Finger Vein Recognition Based on Multi-Features Fusion

The aspects of the research conducted in finger vein recognition processes that are examined in our study can be divided into two categories. The first category encompasses studies aimed at enhancing the quality of obtained finger vein images. The second category encompasses studies that concentrate on increasing the success rate through the development of new feature vectors and the application of fusion techniques.

Numerous studies have been conducted to enhance the quality of images, with respect to the poor quality of FV images. Fu et al. [23] combined Fuzzy and Retinex theory to enhance the contrast between the vein patterns and their surrounding areas in vascular images obtained by NIR sensitive cameras. The optimal fuzzy transformation was utilized to improve the overall contrast of the image. To address the limitation of the optimal fuzzy transformation in preserving image details, this method was applied to enhance the contrast between the finger vein patterns and the surrounding areas . In the study of Gao et al. [24], a filter is utilized in the image to increase the high frequency, and the histogram equalization method is used to boost contrast. Oh and Hwang [25] proposed a novel morphology-based HF technique to improve the contrast of the features in medical images. This approach involves breaking down an image into morphological sub-bands, which are enhanced using a HF. In order to optimize the image enhancement, they utilized a differential evolution algorithm to determine the most suitable gain and structuring element for each sub-band. In their study, Yang et al. [26] proposed an image restoration technique that utilizes scattering removal to enhance finger vein images. To describe the degradation of finger vein images, the authors used a biological optical model based on light scattering in living tissues. This method enhances the contrast of finger vein images and improves the accuracy of finger vein image matching. In the studies of Yang and Shi [27], an approach for enhancing the venous region and segmenting FV images using a Directional Filter method based on the Gabor Filter is proposed. Their approach also includes a matting-based segmentation method that can accurately extract the finger vein networks while accounting for variations in vein intensity and diameter. Shin et al. [28] presented a fuzzy-based fusion technique to improve the FV visibility by using the Gabor Filter in four directions and the Retinex Filter. A fuzzy-based method is used to decide optimal weights for combining the two filtered images produced after Gabor and Retinex filtering. The input features for the fuzzy rule and membership function are derived from local windows, and specifically, the means and standard deviations (SD) of the images within those windows are used. This method eliminates the need for additional training data in image enhancement.

