A discriminant binarization transform using genetic algorithm and error-correcting output code for face template protection

Abstract

In a biometric cryptosystem, biometric templates are encrypted before being stored in the database. This approach often requires a binarization phase to transform the original real-valued templates into their binary versions. Most of the existing binarization schemes do not take into consideration the need to discriminate binary templates; which in turn degrade the system recognition accuracy. This paper proposes a new binarization transform for face template protection using Error-Correcting Output Code (ECOC) and genetic algorithm (GA) which offsets this drawback. The proposed binarization transform novelty lies in the fact that the specific crossover, mutation and extension operators are defined by considering the theoretical properties of the ECOC framework in order to find an optimized coding matrix. The specification of these operations reduce the search space and speed up the convergence rate of GA. In addition, the proposed method is immune from being trapped in local minima. The quality of our method is evaluated using three well-known face databases: CMU PIE, Extended Yale B and FEI. The results demonstrate that the proposed binarization transform can result in a higher recognition accuracy without sacrificing the security of the protocol compared to alternative methods, such as binary discriminant analysis (BDA), discriminability-preserving (DP), chaos biohashing, and Discriminant-Genetic.

Appendices

Appendix

In the following, we use induction to prove the correctness of specific crossover (Algorithms 1 and 2), specific mutation (Algorithm 3), and specific extension (Algorithm 4) algorithms.

Specific crossover algorithm

To prove the correctness of specific crossover, we should prove the correctness of the main function of this algorithm, i.e. ECOC converter. This function prevents the creation of invalid off- springs.

Input: $Cros{s_{{N_c} \times n}}$ = binary matrix, ${N_c}$ = The number of classes, n = The number of columns.

Output: penalty, $ECOC\_Cross$ = Transformed binary matrix.

Hypothesis: ${\text{For}}\;i\,=\,k\left( {1\, \leq \,k \leq {N_c}} \right)$, H(k): when the control reaches in line ${A_1}$ for kth time: $penalty{\text{ }}\left( k \right)\, \geq \,penalty{\text{ }}(k - 1)\, \geq \,0$

Prove the above hypothesis by induction:

• ${\text{H}}\left( 1 \right){\text{ is}}\;{\text{true}} \to penalty\,=\,0,{\text{ and}}\;ECOC\_Cross\,=\,Cross$

• ${\text{Let H}}\left( k \right){\text{ be true}} \to penalty\left( k \right)\, \geq \,penalty{\text{ }}(k - 1)\, \geq \,0$

• We will prove that H (k + 1) is also True

Proof

In iteration i = k + 1, the control reaches in line ${A_1}$; in this case, the penalty value is equal to penalty in previous iteration i.e. penalty(k + 1) = penalty(k).

• Case 1: If 1 ≤ i ≤${N_c}$, the algorithm sets ${u_c}$= {1,…,n} and checks rows equality using steps 1.1 and 1.2 $\left( {\forall~~ i\,+\,1\, \leq \,j \leq {N_c}} \right).$

  • 1.1 When ECOC_Cross(i,:) = ECOC_Cross(j,:) and length( ${u_c}$ ) > 0

    Remove the rows equality using ${u_c}$, and update ECOC_Cross

  • 1.2 When ECOC_Cross(i,:) = ECOC_Cross(j,:) and length(u) = = 0

    penalty (k + 1)  =  penalty(k + 1) + 1

• Case 2: If $i>{N_c}$, the control reaches in line ${A_2}$ to check columns equality using step 1.1. $\left( {\forall ~~ 1\, \leq \,m \leq n,\;m+1 \leq l \leq n} \right).$

  • 1.1 When ECOC_Cross (:,m)  =  ECOC_Cross (:,l)

    penalty (k + 1)  =  penalty(k + 1) + 1

  • 2. Also, by induction hypothesis $\to$ penalty(k) ≥ 0

From the steps (1) and (2), we get: penalty (k + 1)  ≥  penalty (k) ≥ 0.

Termination: The algorithm terminates when $i>{N_c}$ and m > n.

Specific mutation algorithm

Input: ${I^s}=\left\langle {P_{{{N_c} \times n}}^{s},{C^s},~fitnes{s^s}} \right\rangle$: Individual, n_mut: mutation control value

Output: Mutated: mutated offspring

Hypothesis: For 1 ≤  k ≤  n_mut, H(k): when the control reaches in line ${B_1}$ for kth time:

• $u\left( k \right)~ \geq ~u\left( {k - 1} \right) \geq ~0$, $P_{{{N_c} \times n}}^{s}(k)$ is ECOC, and $d_{{HD}}^{k}\left( {{C_i},{C_j}} \right) \geq d_{{HD}}^{{k - 1}}\left( {{C_i},{C_j}} \right)$.

