![Security Analysis
Correlation
Correlation is used to determine the similarity between images, Mathematically,
𝐶𝑜𝑟𝑟 = 𝑖 − µ 𝑗 𝑗 − µ 𝑗 𝑝(𝑖, 𝑗)/𝜕𝑖 𝜕𝑗
Mean Square Error
MSE is used to measure the difference between values implied by an
estimator and the true values of the quantity being estimated, Mathematically,
𝑀𝑆𝐸 =
1
𝑀 ∗ 𝑁
[𝐶1 𝑖, 𝑗 − 𝐶2(𝑖, 𝑗)]2
𝑁
𝑗=0
𝑀
𝑖=0
PSNR
Mathematically,
PSNR=10 ∗ 𝑙𝑜𝑔10 𝑀𝐴𝑋2
/MSE
Histogram testing
Histogram testing is used to check the quality of Encryption.
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Presentation
- 1. PRESENTED BY T A B I S H F A W A D 1 3 - M S - E E - 0 1 4 R I Z W A N A L I 1 3 - M S - E E - 0 1 9 “Comparison of image Encryption using different transforms/techniques” 1
- 2. Contents Introduction Discrete cosine transforms Discrete Wavelet transforms Security Analysis Proposed Algorithm Security table Conclusion 2
- 3. Introduction Encryption is the conversion of data or information from its original form to some other form that basically hides the information in it. The protection of image data from unauthorized access is important. Encryption is employed to increase the data security. Decryption is the inverse process of Encryption. In decryption, Original image is recovered from the encrypted image. 3
- 4. Discrete Cosine Transform The Discrete Cosine Transform (DCT) is a widely used transform coding technique. The DCT represents an image as a sum of sinusoids of varying magnitudes and frequencies. The role of the DCT is to decompose the original signal into its DC and AC components. DCT is a linear transformation it transforms the function f(i) into a function f (u). (1Dimensional DCT) The role of the IDCT is to reconstruct the original signal. 4
- 5. Cosine Basis Function The basis functions should be orthogonal 𝐵𝑝 𝑖 . 𝐵𝑞 𝑖 = 0 𝑖𝑓 𝑝#𝑞 The basis functions should be orthonormal if they are orthogonal 𝐵𝑝 𝑖 . 𝐵𝑞 𝑖 = 1 𝑖𝑓 𝑝 = 𝑞 cos 2i + 1/16 ∗ 𝑝𝜋 . (cos 2i + 1/16 . qπ = 0 𝑀−1 𝑖=0 𝑖𝑓 𝑝 # 𝑞 ( C p 2 ∗ cos 2i + 1 16 ∗ pπ ∗ C q 2 ∗ cos 2i + 1 16 ∗ qπ = 1 𝑖𝑓 𝑝 = 𝑞 𝑀−1 𝑖=0 5
- 6. 2D DCT & IDCT DCT 𝐹 𝑢, 𝑣 = 2𝐶 𝑢 𝐶 𝑣 𝑀 ∗ 𝑁 . . 𝑀−1 𝑖=0 cos 2𝑖 + 1 /2𝑀 ∗ 𝑁−1 𝑗=0 𝑢𝜋 ∗ cos 2𝑗 + 1 2𝑁 ∗ 𝑣𝜋 𝑓(𝑖, 𝑗) IDCT 𝑓 𝑖, 𝑗 = 2𝐶 𝑢 𝐶 𝑣 𝑀 ∗ 𝑁 . . 𝑀−1 𝑢=0 cos 2𝑖 + 1 /2𝑀 ∗ 𝑁−1 𝑣=0 𝑢𝜋 ∗ cos 2𝑗 + 1 2𝑁 ∗ 𝑣𝜋 𝑓(𝑢, 𝑣) 6