The success rate of the studies in the Biometric Recognition Systems field by using the FV shows the effectiveness of biometric recognition processes using FV systems. Many methods have been employed to extract illustrative features from the finger vein databases and utilization of fusion techniques. A new feature extraction method, called principal component local conservation projections, based on a combination of principal component analysis and locality preserving projections (PCLPP) techniques, was proposed in 2022 by Feng et al. [29] The experiments revealed that the accuracy of classification using the feature vector extracted by the PCLPP technique of finger vein recognition outperforms both Principal Component Analysis (PCA) and Locality Preserving Projections (LPP). Additionally, PCLPP showed improved recognition rates across various categories compared to PCA and LPP. Lastly, the results indicated that PCLPP is less affected by noise compared to PCA and LPP. The highest recognition rate of the PCLPP technique is 92.33% using 600 classes of finger vein data. Zhang et al. [30] proposed an Adaptive Gabor Convolutional Neural Network (AGCNN) with receptive field properties to design Gabor convolutional layer in 2022. In the study, it is claimed that AGCNN possesses both characteristics of the Gabor Filter and those of a neural network. The study performed on the MMCBNU_6000 dataset revealed that the highest classification accuracy using the CNN-RAW model is 91.72%. In the study performed by Lu et al. [31] in 2021, a new region of interest extraction method was proposed depending on the characteristics of the FV image and compared with some representative methods. The equal error rates calculation of the proposed method on the MMCBNU_6000 dataset is 5.49% by using maximum curvature, and 3.33% by using repeated line tracking feature extraction method. A curvature algorithm is proposed to extract the feature by calculating the eigenvalues of the image’s Hessian matrix, Yong [32] in 2020. In 2019, a new method for feature extraction was proposed by Yang et al. [33] It is known as polarized depth-weighted binary direction coding, and it involves three components: polarized direction extraction, extended normalized angular binary coding, and self-adaptive depth-dependent weighting. On the other hand, Ma and Zhang [34] proposed a new technique involving the extraction of interested regions and oriented elements, based on finger vein patterns. The study employs a region of interest based on rotation rectification to mitigate the effects of rotation and translation during the acquisition of finger vein images. To extract the direction feature, the texture and stable orientation properties of finger vein images are utilized. This is accomplished by computing the gradient magnitude and orientation of points along the finger vein lines in the fuzzy segmentation region. The use of the extracted feature proves effective in addressing the challenge of geometric deformation. JosephP and Ezhilmaran [35] conducted a study in 2018, which examined the effectiveness of affine invariant attributes in FV images through the use of fuzzy image retrieval. To minimize the computational time of the affine invariant features, the database size is reduced through fuzzy-based image retrieval. According to the study, the experimental results demonstrate improved performance and significantly lower error rates compared to conventional feature matching algorithms. In 2017, Babu et al. [36] investigated two new score-level combinations, holistic and non-linear fusion, and check their effectiveness against more popular score-level fusion approaches. In 2016, Sikarwar and Manmohan [37] analyzed various techniques of local directional patterns to enhance the reliability of finger vein recognition. The local directional pattern was utilized to compute the edge response in eight directions and to assess each pixel position. The accuracy was evaluated using finger vein image samples, yielding a success rate of 86.1635%. Matsuda et al. [38] proposed a method for FV Authentication based on Feature Point Matching. The efficiency of this method evaluated the robustness against irregular shading and deformation. Matsuda et al. compared the accuracies of the study with some conventional methods based on template matching. Khellat-Kihel et al. [39] proposed an identification system using a multimodal-fusion technique by adopting several techniques at different levels. This multimodal-fusion technique utilizes the fusion of finger vein, finger-knuckle print, and fingerprint. The system employs a combination of methods at different levels for the multimodal-fusion, including feature-level fusion and decision-level fusion. An optimization method is introduced to improve the feature level fusion, which involves reducing the feature space by applying various methods. In 2015, Ma et al. [40] proposed a new FV and fingerprint authentication system based on multi-route detection. This study introduces a new biometric identity authentication system that combines fingerprint and finger vein recognition using a multi-route approach. The system first designs separate classifiers for fingerprint and finger vein images, then fuses the feature vectors extracted from the first stage to form a third classifier. The last recognition outcome is obtained by aggregating the results of the three classifiers at the decision level. The conclusion of the research shows that the algorithm surpasses the limitations of single-modal biometrics and enhances the overall recognition performance. Kaur and Mishra [41] emphasized improving the effectiveness of the finger vein networks by merging the repeated line tracking, Gabor Filter, and segmentation with Neural Networks. Rosdi et al. [42] used a feature extraction algorithm called local line binary pattern as a new texture descriptor. Unlike the local binary pattern that uses a square shape as the neighborhood configuration, the local line binary pattern uses a straight line configuration. In the study of Yang et al. [43], a novel approach is proposed that combines Gabor wavelets and circular Gabor filter. This method highlights the finger-vein networks and removes non-vascular regions. The process begins with the application of Gabor wavelets to improve the vascular regions in the image, followed by image restoration by a combination rule. Lastly, the extraction of finger veins is accomplished using a circular Gabor filter. Yang and Li's [44] study addressed the challenges of finger vein localization and feature extraction. The physical characteristics of human fingers are utilized to locate the Region of Interest (ROI) in the finger vein images based on inter-phalangeal joint information and eliminate non-informative vein content. Then, the vein images are characterized using a series of energy features obtained through the application of steerable filters and classified using the Nearest Neighbor Classification method. Despite the extensive research conducted in the literature, there is still a need for further efforts to obtain more discriminative features and new fusion techniques should be applied to achieve better results.

