Table 3 Mathematical definitions of the performance assessment metrics.

From: A secure and efficient image encryption scheme based on chaotic systems and nonlinear transformations

Metric

Descriptive equation(s)

SSIM

\(SSIM(f,g) = l(f,g)^\alpha \cdot c(f,g)^\beta \cdot s(f,g)^\gamma\) where    \(l(f,g) = \frac{{2\mu _{x} \mu _{y} + C_{1} }}{{\mu _{x}^{2} + \mu _{y}^{2} + C_{1} }},\)\(c(f,g) = \frac{{2\sigma _{x} \sigma _{y} + C_{2} }}{{\sigma _{x}^{2} + \sigma _{y}^{2} + C_{2} }},\)\(s(f,g) = \frac{{\sigma _{{xy}} + C_{3} }}{{\sigma _{x} \sigma _{y} + C_{3} }}.{\text{ }}\)

MSE

\(MSE = \frac{\sum _{i=0}^{M-1} \sum _{j=0}^{N-1} (I_{i,j} - I'_{i,j})^2}{M \times N}\)

PSNR

\(PSNR = 10 \log _{10} \left( \frac{I_{max}^2}{MSE}\right) , \quad I_{max} = 255\)

MAE

\(MAE = \frac{1}{M \times N} \sum _{i=0}^{M-1} \sum _{j=0}^{N-1} |P_{i,j} - E_{i,j}|\)

Entropy

\(H(m) = \sum _{i=1}^M p(m_i) \log _2 \frac{1}{p(m_i)}\)

DFT

\(F(k,l) = \sum _{i=0}^{N-1} \sum _{j=0}^{N-1} f(i,j) e^{-i 2\pi \left( \frac{ki}{N} + \frac{lj}{N}\right) }\)

CC

\(\rho (x,y) = \frac{\text {cov}(x,y)}{\sqrt{\sigma (x)} \sqrt{\sigma (y)}}, \quad \text {cov}(x,y) = \frac{1}{N} \sum _{i=1}^N (x_i - \mu _x)(y_i - \mu _y)\)

NPCR

NPCR = \(\frac{{\sum\nolimits_{{x = 1}}^{M} {\sum\limits_{{y = 1}}^{N} D } (x,y)}}{{M \times N}} \times 100,\) where    \(D(x,y) = {\left\{ \begin{array}{ll} 0, & I(x,y) = I'(x,y) \\ 1, & \text {otherwise} \end{array}\right. }\)

UACI

\(UACI = \frac{1}{M \times N} \sum _{x=1}^M \sum _{y=1}^N \frac{|I_1(x,y) - I_2(x,y)|}{255} \times 100\)