Wavelet transform 2D, all as the other transformed Fourier 2D, cosine 2D is used extensively in signal processing, its application took a remarkable flight from the years 1980 with the works of Morlet, Meyers, Daubechies and Mallat. Discrete wavelet transform is characterized at time by its limit temporal and frequential (compact support). The decomposition of a signal x(l) to place through filters and gets itself in a following manner

Y h ( l ) = ∑ l = − ∞ + ∞ x ( l ) h ( 2 i − l ) (1)

Y g ( l ) = ∑ l = − ∞ + ∞ x ( l ) g ( 2 i − l ) (2)

h(l) and g(l) are respectively High and low filters

H ( ω ) = ∑ l = − ∞ + ∞ h ( l ) e − j ω G ( ω ) = ∑ i + ∞ g ( l ) e − j ω (3)

H ( ω ) and G ( ω ) will be orthogonal

| H ( ω ) | 2 + | G ( ω ) | 2 = 1 (4)

The obtained coefficients are:

c [ j − 1 , k ] = ∑ l = − ∞ + ∞ h [ n − 2 k ] c [ j , n ] (5)

d [ j − 1 , k ] = ∑ l = − ∞ + ∞ g [ n − 2 k ] c [ j , n ] (6)

The image which is a signal of two dimension, its application of 2D wavelet transform gives a dyadic decomposition of this one with the help of a couple of quadratic mirrors filters (QMF)a being a High pass filter (HS) and the other a low pass (L). The use of does one coins sampling by a factor 2, permitting to get four sub- bands, one under-strip low frequency (LL) and three under-strips high respective frequencies (LH, HL,HH),representing, orientations with very specific vertical, horizontal and diagonal Face [26].

It permits to separate the low frequencies from the high frequencies. This decomposition is done by dividing in blocks of 8 × 8 pixels according to the following equation:

F ( u , v ) = a ( u ) a ( v ) ∑ x = 0 N − 1 ∑ y = 0 N − 1 f ( x , y ) cos ( ( 2 x + 1 ) j π 2 N ) cos ( ( 2 y + 1 ) j π 2 N ) (7)

The inverse transform is obtained as:

f ( x , y ) = ∑ x = 0 N − 1 ∑ y = 0 N − 1 a ( u ) a ( v ) F ( u , v ) cos ( ( 2 x + 1 ) j π 2 N ) cos ( ( 2 y + 1 ) j π 2 N ) (8)

a ( u ) = a ( v ) = 1 2

With for u = v = 0

a ( u ) = a ( v ) = 1 for u , v = 1 , 2 , ⋯ , n − 1

Measurement of relative smoothness is one of selected applications; it is made by the following relation:

R = 1 − 1 1 + σ 2 ( x ) (9)

where, σ is the standard deviation of the gray values. x represent sub block of the image; biggest is relative smoothness better is the smooth of chosen block. In this paper relative smoothness is use as selected criteria for the four sub-images of second level decomposition of host image by wavelet discrete transform [27].

The watermarking image consists of inserting in an indelible manner, information in a host image then to tempt to recover this information after transfer [28]. The diagrams of watermarking are varied according to their application domain. The insertion of the signature will make itself either in the spatial domain either in the transform domain. In additive diagram, information to insert is added in a picture whereas in a substitutive diagram information to insert is substituted for features of the picture. In our work we use an additive diagram in frequential domain. Equation gives an example:

X W , K = X K + α W K (10)

With W_K marked bock image, X_K, original image block,α mark strength, X_W,K watermarked image.

The model of Vershult is a model of growth proposed by Pierre François Vershult [29]. This model uses the functions refine to explain the birth rate and the death rate of a population. Puttingx as carves of the population; m(x) the death rate and n(x) the birth rate. The size of the population follows the differential equation:

d x d t = x ( n ( x ) − m ( x ) ) (11)

If m and n are affine functions respectively increasing and decreasing functions. If, on the other hand, for x tending towards 0 growths is positive, the equation becomes:

d x d t = x ( a − b x ) (12)

with a and b two real positive.

