# Error Correcting Codes

Priyank Deshpande

Abstract Here we present some revised arguments to a randomized algorithm
proposed by Sudan to find the polynomials of bounded degree agreeing on a
dense fraction of a set of points in $\mathbb{F}^{2}$ for some field
$\mathbb{F}$.

##  1 Introduction

Here we will discuss some concepts in the field of error-correcting codes.

######  Definition 1.

Given $\Sigma$ a collection of symbols, and $x,y\in\Sigma^{n}$. We define the
hamming distance between $x$ and $y$ denoted as $HD(x,y)$ as
$\left|\\{i\in[n]:(x)_{i}\neq(y)_{i}\\}\right|$. That is, the number of
indices at which $x$ and $y$ differ.

######  Example 1.

Given $\Sigma=\\{0,1,2\\}$ and $x=``201"$, and $y=``222"$. We have that
$HD(x,y)=1$.

######  Definition 2.

Let $\Sigma$ be a collection of symbols and $n,k,\delta\in\mathbb{Z}$. We say
$\mathcal{C}\subset\Sigma^{n}$ is a $[n,k,\delta]$ code if
$\left|\mathcal{C}\right|=\left|\Sigma\right|^{k}$, and $\forall
x,y\in\mathcal{C},HD(x,y)\geq\delta$.

######  Definition 3.

Let $\Sigma$ be a collection of symbols and $\mathcal{C}$ a $[n,k,\delta]$. If
$\tau\in\mathbb{Z}:2\tau+1\leq\delta$, then we say $\mathcal{C}$ is a $\tau$
error correcting code.

######  Definition 4.

Let $F$ be a finite field of cardinailty $n$. Let $\mathcal{C}$ be a
$[n,d+1,n-d]$ code of an alphabet $\Sigma$, we say $\mathcal{C}$ is a Reed-
Solomon Code if $\mathcal{C}=\\{``p(0)|p(w)|\dots|p(w^{|F|-1})":p(x)\in
F[x],\deg(p)\leq d\\}$. Here $|$ denotes string concatenation, and $w\in F$,
is a generator of $F^{*}$.

######  Definition 5.

We will refer the maximum-likelihood decoding problem as the following task:
Given a $[n,k,\delta]$ code, a string $s\in\Sigma^{n}$, a string in
$c\in\mathcal{C}$ such that $HD(s,c)\leq HD(s,x),\forall x\in\mathcal{C}$. We
will refer to the list decoding problem as: Given a string $s\in\Sigma^{n}$, a
$[n,k,\delta]$ code, and a parameter $\tau\in\mathbb{N}$, return all
$c\in\mathcal{C}$ such that $HD(s,c)\leq\tau$,

##  2 Algorithm

######  Remark 1.

We will present a randomized algorithm by Sudan-’96, for the following
problem: Given a field $F$, $\\{(x_{i},y_{i})\\}_{i=1}^{n}\subset F^{2}$ and
parameters $t,d\in\mathbb{N}$, find all $f(x)\in F[x]$ such that
$\left|\\{i\in[n]:f(x_{i})=y_{i})\right|\geq t$, and $\deg(f)\leq d$.

######  Remark 2.

We define concept of weighted degree which will be relevant to the randomized
algorithm to be presented. Given $(w_{x},w_{y})\in\mathbb{Z}^{2}$ which we
will call weights and a bivariate monomial in $x,y$, $c_{ij}x^{i}y^{j}$, we
say the weighted degree of such a monomial is $i\cdot w_{x}+j\cdot w_{y}$.
Given a bivariate polynomial $Q(x,y)\in F[x,y]$, we say the weighted degree of
$Q(x,y)=\sum_{i,j}c_{ij}x^{i}y^{j}$ to be the maximum of the weighted degrees
of its monomials.

######  Algorithm 1.

Define the following randomized algorithm: Let
$\\{(x_{i},y_{i})\\}_{i\in[n]}$, $d,t\in\mathbb{N}$ be inputs to the
algorithm, and $m,l$ parameters to be determined to optimize the algorithm.
Then:

  * •

Find a $P(x,y)\in F[x,y]$ such that $P(x,y)$ has weighted degree with weights
$(1,d)$ at most $m+l\cdot d$, $P(x,y)$ is not identically zero, and $P(x,y)$
vanishes on $\\{(x_{i},y_{i})\\}_{i\in[n]}$. That is,
$P(x_{i},y_{i})=0,\forall i\in[n]$. $(1)$

  * •

Factor $P(x,y)$ into irreducible polynomials in $F[x,y]$. $(2)$

  * •

Check all functions $f(x)\in F[x]$ of degree at most $d$, such that
$(y-f(x))\mid P(x,y)$, and $f(x_{i})=y_{i}$ for at least $t$ distinct choices
of $i\in[n]$. $(3)$

######  Remark 3.

We will justify that this algorithm runs in polynomial time.

######  Proposition 1.

The polynomial as described in step $(1)$ can be found in polynomial time,
with respect to the size of the field, if such a polynomial exists.

###### Proof.

By the conditions imposed by the weighted degree constraint, we can write
$P(x,y)\in F[x,y]$ as
$P(x,y)=\sum_{j=0}^{l}\sum_{i}^{m+(l-j)d}c_{ij}x^{i}y^{j}$ because $j\leq l$,
$i\leq m+(l-j)d$ implies that
$(i,j)\cdot(1,d)\leq(m+(l-j)d,j)\cdot(1,d)=m+ld$, which is the weighted degree
of $P(x,y)$. To find the polynomial which satifies the conditions in $(1)$, we
require that
$\sum_{j=0}^{l}\sum_{i=0}^{m+(l-j)d}c_{ij}(x_{k})^{i}(y_{k})^{j}=0,\forall
k\in[n]$. Let $|F|=N$. Using a brute force approach to determine the
appropriate values of $c_{ij}$, we can obtain a solution in $O(n\cdot
N^{(m+ld)l})$, which is polynomial in $N$ for fixed parameters $l,m,d$.
However, this can be solved in polynomial time with respect to the number of
constraints $n$. ∎

######  Proposition 2.

If the parameters $m,l$ are such that $(m+1)(l+1)+d\binom{l+1}{2}>n$, then a
function $P(x,y)\in F[x,y]$ as described in $(1)$ exists.

###### Proof.

Let $\eta=(m+1)(l+1)+d\binom{l+1}{2}$ Note that if $P(x,y)\in F[x,y]$ is
defined as $P(x,y)=\sum_{j=0}^{l}\sum_{i=0}^{m+(l-j)d}c_{ij}x^{i}y^{j}$, then
there are $\eta$ many $x_{ij}$’s. To find the polynomial $P(x,y)$, we need to
solve the system $A\vec{x}=0$, where $\vec{x}$ represents the $c_{ij}$, and
$A$ has dimensions $n\times\eta$. So this amounts to finding the null space of
$A$. Under the assumption that $\eta>n$, we have that $\dim(N(A))\geq 1$,
where $N(A)$ is the null space of $A$. Hence, we may choose a $y\in
N(A)\setminus\\{0\\}$ to obtain the desired $c_{ij}$’s. ∎

######  Proposition 3.

If $P(x,y)\in F[x,y]$ satisfies (1), and $f(x)\in F[x]$ satisfies
$\left|\\{i\in[n]:f(x_{i})=y_{i}\\}\right|\geq t$, and $t>m+ld$, then $y-f(x)$
divides $P(x,y)$.

