Hessenberg factorization and firework algorithms for optimized data hiding in digital images

2.1 Hessenberg factorization

A Hessenberg matrix is a 2D array that has been introduced by Karl Hessenberg, which is almost triangular in terms of linear algebra. It could be either lower or upper. The values of the elements above the first sub-diagonal in the former matrix are zeros. Likewise, the elements below the first sub-diagonal in the latter matrix are also zeros [14]. It can be noticed that many linear algebra procedures can be effectively applied on triangular matrices with substantially less computational power required. This advantage could be extended to include Hessenberg matrices. Therefore, if a typical complete matrix is hard to convert into a triangular to manipulate it in terms of computational power effectively, the next best alternative is to convert it into a Hessenberg matrix. In general, any matrix can be transformed into a Hessenberg matrix. This conversion can be done in a finite number of steps that do not require high computational power [15,16]. One of the conversion procedures is the shifting QR factorization, which is an iterative algorithm that can be employed to transform any matrix into a Hessenberg matrix.

2.2 FWA

An FWA is a swarm-based approach that simulates the explosion of fireworks at night. It produces sparks around each firework [17], as illustrated in Figure 1. Moreover, it can be used as a local search since each firework that sparks around some positions in the range is called amplitude. The global optimum can be reached from the cooperation of evolved populations. FWA is distinguished from other swarm and population-based methods by its distributed parallelism implementation, providing diversity, simplicity, and easy extension [18].

Figure 1

Firework search algorithm.

The general procedure of FWA can be described as follows [19]: it first randomly generates an initial population that contains N fireworks. It then computes their fitness values. The number of sparks and the explosion (explosion amplitude) range are determined by evaluating each firework. The new populations were obtained from exploding the previous fireworks and applying a local search on each one. The amplitude makes sure that the space between the local and global optimum is balanced. As a result, the diversity is maintained, and the chances of escaping from local minima are increased. Furthermore, FWA applies a mutation operator, which is known as Gaussian mutation, and a selection scheme that iteratively chooses a subset of the whole population [20]. The details of each step in FWA are explained as follows:

• Explosion operator: An explosion operator is applied to select multiple sparks from each firework in fireworks. This operator can be calculated by the following equation:

  (1) S i = m × y max − f ( x i ) + ε ∑ i = 1 N ( y max − f ( x i ) ) + ε ,

  where S i denotes the number of sparks, each firework contains, m represents the total number of solutions in a firework, N denotes the total number of sparks, y max is the value of the worst fitness in the fireworks, f ( x i ) is the fitness value of x _i , and ε is a small value to avoid division by zero. The amplitude of explosion A _i is in the following equation:

  (2) A i = A ^ × f ( x i ) − y min + ε ∑ i = 1 N ( f ( x i ) − y min ) + ε ,

  where A ˆ is the sum of all amplitudes and ε is a small value to avoid division by zero.

  To specify the displacement on every dimension in a firework, the aforementioned parameters are utilized as described in the following equation:

  (3) x i k = x i k + U ( − A i , A i ) ,

  where U ( − A i , A i ) is a random number in the uniform range of the amplitude A _i .

• Mutation operator: The mutation operator is applied to the current individual. Let x i k be the position of the current solution, where i belongs to the interval (1 to N), and k represents the current dimension. The neighbors of this position, which are called sparks of Gaussian explosion, are computed by the following equation:

  (4) x i k = x i k × RG ( 1 , 1 ) ,

  where RG (1, 1) is a random number obtained from the Gaussian distribution.

• Mapping rule: This rule guarantees that all solutions in a population are kept in an accepted domain. It makes sure that the resulting solutions after applying any FWA operation will be within the set of feasible solutions. This rule makes use of the modular operation as stated in the following equation:

  (5) x i k = X LBound, k + x i k AMod ( X UBound, k − X LBound, k ) ,

  where x i k represents the positions of sparks, which are laying outside the boundaries, whereas X UBound and X LBound are the boundary limits of the position of a spark, and “AMod” is the arithmetic modular [21].

• Selection method: The selection method might be required to calculate the distance between solutions. To serve this goal, the Euclidean distance measurement could be utilized to find the closest solution, as depicted in the following equation:

where d is the distance, which may be calculated using the Euclidean distance rule, between the two given individuals x i and x j . Sparks, which are obtained from performing the explosion or mutation operators, could be included in this distance.