Prove the above hypothesis by induction:

• H(1) is true $\to$ ${P^s}$ (1) is ECOC matrix since no mutation happened; u(1) = 0, and $({C_i}$, ${C_j})$ are the most confused classes

• Let H(k) be true $~ \to$ $u\left( k \right)~ \geq ~u\left( {k - 1} \right) \geq ~0$, ${P^s}(k)$ is ECOC, and $d_{{HD}}^{k}\left( {{C_i},{C_j}} \right) \geq d_{{HD}}^{{k - 1}}\left( {{C_i},{C_j}} \right)$

• We will prove that H (k + 1) is also True.

Proof

• 1. In iteration k + 1,the control reaches in line ${B_1}$:

  • 1.1. If $P_{{i,w}}^{s}(k+1)$= = $P_{{j,w}}^{s}(k+1)$ and u < n_mut ($\forall {\text{ }}$1 ≤ i, j ≤${N_c}$).

Case 1: when 1 ≤  w ≤  n, the ${(k+1){th}}$ mutation in the code-word of ${C_i}$ will happen based on its confusion with the code-word of ${C_j}$; and also the value of u is incremented by 1, if the ECOC validity of ${P^s}$ (k + 1) is not compromised. Thus, the hamming distance between ${C_i}$ and ${C_j}$, and the number of mutations will be $d_{{HD}}^{{k+1}}\left( {{C_i},{C_j}} \right) \geq d_{{HD}}^{k}\left( {{C_i},{C_j}} \right)$ and u(k + 1) $\geq$ u(k), respectively.

Case 2: when w > n, the new most confused classes $\left( {{C_i},{C_j}} \right)$ are found and then the ${(k+1){th}}$ mutation in the code-word of ${C_i}~$ will occur based on its confusion with ${C_j}$; and the value of u is increased, if the ECOC validity of ${P^s}$ (k + 1) is preserved. The outcomes of this step are $d_{{HD}}^{{k+1}}\left( {{C_i},{C_j}} \right) \geq d_{{HD}}^{k}\left( {{C_i},{C_j}} \right)$ and u(k + 1) $\geq$ u(k).

• 1.2 when u > n_mut $\to$ mutated =${P^s}$ (k + 1)

• 2. Also, by induction hypothesis $\to$ u(k) $\geq$ 0,

From the steps (1) and (2), we get: u(k + 1) $\geq$ u(k) $\geq$ 0 $,~~d_{{HD}}^{{k+1}}\left( {{C_i},{C_j}} \right) \geq d_{{HD}}^{k}\left( {{C_i},{C_j}} \right)$, and ${P^s}$ (k + 1) is ECOC matrix $\to$ H(k + 1) is true.

Termination: The algorithm terminates when u exceeds n_mut, i.e. u = n_mut + 1.

Specific extension algorithm:

Input: ${I^s}~=\left\langle {P_{{{N_c} \times n}}^{s}~\!,~{C^s}~\!,~fitnes{s^s}} \right\rangle$

Output: Extended: offspring matrix.

Hypothesis: For b = k (1 ≤ k ≤ ${N_c}$ + 1), H(k): when the control reaches in line ${C_1}$ for kth time:

${P^s}(k)$ is ECOC matrix.

Prove the above hypothesis by induction

• H(1) is true $\to$ $\left( {{C_i},~{C_j}} \right)$ are the most confused classes, and ${P^s}$ (1) is ECOC matrix.

• Assume H(k) is true$\to$ ${P^s}$ (k) satisfies ECOC conditions.

• We will prove that H (k + 1) is also True.

Proof

• 1. In iteration b = k + 1, the control reaches in line ${C_1}$:

  
  

  Case 1: when b≤$~{N_c}$, the ${(k+1){th}}$ code-word in extension matrix ${E_{Nc \times L}}$ is initialized based on its more confusion with ${C_i}$ ${\text{or }}{C_j}$. In this case, ${P^s}~\left( {k+1} \right)$ will satisfy ECOC conditions since it is not altered in the current loop.$\to {P^s}~\left( {k+1} \right)={P^s}~\left( k \right)$.

  
  

  Case 2: when $b>{N_c},$ the algorithm checks the columns equality in two conditions: (1) the equality of any two columns of ${E_{Nc \times L}}$, and (2) the equality of each column of ${E_{Nc \times L}}$ with ECOC matrix ${P^s}(k).$ To derive a valid ECOC matrix, if the columns are equal, the algorithm changes the code-words of E with the exception of those code-words showing ${C_i}$, ${C_j}$and the ones representing the confused classes with ${C_i}$ or ${C_j}$. Thus, E is a binary matrix with distinct columns which does not have equal column in ${P^s}\left( k \right).$ Then, ${P^s}\left( {k+1} \right)$ is constructed through joining E to ${P^s}$ (k) as ${P^s}(k+1)$=$~{P^s}$ (k) $\mathop \cup \nolimits^{} E$ to enhance discriminability between the confused classes.$\to Extended={P^s}~\left( {k+1} \right)$.

• 2. Also, by induction hypothesis $\to$ ${P^s}$ (k) is ECOC

From the steps (1) and (2), we get: ${P^s}\left( {k+1} \right)$ is ECOC matrix $\to$ H(k + 1) is true.

Termination: The algorithm terminates when b > ${N_c}$.