Horaud et al. Duchenne et al. First-order tensor describes similarities of each feature pair.

Second-order tensor describes compatibilities between each two feature pairs. Leordeanu et al. They recovered the matching relationship based on spectral matching methods, by using the principal eigenvector of adjacency matrix and imposing the one-to-one mapping constraints. This notion can be introduced into minutia matching. Based on these insights, we propose a tensor matching strategy. We construct the minutia tensor matrix simplified as MTM for fingerprint minutiae. It unifies both the first-order features and the second-order features.

The diagonal elements in MTM indicate similarities of each minutia pair and other elements indicate pairwise compatibilities between minutia pairs. Correct minutia pairs are more likely to establish both large similarities within them and large compatibilities among them, thus they form a dense sub-block.

Incorrect pairs establish links with the other pairs accidentally, so they are unlikely to belong to dense sub-blocks The definition of dense sub-block can be seen in section 2. Minutia matching is formulated as recovering the main dense sub-block in the MTM. Our previous papers tried to apply the tensor idea to fingerprint matching [ 24 ] [ 25 ].

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We applied the spectral matching strategy to fingerprint global matching [ 24 ]. The disadvantage was that it was only applied to global matching, while other strategies were still required for local matching. We proposed extended clique models to deal with local matching and global matching [ 25 ]. The disadvantage is that the local matching process is time-consuming. Meanwhile, local matching is not associated with global matching. This paper is innovative on the basis of previous papers.

First, the tensor strategy is used in local matching for the first time. Main contributions of this paper are: First, the proposed MTM unifies both similarities and compatibilities appropriately. Second, minutia matching is formulated as recovering the dense sub-block in the MTM. Optimal matching relationship corresponds to the dense sub-matrix in the MTM.

Third, spectral matching methods are then used for recovering the dense block.

It is efficient and effective , which can be seen in the following experiments. Forth, two MTM s with different constraints are constructed, which makes use of local rigidity and global compatibility, respectively. In the local matching level, we build the local minutia topologic structures and construct local MTM for each minutia structure pair.

Calculating similarities of minutia pairs corresponds to recovering the dense sub-block in the local MTM. In the global matching level, we construct the global MTM for entire minutia sets. Calculating similarities of minutia sets corresponds to recovering the dense sub-block in the global MTM. Proposed approach has stronger description ability and better robustness to non-linear deformation and noise.

Suppose there are two fingerprint minutia sets P and P' , with N p and N p' minutiae, respectively.

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We represent node attributes as special edge attributes, i. A ii for node i. So it is with P'. Let S A ii' , jj' indicate the similarity score between attributes A ij and A' i'j'. We want to find a mapping that best preserves the attributes of nodes and edges between attributed graphs G P and G P'. High-order tensors are constructed to represent high-order affinities [ 22 ]. Inspired by this notion, we propose the minutia tensor matrix MTM.

## A Fingerprint Matching Technique Based on Phase-Only Correlation

It is proposed based on these considerations: First-order tensor matrix describes similarities of each minutia pair. Second-order tensor matrix describes compatibilities between each two minutia pairs. For fingerprint minutia sets P and P' , each minutia pair i , i' is assigned an similarity attribute T 1 i , i' and each two minutia pairs ii' , jj' is assigned a compatibility attribute T 2 ii' , jj'. Calculation of T 1 i , i' and T 2 ii' , jj' can be seen in section 2.

Thus we can build the first-order tensor matrix T 1 and the second-order tensor matrix T 2.

The fusion rule is shown in the formula:. Where T 1 is the first-order tensor matrix, whose element T 1 i , i' describes the similarity of minutia pair i , i'. T 2 is the second-order tensor matrix, whose element T 2 ii' , jj' describes compatibility between i , i' and j, j'. It unifies both similarities and compatibilities among minutia pairs. As minutia matching is one-to-one matching, minutia i in P mapped to both minutia i' and j' in P' is impossible.

We use this priori knowledge to prune and can get a very sparse MTM. It is set manually. T F is a symmetric matrix as T F ii' , jj'. As shown in Fig. Correct minutia pairs are likely to establish links among each other and thus form a strongly connected subgraph, which is also the dense subgraph. Incorrect pairs establish links with the other pairs only accidentally.

Minutia matching can be formulated as seeking for the dense subgraph in the correspondence graph. For clarity, only a small subset of minutia pairs are shown.

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Candidate minutia pairs shown in a form the correspondence graph in b and the minutia tense matrix MTM in c. Genuine minutia pairs corresponds to the dense subgraph of correspondence graph, and also the dense block of MTM. Minutia matching is formulated as recovering the dense sub-block in the MTM. It can be solved by the spectral correspondence methods. In T F , the dense sub-block T D is the biggest sub-matrix in T F , whose elements in all the rows and all the columns have big values.

The element values in the dense sub-block T D should be greater than a certain threshold. Correct pairs are likely to establish elements with big values among each other and thus form a strongly connected cluster, which is also the dense sub-block T D. Minutia matching corresponds to recovering the dense sub-block. The main idea is illustrated in Fig.

The optimization problem 3 can be solved by spectral matching method, which will be discussed in section 2. Formula 3 yields to the following binary optimization problem:. It can be derived by using the principal eigenvector of T F and imposing the one-to-one mapping constraints, which is proposed in [ 23 ].

Here we make some improvements. Firstly we prune the elements with small similarities or low compatibilities. Secondly we relax both the mapping constraints and the integral constraints on m , so that its elements can take real values in [ 1 ]. As minutia pairs with big local structural similarities are more likely to have large compatibilities, we initialize m 0 by normalized first-order tensor matrix T 1. It converges much faster using this strategy. Input: minutia tensor matrix T F. Indeed, each step of the iteration algorithm requires only O z operations, where z is the number of non-zero elements of the matrix.

Also, typically, in our situation, the algorithm converges in a few dozen steps. Meanwhile, minutia matching is one-to-one matching, minutia i in P mapped to both minutia i' and j' in Q is impossible. We add this compatibility constraint during the iteration process. It is summarized as Algorithm 2. Corrected matched pairs can be gained and similarities of minutia sets can be evaluated. We can get the number N m of matched pairs. After that we adjust the similarity score with the number of matched pairs. Take minutia sets P and P' for example, the similarity S PP' is calculated using the following formula:.

S ii' measures the similarity of matched pairs ii'. N m indicates the number of matched pairs, and N p and N p' indicate the minutia number of P and P'. In practice, we use the tensor matching strategy both in local matching level and global matching level. There are some differences in these two levels.

We will give detailed descriptions in section 2. Local minutia topologic structure simplified as LMTS is firstly introduced in [ 12 ].

## Minutia Tensor Matrix: A New Strategy for Fingerprint Matching

A typical LMTS is constructed from a center minutia and a list of neighboring minutiae in a specified area. Fingerprint images show rigidity within the range of LMTS. Each LMTS can be seen as a small minutia set.

Minutia pairs within two local minutia topologic structures shown in a form the local correspondence graph in b and the local minutia tense matrix local MTM in c. Correct pairs corresponds to the dense subgraph of local correspondence graph, and also the dense block of local MTM. Minutia matching is formulated as recovering the dense sub-block in the local MTM. According to the coordinates and orientations of minutiae in L a , we calculate the distance vector v - between each two minutiae within L a , so it is with L a'.