In this study, the effect of the HVTP feature, which was proposed as a novel feature extraction method in the study of Titrek and Baykan [45], on the success rate after using the fusion technique with GLRLM, GLCM, and SFTA feature extraction is investigated. HF and PMAD are employed to address the noise and light scattering issues in our FV image databases. Texture features are described using GLRLM, GLCM, SFTA, HTP, and VTP methods. The fusion of multiple features, as opposed to relying on a single feature, is expected to enhance the accuracy of FV recognition systems. The study's contribution lies in the fusion of HTP and VTP with GLRLM, GLCM, and SFTA features.

4.1 Finger vein image databases

Experiments are carried out on two different available databases for this study. First of them is the database used in [47] which will be referred to as Database_1 in this study. And the second one is the MMCBNU_6000 finger vein database which will be referred to as Database_2 [48].

Yang and Zhang [47] from Tianjin Key Lab for Advanced Signal Processing were asked to submit a sample of the FV image databases they used in their study. The samples’ resolutions in the finger vein database, which consists of 15 index finger data of 64 individuals, are generally 170x80 pixels and stored in JPG format. Database_1 consists of 960 images total and FV images in this database were captured using a NIR sensitive CCD camera over a 760 nm wavelength NIR light source [44]. Figure 4 shows some FV images from Database_1.

4.png

Figure 4. Original images in Database_1 for separate classes

Database_2, also known as MMCBNU_6000, is a finger vein image database that contains images from 100 people. The images were collected by the Division of Electronic and Information Engineering at Chonbuk National University. Each person was requested to provide images of the index finger, middle finger, and ring finger from both the left and right hands. Thus, obtained 60 FV images by taking ten pictures from each of the six fingers of every volunteer. Hence, a total of 6000 FV images were obtained from 100 volunteers. The samples' resolutions in the finger vein database are 480x640. The extracted ROI images were normalized to 60x128 pixels and stored in BMP format [48]. The images taken from six separate fingers of both hands were accepted as different classes and increased the number of classes from 100 to 600. Figure 5 shows some FV images taken from six different fingers of a volunteer.

5.png

Figure 5. Original ROI images in different classes of each finger of a volunteer in Database_2

4.2 Homomorphic filter

The visibility of FV is significantly challenged by image distortion resulting from light scattering in the tissue. To address this issue, HF is applied to enhance the vein region visibility by removing the light scattering present in the image. HF works by eliminating local imbalances in the image exposure. The distribution of light in an image is determined by multiplying the reflectance of objects with the illumination of the scene, and this theory forms the basis of the HF.

$\operatorname{Img}(i, j)=\operatorname{LightSource}(i, j) \cdot \operatorname{RefObject}(i, j)$                    (1)

Img(i,j) is the resulting image while LightSource(i,j) is the intensity of the illuminating Light source. RefObject(i,j) is the reflectance of the object scaled between 0 and 1. It is assumed that the illumination in the image possesses low pass characteristics [49]. That's why a high pass filter is needed after taking a Fourier Transform of the image. This process is followed by the inverse Fourier transform and the contributions of the light source are effectively separated from the image based on the quality of the filter choice. The application of the HF to the finger vein image eliminates light scattering from the image. Figure 6 shows a sample result of a HF.

6.png

Figure 6. Result of HF

4.3 Perona-Malik anisotropic diffusion

To achieve more successful segmentation of the veinous area in the image, which is free from light scattering, it is necessary to remove the noise in the image. This can be accomplished by applying a smoothing process to the image. The process of smoothing an image can be explained as the outcome of a diffusion process that does not compromise the important data content, which is based on Fick's law [50]. PMAD is a space-variant smoothing filter that employs non-linear anisotropic diffusion based on the content of the data. It is designed to reduce high frequency components while preserving important parts of the data. The filter achieves this by adjusting the diffusivity signal according to the content of the data. The diffusivity will change according to the selected diffusion coefficient (DC) between 0 and 1. When the DC is set to 1, the output obtained from PMAD will be the same as the Linear Diffusion filter [51]. However, as the process nears the significant parts of the data, the DC approaches to zero. The DC is calculated in Eq. (2) by utilizing the magnitude of the first derivative.

