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Reversible Data Hiding

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Reversible Data Hiding

ECE643 Digital Image Processing (I) Course Project

Professor: Yun Q. Shi

Su Yu

12/02/2011

- Introduction
- Applications
- Methods
- Histogram Pair
- Optimum Histogram Pair

- Conclusion
- Simulation

- Introduction
- Applications
- Methods
- Histogram Pair
- Optimum Histogram Pair

- Conclusion
- Simulation

- What’s Data Hiding?
- A process to embed useful data (information) into a cover media.
- Data invisibility is the major requirement.

Data

1

1

…

…

1

0

Cover Media

Marked Media

+

=

- Distortion happens in embedding process:

Data

1

1

…

…

1

0

So Bad

Unacceptable

+

=

- Distortion happens in embedding process:
- First Requirement:
Minimize the distortion and maximize the data payload

Data

OK

Acceptable

+

=

1

1

…

…

1

0

- What’s Reversible Data Hiding?
- A process to reverse the marked media back to the original cover media after the hidden data are extracted.
- Reversible or lossless ability is required.

Data

1

1

…

…

1

0

Cover Media

Marked Media

+

- Errors in reverse process are not allowed:
- Second Requirement:
No error in data and cover media

Data

0

1

…

…

1

1

Data Error

Unacceptable

Not Original

Unacceptable

+

- Introduction
- Applications
- Methods
- Histogram Pair
- Optimum Histogram Pair

- Conclusion
- Simulation

- Secure medical image data system
- Law enforcement
- E-government
- Image authentication
- Covert Communication

G. Xuan, C. Yang, Y. Zhen, Y. Q. Shi, and Z. Ni; Reversible Data Hiding Using Integer Wavelet Transform and Companding Technique

- Introduction
- Applications
- Methods
- Histogram Pair
- Optimum Histogram Pair

- Conclusion
- Simulation

- Histogram Pair
- Based on Paper:
- Z. Ni, Y. Q. Shi, N. Ansari and W. Su, Reversible Data Hiding

- Optimum Histogram Pair
- Based on Papers:
- G. Xuan, C. Yang, Y. Zhen, Y. Q. Shi, and Z. Ni, Reversible Data Hiding Using Integer Wavelet Transform and Companding Technique
- G. Xuan, Y. Q. Shi, P. Chai, X. Cui, Z. Ni, and X. Tong, Optimum Histogram Pair Based Image Lossless Data Embedding

- Introduction
- Applications
- Methods
- Histogram Pair
- Optimum Histogram Pair

- Conclusion
- Simulation

- PSNR (Peak Signal-to-Noise Ratio)
- An engineering term for the ratio between the maximum possible power of a signal and the power of corrupting noise that affects the fidelity of its representation
- The PSNR is most commonly used as a measure of quality of reconstruction of lossy compression (e.g., for image compression).

http://en.wikipedia.org/wiki/Peak_signal-to-noise_ratio

- PSNR (Peak Signal-to-Noise Ratio)
- Mathematical definition
- , in dB
- I = cover image, K = marked image
- MAXI = maximum gray value 255

- PSNR represent the distortion level between marked image and cover image

http://en.wikipedia.org/wiki/Peak_signal-to-noise_ratio

- PSNR (Peak Signal-to-Noise Ratio)
- Typical values in lossy image and video compression are between 30 and 50 dB, where higher is better.

Original Image

PSNR=31.45dB

http://en.wikipedia.org/wiki/Peak_signal-to-noise_ratio

- Histogram Pair
- Histogram h(x) is the number of occurrence as the variable X assumes value x, i.e. X is number of pixels on one certain gray value in an image.
- Only two consecutive integers a and b assumed by X are considered, i.e. x ∈ a, b.
- Furthermore, let h(a) = m and h(b) = 0. We call these two points as a histogram pair.
- And sometimes denote it by, h = [m, 0], or simply [m, 0].

G. Xuan, Y. Q. Shi, P. Chai, X. Cui, Z. Ni, and X. Tong, Optimum Histogram Pair Based Image Lossless Data Embedding

- Histogram Pair
- Example: in a histogram of an image, a and b are adjacent integers, h = [m, 0] is a histogram pair.