######  Remark 4.

Let $f(x)\in F[x]$. Denote the condition
$\left|\\{i\in[n]:f(x_{i})=y_{i}\\}\right|\geq t$ as (*), and say $f(x)$
satifies (*) should it be the case

###### Proof.

Let $f(x)\in F[x]$ satisfy (*). We claim that $P(x,f(x))$ is identically zero.
Since $P(x,y)$ has $(1,d)$ weighted degree at most $m+ld$, we have that
$P(x,f(x))$ (as a uni-variate polynomial) has degree at most $m+ld$ since
$f(x)$ has degree at most $d$. However $P(x,f(x))=0$ whenever $x=x_{i}$ for
some $i\in[n]$. If $f(x)$ satisfies (*), then there are at least $t$ zeros.
Under the assumption that $t>m+ld$, we have that the number of roots of
$P(x,f(x))$ is greater than its degree, so $P(x,f(x))\equiv 0$. Consider
$P(x,y)=P_{x}(y)=\sum_{j=0}^{l-1}P_{j}(x)y^{j}:P_{j}(x)\in F[x]$. Since
$P_{x}(f(x))=0$, we have that $P_{x}(y)$ has a root $f(x)$. By the division
algorithm, $(y-f(x))$ divides $P_{x}(y)=P(x,y)$, which is the claim. ∎

######  Remark 5.

It remains to choose the parameters $m,l$ such that $t>m+ld$ and
$(m+1)(l+1)+d\binom{l+1}{2}>n$, We can rephrase the condition to be
$(m+1)(l+1)+d\binom{l+1}{2}\geq n+1$ Observe that this condition yields
$m\geq\frac{n+1-d\binom{l+1}{2}}{l+1}-1$. Suppose that we want $t\geq
m+ld+1\implies
t\geq\frac{n+1-d\binom{l+1}{2}}{l+1}+ld=\frac{n+1}{l+1}-\frac{dl}{2}+dl=\frac{n+1}{l+1}+\frac{dl}{2}$.
To find the minimum of this function with respect to $l$, we perform a first
derivative which yields that $\frac{-(n+1)}{(l+1)^{2}}+\frac{d}{2}=0\implies
l=\sqrt{\frac{2(n+1)}{d}}-1$. Substituting the expression for $l$ in for the
expression on $m$, we obtain that
$m\geq\frac{n+1-d\binom{l+1}{2}}{l+1}-1=\frac{n+1}{l+1}-\frac{dl}{2}-1=\sqrt{\frac{d(n+1)}{2}}-\left(\sqrt{\frac{d(n+1)}{2}}-\frac{d}{2}\right)-1=\frac{d}{2}-1$.
This yields for the condition on $t$ that $t\geq
m+ld+1\geq\frac{d}{2}+d\cdot\left(\sqrt{\frac{2(n+1)}{d}}-1\right)=\frac{d}{2}+\sqrt{2(n+1)d}-d=\sqrt{2(n+1)d}-\frac{d}{2}$.
This will allow us to make the following claim, which follows from the
previous propositions.

######  Corollary 1.

Given a field $F$ and a set of points $\\{(x_{i},y_{i})\\}_{i\in[n]}\subset
F^{2}$, and paramters $d,t\in\mathbb{N}$ such that $t\geq
d\cdot\lceil\sqrt{\frac{2(n+1)}{d}}\rceil-\lfloor\frac{d}{2}\rfloor$, then
there is a polynomial time algorithm in $n$ which finds all polynomials
$f(x)\in F[x]$ which satisfy (*), and have degree at most $d$.

###### Proof.

Setting $m=\lfloor\frac{d}{2}\rfloor-1$ gives and
$l=\lceil\sqrt{\frac{2(n+1)}{d}}\rceil-1$ gives that
$(m+1)(l+1)+d\binom{l+1}{2}\geq n+1$. By proposition 2, a function $P(x,y)$
not identically zero satisfying that $P(x_{i},y_{i})=0,\forall i\in[n]$
exists. Under the assumption that $t\geq
d\cdot\lceil\sqrt{\frac{2(n+1)}{d}}\rceil-\lfloor\frac{d}{2}\rfloor>m+ld$, we
have that $(y-f(x))$ divides $P(x,y)$ should such an $f(x)\in F[x]$ satisfy
(*). By step 3 in the algorithm, $f(x)$ will be reported as output. ∎

######  Definition 6.

Denote the tuple of $k$ variables $(x_{1},\dots,x_{k})=\vec{x}$. Let $F$ be a
field, $H\subset F$, and $g:H^{k}\xrightarrow{}F$. Given parameters
$t,d\in\mathbb{N}$, output all polynomials $f$ of degree at most $d$ such that
$\left|\\{\vec{x}\in H^{k}:f(\vec{x})=g(\vec{x})\\}\right|\geq t$. Here define
the degree of $f$ to be the maximum degree of its monomials.

######  Remark 6.

Let $f(x)\in F[\vec{x}]$. We say $f(x)$ satisfies the condition (*) if
$\left|\\{\vec{x}\in H^{k}:f(\vec{x})=g(\vec{x})\\}\right|\geq t$.

######  Definition 7.

Generalize the definition of weight degree to an $n$-variate polynomial.
First, the $(w_{1},\dots,w_{n})$ weighted degree of a monomial
$\prod_{i=1}^{n}x_{i}^{d_{i}}$ is defined to be $\sum_{i=1}^{n}w_{i}d_{i}$.
Define the $(w_{1},\dots,w_{n})$ weighted degree of an $n$-variate polynomial
to be the maximum of the weighted degrees of its monomials (which have non-
zero coefficients).

######  Algorithm 2.

Define the following algorithm. Let $F,H,k,t,d,g$ be as in definition $6$, and
$m,l\in\mathbb{N}$ be parameters to be determined.

  * •

Find a $P(x_{1},\dots,x_{k},y)\in F[x_{1},\dots,x_{k},y]$ such that
$P(\vec{x},y)$ has weighted degree with weights $(1,\dots,1,d)$ at most
$m+l\cdot d$, $P(\vec{x},y)$ is not identically zero, and $P(\vec{x},y)$
vanishes on $\\{(\vec{x},g(\vec{x})):\vec{x}\in H^{k}\\}$. That is,
$P(\vec{x},g(\vec{x}))=0,\forall\vec{x}\in H^{k}$.

  * •

Factor $P(\vec{x},y)$ into irreducible polynomials in $F[\vec{x},y]$. $(2)$

  * •

Check all functions $f(\vec{x})\in F[\vec{x}]$ of degree at most $d$, such
that $(y-f(\vec{x}))\mid P(\vec{x},y)$, and $f(\vec{x})=g(\vec{x})$ for at
least $t$ distinct choices of $\vec{x}\in H^{k}$. $(3)$

######  Remark 7.

This is more or less a generalization of the uni-variate case. We will state
conditions for the existence of $P(\vec{x},y)$, and show that a polynomial
$f(\vec{x})\in F[\vec{x}]$ satisfying (*) will be such that $y-f(\vec{x})\mid
P(\vec{x},y)$. Let $|H|=h$.