- 7. Wavelet Transform In multi resolution analysis (MRA), a scaling function is used to create a series of approximations of an the image. Each differing by the factor of 2 in resolution from its nearest neighbouring approximations. It seeks to represents a signal with good resolution in both time and frequency by using a set of basis functions called wavelets {𝛷 𝑘(𝑥)}. Wavelets are used to encode the difference in information between adjacent approximations. 𝑓 𝑥 = ⍺ 𝑘 ∗ 𝛷 𝑘(𝑥) 𝑀−1 𝑘 7
- 8. Discrete Wavelet transform DWT is a mathematical tool for decomposing an image. The DWT splits the signal into high and low frequency parts. The low frequency part is split again into high and low frequency parts. For each level of decomposition we first perform the DWT in the vertical direction followed by the DWT in the horizontal direction. 8
- 9. Wavelet Decomposition For 2nd level of decomposition, there are 4 sub-bands LL1, LH1, HL1, and HH1. For each successive level of decomposition, the LL sub band of the previous level is used as the input. To perform second level decomposition, the DWT is applied to LL1 band which decomposes the LL1 band into the four sub- bands LL2, LH2, HL2, and HH2. 9
- 10. Wavelet Functions 𝛹𝑗,𝑘 𝑥 = 2 𝑗 2 𝛹 2 𝑗 𝑥 − 𝑘 𝑊𝑖 = 𝑠𝑝𝑎𝑛{𝛹𝑗,𝑘(𝑥)} f (x) = ⍺ 𝑘 𝛹𝑗,𝑘(𝑥)𝑘 Orthogonality and orthonormality condition must be met. 10
- 11. Discrete wavelet transform Forward DWT 𝑊𝛷 𝑗𝑜, 𝑘 = 1 𝑀 𝑓(𝑛)𝛷𝑗𝑜,𝑘(𝑛) 𝑛 𝑊 𝛹 𝑗, 𝑘 = 1 𝑀 𝑓(𝑛)𝛹𝑗,𝑘(𝑛)𝑛 j ≥ jo Inverse DWT 𝑓 𝑛 = 1/ 𝑀 𝑊𝛷 𝑗𝑜, 𝑘𝑘 𝛷𝑗𝑜,𝑘(𝑛) +1/ 𝑀 𝑊 𝛹 𝑗, 𝑘𝑘 𝛹𝑗,𝑘(𝑛) 𝑤ℎ𝑒𝑟𝑒 𝑀 = 2 𝑗 11
- 12. Security Analysis Correlation Correlation is used to determine the similarity between images, Mathematically, 𝐶𝑜𝑟𝑟 = 𝑖 − µ 𝑗 𝑗 − µ 𝑗 𝑝(𝑖, 𝑗)/𝜕𝑖 𝜕𝑗 Mean Square Error MSE is used to measure the difference between values implied by an estimator and the true values of the quantity being estimated, Mathematically, 𝑀𝑆𝐸 = 1 𝑀 ∗ 𝑁 [𝐶1 𝑖, 𝑗 − 𝐶2(𝑖, 𝑗)]2 𝑁 𝑗=0 𝑀 𝑖=0 PSNR Mathematically, PSNR=10 ∗ 𝑙𝑜𝑔10 𝑀𝐴𝑋2 /MSE Histogram testing Histogram testing is used to check the quality of Encryption. 12
- 13. Proposed Algorithm Discrete Cosine transform CorrelationPSNRMSE Discrete Wavelet transform MSE PSNR Correlation Image Size (M*N) Histogram testing Histogram testing 13
- 14. Security table (Comparison) Correlation MSE PSNR -0.0161 -0.0882 18.6475dB Discrete Cosine Transform Decomposition level Correlation MSE PSNR L1 0.0046 1.0773 17.8076dB L2 0.0049 1.0779 17.8092dB L3 0.0049 -1.3339 16.8795dB L4 0.0034 0.8831 18.6705dB Discrete Wavelet Transform 14
- 15. Conclusion We applied DCT & DWT on a 256*256 size image. DWT has been a better transform and is a better decomposition technique. Each Level of decomposition has a different correlation and PSNR. 15
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