Page 52 - 2024-Vol20-Issue2

48 | Alobaidi & Mikhael

ficients are [20]: Pass-High Pass) ?D(i, j), all combined within a matrix. When
applied to 2D signals like images, a single-level DWT decom-
v2 M-1 N-1 position involves the utilization of a scaling function called
M×N ?(i, j) and 3 wavelets referred to as ?(i, j). The computation
g(u, v)cm of these wavelets is performed as follows:

u=0 v=0
? ?A(m,n)= (1)
,
cos m(2u + 1)p cn cos n(2v + 1)p
2M 2N
?(i, j) = ?(i)?( j) (4)
where g(u, v) is the signal in the time domain and G(m, n) is ?H (i, j) = ?(i)?( j) (5)
the mth row, nth column DCT coefficient for u = 0, 1, . . . M - 1 ?V (i, j) = ?(i)?( j) (6)
?D(i, j) = ?(i)?( j) (7)
and v = 0, 1, . . . N - 1.

= v2 M-1 N-1
M×N
A(m, n)cm

u=0 v=0
? ?g(u,v) (2) The 2D-DWT of an image g(i,j) of size M × M is:

cos m(2u + 1)p cn cos n(2v + 1)p = v1 M-1 M-1 j)?t0,m,m(i, j)
2M 2N MM
g(i,

i=0 j=0
? ?W? (t0, m, m) (8)

where cm, and cn are:

v1 for m = 0 =v 1 M-1 M-1
cm = 2 MM
g(i,
1 otherwise
i=0 t=0
(3) ? ?W?r (t, m, m) j)?tr,m,m(i, j) (9)

In steganography, the DCT is applied to blocks or segments r = {H,V, D}
of the cover image. Steganography algorithms that utilize
DCT often choose certain frequency coefficients for hiding t0 is an arbitrary initial scale and the W? (t0, m, m) coeffi-
information. These coefficients are typically selected based
on their perceptual importance, meaning coefficients that are cients is the Approximation of the g(i,j) at scale t0. The
less noticeable to the human eye are preferred. The most W?r (t, m, m) coefficients add horizontal, vertical, and diagonal
commonly used coefficients for steganography are usually details for scales t = t0. Practically, t0 = 0, M = 2Jso that
those corresponding to low-frequency components. The se- t = 0, 1, 2, ....T - 1 and m = 0, 1, 2, ..., 2t - 1.
cret data is usually in the form of binary bits. These bits are
then embedded by modifying the selected DCT coefficients. D. Adaptive Algorithm
The modification can be achieved by adding or subtracting The steps in the adaptive algorithm detailed in [17] can be
small values to the coefficients, thereby encoding the secret summarized as follows:
information. After embedding the secret data, the modified
DCT coefficients are quantized and compressed. Quantization 1. the total cover image energy is calculated;
reduces the precision of the coefficients, making the changes
introduced by embedding less noticeable. Compression fur- 2. The 2D DCT is applied to get the first 2D DCT repre-
ther helps in reducing the size of the stego image. At the sentation and the energy in this domain is calculated;
receiver end and in order to retrieve the hidden information,
the stego image undergoes the reverse process. The DCT 3. Predefined number of coefficients are chosen and the
coefficients are inversely transformed to the spatial domain, rest are transformed back to the spatial domain;
resulting in the reconstructed image. The hidden data is ex-
tracted by examining the modified coefficients. Finally, it’s 4. The current version of the image is transformed to the
worth noting that the specific techniques and algorithms used Haar domain using 2D DHT. Then, predefined number
in steganography can vary, and there are numerous variations of coefficients are chosen and the rest are transformed
and refinements to the process described earlier. back to the spatial domain;

C. Discrete Haar Transform (DHT) 5. The current total energy is calculated. If the calculated
value is less than 0.05% of the value in step 1, go to
The process of Wavelets [21] results in 4 frequency bands: step 2 ;otherwise the algorithm halts;

LL (Low Pass-Low Pass) ?(i, j), LH (Low Pass-High Pass) 6. The final outputs are the weights of each coefficient in
?H (i, j), HL (High Pass-Low Pass) ?V (i, j), and HH (High each domains (Cosine, and Haar).