Then, by setting K = a/b, the Equation (12) becomes:

d x d t = α X n ( 1 − x k ) (13)

The constant K affects solution of this Equation (13):

It is shown that:

· The constant function K is a solution of this equation

· If x < K then the population grows

· If x> K then the population decreases.

The discrete resolution of the transformed Equation (13) is:

X n + 1 − X n = α X n ( 1 − X n k ) (14)

if we put α + 1 = μ Equation (14) becomes

Y n = α X n μ k (15)

Y n + 1 = μ Y n ( 1 − Y n ) (16)

Equation (16) is call logistic map equation; its behavior varies according to the values of μ :

For μ included between 1 and 3, that means understood between 0 and 2, does the Y_n continuation converge toward ( 1 − μ ) / μ and does one well recover a continuation (Y_n) convergent toward K

For μ superior to 3, the continuation (Y_n) can, according to the values of oscillation between 2, 4, 8, 16… goes.

For μ equal to 4, the continuation (Y_n) can, according to the values of μ , to oscillate in all senses, producing the chaos thus.

Watermark embedding and encryption are described as follows:

Input: Host_object, Watermark

Output: Encrypted_Object

Step 1: Read grey scale Host_object of size M × N.

Step 2: Decompose Host Object using Daubichies wavelet [LL1, HL1, LH1, HH1] = dwt2 (Host_Object, “db”)

Step 3: Apply DWT to HL1 sub-band of Host Object, [LL2, HL2, LH2, HH2] = dwt2 (HL1, “db”).

Step 4: Calculate relative smoothness (R) of four LL2, HL2, LH2, and HH2, choose the sub-band that has the biggest value of smoothness, HH2.

Step 5: Read audio signal, converted in 2D, call watermark of size m × n.

Step 6: Apply DCT to Watermark W, W_DCT = dct2 (“Watermark”).

Step 7: Perform embedding Watermark W_DCT by sub-band (HH2) of Host_object find in step 4, HH2new = HH2 + αW_DCT after resizing.

Step 8: Apply inverse IDWT to get new HL1_new, HL1new = idwt2 (LL2, HL2, LH2, HH2new, “db”);

Step 9: Apply now inverse IDWT to get Watermarked_Objet, Watermarked_Objet = idwt2 (LL1, HL1new, LH1, HH1, “db”);

Display Host_Object, Watermarked_Object, PSNR.

Step 10: Generate chaotic sequence with following parameters: µ = 3.93695629844, and initial x₀ = 0.456

Step 11: Generate n-iterations with logistic map to get h(i) different orbit: y n + 1 = μ y n ( 1 − y n ) ;

Step 12: Associate the chaotic sequence with pixels of watermarked;

Step 13: Make permutation in lines, by N-iterations;

Step 14: Make permutation in columns by M-iterations and get encrypted image;

Step 15: Display Encrypted_objet.

Input: Encrypted Object, Host object,

Output: Extracted Watermark;

Step 1: Read Encrypted Object.

Step 2: Generate chaotic sequence with following parameters: µ = 3.93695629844, and initial x₀ = 0.456

Step 3: Generate n-iterations with logistic map to get h(i) different orbit: y n + 1 = μ y n ( 1 − y n ) ;

Step 4: Associate the chaotic sequence with pixels of watermarked;

Step 5: Make permutation in lines and columns, by N-iterations then M-iterations;

Step 6: GetDecrypted_Objet;

Step 7: Decompose Decrypted_Objet using Daubichie wavelet

[LLr, Recover_HLr, LHr, HHr] = dwt2 (Decrypted_Object, “db”),

Step 8: Apply DWT to Recovered_HLr sub-band of Decrypted_Objet, [LL2r, HL2r, LH2r, and HH2r] = dwt2 (Recoverd_HLr, “db”).

Step 9: Read HH2 sub-band of Host_object, step 3; step 4 of watermark embedding algorithm.

Step 10: Find Scrambled_Watermark using scale factor α used in watermark embedding algorithm as, Scrambled_Watermark = (HH2r-HH2)/α, get 2D audio son.

