slope of CDF, thus, the higher the ability to restrain the contrast, whereas, higher values of the clip limit result in more contrast enhancement. Figure 4d shows the higher limit ability with a smaller β. From the corresponding enhancement image in Figure 3d, it effectively overcomes the over-enhancement problem appeared in Figure 3b, while a very slight over-enhancement phenomenon still occurs with a bigger β in Figure 3c. Since this clipped histogram equalization is still a global approach that is useful to enhance the content of the entire input image, it fails to highlight details of the local regions.

#### 3.2. Optimum Clip-Limit Strategies

_{x}and μ

_{y}, are the mean intensity, σ

_{x}and σ

_{y}are the standard deviation and σ

_{xy}is the covariance of images x and y, respectively, and c

_{1}, c

_{2}are the constant values. The local parameters μ

_{x}, μ

_{y}, σ

_{x}, σ

_{y}and σ

_{xy}are calculated within a local 8 × 8 square window, and the square window slides from pixel to pixel over the whole image. At each step, the SSIM index together with the local parameters are computed within the local window. Based on the value of SSIM, MSSIM can be calculated by

_{i}, y

_{i}are used to describe the image contents for the ith local window; and N represents the total number of local windows of the image. In general, a high MSSIM value for an enhanced image represents a good similarity index.

_{av}before performing sub-block equalization in the proposed POSHEOC algorithm according to the analysis in Section 3.1 and the minimum value of n is one. It is obvious that the POSHEOC algorithm cannot enhance the image sufficiently if n is set to a small value whereas it may result in over-enhancement when n is too big; therefore, n is set to be in the range of [1, 10]. Figure 5 shows an example of the relationships between the two mentioned measures and n. Figure 5a,b plots the metrics MG and MMSIM of enhanced image with different n, respectively. Opposite varying tendencies of the two measures with the clip-limit value are observed. Considering the effects of the two factors, finally, we combine the two metrics, and a new index can be built by means of the product of the mean gradient and mean structural similarity (PMGSIM), which is defined by the following Equation (17). It is used to calculate the optimal n value. The optimal value of n in the range of [1, 10] can be obtained by Equation (18).

## 4. Experimental Results and Discussion

#### 4.1. Subjective Evaluation

#### 4.2. Objective Evaluation

_{max}is the maximum intensity of the input image, for the common 8-bit gray-level image with 256 possible gray level values, as it is known that f

_{max}= 255. Generally, PSNR is used to estimate the artifacts or noise produced in the process of contrast enhancement. It is expected that a good enhancement method will generate a high PSNR value. AMBE is the absolute mean brightness error between the input and output image, which is described as:

_{X}and μ

_{Y}are the mean intensity values of X and Y, respectively. The lower the value of AMBE, the better is the brightness preservation and vice versa. IE is an effective way to evaluate the amount of information content within an image; for the image with the gray level in the range [0, L − 1], the entropy of the image can be expressed as

_{i}) represents probability density function for a given image at gray level s

_{i}. In general, a larger value of the entropy indicates more richness of details is available in the image. The local contrast criterion is defined in Equation (22)

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**The example of enhanced results by the HE and the clipped HE method. (

**a**) Original input image; (

**b**) Image enhanced by HE; (

**c**) Image enhanced by clipped HE with β = 2.5 N

_{av}; (

**d**) Image enhanced by clipped HE with β = 1.5 N

_{av}.

**Figure 4.**Clipping and redistribution of clipped histogram equalization. (

**a**) Histogram of the original input image; (

**b**) Histogram of the original input image and the modified histogram after redistribution; (

**c**) The cumulative density function of original and modified histograms with β = 2.5 N

_{av}; (

**d**) The cumulative density function of original and modified histograms with β = 1.5 N

_{av}.

**Figure 5.**The relationships between the two measures and n. (

**a**) MG of the enhanced image with different n values; (

**b**) MMSIM of the enhanced image with different n values.

**Figure 6.**Examples of the proposed POSHEOC with different n values. (

**a**) The original ultrasonic logging image; (

**b**) PMGSIM of the enhanced image with different n values; (

**c**) The enhanced result of the proposed POSHEOC with n = 1.5; (

**d**) The enhanced result of the proposed POSHEOC with n = 3; (

**e**) The enhanced result of the proposed POSHEOC with n = 6.

**Figure 7.**Comparison of enhancement results with corresponding statistical histogram using various techniques for ultrasonic logging image of the model well. (

**a**) Original image; (

**b**) HE; (

**c**) BOHE; (

**d**) POSHE; (

**e**) MLBOHE; (

**f**) BBHE; (

**g**) RMSHE; (

**h**) CLAHE-PL; (

**i**) Proposed POSHEOC.

**Figure 8.**Comparison of enhancement results with corresponding statistical histogram using various techniques for ultrasonic logging image of Changqingyi well. (

