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The energy residual, F(a, ß ), the distinction lies in minimiz-
ing the cost function, which is calculated as the discrepancy
between the initial energy and the energies preserved within
each domain. In particular, F(a, ß ) is calculated as follows:
F(a, ß ) = [C1]2 - [T2,1(C2)]2 - [T3,1(C3)]2
(10)
where [ ]2 is the element-wise square. The process begins by
utilizing a Steepest Descent Algorithm [22] to decrease the
remaining error. Once the iteration concludes, a designated
count of coefficients is preserved in two distinct domains: the
Cosine and the Haar domains. The resulting feature vector
for each signal (here, it is the cover image) is obtained by
combining these retained coefficients together.
The parameters for the Training phase are as follows. The
weight matrices a is populated with 0.5 while ß is initialized
with 0.3. The updating equations in every iteration are as
follows [23]:
ax,y(n + 1) = ax,y(n) - µax,y ?ax,y F (11)
ß x, y(n + 1) = ß x, y(n) - µßx,y?ßx,yF (12)
where x, and y span the entire domain and depending on
ax,y and ßx,y are elements in [a] and [ß ] respectively, n is the
iteration index, and µ is the converging factor. The converging
factors, µax,y and µßx,y, are calculated in the following fashion:
µa = F(n) (13)
(14)
N-1 N-1 [?ax,y F]2
?x=0 ?y=0
µß = F(n)
?xN=-01 ?Ny=-01[?ßx,y F]2
E. Performance Metrics
Performance metrics [24] are measurements used to evaluate
the effectiveness, efficiency, accuracy, or quality of a system,
process, algorithm, or model. The choice of performance
metrics depends on the specific task or application.
1) Structural Similarity Index (SSIM)
To measure similarity between parts of same or different im-
ages, SSIM is employed. Post-processing quantitative judg-
ment of the change of the structure of these parts is measured
by SSIM. Structure, contrast, and luminance are the segment
ofSSIM. SSIM ? [-1, 1], and the maximm limit is reached
when image parts are identical. The calculation of SSIM is:
SSIM(a, b) = [s(a, b) * c(a, b) * l(a, b)] (15)
where a and b are input images (or blocks) under comparison,
s(a, b) is structure component, c(a, b) equals contrast, and l(a,
b) is luminance. These factors are calculated in the following
Alobaidi & Mikhael
manner:
s(a, b) = sab +C3)
sasb +C3
c(a, b) = 2sasb +C2
sa2 + sb2 +C2
C2 = (K2 * L)2 (16)
l(a, b) = 2µaµb +C1
µa2 + µb2 +C1
C1 = (K1 * L)2
where s depicts standard deviation, µ represents mean, s 2
represents variance, K1 = 0.01, L equals to one, C3 is a small
constant, and K2 = 0.03.
2) Root Mean Squared Error (RMSE)
Mean Squared Error, one of the Regression metrics, measures
the average squared difference between the pixel values of
the original signal and the reconstructed/compressed signal.
It provides a quantitative measure of the overall distortion
between the two signals. The Root MSE (RMSE), the square
root of the MSE, is calculated as follows:
RMSE = m 1 n ? ?(I(x, y) - K (x, y))2 (17)
×
m n
where I(x, y) represents the pixel value of the original signal
at position (x, y), K(x, y) represents the pixel value of the
reconstructed/compressed signal at the same position, and (m
* n) is the total number of pixels in the image. A lower RMSE
value indicates a smaller average difference and, therefore,
better reconstruction or compression quality. However, RMSE
alone may not provide a perceptually meaningful measure of
quality, as it does not consider the human visual system’s
sensitivity to different image characteristics.
3) Peak Signal-to-Noise Ratio (PSNR)
PSNR is a logarithmic measure that relates the maximum
possible power of a signal (in this case, the maximum pos-
sible pixel value) to the power of the noise (the difference
between the original and reconstructed/compressed signals).
It is expressed in decibels (dB). The formula for PSNR is:
PSNR = 10 * log10(MAX2/MSE) (18)
where MAX is the maximum pixel value (e.g., 255 for an
8-bit grayscale image). PSNR provides a more perceptually
relevant measure of quality because it takes into account the
dynamic range of the pixel values and is logarithmic. A higher
PSNR value indicates better quality, as it indicates a smaller
ratio of noise to the maximum signal power.