######  Proposition 4.

If $m+ld\geq k(h-1)$, then a non-trivial polynomial $P(\vec{x},y)\in
F[\vec{x},y]$ vanishing on $S=\\{(\vec{x},f(\vec{x}))\in F^{k+1}:\vec{x}\in
H^{k}\\}$ exists.

###### Proof.

We want to show that the number of monomials of a $k+1$ variate weighted
degree polynomial is greater than $|H|^{k}$. Then we can apply a similar
argument for there being a non-trivial solution to the system of linear
equations $A\vec{z}=\vec{0}$ for finding the coefficients of such a polynomial
$P(\vec{x},y)$. We observe that a polynomial of $(1,\dots,1,d)$ weighted
degree $m+ld$ contains $\sum_{j=0}^{l}\binom{m+(l-j)d+k}{k}$ monomials. This
is because $P(\vec{x},y)=\sum_{j=0}^{l}P_{j}(\vec{x})y^{j}$, where $P_{j}$ has
total degree at most $m+ld-jd=m+(l-j)d$. Hence, let $M(Q)$ denote the number
of distinct monomials of a polynomial $Q$. Then
$M(P)=\sum_{j=0}^{l}M(P_{j})=\sum_{j=0}^{l}\binom{m+(l-j)d+k}{k}$, which
implies the claim. Now we would like $M(P)>h^{k}$. To do this, we provide some
lower bounds. Observe that
$M(P)=\sum_{j=0}^{l}\binom{m+(l-j)d+k}{k}\geq\sum_{j=0}^{l}\left(\frac{m+(l-j)d+k}{k}\right)^{k}\geq\left(\frac{m+ld+k}{k}\right)^{k}+l\cdot\left(\frac{m+k}{k}\right)^{k}>\left(\frac{m+ld+k}{k}\right)^{k}\geq\left(\frac{k(h-1)+k}{k}\right)^{k}=h^{k}$,
which proves the proposition. ∎

######  Proposition 5.

If $t>(m+ld)h^{k-1}$, where $t$ is the number of agreements of a $k$-variate
polynomial $f$ on the set $S=\\{(\vec{x},g(\vec{x}))\in F^{k+1}:\vec{x}\in
H^{k}\\}$, then $y-f(\vec{x})\mid P(\vec{x},y)$.

###### Proof.

We observe that $\theta_{f}(\vec{x})=P(\vec{x},f(\vec{x}))$ is a $k$-variate
polynomial of total degree $m+ld$. Let $Z(Q,S)=\\{x\in S:Q(x)=0\\}$. By the
Schwartz-Zippel Lemma, if
$\left|Z(\theta_{f},H^{k})\right|>\deg(\theta_{f})\cdot|H|^{k-1}$, then
$\theta_{f}=P(\vec{x},f(\vec{x}))\equiv 0$. But
$t=\left|Z(\theta_{f},H^{k})\right|>\deg(\theta_{f})\cdot|H|^{k-1}=(m+ld)h^{k-1}$
by assumption so $P(\vec{x},f(\vec{x}))\equiv 0$, and $y-f(\vec{x})\mid
P(\vec{x},y)$, since it is a root of $P(\vec{x},y)$. ∎

######  Lemma 1.

(Schwartz-Zippel) Let $p(x_{1},\dots,x_{n})\in F[x_{1},\dots,x_{n}]$ be a
polynomial of total degree $d$ that is not equivalently $0$. Let $|S|\subset
F$ be an arbitrary finite subset of the field $F$. Then
$\mathbb{P}_{\vec{x}\in_{R}S^{k}}[p(\vec{x})=0]\leq\frac{d}{|S|}$.

###### Proof.

We proceed by induction. Considering the uni-variate case yields that
$\mathbb{P}_{x\in_{R}S}[p(x)=0]\leq\frac{d}{|S|}$. This is true because $p$
has degree $d$ and since $p$ is not identically $0$ we have that there are at
most $d$ roots in $F$, and hence there are at most $d$ roots of $p$ in a
finite subset $S\subset F$. Let $k=\deg_{x_{n}}(p)$. Then we may write
$p(x)=x_{n}^{k}q(x_{1},\dots,x_{n-1})+r(x_{1},\dots,x_{n})$, where
$q(x_{1},\dots,x_{n-1})$ has total degree at most $d-k$ and
$r(x_{1},\dots,x_{n})$ has $x_{n}$ degree strictly less than $k$. For a
$\vec{x}\in_{R}S^{k}$, we have that

$\mathbb{P}[p(\vec{x})=0]=\mathbb{P}[p(\vec{x})=0\mid q(\vec{x})\neq
0]\mathbb{P}[q(\vec{x})\neq 0]+\mathbb{P}[p(\vec{x})=0\mid
q(\vec{x})=0]\mathbb{P}[q(\vec{x})=0]$

by Bayes Formula. But $\mathbb{P}[p=0]\leq\mathbb{P}[q=0]+\mathbb{P}[p=0\mid
q\neq 0]\leq\frac{d-k}{|S|}+\frac{k}{|S|}=\frac{d}{|S|}$, by the inductive
hypothesis. This completes the proof. ∎

######  Remark 8.

There is some subtlety to the fact that $\mathbb{P}[p=0\mid q\neq
0]\leq\frac{k}{|S|}$. This is because if we are given that $q\neq 0$, then
$p(x_{1},x_{2},\dots,x_{n})$ considered in that regard becomes a uni-variate
polynomial of degree $k$ in the variable $x_{n}$ that is not identically $0$,
as we are fixing that $x_{1},\dots,x_{n-1}$, and we then may apply the
inductive hypothesis. The application of the Schwartz-Zippel Lemma as
presented here to proposition $5$ is that the contrapositive of the statement
of lemma $1$ is sufficient to deduce that $\theta_{f}\equiv 0$ as it has more
than $\deg(f)\cdot h^{k-1}$ roots on $H^{k}$.

######  Theorem 1.

If the parameters $d,t,|H|=h,k\in\mathbb{N}$, are such that
$\frac{t}{dh^{k-1}}>\frac{k(h-1)}{d}$, and the open interval
$\left(\frac{k(h-1)}{d},\frac{t}{dh^{k-1}}\right)$ contains a positive
integer, then Algorithm 2 as described above outputs all the desired
polynomials.

