**5. Feature Fusion and Matching**

In this section, a gray histogram-based feature matching method is used for finger trimodal fusion recognition, as shown in Figure 10. First, the coded finger trimodal images are uniformly separated into *M* non-overlapping division blocks. Then, the *M* local histograms corresponding to each sub-block are established, respectively. Assuming that *Hfv<sup>i</sup>* (*I* = 1, 2, ... , *M*) represents the histogram of the *i*th division block in a coded finger-vein image, the global histogram *Hfv* is defined as

$$H\_{fv} = \left(H^1, H^2, \dots, H^M\right) \tag{10}$$

**Figure 10.** The fusion of finger trimodal features.

Similarly, *Hfp* and *Hfkp* represent the global histogram of a coded fingerprint image and finger-knuckle-print image. Then, the final feature histogram *H* of a finger trimodal image can be expressed by

$$H = \left( H\_{f\flat\prime} H\_{f\flat\prime} H\_{f\flat p} \right) \tag{11}$$

After the above calculation, we can obtain the feature histogram of each finger sample. Here, we can use various classification algorithms, such as SVM, ELM and *k*-NN [34]. In this section, for convenience, the intersection coefficient between two feature vectors is calculated to determine the similarity of two individuals [29]. Assuming *H1(i)* and *H2(i)* denote the histograms of two samples to be matched, the similarity can be computed by

$$\text{sim}(H\_1(i), H\_2(i)) = \frac{\sum\_{i=1}^{L} \text{min}[H\_1(i), H\_2(i)]}{\sum\_{i=1}^{L} H\_1(i)} \tag{12}$$

where *L* denotes the dimension of a feature vector to be matched. In the matching process, if the intersection coefficient *sim*(·) is >T (similarity decision threshold), it means that the two samples are similar and are able to be matched. But if the intersection coefficient *sim*(·) is ≤T, it means that the two samples are not matched. Thus, two samples will tend to be more similar as the intersection coefficient increases. The similarity decision threshold T corresponds to the threshold value when the false rejection rate (FRR) is the same as the false accept rate (FAR).