$\text{DC}=g{{(\frac{\partial }{\partial x}u)}^{2}}$                     (2)

Through the use of Eq. (2), the PMAD equation [52] is obtained in Eq. (3) where u is the input data.

$\frac{\partial u}{\partial t}=\frac{\partial }{\partial x}(DC)$                      (3)

The result is obtained in Eq. (4) where the λ is a contrast parameter that distinguishes between areas of forward diffusion and those of backward diffusion.

$g(\frac{\partial }{\partial x}u)=\frac{1}{(1+(\frac{{{(\frac{\partial }{\partial x}u)}^{2}}}{{{\lambda }^{2}}}))}$                      (4)

Figure 2 depicts the resultant image obtained through the application of the PMAD to the output image of the HF.

4.4 Niblack segmentation

The FV images often exhibit light scattering and contrast issues, therefore, the use of a multi thresholding method is required. One of the locally adaptive multi thresholding algorithms is Niblack's Segmentation [53]. Threshold values are calculated at each pixel depending on the local mean and local SD of the neighboring pixels' intensity values by using Eq. (5).

$T(i, j)=m(i, j)+k \cdot \sigma(i, j)$                       (5)

The algorithm uses a fixed-size window to calculate the mean of the sum of pixels (m) and the SD (σ) of the window. A constant value k is chosen by the user to adjust the threshold value. If the k is chosen as 0 then the threshold value is determined only by the local mean. The window dimension should be chosen to balance between preserving image details and reducing noise. An output image of Niblack’s Segmentation is shown in Figure 3.

4.5 Feature extraction methods

Textural features are a crucial component to characterize the spatial relationship between each pixel and its neighboring pixels. This technique characterizes images to determine changes in functional characteristics. In the present study, the feature sets were generated by using the gray level run length matrices, the gray level co-occurrence matrices, the segmentation-based fractal texture analysis, and the Horizontal and Vertical Total Proportion.

4.5.1 Gray level run length matrices method

GLRLM is a method of trying to obtain information about the texture feature of an image. This method is based on counting the number of pairs of gray level values and the length of runs in the ROI area with their directions. A gray level run can be expressed as a set of pixels with the same gray value distributed as successively and collinearly in the ROI along with some defined directions to the system. Thus, it is tried to obtain information about the texture feature of an image. The length of the gray level run refers to the number of pixels counted within a specific ROI [54, 55]. This value is also associated with each run, providing further information about the properties of the image. For a given image, gray level run length matrice P(i, j | θ) gives the number of occurrences for the pixel gray level ‘i’ in the direction ‘θ’ repeated consecutively for ‘j’ times. GLRLM is computed for each image in four directions, θ=0º, 45º, 90º, 135º. From each of the matrices, we have considered 11 statistical features derived in the study of Belur et al. [56-58]. Therefore, 4x11 feature vectors are created in total for every direction.

Five run length statistical features are derived by Galloway [58] using GLRLM P(i, j | θ) as follows. The equations use abbreviations for the terms Run Emphasis (RE), Gray Level (GL), and Gray Level Emphasis (GLE).

$Short\text{ }RE\text{ = }\frac{1}{{{n}_{r}}}\sum\limits_{i=0}^{{{N}_{GL}}-1}{ \sum\limits_{j=1}^{{{N}_{RL}}}{\frac{P(i,j\text{ }|\text{ }\theta )\text{ }}{{{j}^{2}}}}}$                     (6)

$Long\text{ }RE\text{ = }\frac{1}{{{n}_{r}}}\sum\limits_{i=0}^{{{N}_{GL}}-1}{ \sum\limits_{j=1}^{{{N}_{RL}}}{P(i,j\text{ }|\text{ }\theta )\cdot {{j}^{2}}}}$                     (7)

