An effective fingerprint orientation field estimation method using differential values of grayscale intensity

Differential values of grayscale intensity in each orientation

For each fingerprint image, we drew straight lines through the center of the image in N (4, 8, 16, 32, etc.) orientations, and plotted line graphs of the grayscale values of the image along each straight line. For example, Figure 1 shows 16 straight lines with 16 orientations being used for the fingerprint image. If the straight lines are drawn more orthogonal to the orientation of the fingerprint, the grayscale values are more sensitive along the orientation of the lines. In contrast, if the straight lines are drawn close to the orientation of fingerprint, the grayscale value is less sensitive to changes. Figure 2 shows the line graphs of the grayscale values of the 16 orientations, where the x-axis is the index of each pixel and the y-axis is the corresponding grayscale value of the pixel. We then differentiated the grayscale value of each orientation and calculated its absolute value to represent the degree of change in the grayscale value, with smaller values indicating that the orientation of the straight line is closer to the orientation of the fingerprint . Because of smaller variances in grayscale values, we estimated that the fingerprint orientation of the fingerprint image in Finger 2 is around 56.25°. The aim of this study is to create an efficient and dependable algorithm for OF estimation using the aforementioned concept. To simplify the calculations, differential operations will be substituted with convolution calculations. Figure 3 provides the calculation steps used in this proposed method. In our experimental results, the performance of the proposed OF estimation increased as the number of orientations increased. However, because the convolution kernel size should increase with the number of orientations, the computational demand of 32 orientations is much higher than 16 orientations, and the convolution kernel size should also not be larger than the pixels between fingerprint ridges (10 pixels), so we selected 16 orientations for this study.

Image slicing

The first step of the proposed OF estimation algorithm is slicing each fingerprint image into 10 equal parts along the x and y axes (Cartesian coordinates), resulting in 100 blocks of 16 x 16 pixels from each fingerprint image. We decided on 10 divisions on each axis because each small block should not be smaller than the kernel size of convolution. We then estimated the orientation field for each individual block. Figure 4 depicts image slicing.

Image convolution

The second step of the proposed algorithm is performing a convolution calculation on each block, as shown in Eq. (1), where x is the grayscale value of the image, h is the convolution kernel, y is the convolution result, m and n denote the column and row of the block, and i and j denote the column size and row size of the kernel, respectively. Our method performs a convolution calculation on each block with 16 kernels of 5 × 5 each, and each kernel representing a different orientation. Figure 5 shows the calculation results of the 16 kernels. The angular difference of each direction was 11.25 degrees, which is in line with the common Gabor filter bank (Medina et al., 2017; Mukherjee & Das, 2021), and the result of the convolution calculation was related to the differential grayscale value in each orientation. For 2D function f (x,y), the partial differential equation is shown in Eq. (2) and for discrete data, we can approximate using finite differences with Eq. (3). Then, we discuss about the Prewitt filter, which comprises two convolution kernels of size 3 × 3 each (shown in Fig. 6). These kernels are specifically designed to detect horizontal and vertical edges and can also be applied independently to determine the gradient component in each orientation (as Gx and Gy), which is equivalent to the partial differential results for 2D images (Eq. (3)). Therefore, in this article, we utilize this concept to calculate the differential using convolution.

(1)${\displaystyle y\left[m,n\right]=x\left[m,n\right]\ast h\left[m,n\right]=\sum _j^{}\sum _i^{}x\left[i,j\right]\cdot h\left[m-i,n-j\right]}$ (2)${\displaystyle \frac{\partial f\left(x,y\right)}{\partial x}=\underset{\varepsilon \to 0}{lim}\frac{f\left(x+\varepsilon ,y\right)-f\left(x,y\right)}{\varepsilon }\frac{\partial f\left(x,y\right)}{\partial y}=\underset{\varepsilon \to 0}{lim}\frac{f\left(x,y+\varepsilon \right)-f\left(x,y\right)}{\varepsilon }}$ (3)${\displaystyle \frac{\partial f\left(x,y\right)}{\partial x}=G_x\approx \frac{f\left(x+1,y\right)-f\left(x,y\right)}{1}\frac{\partial f\left(x,y\right)}{\partial y}=G_y\approx \frac{f\left(x,y+1\right)-f\left(x,y\right)}{1}.}$

Orientation field estimation (index, matrices)

After image convolution, the third step entails computing the sum of the absolute grayscale values for each block (represented as Y) for all 16 orientations (kernels). Then, the minimum Y value is determined using Eq. (4), whereby a higher Y value indicates a more substantial change in grayscale value in that orientation and a more orthogonal orientation relative to the fingerprint orientation. Conversely, a lower Y value implies greater parallelism with the fingerprint orientation. To estimate the orientation field, the minimum two Y values are weighted, and their weighted average is computed using Eq. (5). (4)${\displaystyle Y=\sum _n^{}\sum _m^{}\left|y\left[m,n\right]\right|Y_{\min }=min\left(Y_1,Y_2,Y_3,\dots ,Y_{16}\right)}$ (5)${\displaystyle O{F}_{\text{weighted average}}=O{F}_{min1}\times \frac{\frac{1}{Y_{min1}}}{\frac{1}{Y_{min1}}+\frac{1}{Y_{min2}}}+O{F}_{min2}\times \frac{\frac{1}{Y_{min2}}}{\frac{1}{Y_{min1}}+\frac{1}{Y_{min2}}}.}$

Image stitching

The fourth step of the proposed algorithm is stitching the 100 blocks back together to display the estimated fingerprint OF for the whole fingerprint image. Figure 7 shows the fingerprint image and resulting OF.