0

Number of Pixels

m

b

a

Gray Value

- Advantages
- Large data payload
- 5k-60k bits for 512*512*8 grayscale image
- High visual quality
- PSNR > 48 dB

- Method
- Histogram Pair

Z. Ni, Y. Q. Shi, N. Ansari and W. Su, Reversible Data Hiding

- Use “Lena” image as an example
- Step 1:
- In the histogram find zero point (e.g. 255 no pixel on the gray value of 255);
- Then find peak point (e.g. 155 maximum number of pixels on the gray value of 155);
- The objective to find the peak point is to increase the embedding capacity as large as possible, which will be further explained.

Z. Ni, Y. Q. Shi, N. Ansari and W. Su, Reversible Data Hiding

- Step 1:

- Step 2:
- The whole image is scanned;
- The gray value of pixel with gray value between 156 and 254 is incremented by one;
- This step is equivalent to shifting the range of histogram [156,254] one unit towards the right hand side leaving the gray value 156 empty;
- Then a=155 and b=156 are adjacent integers, h = [2785, 0] is a histogram pair.

Z. Ni, Y. Q. Shi, N. Ansari and W. Su, Reversible Data Hiding

- Step 2: h = [2785, 0] is a histogram pair

- Step 3:
- The whole image is scanned once again;
- Once a pixel with gray value of 155 is encountered, we check the data to be embedded;
- If the to-be-embedded bit is “1”, the pixel value is added by 1. Otherwise, the pixel value is kept intact.
- The capacity of this algorithm equals to the maximum number of pixels (2785 bits)

Z. Ni, Y. Q. Shi, N. Ansari and W. Su, Reversible Data Hiding

- Step 3: Embedded data

- Step 3: Embedded data
- PSNR = 53.8 dB

- Step 1:
- The whole marked image is scanned;
- The order must be same as embedding;
- Once the gray value of the maximum point is met, if the value is intact, e.g., 155, the “0” is retrieved;
- If the value is altered, e.g., 156, the “1” is retrieved;
- In this way, the data embedded can be retrieved.

Z. Ni, Y. Q. Shi, N. Ansari and W. Su, Reversible Data Hiding

- Step 2:
- The whole image is scanned once again;
- Once the pixels whose gray value is between the peak point (e.g. 155) and the zero point (e.g. 255) is met (e.g. interval [156,255]), the gray value of those pixels will be subtracted by 1;
- In this way, the original image can be recovered without any distortion.

Z. Ni, Y. Q. Shi, N. Ansari and W. Su, Reversible Data Hiding

- Result: Data error rate=0, Image error rate=0

Z. Ni, Y. Q. Shi, N. Ansari and W. Su, Reversible Data Hiding

- PSNR of all marked images is above 48 dB;
- Because the pixels whose gray value is between the zero point and the peak point will add by 1 or minus by 1;
- In the worst case, all pixels of the image will add or minus by 1. That means MSE=1;
- Hence the PSNR=10xLog10(255x255/MSE)=48.13 dB.

Z. Ni, Y. Q. Shi, N. Ansari and W. Su, Reversible Data Hiding

- Introduction
- Applications
- Methods
- Histogram Pair
- Optimum Histogram Pair

- Conclusion
- Simulation

- Companding
- The process of signal compression and expansion.

- Compression and Expansion
- Compression: mapping large range of original signals x, into narrower range, y=C(x).
- Expansion: reverse process of compression, x=E(y).
- After expansion, the expanded signals are close to the original ones.

G. Xuan, C. Yang, Y. Zhen, Y. Q. Shi, and Z. Ni, Reversible Data Hiding Using Integer Wavelet Transform and Companding Technique

- Companding
- Assume the original signals are x,
- If the compression function is y=C(x);
- If the expansion function is x=E(y);
- If the equation E[C(x)]=x is satisfied, then this kind of companding could be applied into reversible data hiding.