**a**) Original image; (

**b**) HE; (

**c**) BOHE; (

**d**) POSHE; (

**e**) MLBOHE; (

**f**) BBHE; (

**g**) RMSHE; (

**h**) CLAHE-PL; (

**i**) Proposed POSHEOC.

**Figure 9.**Comparison of enhancement results with corresponding statistical histogram using various techniques for ultrasonic logging image of Changqingli well. (

**a**) Original image; (

**b**) HE; (

**c**) BOHE; (

**d**) POSHE; (

**e**) MLBOHE; (

**f**) BBHE; (

**g**) RMSHE; (

**h**) CLAHE-PL; (

**i**) Proposed POSHEOC.

Methods | Objective Indexes | ||||
---|---|---|---|---|---|

PMGSIM | PSNR | IE | AMBE | LC | |

HE | 41.1477 | 7.8129 | 5.2915 | 91.3723 | 0.3517 |

BOHE | 32.7773 | 7.1055 | 7.9546 | 97.4517 | 0.5681 |

POSHE | 41.5468 | 7.7988 | 7.9259 | 91.5251 | 0.6086 |

MLBOHE | 45.2901 | 15.7386 | 6.9873 | 38.0596 | 0.1769 |

BBHE | 55.4418 | 14.5983 | 5.9073 | 25.1141 | 0.3563 |

RMSHE | 52.2859 | 25.1331 | 5.9667 | 1.7287 | 0.1913 |

CLAHE-PL | 60.1273 | 13.6930 | 7.4505 | 41.4485 | 0.2067 |

POSHEOC | 62.5286 | 16.6169 | 7.2052 | 27.7857 | 0.2773 |

**Table 2.**Quantitative results for ultrasonic logging image of Changqingyi well with various methods.

Methods | Objective Indexes | ||||
---|---|---|---|---|---|

PMGSIM | PSNR | IE | AMBE | LC | |

HE | 52.4388 | 13.7409 | 5.8441 | 31.0693 | 0.2991 |

BOHE | 48.6238 | 10.8948 | 7.9774 | 54.0931 | 0.5068 |

POSHE | 54.3196 | 12.7994 | 7.9665 | 38.2192 | 0.5249 |

MLBOHE | 40.7913 | 16.7165 | 7.2906 | 34.4128 | 0.1624 |

BBHE | 52.9611 | 14.4309 | 6.8055 | 24.2409 | 0.4237 |

RMSHE | 47.6993 | 24.7405 | 6.7605 | 2.7983 | 0.2098 |

CLAHE-PL | 58.4960 | 14.5198 | 7.7645 | 35.4124 | 0.2783 |

POSHEOC | 61.1664 | 17.2683 | 7.6481 | 19.6878 | 0.3606 |

**Table 3.**Quantitative results for ultrasonic logging image of Changqingli well with various methods.

Methods | Objective Indexes | ||||
---|---|---|---|---|---|

PMGSIM | PSNR | IE | AMBE | LC | |

HE | 49.2776 | 17.8113 | 5.9706 | 8.2204 | 0.2409 |

BOHE | 53.4988 | 14.5454 | 7.9473 | 29.0144 | 0.4281 |

POSHE | 54.6502 | 16.6845 | 7.9561 | 8.4551 | 0.2990 |

MLBOHE | 40.1781 | 19.3844 | 7.5268 | 24.8180 | 0.1898 |

BBHE | 49.7495 | 18.2670 | 7.2280 | 4.1289 | 0.3378 |

RMSHE | 44.0206 | 29.3174 | 7.2733 | 0.7827 | 0.2234 |

CLAHE-PL | 56.4173 | 16.8092 | 7.8462 | 16.0812 | 0.2857 |

POSHEOC | 58.2492 | 18.5768 | 7.6870 | 2.3548 | 0.3450 |

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## Share and Cite

**MDPI and ACS Style**

Fu, Q.; Zhang, Z.; Celenk, M.; Wu, A.
A POSHE-Based Optimum Clip-Limit Contrast Enhancement Method for Ultrasonic Logging Images. *Sensors* **2018**, *18*, 3954.
https://doi.org/10.3390/s18113954

**AMA Style**

Fu Q, Zhang Z, Celenk M, Wu A.
A POSHE-Based Optimum Clip-Limit Contrast Enhancement Method for Ultrasonic Logging Images. *Sensors*. 2018; 18(11):3954.
https://doi.org/10.3390/s18113954

**Chicago/Turabian Style**

Fu, Qingqing, Zhengbing Zhang, Mehmet Celenk, and Aiping Wu.
2018. "A POSHE-Based Optimum Clip-Limit Contrast Enhancement Method for Ultrasonic Logging Images" *Sensors* 18, no. 11: 3954.
https://doi.org/10.3390/s18113954