###### Proof.

Note that to obtain the non-trivial polynomial $P(\vec{x},y)\in
F[x_{1},\dots,x_{k},y]$ which vanishes on $S$, we need by proposition $4$ that
$m+ld\geq k(h-1)$. Similarly, to ensure that $f(\vec{x})\in
F[x_{1},\dots,x_{k}]$ of degree at most $d$ having at least $t$ agreements on
$S$ satisfies that $y-f(x)\mid P(\vec{x},y)$, we require that
$(m+ld)<\frac{t}{h^{k-1}}$. Hence, we want $m,l\in\mathbb{N}$ such that
$\frac{t}{h^{k-1}}>m+ld>k(h-1)$. Considering simpler case of setting $m=0$,
and finding the appropriate $l$, we obtain that such an $l$ exists precisely
when $I=\left(\frac{k(h-1)}{d},\frac{t}{dh^{k-1}}\right)$ contains a positive
integer. Assuming that the given parameters are such that $l$ exists, by
proposition 4, we obtain a non-zero polynomial vanishing on $S$. and by
proposition $5$, we have that a polynomial $f\in F[\vec{x}]$ of degree at most
$d$ having at least $t$ agreements on $S$ will have $y-f$ divide
$P(\vec{x},y)$. Hence, Algorithm 2 will return the desired polynomials. ∎

##  3 References

  * •

M. Sudan, Decoding of Reed-Solomon Codes Beyond the Error Correction Bound.
http://people.csail.mit.edu/madhu/papers/1996/reeds-journ.pdf


# FV-UPatches: Enhancing Universality in Finger Vein Recognition

Ziyan Chen1, Jiazhen Liu2, Changwen Cao1, Changlong Jin1, and Hakil Kim3
1 School of Mechanical $\&$ Information Engineering, Shandong University
2 School of Information, Renming Universality of China
3 School of Information $\&$ Communication Engineering, INHA University

Ziyan Chen, Jiazhen Liu, Changwen Cao, Changlong Jin and Hakil Kim

###### Abstract

Many deep learning-based models have been introduced in finger vein
recognition in recent years. These solutions, however, suffer from data
dependency and are difficult to achieve model generalization. To address this
problem, we are inspired by the idea of domain adaptation and propose a
universal learning-based framework, which achieves generalization while
training with limited data. To reduce differences between data distributions,
a compressed U-Net is introduced as a domain mapper to map the raw region of
interest image onto a target domain. The concentrated target domain is a
unified feature space for the subsequent matching, in which a local descriptor
model SOSNet is employed to embed patches into descriptors measuring the
similarity of matching pairs. In the proposed framework, the domain mapper is
an approximation to a specific extraction function thus the training is only a
one-time effort with limited data. Moreover, the local descriptor model can be
trained to be representative enough based on a public dataset of non-finger-
vein images. The whole pipeline enables the framework to be well generalized,
making it possible to enhance universality and helps to reduce costs of data
collection, tuning and retraining. The comparable experimental results to
state-of-the-art (SOTA) performance in five public datasets prove the
effectiveness of the proposed framework. Furthermore, the framework shows
application potential in other vein-based biometric recognition as well.

##  1 Introduction

Recent years have seen the rapid development of security systems, especially
the ones involving biometric traits[1]. The popular identification or
verification system based on extrinsic biometric modalities, e.g.,
fingerprint, face, and iris, have exposed much to the public eye. However,
they are considered vulnerable to malicious attacks such as forgery[2]. While
finger vein biometrics, as one of the intrinsic biometric traits, is more
desirable for users and has drawn the attention of many researchers due to its
excellent advantages: liveness-effective [3], immune to counterfeit [4] and
user-friendly [5]. Therefore, Finger Vein Recognition(FVR) has been greatly
valued in practical application and deployment.

Figure 1: Pipeline of the FV-UPatches. The sign ${}^{\textbf{*}}$ in the
diagram indicates a descriptor matching process. The right-hand side of the
dotted boxes represents the target domain while the left side represents the
raw feature domain.

Recent advances in FVR are mainly focused on recognition challenges posed by
the poor image quality and large intra-class distance. Among these solutions,
many deep learning-based approaches have demonstrated their powerful
capability of discriminative feature representation. For example, FV-Net [9]
leveraged pretrained convolutional neural network to extract regional features
of finger veins. FV-GAN [8] employed the powerful Generative Adversarial
Network (GAN) to generate vein patterns, and JAFVNet [10] introduced joint
attention module to further improve the ability of the network to extract
discriminative identity features.

The learning-based methods have achieved superior performance when trained and
tested in the same dataset. However, most models can only perform well in a
single dataset and suffer from poor generalization. It means the tuning or
retraining effort is often required when given a different dataset. The root
cause for this problem lies in the base of traditional machine learning
methodologies: assuming that the distribution of training and testing data are
from the same domain, such that inputs can be embedded into the same feature
space [11]. Away from the assumption, the performance of the learner will be
compromised. In this work, we propose a universal framework in which different
data distributions are transformed to mitigate the data dependency of the
model without deviating from the assumption.

As images of the same object could fall into different marginal or conditional
distributions when captured in different poses, illuminations, and user states
[27], the differences in the data distribution of public datasets lead to
difficulties in achieving universality with existing models. To improve
generalization, some researchers had made attempts to leverage fusion training
strategies [12]. These strategies, however, are only to obtain a fused
marginal data distribution which is not necessarily indicative of the
underlying distribution and can be still biased.

Therefore, it would be natural to come up with an idea: whether it is possible
to train a framework that can be universal enough and retraining-free to
transfer to a different dataset? Many traditional recognition algorithms are
data distribution-independent as usually the morphological features extracted
in vein pattern maps. We leverage the characteristic of traditional methods to
help the learners generalize. Drawing on the idea of domain adaptation in
transfer learning [11], the vein pattern extraction can be considered as
mapping and concentration of feature distributions. Based on the work in [28],
we introduce a compressed U-Net as a pattern extraction model, and meanwhile
as a domain mapper to help unify different marginal data distributions into a
target feature domain. It allows the subsequent matching network to always
behave in the same feature domain; thus, the data dependency can be mitigated.
The hand-crafted morphological features are used to supervise the training of
the U-Net and the approximation to the traditional extraction function allows
the network to generalize well. The model shows a good generalization and can
be trained only as a one-time effort.

We then use the image local feature matching to accomplish recognition in the
target feature domain since the minutia matching has been proven successful in
[6, 7]. A powerful local feature descriptor SOSNet is employed to perform
further feature extraction. As it is the similarity between features to
measure, the model can be trained to be representative enough by simply
feeding a public dataset of non-finger-vein images, rather than consuming the
already sparse finger vein data. The main contributions of this work are
summarized as follows:

  * •

To the best of our knowledge, this work makes the first attempt to enhance
universality using a learning-based method in FVR. We propose a universal
framework that can reach comparable performance to SOTA with limited data for
training and without any retraining or tuning effort to transfer to other
datasets.

  * •

The U-Net is innovatively introduced to achieve domain mapping and
probabilistic pattern extraction in FVR. The hand-crafted features are
leveraged as soft-label information to supervise the training of the U-Net.

The rest of the paper is organized as follows: after a review in recent
advances in FVR (section 2), we elaborate the proposed FV-UPatches framework
(section 3) and explore its effectiveness extensively (section 4). The effects
of involved parameters are further discussed in section 5 concerning a balance
between computation, storage memory, and performance. Then, the conclusion and
future work are given in section 6.

##  2 Related Work

The proposed work draws on recent successes of deep neural networks in vein
pattern extraction, and the development of local feature matching methods in
FVR, which are introduced in detail in this section.