Database

In this study, we used the fingerprint data of volunteers, which were captured with an OLED panel display image sensor. These fingerprint images were only used for orientation field estimation, which was explained to the study participants. No personal information such as age, gender, or name, was collected from the volunteers. The final study dataset included 2520 images (126 finger samples, with 20 elements each). The resolution of each sample is 813 DPI and 160 × 160 pixels.

Experiment 1: accuracy assessment

To validate the accuracy of our proposed orientation field (OF) estimation algorithm, we conducted an experiment with a “control group” using the commonly-used gradient-based method. We selected 15 fingerprint images from our study database and computed the OF using the proposed algorithm, the gradient-based method, and the PSD-based method. The results were then compared and recorded. Our proposed algorithm’s estimated OF was found to be similar to that of the gradient-based method, whereas the PSD-based method had more differences with the gradient-based method. Figure 9 shows that the proposed algorithm had a deviation of more than 20 degrees in 67 blocks, while the PSD-based method had a deviation of more than 20 degrees in 119 blocks. We considered these deviations as misinterpretations, and the 67 blocks were divided by the total number of 1,500 blocks from the 15 fingerprint images to obtain an accuracy rate of 95.53% for our proposed algorithm, which is higher than the 92.06% accuracy rate for the PSD-based method. The error blocks in the fingerprints were primarily concentrated around the edges (as shown in Fig. 10) and there is a likelihood that error blocks were also present in the core area of the fingerprints. This could be attributed to the fact that the edges of the fingerprints tend to have the lowest image quality, which makes it more challenging to obtain accurate data. Furthermore, the core area of the fingerprint is a semi-circle or arc, which lacks obvious orientations, making it difficult to define the orientation of the overall fingerprint in those core blocks. According to the results of the correlation coefficient analysis, the proposed algorithm showed a strong positive correlation (r = 0.909) with the gradient-based method, indicating that the two methods are highly correlated. In comparison, the correlation coefficient between the PSD-based method and the gradient-based method was found to be 0.755, which is lower than the correlation coefficient between the proposed algorithm and the gradient-based method. These findings indicate that the proposed algorithm a more reliable and accurate method compared to the PSD-based method.

Experiment 2: accuracy assessment in blurred fingerprint images

To verify the reliability of the proposed OF estimation algorithm on blurred images, we performed Gaussian blurring on the 15 fingerprint images selected in Experiment 1. The Gaussian blur (the Gaussian smooth) is a nonuniform, low-pass filter that blurs the details of an image, but preserves low spatial frequencies and reduces image noise. This is done by convoluting an image with a Gaussian kernel; this study used a 7 × 7 kernel, as shown in Fig. 11. We calculated the estimated OF of the original images and the Gaussian blurred images using the proposed algorithm, the gradient-based method, and the PSD-based method, and compared the results of all three methods between the original and Gaussian blurred fingerprints to determine the reliability of the proposed algorithm in blurred fingerprints. The histograms of the results (shown in Fig. 12) show that there was only a small difference between the estimated OF results of the clear and blurred fingerprint images calculated by the proposed algorithm. Seven blocks had an orientation deviation of more than 20 degrees, indicating an accuracy rate of 99.53%, the average orientation deviation was 1.81°, and the correlation coefficient was 0.995, which shows that the proposed algorithm is reliable for estimating OF in blurred fingerprint images. The OF estimation results of the clear and blurred fingerprint images calculated by the gradient-based method resulted in 13 blocks with an orientation deviation of more than 20 degrees, indicating an accuracy rate of 99.13%, the average orientation deviation was 2.18°, and the correlation coefficient was 0.978. The PSD-based OF estimation results of the clear and blurred fingerprint images resulted in 126 blocks with an orientation deviation of more than 20 degrees, an accuracy rate of 91.6%, an average orientation deviation of 6.92°, and the correlation coefficient was 0.868. These results indicate that the OF estimation performance of the proposed method was slightly better than the OF estimations of the gradient-based and PSD-based methods in blurred fingerprint images.

Experiment 3: accuracy assessment in noisy fingerprint images

To verify the reliability of the proposed OF estimation algorithm in noisy fingerprint images, we performed the Gaussian white noise process on the 15 selected fingerprint images. The Gaussian white noise process is a signal processing technique used to add random noise values from a Gaussian distribution to the original pixel values of an image. The probability distribution function for a Gaussian distribution has a bell shape (or normal distribution). We then calculated the estimated OF of the original images and the images with Gaussian white noise using the proposed algorithm, the gradient-based method, and the PSD-based method, and compared the results to determine the reliability of the proposed algorithm in noisy fingerprint images. The experimental results (shown in Fig. 13) showed that the differences in estimated OF in the clear and noisy fingerprint images were much larger than in blurred fingerprints (Experiment 2) among all three algorithms. The proposed algorithm resulted in 383 blocks with an orientation deviation of more than 20 degrees, giving it an accuracy rate of 74.46%, the average orientation deviation was 13.86°, and the correlation coefficient was 0.898. The OF estimation results of the gradient-based method produced 480 blocks with an orientation deviation of more than 20 degrees, an accuracy rate of 68%, an average orientation deviation of 21.79°, and the correlation coefficient was 0.661. The OF estimation results of the PSD-based method produced 877 blocks with an orientation deviation of more than 20 degrees, an accuracy rate of 41.53%, an average orientation deviation of 42.19°, and the correlation coefficient was 0.236. These results indicate that the proposed method outperforms both the gradient-based and PSD-based methods in OF estimation in noisy fingerprint images. A comparison of the performance and positive/negative aspects of the proposed algorithm and the two classic OF estimation methods is shown in Tables 1 and 2.