$GL\text{ }Nonuniformity\text{ = }\frac{1}{{{n}_{r}}}{{\sum\limits_{i=0}^{{{N}_{GL}}-1}{ \left( \sum\limits_{j=1}^{{{N}_{RL}}}{P(i,j\text{ }|\text{ }\theta )} \right)}}^{2}}$                 (8)

$Run\text{ }Length\text{ }Nonuniformity\text{ = }\frac{1}{{{n}_{r}}}{{\sum\limits_{j=1}^{{{N}_{RL}}}{ \left( \sum\limits_{i=0}^{{{N}_{GL}}-1}{P(i,j\text{ }|\text{ }\theta )} \right)}}^{2}}$             (9)

$Run\text{ }Percentage\text{ = }\frac{{{n}_{r}}}{{{n}_{p}}}$                    (10)

where, N_GL  is the number of gray levels, N_RL  is the maximum run length, n_r  is the total number of runs, and n_p  is the total number of pixels in the image. The two more features of run length statistics based on the idea to highlight further the effect of gray level information derived by Chu et al. [57] are as follows.

$Low\text{ }GL\text{ }RE\text{ = }\frac{1}{{{n}_{r}}}\sum\limits_{i=0}^{{{N}_{GL}}-1}{ \sum\limits_{j=1}^{{{N}_{RL}}}{\frac{P(i,j\text{ }|\text{ }\theta )\text{ }}{{{i}^{2}}}}}$                    (11)

Four more run length statistical features based on increasing the effect of gray level and run length information on the features are derived by Dasarathy and Holder [56] and presented as follows.

$Short\text{ }Run\text{ }Low\text{ }GLE\text{ = }\frac{1}{{{n}_{r}}}\sum\limits_{i=0}^{{{N}_{GL}}-1}{ \sum\limits_{j=1}^{{{N}_{RL}}}{\frac{P(i,j\text{ }|\text{ }\theta )\text{ }}{{{i}^{2}}\cdot  {{j}^{2}}}}}$             (13)

$Short\text{ }Run\text{ }High\text{ }GLE\text{ = }\frac{1}{{{n}_{r}}}\sum\limits_{i=0}^{{{N}_{GL}}-1}{ \sum\limits_{j=1}^{{{N}_{RL}}}{\frac{P(i,j\text{ }|\text{ }\theta )\cdot {{i}^{2}}\text{ }}{{{j}^{2}}}}}$           (14)

$Long\text{ }Run\text{ }Low\text{ }GLE\text{ = }\frac{1}{{{n}_{r}}}\sum\limits_{i=0}^{{{N}_{GL}}-1}{ \sum\limits_{j=1}^{{{N}_{RL}}}{\frac{P(i,j\text{ }|\text{ }\theta )\cdot {{j}^{2}}\text{ }}{{{i}^{2}}}}}$              (15)

$Long\text{ }Run\text{ }High\text{ }GLE\text{ = }\frac{1}{{{n}_{r}}}\sum\limits_{i=0}^{{{N}_{GL}}-1}{ \sum\limits_{j=1}^{{{N}_{RL}}}{P(i,j\text{ }|\text{ }\theta )\cdot {{i}^{2}}\cdot {{j}^{2}}}}$             (16)

4.5.2 Gray level co-occurrence matrices method

Statistical features extracted from the relationship between the intensity values of the pixels are one of the methods used in recognizing and distinguishing objects. The features extracted by Haralick and Shanmugam [59] are applied to FV recognition systems, and their distinctiveness is examined in this study. These features make quadratic statistical inferences about the properties of the image by calculating the spatial relationship of the pixels. In general, the relationship between the structure of an image and the objects is assumed to be hidden in the spatial relationship of the gray tones in the image to each other. This texture-context information, expressed as P(i, j | d, θ) in our grayscale image, is assumed to be determined as the distance d at an angle θ for the i and j, which are two neighboring resolution pixel values. Matrice obtained from the grayscale image gives a function of the distance and angular relationship between neighbor resolution cells.