Typically, an FVR system includes stages of image acquisition, pre-processing,
feature extraction and matching [25]. The structural characteristics of the
finger and uneven illumination in the data acquisition stage have led to
reduced image quality. Many studies employed some strategies to combat the
recognition challenges posed by these problems, including vein enhancement and
segmentation with traditional methodologies. Miura et al. [17] firstly
presented a solution for vein segmentation using Repeated Line Tracking (RLT)
to address the problem of degraded performance of the minutiae detection
algorithm at vein ridges. However, the RLT is computationally demanding and
tends to introduce line-like trajectory noise that leads to poor results. To
address these problems, Song et al. [20] extracted fine vein patterns using
mean curvature. Similarly, in the work of Kumar et al. [15], non-linear image
enhancement and the use of Gabor filters also allowed the veins to be
segmented.

Based on the above methods, Qin and El-Yacoubi [16] proposed to achieve
automatic labeling to avoid costly intensive manual annotation, which is also
prone to errors. These traditional methods have met with some success but tend
to be computationally expensive and often suffer from a problem that many
thresholds involved need to be manually and iteratively tuned when given a
different dataset. With the development of deep learning, Yang et al. [8]
leveraged the powerful Generative Adversarial Network (GAN) to generate vein
patterns, with segmentation results obtained from traditional algorithms
serving as hard labels to supervise the training. However, the vein patterns
extracted by this framework are coarse-grained, since the based annotation
algorithms extract binary segmentation and tend to introduce noise. Other
methods that achieved fine probabilistic segmentation, i.e. vein-pattern
extraction, are usually based on second derivatives of the image, such as the
Hessian matrix. Meng et al. [6] calculated the Hessian matrix and a chief
curvature map (CCM), to capture the gradient information of the veins
effectively by indicating the concave and convex parts of the image.

In most recognition algorithms, vein-pattern extraction is considered not only
a way to reduce sample variation but serves for subsequent feature extraction
and matching. For traditional methods, the hand-crafted features and their
combinations are employed for matching. Matsuda et al. [7] detected feature
points by eigenvalue analysis based on Minimum-Curvature Map (MCM) and
combined the vein directions and vein patterns to calculate descriptors. For
learning-based methods, there are generally two main streams according to the
recognition ability: closed-set mode and open-set mode[14]. Models working in
closed-set mode can only recognize the identity which has been included in the
training set, the algorithm with a CNN classifier are typical examples [13,
22, 23, 24]. Though the close-set models are simpler and perform better, they
still have difficulties generalizing and have to be retained to recognize a
new identity. Models working in open-set mode tend to involve a metric
learning task, i.e. learning to measure similarities between matching pairs
instead of predicting the class. Therefore, such methods can be used in a
wider range of scenarios but suffer from performance degradation, which can be
reflected in the work of Hu et al. [9].

In this work, we try to achieve better universality of the FVR based on deep
learning methods. Ronneberger et al. [28] had proposed to use U-Net to extract
high-precision biometric patterns. Based on this work, we introduce a
compressed U-Net as a vein extraction network, and meanwhile as a domain
mapper to unify different marginal data distribution into a target
probabilistic segmentation domain, which allows the framework work in a much
wider range of application scenarios.

(a) Input ROI (b) Vein pattern

(c) Gassian filtered (d) Skeleton

Figure 2: Vein extraction and ridge keypoint detection. (a) ROI of an example
from MMCBNU [44]. The three remaining images demonstrate the pre-processing
process. (b) is the vein extraction using a traditional second-order central
moment method, which is regarded as soft labels of the U-Net. (c) (d) are
further processed based on (b) to feed to next module.

##  3 Proposed Method

In this section, we firstly focus on preprocessing, in which the region of
interest (ROI) of the finger-vein image is aligned and located. Then the
domain mapping and vein pattern extraction based on a compressed U-Net is
introduced. We leverage the hand-crafted features as soft-label information to
supervise the training of the U-Net, which mitigates label noise problems
caused by automatic annotation. Finally, a pre-trained local descriptor model
is employed to achieve high-performance minutia matching.

The proposed framework is named as FV-UPatches, as it is mainly based on a
probabilistic U-Net and a patch descriptor model. The pipeline of this
framework is shown in Figure 1. The models involved can be retraining-free and
enhance universality across datasets which are elaborated on in this section
and discussed more extensively in section 4 and section 5 as well.

###  3.1 Preprocessing

The preprocessing in the proposed framework consists only of ROI extraction
and Gaussian filtering. The ROI extraction is to align the centerline of the
finger and filter out the background, which facilitates more stable
recognition.

Different ROI extraction algorithms are often required due to large
differences between datasets. A dataset with a strong light background like
HKPU is well served by the fixed threshold segmentation method to extract its
ROI. Since the method is not general to different datasets, many studies had
employed an edge enhancement and extraction-based method. However, this method
can suffer from problems of edge breakage and missing during extraction due to
poor image quality such as overexposure or underexposure.

To solve this problem, we designed a more general ROI extraction algorithm.
Firstly, a second-order central moment method is used to calculate the pixel-
wise orientation intensity and the maximum responses will appear on the finger
edge. A horizontal Gabor filter is then employed to ensure subsequent fixed-
threshold binarization can determine the edges more robustly. Furthermore, the
quadratic curve fitting is used for noise abatement and edge breakage caused
by the image quality. The image is aligned by rotating the centerline of the
finger to horizontal, and the ROI is cropped out from the central area of the
image.

Figure 3: Design of the compressed U-Net architecture. The basic convolutional
block selected with black dotted box consists of two $3\times 3$ convolutions
(padding=1) forms the most part of the network.

###  3.2 Domain mapping and vein-pattern extraction

(a) Raw (b) Histogram equalized (c) Augmented (d) Domain mapped

Figure 4: Visualization of data distribution using T-SNE on five public
datasets. (b) is from data processed with Gamma transformation and CLAHE. (c)
is data augmented by random contrast, brightness, Gaussian blur, and
saturation. (d) is from the target domain data.

Drawing on the idea of domain adaptation in transfer learning, we argue that
different data distributions can be mapped into the same target feature
domain. The reduced variation between datasets enables the model to be
universal and retraining-free. The mapping can be achieved by feature
transformers with an optimization function. Based on the characteristics of
traditional methods, we consider the vein-pattern extraction itself as an
intuitive mapping process to help unify different data distributions.

Most of the current pattern extraction methods are based on traditional
algorithms with manual tuning efforts involved when given a different dataset.
To achieve vein-pattern extraction in a more efficient and fast way, we modify
the U-Net [28] to a compressed variant and define the supervised learning as a
task of probabilistic segmentation, i.e. a pixel-wise $0$-$1$ classification.

We first obtain the fine vein-pattern map from a traditional method based on
second-order central moments. The vein-pattern map is further filtered by a
Gaussian kernel, which is then taken as soft-label information to supervise
the training of the U-Net. It means that the normalized gray level of each
pixel is regarded as a probability that it contains a vein.

