**3. Results and discussion**

*Cutaneous Melanoma*

the calculation of the FD.

**Figure 5.**

frequency *f* is given by *f* = √

where *c* is a constant.

image *I*(*i*,*j*) is given by

FD. Therefore, a more accurate and robust methodology, the 2D FAA, is utilized for

*Koch curve. The initiator (A) and generator (B) are used for constructing the Koch curve. Curves C, D, and E* 

The method for the calculation of the 2D Fourier FD is applied to a 2D grayscale

*I*(*k*,*l*) exp

where *u* and *v* are the horizontal and vertical frequency, respectively. The total

−β

8 − β 2

Another methodology the 2D DBCM will be used in this paper to calculate the

FD. The detail of the 2D DBCM was introduced in Sarkar's paper [22].

*β* can be calculated by fitting the function in Eq. (3) by calculating the slope of the curve *lnP* × *lnf*. The least square method was used to obtain the slope in this paper. The 2D Fourier FD was then calculated by using the following equation [21]:

[− \_*i*2*π*

*<sup>N</sup>* (*uk* <sup>+</sup> *vl*)

. Then, the power spectrum of the 2D grayscale

, (3)

. (4)

*I* can be expressed as follows:

], (2)

image *I*(*k*,*l*) with the size *N* × *N*. The Fourier transform ¯

*are levels 2, 3, and 4 in the construction of the Koch curve, respectively.*

*I* (*u*, *v*) = ∑

*u*2 + *v*<sup>2</sup>

*k*=0 *N*−1 ∑ *l*=0 *N*−1

\_

¯

*P* = *c f*

*FD* <sup>=</sup> \_

The range of possible values is between 2 and 3.

**82**

In our previous paper, the quantitative image features including the FD have already been studied to differentiate the skin tumors. However, the FD was extracted from OCT images by using the 2D DBCM. Generally speaking, the 2D DBCM is a time-consuming methodology. In order to quickly detect and classify the skin tumors, the 2D FAA was introduced in this paper. Twenty OCT images per type of skin tumors were randomly chosen from the database. The 2D FAA as well as the 2D DBCM was used to calculate the 2D FD. The FD calculated by using the 2D FAA and the statistical analysis between study groups were showed in **Table 1**. The FD that was obtained by employing the 2D DBCM and the statistical analysis between study groups were showed in **Table 2**. The averaged time for extracting the FD by using the two methodologies was showed in **Table 3**. The results in **Table 1** indicated that the Fourier FD of the basal cell carcinomas is significantly smaller than FD of melanomas. Compared to the FD value of melanomas, the Fourier FD of the basal cell carcinomas has a 2.79% decrease. The results also indicated that the Fourier FD of the benign melanocytic nevi is significantly smaller than FD of melanomas. Compared to the FD value of melanomas, the Fourier FD of the benign melanocytic nevi has a 2.69% decrease. The results in **Table 2** indicated that the FD of the basal cell carcinomas by using the 2D DBCM is significantly smaller than the FD of melanomas. Compared to the melanomas, the DBCM FD of the basal cell carcinomas has a 1.76% decrease. Compared to the melanomas, the DBCM FD of the benign melanocytic nevi showed the same tread. Specifically, the FD (calculated by using the 2D DBCM) of the benign melanocytic nevi decreased 1.38% as compared to the melanomas. In order to compare the computational time between the two methods, we run the two MATLAB codes (ver. R2007b) in the same laptop (i5-4210 CPU, 8GB RAM). In **Table 3**, the computational time was shorter by 91.71% for FAA than 2D DBCM.

Our results showed that the melanomas had a larger FD than the basal cell carcinomas and the benign melanocytic nevi when both of the two methodologies were utilized in the calculations. As the FD is used to express the abnormality of the biological tissue, our results suggested that the melanomas had more irregularity than the basal cell carcinomas and the benign melanocytic nevi. Melanomas feature heavily disorganized vessels with chaotic branching, which might be the explanation for that finding. These specific results indicated that both the Fourier FD and the differential box counting dimension could be used as an indicator to differentiate the melanomas from the basal cell carcinomas and the benign melanocytic nevi. It is worth noting that the Fourier FD is bigger than the differential box counting dimension in our calculations. The Fourier FD was calculated in the frequency domain, while the differential box counting dimension was calculated in the spatial domain. One possible reason to explain the difference is due to the undercount of the number of the boxes in the 2D DBCM which resulted in a small differential box counting dimension in the calculations. Moreover, our results also


#### **Table 1.** *Distribution of FD (mean ± SD) values calculated by performing the FAA.*


*p < 0:001 (ANOVA followed by Newman-Keuls post hoc analysis) between melanomas and benign melanocytic nevi (see nevi column) and between melanomas and basal cell carcinomas (see basal column)*

#### **Table 2.**

*Distribution of FD (mean ± SD) values calculated by using the DBCM.*


#### **Table 3.**

*Comparison of the computational time for calculating the FD by using the two methods.*

showed that the differences of the Fourier FDs between the melanomas and the basal cell carcinomas are bigger than the differences of the differential box counting dimension, which could lead to a conclusion that the 2D Fourier FD could be better to classify the melanomas from the basal cell carcinomas. Our results also showed that the computational time for calculating 2D Fourier FD is much less than the computational time for calculating the 2D differential box counting dimension. This particular result suggested that the 2D FAA is more efficient to differentiate the skin tumors than the 2D DBCM.

There are several potential shortcomings of our study. The custom-built SD-OCT technology has some limitations as compared to the more pioneering OCT technology. In addition, current OCT devices include different algorithms and methodologies for the removal of the speckle noise. Therefore, data analysis is influenced by special assumptions and technological specifications that are in place for each individual OCT device. Another limitation is that only 20 scans were randomly selected for each type of skin tumors. Thus, more scans would be beneficial for extracting the more accurate FD and find the diagnostic parameter to differentiate the skin tumors.

### **4. Conclusion**

In summary, we have described an efficient approach to calculate the 2D FD form OCT images for classifying the basal cell carcinomas, melanomas, and benign melanocytic nevi in this paper. The preliminary results presented have indicated that the 2D FAA is more efficient for extracting the FD than the 2D DBCM. Particularly, the change in the fractal dimension may reflect the pathological metabolic changes in melanomas. More research studies are needed to determine the accuracy, repeatability, and full capability of this methodology with more OCT images.

### **Acknowledgements**

This research was supported in part by the research grant D2016009 from the Ningbo University of Technology of China and the research grant nos. 2017A610239, 2018A610249, and 2018A610362 from the Ningbo Natural Science Foundation.

**85**

**Author details**

, Bingjiang Lin2

2 Ningbo First Hospital, Ningbo, China

University, Samara, Russian Federation

provided the original work is properly cited.

\*, Valery P. Zakharov3

1 School of Safety Engineering, Ningbo University of Technology, Ningbo, China

3 Department of Laser and Biotechnical Systems, Samara National Research

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: bingjianglin@foxmail.com

and Oleg O. Myakinin3

Wei Gao1

*2D Fourier Fractal Analysis of Optical Coherence Tomography Images of Basal Cell Carcinomas…*

*DOI: http://dx.doi.org/10.5772/intechopen.89196*

*2D Fourier Fractal Analysis of Optical Coherence Tomography Images of Basal Cell Carcinomas… DOI: http://dx.doi.org/10.5772/intechopen.89196*
