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Lossy Compression and Iterative Reconstruction for

Encrypted Image

Source: IEEE Transactions on Information Forensics and Security, vol.6, no.1, pp.53-58, March 2011

Author: Xin-Peng Zhang

Speaker: Le Hai Duong

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Introduction

Securely and efficiently transmitting dataCompress the dataEncrypt the compressed data

New sequenceEncrypt the dataCompress the encrypted data for transmission

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Goals

Novel scheme for lossy compression of encrypted image with flexible compression ratio

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The proposed scheme

Image Encryptionpsuedo-random permutation

Compression of Encrypted Imagediscarding the excessively rough and fine information of coefficients in the transform domain

Image Reconstructionreconstruct the priciple content of the original image by iteratively updating the values of the coefficients, with the help of spatial correlation in natural image

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Image Encryption

Number of pixelsNumber of bits is 8.NEncrypted data = permuted pixel-sequence

1 2 3 4 5 6 7 8 9 10

13 14 12 68 59 100 111 42 39 36

1 2 3 4 5 6 7 8 9 10

42 111 12 13 100 14 39 68 36 59

Permutation order 8, 7, 3, 1, 6, 2, 9, 4, 10, 5

Original data

Encrypted data

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Compression (1/5)

1. Decomposition q2

q1p

α.Np2

p1

q(1-α).N

rigid pixels elastic pixels

. . . . . .permuted pixels

Encrypted data

rigid pixels elastic pixelse.g., α = 0.3

p1 p2 p3 q1 q2 q3 q4 q5 q6 q7

42 111 12 13 100 14 39 68 36 59

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Compression (2/5)

2. Perform an orthogonal transform in the elastic pixelswhere H is a public orthogonal matrix size of

e.g.,p1 p2 p3 q1 q2 q3 q4 q5 q6 q7

42 111 12 13 100 14 39 68 36 59Encrypted data

rigid pixels elastic pixelse.g., α = 0.3

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Compression (3/5)

3. Compute

Rewritten the formula (2)

wherer

k rough information

tk fine information

Total length of bits to represents all sk

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Compression (4/5)e.g., M =4, Δ = 50

thus,

Total length of sk (bits)

Each value of sk is represented by bits

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Compression (5/5)

4. The compressed data including the rigid pixels, the bits of s

k, and the parameters N

1, N2, M, α, Δ

The compression ratio R

For our example,

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Image Reconstruction (1/8)

1. Obtain sk and all the parameters from compressed data

2. Use the secret key to retrieve the original position of rigid pixel

1 2 3 4 5 6 7 8 9 10

42 111 12

Permutation order 8, 7, 3, 1, 6, 2, 9, 4, 10, 5

e.g., rigid pixels in encrypted data

1 2 3 4 5 6 7 8 9 10

12 111 42

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Image Reconstruction (2/8)

1 2 3 4 5 6 7 8 9 10

12 111 42

3. Estimate the values for elastic pixels

1 2 3 4 5 6 7 8 9 10

12 12 12 12 62 111 111 42 42 42

average of 12 and 111value of nearest rigid pixel

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Image Reconstruction (3/8)

1 2 3 4 5 6 7 8 9 10

12 12 12 12 62 111 111 42 42 42

4. Rearrange the elastic pixels using the permutation

Permutation order 8, 7, 3, 1, 6, 2, 9, 4, 10, 5

p1 p2 p3 q'1 q'2 q'3 q'4 q'5 q'6 q'7

42 111 12 12 111 12 42 12 42 62

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Image Reconstruction (4/85. Calculate

e.g.,

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Image Reconstruction (5/8)6. Calculate the differences

e.g.,

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Image Reconstruction (6/8)7. Modify the coefficients to closet values consistent with the corresponding sk

e.g.,

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Image Reconstruction (7/8)8. Perform inverse transform

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Image Reconstruction (8/8)9. Calculate the average energy of difference between the two versions of elastic pixels

e.g.,

If D is greater than a threshold T (recommended 0.05), go back to step 5 – iterating Otherwise, terminate the iteration and output the image

made up of the rigid pixels and the final version of elastic pixels

go back to step 5.

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Experimental results (1/5)

(a) Original image Lena, (b) encrypted version, (c)the medium reconstructed image from compressed data with PSNR 27.1 dB, and (d) the final reconstructed image with PSNR 39.6 dB.

(a) (b)

(c) (d)

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Experimental results (2/5)

Smaller M and α result lower RSmaller Δ result higher PSNR

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Experimental results (3/5)

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Experimental results (4/5)

Smaller Δ result more iteration

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Experimental results (5/5)

[7] Kumar, A.A.; Makur, A.; , "Lossy compression of encrypted image by compressive sensing technique," TENCON 2009 - 2009 IEEE Region 10 Conference , vol., no., pp.1-5, 23-26 Jan. 2009

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Conclusions

Novel idea for compressing encrypted imageFlexible in setting system's parameters Good compression ratio down to 0.25 (Huffman: 94% for image)Justification for the use of orthogonal transform?