This task is usually highly class-unbalanced since the foreground pixels take
up only a small fraction. Thus a combination of Dice coeffient [29] and Binary
Cross Entropy (BCE) loss [30] is introduced. Formally, for an input
$z\in\mathbb{R}^{m\times n\times 1}$, the expected output probability map
denotes as $\mathcal{R}(z)\in\mathbb{R}^{m\times n\times 1}$. The optimization
problem we solve as:

$\begin{gathered}\mathcal{L}_{S}=1-\frac{2\cdot{\sum_{i,j}^{m,n}(\hat{y}_{ij}\cdot
y_{ij})}+s}{\sum_{i,j}^{m,n}(\hat{y}_{ij}+y_{ij})+s}\\\ \end{gathered}$ (1)

where $s$ represents the smoothing factor introduced for further label
smoothing.

The design of the modified architecture is illustrated in Figure 3. Compared
to the original U-Net, the encoder-decoder block with a skip connection is
reduced by one layer, and all the channels of the feature map are reduced by a
quarter, i.e. $16$, $32$, $64$, $128$ as the channels of the intermediate
feature maps at the horizontal encoder-decoder blocks, in which $128$ is the
output dimension of the bottleneck. Even if the model parameters are cut down
from $118M$ to $1.86M$ for a lightweight architecture, it is still proven
powerful enough to achieve vein pattern extraction properly. The final
$1\times 1$ convolution and sigmoid layer is to map all the 16-dimensional
vectors to the class probability of each pixel.

The pattern extraction itself can be considered as domain mapping. To
illustrate more intuitively, we employ T-SNE for the observation of different
data distributions in two-dimensional European space. The visualization of
five public datasets, namely MMCBNU[44], SDUMLA[45], UTFVP[46], FVUSM[47] and
HKPU-FV[43], is shown in Figure 4. To observe the effect of environmental
changes on data distributions, we apply histogram equalization and data
augmentation to the original ROI data. The clustering results shown in Figure
4b are from data processed with Gamma transformation and Contrast-Limited
Adaptive Histogram Equalization (CLAHE), and the one showed in Figure 4c is
from data augmented by random contrast, brightness, Gaussian blur, and
saturation. It can be discovered a significant difference between datasets
enables these data to be clustered properly and sufficiently dispersed. In
contrast, the domain mapped data tend to confuse and fail the clustering.
Figure 4d shows all feature points are clustered in the central zone as the
variation between distributions decreases after domain mapping, which is
consistent with the observations of the human eye system.

Meanwhile, training on the U-Net can be a one-time effort with limited data.
Specifically, we select MMCBNU with clearer vein-pattern as the training data
and it can generalize well without retraining or fine-tuning effort. Since the
model is an approximation to the exact traditional extraction function, it can
easily transfer to other datasets to extract specific features without
retraining or fine-tuning effort. Though the compressed network is modified
simply without a complicated model-pruning strategy applied, it is still
powerful enough to be competent in the task of the specific function
approximation, which is validated in section 4.

###  3.3 Keypoint detection

In the last section, we elaborate on the vein-pattern maps that can fall on a
unified target domain, which allows the subsequent local descriptor model to
behave in the same feature space despite being given a different dataset.

To achieve descriptor-level matching, the keypoint detection is needed to
obtain vein patches as is shown in Figure 5. The minutia area is cropped as
standard patches and will be embedded into descriptors for matching.
Traditional methods used for keypoint detection, such as detector in SIFT [31]
and SURF [32], are all proposed as generic detectors for real-world scenarios
and typically include keypoint localization, scaling, and orientation
assignment. It means most of the keypoints detected by these methods are not
necessarily located on the vein, where valid identity Information is included.
Therefore, we propose a more straightforward method for keypoint extraction
based on the vain-pattern map.

For a vein-pattern map $I$ and a candidate interest point set $p_{c}(I)$ from
the skeleton, the detection process can be elaborated as Algorithm sheet 1.
Since the scaling and orientation are trivial in recognition thanks to the
alignment of the finger image in preprocessing, we left out the estimation of
them for efficient calculations. It may be counterproductive when these
estimations are prone to introduce noise dealing with the linear features.

The proposed method makes the most of the location information of the vein
skeleton, enabling most of the patches transformed to contain valid identity
information.

Data: $length(p_{c}(I))\geq 20$, $I$, $c$

Result: Storing $p(I)$

$\mathcal{G}_{I}\leftarrow GassianFilter(I)$;

$\mathcal{S}_{I}\leftarrow Skeletonize(G_{I})$;

$pc_{I}\leftarrow argwhere(S_{I}>20)$;

$mask\leftarrow Ones(size(I))$;

$p_{I}\leftarrow list\\{\varnothing\\}$;

for  _point $(x,y)$ in $pc$_ do

if  _mask $(x,y)$ is nonzero_ then

$left\leftarrow x-c$;

$right\leftarrow x+c$;

$up\leftarrow y-c$;

$down\leftarrow y+c$;

$mask[up:down,left:right]\leftarrow 0$;

append $(x,y)$ to $p_{I}$;

end if

end for

Algorithm 1 Algorithm for point detection and reduction

###  3.4 Local feature descriptor

Figure 5: Locate the interest keypoints and rectifing patches around them from
the vein pattern map.

The patch-based local feature extraction and representation are introduced in
this section.

The methods used to describe local features of images have been widely
studied, early traditional methods are mainly based on hand-crafted
representations, e.g. gradient distribution, spatial frequency, and
differential techniques [34]. The emergence of annotated patch datasets has
led to many data-driven learning descriptor models being proposed, including
L2Net [35], TFeat [36] and HardNet [37], which have achieved encouraging
results in image matching. In the proposed framework, the L2Net architecture
is introduced to embed the patches to $\alpha$-dimention descriptors.

Formally, for the training set with $N$ pairs of patches, the network should
embed them into $N$ pairs of descriptors, denoting that the set of descriptors
is $\left\\{\boldsymbol{x}_{i}\right\\}_{i=1\ldots N}$ and
$\\{\boldsymbol{x}_{i}^{+}\\}_{i=1\ldots N}$ as its positive correspondence.

The first-order similarity (FOS) loss comes in the form of a Quadratic Hinge
Triplet (QHT) loss, enforcing the negatives to be farther away from the
positives by a margin:

$\begin{gathered}\mathcal{L}_{FOS}=\frac{1}{N}\sum_{i=1}^{N}\max\left(0,t+d_{i}^{\mathrm{pos}}-d_{i}^{\mathrm{neg}}\right)^{2}\\\
d_{i}^{\mathrm{pos}}=d\left(\boldsymbol{x}_{i},\boldsymbol{x}_{i}^{+}\right)\\\
d_{i}^{\mathrm{neg}}=\min_{\forall j,j\neq
i}\left(d\left(\boldsymbol{x}_{i},\boldsymbol{x}_{j}\right),d\left(\boldsymbol{x}_{i},\boldsymbol{x}_{j}^{+}\right),d\left(\boldsymbol{x}_{i}^{+},\boldsymbol{x}_{j}\right),d\left(\boldsymbol{x}_{i}^{+},\boldsymbol{x}_{j}^{+}\right)\right)\end{gathered}$
(2)

where $d(x_{i},x_{j})$ is the $L_{2}$ distance $||x_{i}-x_{j}||_{2}$, $t$
represents the margin.