One of the approaches that FWA could use for selecting a new population is the classical Roulette Wheel procedure. It depends on P ( x i ) , which can be computed by the following equation:

The resulting generation’s diversity can be increased due to maintaining solutions with high distances to each other.

3.1 Split to RGB color band

In this step, a color image is split into three color bands (RGB) [23], as shown in Figure 4. Each of these bands is used in the hiding procedure.

Figure 4

The splitting of the color image.

3.2 Applying 2D-DWT

In this step, the 2D-DWT procedure is applied to the three-color bands. The result of this conversion will be (high–high, high–low, low–high, and low–low) for each color band, as shown in Figure 5.

Figure 5

The 2D-DWT of the three-color band.

3.3 Reordering DWT parameters

After the conversion process, a new matrix (embedding matrix) is formed by taking all the conversion parameters except the low–low band, which retains the basic characteristics of the image. As a result, the new matrix consists of the merge of the three conversion parameters high–high, low–high, and high–low for the red band as a first row, the three conversion parameters of the green band as a second row, and the three conversion parameters of blue as a third row, as shown in Figure 6 [24,25,26,27,28]. The minimum value in the embedding matrix is found, and its absolute value is added to all parameters to process the negative values, i.e., all values will be positive.

Figure 6

Embedding matrix.

3.4 Key generation

Two chaotic map methods, namely Gaussian map and exponential map, are used for the key generation process to produce numbers that will be added to the embedding matrix. The generated numbers are equal to the number of parameters in the embedding matrix. The initial parameters of these chaotic map functions are carefully selected later through the optimization process. The parameters of FWA are as follows: population size (N = 100), y _min = 3, y _max = 50, A _i = 40, and mutation spark number x i k = 5.

3.5 Partitioning embedding matrix

The embedding matrix is partitioned into blocks with a specific size, such as 4 × 4. Each block contains secret information (secret bit). The Hessenberg transform is applied on each block in the next stage, as shown in Figure 7.

Figure 7

Embedding matrix partitioning.

3.6 Applying Hessenberg transform

Hessenberg transformation (factorization) is applied to each block that was partitioned in the previous step. The result of applying this transformation will be two matrices called P-matrix and H-matrix, as shown in Table 1. The embedding process will only use H-matrix for embedding.

3.7 Embedding method

The embedding method involves converting the secret information (watermark image) into a vector of binary numbers [29,30]. Each binary number (bit) is embedded into one block using the following steps: first, specify one position in H-matrix such as (4,4); second, get the sign of numbers; third, get the integer value of the number; and fourth, get the digit after the floating point and check if it is odd or even. If it is odd and the secret bit is one or even and the secret bit is zero, do nothing. The change is applied when the secret bit is one and the selected digit is even. This is modified to be odd by adding one to it in the same option when the secret bit is zero, and the selected digit is odd. It will be modified by subtracting one from it to be even.

3.8 The inverse of Hessenberg transform

The inverse Hessenberg transform is applied on the P-matrix and the modified H-matrix by multiplying the P-matrix with the modified H-matrix, and the result is multiplied by the transpose of the P-matrix.

3.9 Reorder modifying parameters

The modified embedding matrix is first subtracted from the added key (generated numbers). The minimum value that was added to all blocks is subtracted before applying the inverse DWT (IDWT). The matrix is portioned into 3 × 3 sub-matrices representing the DWT parameters for each color band. The three sub-matrices are in the first row for the red band. The three sub-matrices are in the second row for the green band, and the three sub-matrices are in the third row for the blue band, respectively. The IDWT is applied on the low–low parameters and the three modified high–low, low–high, and high–high parameters and produced a red-color band. Similarly, the same procedure is applied to the other parameters to the produced green-color band and blue-color band.

3.10 Extraction of secret information

To perform the extraction process, the same steps in embedding are applied. It is started with splitting the image into three color bands and applying 2D-DWT on the three color bands. It then reorders parameters into the embedding matrix, adds minimum value, and adds the generated numbers. It is followed by partitioning the embedding matrix into blocks (4 × 4) and applying Hessenberg transform on the block that gets the digit after the floating point. Finally, checking if it is odd for secret bit one and even for secret bit zero and collecting secret bit into array represent watermark image.