Step 11: Apply inverse IDCT to Scrambled Watermark to find final Extracted Watermark, Extracted Watermark = idct2 (“Scrambled Watermark”) equal audio son.

Step 15: Display Extracted Watermark.

In order to show the differences between the original and watermarked images, the signal-to-noise ratio (PSNR) and the structural similarity index (SSIM) are used [30]. The PSNR indicates the destruction rate while the SSIM is used to express the level of similarity ( [31] [32] ). These two metrics are used after embedding process.

Peak-Signal-To-Noise-Ratio (PSNR) is defining as follow:

PSNR = 10 log 10 ( 255 × 255 ∑ m = 0 M − 1 ∑ n = 0 N − 1 [ I 1 ( x , y ) − I 2 ( x , y ) ] 2 ) (17)

where I₂ is watermarked image and I₁ is original image. Bigger the PSNR is better the watermark conceals is [33].

The SSIM metric is defined by the following equation:

SSIM ( O , W ) = ( μ y μ x + C 1 ) ( 2 σ x y + C 2 ) ( μ x 2 + μ y 2 + C 1 ) ( σ x 2 + σ y 2 + C 2 ) (18)

σ_x and σ_y are the mean intensity of x and y. In the same time σ x 2 and σ y 2 are respectively variance of x and y [33]. σ_xy is the covariance of x and y. The averages of x and y are μ_y and μ_x. Variables C₁ and C₂ are used to stabilize the division with weak denominator. The value of SSIM varies between −1 and 1, where the maximum value, i.e., 1 is obtained for two similar images [33].

Based on the results of this Table 1, it can be concluded that the destruction of images is acceptable after embedding process.

Bit Error Rate (BER) is one of many metrics use to verified the robustness of the proposed algorithm, we use it, in our case. It’s calculated between original watermark and the extracted watermark after applying attacks on the watermarked images [33]. The bit error rate is calculated by:

BER ( W , E W ) = ∑ i = 1 m ∑ j = 1 n W ( i , j ) + E W ( i , j ) m × n (19)

where W is the original watermark and EW is the extracted watermark. m and n are the size of the original watermark [34].

Another statistic parameter use is Normalized Correlation (NC) which measures the similarity and difference between original watermark and extracted watermark. Generally, NC obtained between 1 and 0.7 is acceptable [35].

Another statistic parameter use is Normalized Correlation (NC) which measures the similarity and difference between original watermark and extracted watermark. Generally NC obtained between 1 and 0.7 is acceptable.

NC = ∑ i = 0 n Or_watermark × Ex_watermark ( ∑ i = 0 n Or_watermark × ∑ i = 0 n Ex_watermark ) 1 2 (20)

Or_watermark is original watermark, Ex_watermark is extracted watermark.

Table 2 shows SSIM and BER results after applying cropping 20% and salt & pepper attacks on the watermarked images.

In this section we use many different types of attacks (like Gaussian), image processing attacks (resizing, rotation) and hybrid attacks (like salt & pepper and Gaussian filter)to demonstrate the robustness of the suggested watermarking algorithm on the watermarked images and then extracts the embedded watermarks. The average results of these tests are shown in Table 2. In the above table, focused of obtained results SSIM, NC and BER tests, we can conclude that the presented watermarking approach has better robustness against attacks in the embedding and the extracting processes of the proposed approach [35].

Tests results of our scheme are compare with the results of two other researchers scheme ( [36] [37] ), these results are better than the two others methods.

Figure 3 and Table 3 show the results under different scale factor, the results of PSNR in our case are superior to 50 which denoted the robustness and the imperceptibility of the proposed scheme.

The difference and one advantage of our scheme is the use of audio logo which is transform in 2D then DCT transformed. This watermark is not easily recognizable.

In this section, we present the obtained visual results after the conjoint phase of watermarking and encryption of the proposed algorithm. Figures 4-7 show that effectively the watermark is invisible and the encryption is realized. The two mains objectives for medical images security, imperceptibility and confidentiality are satisfied.