Furthermore, the second-order similarity regularization(SOSR) is proposed to
further supervise the training:

$\begin{gathered}\mathcal{R}_{SOS}=\frac{1}{N}\sum_{i=1}^{N}d^{(2)}\left(\boldsymbol{x}_{i},\boldsymbol{x}_{i}\right)\end{gathered}$
(3)

The objective function of training goes as follows:

$\mathcal{L}=\mathcal{L}_{FOS}+\mathcal{R}_{SOS}$ (4)

More comprehensive information can be found in [42].

Training the descriptor model can be based on a dataset of non-finger-vein
images since it is the similarity between patches to learn. The difficult
negatives in the public patch-based dataset enable the model to be powerful
enough to extract discriminative features between pairs.

###  3.5 Matching

Descriptor-level matching. We employ L2-distance to measure distance of a
descriptor pair and design a two-stage filtering strategy to minimize initial
mismatches. A threshold parameter $t_{d}$ is set for the initial filtering of
matches with too large a distance, i.e. only distances less than a specific
distance will be accepted. Then we further filter this matching set to ensure
that each key point appears only once in the matching set, which is essential
for higher accuracy. Finally, a RANSAC method is employed to estimate the best
homography matrix between an image pair based on the descriptor matchings,
with only the correct matching pairs to be retained.

Image-lavel matching. For a probe and gallery image matching pair, the system
needs to make an accept or reject decision comparing the number of descriptor
matches with the decision boundary. We calculate the decision matches based on
the Receiver operating characteristic (ROC) curve in our experiments, and a
threshold is used for image matching decisions. For an image pair with a
descriptor matching number near the decision boundary, the system is prone to
make mistakes. While for consideration of system security, the decision
parameter can generally be selected higher manually, making the system rather
recognize a bad genuine as an imposter in real application scenarios.

(a) MMCBNU (b) SDUMLA (c) UTFVP (d) FVUSM (e) HKPU-FV

Figure 6: Performance in five public datasets.

##  4 Experiments

In this section, we evaluate the effectiveness of the proposed FV-UPatches and
compare the performance against other SOTA recognition algorithms in five
public datasets, whose basic information is introduced in Table 1.

Table 1: Description to the five public datasets

Dataset | Session |  $\\#$ Sample |  $\\#$ Class/Subject
---|---|---|---
MMCBNU | Single | 6,000 | 600/100
SDUMLA | Single | 3,816 | 636/106
HKPU-FV | Semi-dual* | 3,132 | 312/156
UTFVP | Dual | 1,440 | 360/60
FVUSM | Dual | 5,904 | 492/123

  * *

The semi-dual session refers to the fact that only the first 105 subjects in
HKPU-FV are dual-session, with the rest only providing singal-session data.

###  4.1 Evaluation protocol

Since the FV-UPatches framework is proposed to be more universal and data-
independent, the framework can be evaluated in a way similar to traditional
algorithms, i.e. involving all the classes in the dataset instead of using a
portion of it as most learning-based methods do. We introduce two protocols
for mixed-session and cross-session evaluation.

Preliminaries. For a dataset consisting of $z$ classes, with $m$ samples in
each class and especially $m_{s}x$ samples in Session-x within a class, the
number of genuine and imposter matching pairs are denoted as $N_{g}$ and
$N_{i}$, respectively.

FVC2004 protocol. To validate the performance of different algorithms in a
reproducible matching set, we employ the FVC2004 protocol from fingerprint
verification contests [38]. For genuine matching, each sample is compared
against all remaining samples of the same class. While for imposter matching,
only the first sample of the class is compared against the first sample of the
remaining class. The numbers of matching pairs generated with the FVC2004
protocol are:

$\begin{gathered}N_{g}=C_{m}^{2}\cdot z,\\\ N_{i}=C_{z}^{2}.\end{gathered}$
(5)

The above protocol will be employed for all-session performance evaluation
when comparing against other algorithms, and for effectiveness evaluation of
the descriptor model.

Cross-session FVC protocol. We propose a cross-pair protocol for cross-session
evaluation with reference to the ISO/IEC 19795-1 suggestion [39]. The
Session-1 data is considered as enrolled while the Session-2 is the probe in a
matching pair for cross-session evaluation. For genuine matching, all
Session-1 data will combine with all Session-2 data from the same class. While
for imposter matching, only the first Session-1 sample of the class and the
first Session-2 sample of another class will form a matching pair. The genuine
and imposter matching numbers will be:

$\begin{gathered}N_{g}=m_{s1}\cdot m_{s2}\cdot z,\\\
N_{i}=C_{z}^{2}.\end{gathered}$ (6)

This protocol is for cross-session evaluations over the effectiveness and
robustness of the system.

###  4.2 Implementation details

The vein extraction network has to be fed with images in the size of
$128\times 256$, while the image size in other parts can be arbitrary. In our
experiments, we uniformly scale the input ROI images to $(90,225)$ after
preprocessing.

The compressed U-Net is trained with Adam [40] and the involved
hyperparameters are set as $\alpha=0.01$, $\beta_{1}=0.9$ and
$\beta_{2}=0.999$. The smoothing factor $s$ in the loss function is $1$, the
learning rate is set to $1e-4$ and the batch size is $16$. We train the model
in a randomly selected dataset and the model can be transferred to other
datasets properly.

For keypoint detection, the orientation and scaling are left out and are fixed
to $-1$ and $ks=11$, respectively. The point set reduction factor $c$ is fixed
as $4$. Considering the balance between computational speed, storage
consumption, and performance, we further discuss the impact of these
parameters in section 5.

We use the SOSNet model trained with the patch-based dataset UBC-phototour
[41]. All the patches are rescaled to $32\times 32$ whatever value $ks$ is
selected. The batch size is $M=32$ and a descriptor vector dimention is
$\alpha=128$. The threshold $t_{d}$ is set to $1.2$ for matching.

All the experiments are performed on a private LXD container with a 32-Core
Intel Xeon Silver 4110 2.10GHz processor, and NVIDIA GTX 2080Ti GPU of 11GB
graphics memory.

###  4.3 Evaluations

Table 2: Comparison against SOTA methods.

Datasets | EER(%) | Method
---|---|---
| 3.37 | Proposed
| 2.78 | Siamese CNN* [12]
| 2.61 | zone-based minutia matching [6]
| 2.35 | Composite Image [50]
| 1.59 | dual-sliding window [52]
| 1.2 | FV-Net [9]
SDUMLA | 1.18 | JAFV-Net [10]
| 0.63 | Siamese CNN [12]
| 0.3 | FV-Net [9]
| 0.08 | JAFV-Net [10]
MMCBNU | 0.05 | Proposed
| 2.32 | dual-sliding window [52]
| 0.89 | Proposed
| 0.76 | FV-Net [9]
| 0.41 | Siamese CNN* [12]
FVUSM | 0.34 | JAFV-Net [10]
| 0.70 | Proposed
| 0.36 | zone-based minutia matching [6]
HKPU-FV | 0.33 | Composite Image [50]
| 0.46 | Maximum curvature [56]
UTFVP | 0.14 | Proposed

  * Instead of generalising to other datasets, the models of all the listed learning-based methods were trained and tested in the same dataset.