In the present study, 20 GLCM features based on the second-order statistical probability derived in the studies [59-62] are used. Let P(i, j) be the (i, j)th entry in a normalized gray level spatial dependence matrices and N_g  be the number of distinct gray levels in the quantized image. The equations use abbreviations for the terms Inverse Difference (ID), Information Measures of Correlation (IMC), and Maximal Correlation Coefficient (MCC).

Suppose that,

$\begin{gathered}P_{x+y}(k)=\sum_{i=1}^{N_g} \sum_{i=1}^{N_g} \mathrm{P}(\mathrm{i}, \mathrm{j}), \mid i+j=k\left\{k=2,3, \ldots, 2 N_g\right\}, \\ P_{x-y}(k)=\sum_{i=1}^{N_g} \sum_{j=1}^{N_g} \mathrm{P}(\mathrm{i}, \mathrm{j}),|| i-j \mid=k\left\{k=0,1, \ldots, N_g-1\right\}, \\ P_x(i)=\sum_{j=1}^{N_g} \mathrm{P}(\mathrm{i}, \mathrm{j}) \text { and } P_y(j)=\sum_{i=1}^{N_g} \mathrm{P}(\mathrm{i}, \mathrm{j}) .\end{gathered}$

$Angular\text{ }Second\text{ }Moment\text{ = }\sum\limits_{i=1}^{{{N}_{g}}}{{}}\sum\limits_{j=1}^{{{N}_{g}}}{{{\{P(i,\text{ }j)\}}^{2}}}$                       (17)

$Entropy\text{ = -}\sum\limits_{i=1}^{{{N}_{g}}}{{}}\sum\limits_{j=1}^{{{N}_{g}}}{P(i,j)\text{ }(\log \text{ }(P(i,\text{ }j))}$                       (18)

$Dissimilarity\text{ = }\sum\limits_{i}^{{{N}_{g}}}{ \sum\limits_{j}^{{{N}_{g}}}{\left| i-j \right|\cdot \text{ }P(i,j)}}$                   (19)

$ID\text{ = }\sum\limits_{i}^{{{N}_{g}}}{ \sum\limits_{j}^{{{N}_{g}}}{\frac{1\text{ }}{\text{ }1+(i-j)\text{ }}\text{ }P(i,j)}}$                      (20)

$ID\text{ }Moment\text{ = }\sum\limits_{i}^{{{N}_{g}}}{ \sum\limits_{j}^{{{N}_{g}}}{\frac{1\text{ }}{\text{ }1+{{(i-j)}^{2}}\text{ }}\text{ }P(i,j)}}$                      (21)

$Correlation\text{ = }\frac{\sum\limits_{i=1}^{{{N}_{g}}}{{}}\sum\limits_{j=1}^{{{N}_{g}}}{\left. \left\{ (i-{{\mu }_{x}})(j-{{\mu }_{y}})P(i,j) \right. \right\}}}{\text{ }{{\sigma }_{\text{x}}}{{\sigma }_{\text{y}}}\text{ }}$                   (22)

where, $\mu_x, \mu_y, \sigma_x$, and $\sigma_y$ are the means and SD of P_x and P_y.

$Autocorrelation\text{ = }\sum\limits_{i=1}^{{{N}_{g}}}{{}}\sum\limits_{j=1}^{{{N}_{g}}}{\left( i\text{ }j \right)P(i,j)}$                         (23)

$Contrast\text{ = }\sum\limits_{n=0}^{{{N}_{g}}-1}{{{n}^{2}} \left\{ \sum\limits_{i=1}^{{{N}_{g}}}{{}}\sum\limits_{j=1}^{{{N}_{g}}}{P(i,j)} \right\}}\text{          }\left| i-j \right|=n$                    (24)