The secret key sensitivity is an essential characteristic for a good encryption-system; it guarantees the safety and robustness against exhaustive attacks.

The largest key space is very important, because of failure brute-force attacks. Secret key for our proposed scheme is 3.93695629844, representing initial values of Logistic map, having 10⁻¹¹ precision [38]. When a change of 1 digit is made in the secret key, the result is very different in term of encrypted and decrypted image 2 [39]. Figures 8-11 show the results of key space analysis of our tests images.

To get information about statistical properties of encrypted image, histogram analysis is a better way to obtain this information. Histogram of the encrypted image gives the distribution of pixels. Figure 12 shows the histograms of the original images used, and Figure 13 shows the histograms of these same encrypted images, where we note a uniform distribution, reflecting a good encryption.

Correlation analysis is very essential for the encryption phase and is carried out between the different pixel pairs of the original and encrypted images.

The correlation coefficients are calculated by the following formulas:

c o v ( u , v ) = E ( u − E ( u ) ) ( v − E ( v ) ) (21)

r u v = c o v ( u , v ) ( D ( u ) ) ( D ( v ) ) (22)

where u and v are intensities values of two adjacent pixels, r u v is the correlation coefficient. c o v ( u , v ) , E ( u ) and D ( u ) are given as follows

E ( u ) = 1 N ∑ i = 1 N u i (23)

D ( u ) = 1 N ∑ i = 1 N ( u i − E ( u ) ) 2 (24)

c o v ( u , v ) = 1 N ( u i − E ( u ) ) ( v i − E ( v ) ) (25)

The pixels of an original image are strongly correlated in the horizontal,

vertical and diagonal directions. In order to know whether a cryptographic system should produce encrypted images without any correlation between adjacent pixels. Adjacent pixel correlation coefficients for horizontal, vertical and diagonal directions respectively, were calculated. Figure 14 shows the results of the horizontal correlation of the original image and Figure 15 shows the encrypted image.

It’s remarkable in Figure 14. The pixels are concentrated in a particular area of original image. It means that the correlation is important, but in case of Figure 15, the pixels are not concentrated in one part of encrypted image means that the correlation is low. Some correlation coefficients are mentioned in following Table 4. The values of correlation coefficients are very less under 1. These give a conclusion that the encryption algorithm is very robust.

This test gives the sensibility of encrypted image referred to original image. For this, we encrypted both images (I₀₁, I₀₂) which are different by one pixel, and then Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI) are calculated. NPCR represents the amount of pixels changed when one pixel is change [39].

NPCR = ∑ i = 0 H ∑ j = 0 L D j H × L (26)

With D i ; j = { 1         if   I 01 ≠ I 02 0     if   I 01 = I 02

UACI = 1 H × L ∑ i = 0 H ∑ j = 0 L I 01 ( i , j ) − I 02 ( i , j ) 2 8 − 1 (27)

Table 5 gives the values obtained after tests for originals images.

The obtained results of NPCR are around 90% for all four tests encrypted images, in the same time results of UACI are around 40%. We can say that our algorithm is really satisfactory in term of robustness and confidentiality, better aspect of medical image security.

Speed of encryption and decryption is also use to characterize scheme. In our case the time results are obtained with computer having the following specifications Dell core Duo E7200 @2GHz, 1.96 GB RAM. Table 6 shows the time of encryption and decryption. This time varies depending on the image used.

The calculation of the entropy is a statistical parameter of the image. It is given by the following formula:

H ( m ) = ∑ i = 10 r − 1 p ( m i ) log 2 1 p ( m i ) (28)

where p(m_i) represents the probability of pixel m_i. Bits is expression of Entropy. We can have 2⁸ bits, where each pixel in the image is represented by 8 bits. Following tables give the comparison of entropy with other methods.

For above tables the pertinence of the proposed method is proved, our method after the phase of frequential watermarking with adding audio signal make encryption by the technique of permutation and confusion, function of numbers of iterations. The main advantage of the proposed method is the sensitivity to initials conditions which in this case are keys of encryption. The same keys are indispensable for decryption phase.