In this section, firstly the effectiveness of the system is explored. Then the
performance of the proposed method is evaluated by comparing it against both
the baselines and SOTA methods. Finally, the cross-session performance of the
proposed method is explored as well.

Effectiveness of the vein extraction U-Net. The modified U-Net shows a
significant improvement in training efficiency, for each epoch takes only $19$
seconds and the network converges in about $20$ epochs. Training the U-Net is
a one-time effort, and the model can be transferred directly to other datasets
though trained in only a randomly selected dataset MMCBNU. The probabilistic
maps and the labels are binarily segmented with a threshold of $0.5$ for
evaluation of the U-Net. The precision scores reach $90.82\%$, $89.24\%$,
$93.47\%$, $90.82\%$, $88.27\%$ for UTFVP, SDUMLA, MMCBNU, HKPU, and FVUSM
datasets, respectively.

Effectiveness of the local descriptor model. To validate the effectiveness of
SOSNet in the proposed system, TFeat [36], SIFT and SURF are introduced to
conduct comparative experiments. We evaluate them from the perspective of
performance, descriptor matching decision boundary numbers (namely decision
matches), and total matching distance. We firstly conduct controlled
experiments with SOSNet, TFeat, SIFT, and SURF in the UTFVP. For SOSNet and
TFeat, We simply employ the models pre-trained in the public patch-based
dataset. Figure 6 shows the ROC curves of the performance in five public
datasets, in which the SOSNet shows a powerful similarity metric ability and
achieves superior performances.

Comparison against SOTA. Methods working in the closed-set mode often perform
better but suffer from poor generalization and are difficult to reflect
realistic scene performance. To ensure the validity of the comparison, we
compared against only SOTA traditional algorithms and learning-based methods
that can work in the open-set mode. We evaluate the performance of the
proposed universal framework in five public datasets. The experimental results
in Table 2 show it is comparable to the current SOTA methods. However, it is
also worth noting that all the performance of the learning-based methods were
tested with only part of each dataset and a large portion has to be divided
for model training. While the proposed framework is evaluated with all classes
in each dataset and requires no re-training efforts when applied to a new
dataset. It means our method is capable of enhancing the universality of the
recognition while maintaining excellent performance. Specifically, our method
on SDUMLA underperforms but within an acceptable range and is more plausible
from a security point of view. Many of the samples from SDUMLA tend to be
over-rotated, resulting in a significant reduction in the overlap area between
a genuine matching and a much larger intra-class distance than that of other
datasets. In this case, the proposed method tends to reject sample pairs with
few local matched areas to avoid increasing the security risk, since many of
them are hard to discern to be accepted even by human eyes.

Table 3: Cross-session evaluation results. Datasets | FVUSM | HKPU | UTFVP
---|---|---|---
EER(%) | 1.41 | 1.10 | 0.14

Cross-session evaluations. Although extensive evaluations have been carried
out for the above experiments, few algorithms have been evaluated for vein
sampling from different sessions. The FVUSM, UTFVP, and HKPU datasets provided
dual-session finger samples. Among them, the UTFVP consists of images taken in
two distinct sessions in two days. The interval between the two sessions of
FVUSM is more than two weeks. And the separated session in HKPU is in
intervals of one month to six months, with an average interval of 66.8 days.
We employ the cross-session evaluation protocol as is introduced in section
4.1. In the cross-session protocol, only the sample from different sessions
with the referred one could form a matching pair. Compared with the FVC
protocol or the fully-connected one, the cross-session evaluations will better
reflect the performance of the system in real application scenarios. The
results are listed in Table 3. Though it shows some performance degradation
due to variation between images of different sessions, it still proves the
effectiveness and applicability of the proposed system.

Efficiency evaluations. Experiments in this part evaluate the operation time
efficiency and memory of the models. The VGG16 [53], ResNet-152 [55] and
DenseNet-161 [54] had been employed to evaluations in [50], which is very deep
networks with strong power to extract features. But they are generally memory-
consuming without compression and might be hard to be deployed in some mobile
or embedded devices, as is shown in Table 4. The interference time from vein
extraction to recognition decision of the proposed system is $129ms$ with
$38ms$ for vein pattern extraction and $91ms$ for subsequent recognition when
implemented in the Python environment. We have not compared the inference
times directly with those of other algorithms, as differences in experimental
computing devices may introduce significant differences in the estimation of
running times.

Table 4: Memory estimation of different networks.

| Proposed | VGG16 | ResNet-152 | DenseNet-161
---|---|---|---|---
Params(/M) | 1.73 | 132 | 57.5 | 27.7

##  5 Discussion

In this section, we conducted controlled variable experiments to investigate
the effect of two parameters, reduction window size $c$ and keypoint scale
$ks$, on the performance of the system. The impact of these two parameters on
the balance between computation, memory, and performance of the system will be
also further discussed.

###  5.1 Sensitivity analysis

We explore the balance between performance, storage, and computation by
changing some key parameters.

Impact of reduction factor $c$. We keep the keypoint scaling $ks$ fixed to 10,
and change the value of the reduction window size $c$ in the interval of
$[1,12]$. The larger the $c$, the fewer valid descriptors a sample can
eventually be extracted, which means that the average system runtime per run
will be reduced somewhat and performance may be reduced as is shown in Figure
7a.

Impact of keypoint scaling $ks$. To analyze the impact, the reduction factor
$c$ is kept fixed to $4$, and $ks$ is changed in the interval $[5,32]$ in step
$1$. As is shown in Figure 7b, the increase of $ks$ does not bring much
improvement in system operation efficiency, but the performance and number of
matches are affected for more noise may be introduced to the detriment of
matching when a larger keypoint scaling is selected.

(a) Impact of $c$ (b) Impact of $ks$

Figure 7: Sensitivity analysis. descriptors have different intervals of
distance distribution, we use linear regularisers that do not change the
distribution (min-max regularisation) to enable them to be compared.

##  6 Conclusion and Future Work

In this work, we proposed a universal framework for flexible and transferable
finger vein recognition. We argued that the vein-pattern extraction can be
seen as a mapping process of feature domains, transforming different data
distributions into the same target domain. This allowed us to use a learning-
based model based on this target domain without concern about variations
between different datasets, thus being retraining-free and reducing costs of
data acquisition, retraining, and tuning. The experimental results proved the
effectiveness of the proposed framework, which shows it can perform properly
when fed with a new dataset.

Future work can focus on more applications in interoperability between devices
since the proposed universal framework has the potential to enable recognition
across devices. Moreover, it can be further explored about the universality
feature of the proposed framework not only at the cross-dataset level but also
at the cross-biometric level. The universality of the proposed framework may
also lie in its easy applicability to other vein-based biometric recognition.

## References

  * [1] Shaheed, Kashif, et al. ”Recent advancements in finger vein recognition technology: Methodology, challenges and opportunities.” Information Fusion 79 (2022): 84-109.
  * [2] Yang, Jinfeng, Jianze Wei, and Yihua Shi. ”Accurate ROI localization and hierarchical hyper-sphere model for finger-vein recognition.” Neurocomputing 328 (2019): 171-181.
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