$Cluster\text{ }Shade\text{ = }\sum\limits_{i=1}^{{{N}_{g}}}{{}}\sum\limits_{j=1}^{{{N}_{g}}}{\left. \left\{ {{(i+j-{{\mu }_{x}}-{{\mu }_{y}})}^{3}}P(i,j) \right. \right\}}$                      (25)

$Cluster\text{ Prominence =}\sum\limits_{i=1}^{{{N}_{g}}}{{}}\sum\limits_{j=1}^{{{N}_{g}}}{\left. \left\{ {{(i+j-{{\mu }_{x}}-{{\mu }_{y}})}^{4}}P(i,j) \right. \right\}}$                     (26)

$Maximum\text{ Probability = max  }P(i,j)\text{      for all (i, j)}$                        (27)

$Sum\text{ }Of\text{ }Squares\text{ = }\sum\limits_{i=1}^{{{N}_{g}}}{{}}\sum\limits_{j=1}^{{{N}_{g}}}{{{(i-{{\mu }_{{}}})}^{2}}P(i,j)}$                          (28)

$Sum\text{ }Average\text{ = }\sum\limits_{i=2}^{2{{N}_{g}}}{i {{P}_{x+y}}(i)}$                         (29)

$Sum\text{ }Entropy\text{ =  -}\sum\limits_{i=2}^{2{{N}_{g}}}{{{P}_{x+y}}(i)\text{ }(\log \text{ }({{P}_{x+y}}(i))}$                      (30)

$Sum\text{ }Variance\text{ =  }\sum\limits_{i=2}^{2{{N}_{g}}}{{{(i-{{f}_{SE}})}^{2}}\text{ }{{P}_{x+y}}(i)}$                   (31)

$Difference\text{ }Variance\text{ = Var of }{{P}_{x-y}}(i)$                    (32)

$Difference\text{ }Entropy\text{ = -}\sum\limits_{i=0}^{{{N}_{g}}-1}{{{P}_{x-y}}(i)}\text{ }(\log \text{ }({{P}_{x-y}}(i))$                    (33)

$IM{{C}_{1}}\text{ = }\frac{{{H}_{xy}}-{{H}_{xy1}}}{max  \left\{ {{H}_{x}},{{H}_{y}} \right\}}$                    (34)

where,

$\begin{align}  & {{H}_{xy}}=\text{ -}\sum\limits_{i=1}^{{{N}_{g}}}{{}}\sum\limits_{j=1}^{{{N}_{g}}}{P(i,j)\text{ }(\log \text{ }(P(i,j))}, \\ & {{H}_{xy1}}=\text{ -}\sum\limits_{i=1}^{{{N}_{g}}}{{}}\sum\limits_{j=1}^{{{N}_{g}}}{P(i,j)\text{ }\log \text{ }\{{{P}_{x}}(i)\text{ }{{P}_{y}}(j)\}}, \\ & {{H}_{xy2}}=\text{ -}\sum\limits_{i=1}^{{{N}_{g}}}{{}}\sum\limits_{j=1}^{{{N}_{g}}}{{{P}_{x}}(i)\text{ }{{P}_{y}}(j)\text{ }\log \{{{P}_{x}}(i)\text{ }{{P}_{y}}(j)\}}, \\ & {{H}_{x}}\text{ }and\text{ }{{H}_{y}}\text{ are entropies of }{{P}_{x}}\text{ and }{{P}_{y}}. \\\end{align}$

$IM{{C}_{2}}\text{ = }\sqrt{\text{(1- exp }\!\![\!\!\text{  -2}\text{.0 (}{{H}_{xy2}}-{{H}_{xy}})])}$                       (35)

$MMC\text{ = }\sqrt{({{\text{2}}^{\text{th}}}\text{ largest eigenvalue of Q)}}$                         (36)

where,

$Q(i,j)=\sum\limits_{k}^{{}}{\frac{P(i,k)P(j,k)\text{ }}{{{P}_{x}}(i){{P}_{y}}(k)}}$

4.5.3 Segmentation-based fractal texture analysis method

To implement the SFTA algorithm, a gray level image and the number of threshold levels to be used are required. There are two main steps in the implementation of the SFTA algorithm. In the first step, the multi-threshold levels are determined by applying the OTSU multilevel thresholding algorithm to the gray level image with the total number of threshold levels. The found n thresholds are stored in the vector $T=t_1, t_2, t_3, \ldots, t_n$. These thresholds are added in pairs to the vector $T_A=\left(\right.$Min$\left._{\text{Grayvalue}} \quad, t_1\right),\left(t_1, t_2\right) \ldots\left(t_{n-1}, t_n\right)$. After T_A  is completed, in vector T_B  all the threshold levels are stored as $T_B=\left(t_1, Max_{GrayVal}\right)\,\,,\left(t_2, Max_{{GrayVal }} \,\, \right), \ldots,\left(t_n, Max_{ {Grayval }}\,\,\right)$. These threshold pairs are applied separately to the gray image using the Two Threshold Binary Decomposition method, and the gray level image is converted into binary images.

${{I}_{b}}(x,y)\text{ = }\left\{ \begin{align}  & \text{1,    if   }{{\text{t}}_{lower}}<{{I}_{g}}(x,y)\le {{\text{t}}_{upper}} \\ & \text{0,    Otherwise} \\\end{align} \right.\text{ }$                    (37)

Thus, our input image is converted into many different sub-binary images. $I_b: binary \,\,image, I_g: gray \,\,image,\,\,t_{\text {lower }}$ and $t_{u p p e r}$ values represent threshold pairs. In the second step, the features are extracted by using these binary images.

$\Delta (x,y)\text{ = }\left\{ \begin{align}  & \text{1,    }if\text{ }\exists \text{ }(x',y')\in {{N}_{8}}\text{ }\!\![\!\!\text{ }(x,y)]: \\ & \text{       }{{I}_{b}}(x',y')=0\text{    }\wedge \text{   }{{I}_{b}}(x,y)=1, \\ & \text{0,    Otherwise} \\\end{align} \right.\text{ }$                             (38)

N₈[(x,y)] represents the eight pixels connected to (x,y), and $\Delta(x, y)$ represents the binary image boundary. Edge pixels in each of the resulting images are obtained, and the number of pixels is counted. By using the edge pixel coordinates, the average gray level values in the gray level images are calculated. The Fractal dimension information is obtained by using the box-counting method for each binary sub-image, and the feature vector is created [63].

4.5.4 Horizontal and vertical total proportion features

HTP and VTP are two methods that gather statistical information from FV images by using the intersection coordinates of the horizontal and vertical axes. The underlying principle of these methods is to calculate the proportion of distances between veins intersecting a horizontal or vertical plane in order to distinguish between different classes [45]. These features are extracted from the skeletonized FV images. Assuming (A_x, A_y)  is the center of gravity of the FV image: The coordinates of the FV points, intersecting the horizontal and vertical lines, are determined as in Figure 7. In the horizontal plane, the intersection points are denoted as $X_1, X_2$, and $X_3$ with the maximum subscript number indicating the number of intersections (# of INTs) [45].

7.png

Figure 7. Definition of HTP and VTP features

HTP value is obtained by applying the formula in Eq. (39).

$HTP=(\frac{{{X}_{1}}}{{{X}_{2}}}+\frac{{{X}_{2}}}{{{X}_{3}}}+...)*(\#\text{ }of\text{ }INTs)$                           (39)

The points where the veins intersect in the vertical plane are represented by $Y_1, Y_2, Y_3, Y_4$, and $Y_5$. The maximum subscript number indicates the total number of intersections in the vertical plane.

VTP value is obtained by applying the formula in Eq. (40).

$VTP=(\frac{{{Y}_{1}}}{{{Y}_{2}}}+\frac{{{Y}_{2}}}{{{Y}_{3}}}+...)*(\#\text{ }of\text{ }INTs)$                     (40)

The features we obtained from the combination of the HTP and the VTP are called Horizontal and Vertical Total Proportion.