G. Xuan, C. Yang, Y. Zhen, Y. Q. Shi, and Z. Ni, Reversible Data Hiding Using Integer Wavelet Transform and Companding Technique

- Companding in Reversible Data Hiding
- Let y=C(x), y=P1P2P3…Pn, Pi ∈ {0,1};
- Let b ∈ {0,1}, y’=P1P2P3…Pnb, then y’=P(y)=2y+b, b is the hiding data;
- If y’x, modification of signal will hardly be perceived;
- By hiding data extraction, extract LSB of y’, i.e. b=LSB(y’), recover signal y=(y’-b)/2;
- Recover original signal x by expansion x=E(y)

- Companding in Reversible Data Hiding
- So, two conditions must be satisfied:
- y=C(x), x=E[y], => E[C(x)]=x;
- Condition (1): E[C(x)]=x;
- y’=P(y)=2y+bx, => P[C(x)]x;
- Condition (2): P[C(x)]x

- So, two conditions must be satisfied:

- For Condition (1), E[C(x)]=x
- Any one-to-one mapping function can be used;
- For multiple x mapped to single y, still could be used, E.g. , use bit “0” to indicate , and bit “1” to indicate , these overhead data also need to embedded into original signal.

- So, condition (1) is easy to satisfied.

- For Condition (2), P[C(x)]x
- On study of Human Visual System (HVS) points out that slight modification on wavelet high frequency sub band coefficients is hard to be perceived by human eyes.
- The method to compand image is to slightly change wavelet high frequency sub band coefficients for hiding data.

- So condition (2) could be satisfied.

- Sub bands (embedding region) for data hiding in coefficients are three high frequency sub bands HH, HL and LH.
- Question is: How to select the most suitable embedding region?

G. Xuan, Y. Q. Shi, P. Chai, X. Cui, Z. Ni, and X. Tong, Optimum Histogram Pair Based Image Lossless Data Embedding

- Wavelet Transform
- Likes Fourier Transform, is used to analysis image in frequency domain.
- Fourier Transform is based on sinusoid functions;
- Wavelet Transform is based on small waves (wavelets) which are varying in frequency and limited duration.

- Integer Wavelet Transform (IWT) maps integer to integer and can reconstruct the original signal with out distortion.

C. Gonzalez and R. E. Woods, < Digital Image Processing >, Prentice Hall, 3rd (2007) edition

G. Xuan, C. Yang, Y. Zhen, Y. Q. Shi, and Z. Ni, Reversible Data Hiding Using Integer Wavelet Transform and CompandingTechnique

- IWT high frequency sub band coefficients has two features:
- Most coefficients are small in magnitude, so it is convenient to select compression function C(x), could use linear functions to satisfy Condition (2), P[C(x)]x;
- For coefficients are large in magnitude, linear functionsare difficult to satisfy Condition (2), P[C(x)]x;
- A pre-defined threshold T is introduced to treat these two kinds of coefficients differently.

- Threshold T
- For use one to one mapping function;
- For x, x+1, or x-1… are compressed to same y and the recording data need to be embedded into wavelet coefficients as index for distinguish different values mapped to the same y;
- If T is small, a good visual quality of marked image is achieved;
- If T is large, a larger payload can be achieved.

- Question is: How to choose the best threshold T?

- Histogram Modification
- After data embedded in coefficients, some pixel’s gray value may overflow (>255) or underflow (<0);
- Histogram modification is needed to narrow the histogram from both sides by GR and GL;
- Modification G=GR+GL.

- Histogram Modification
- This modification is needed to be recorded and embedded as part of the overhead for recovery the original cover image.

- Question is: How to make adaptive histogram modification?

- Advantages
- Selection of most suitable embedding region
- Selection of best threshold T, leads highest PSNR for a given payload
- Minimum amount of histogram modification

- Method
- Optimum Histogram Pair
- Using Integer wavelet transformation
- Using adaptive histogram modification

- Selection of Optimum Parameters
- Suitable embedding region R
- Best Threshold T
- Adaptive histogram modification value G
- The selection of three arguments is based on generate maximum PSNR.

- Selection of Suitable embedding region R
- In order to improve PSNR,
- When the payload is small, R=HH, only embed data into HH sub band;
- When the payload is large, R=HH,HL,LH all three high frequency sub bands are used.

- Selection of Best Threshold T
- By experiment, for certain embedding capacity 0.02 bpp and three different cover image, the best threshold T does exist.

- Selection of Adaptive histogram modification value G
- After data embedding into each coefficient, underflow and overflow are checked;
- By experiment, only when the payload is larger than certain level, it needs histogram modification (G>0), otherwise, there is no need for histogram modification.
“Lena”, if payload > 1.0873 bpp (285027 bits)

“Barbara”, if payload > 0.5734 bpp(150320 bits) “Baboon”, if payload > 0.0080 bpp(2089 bits)

- Use as an example:
- Assume the to-be-embedded bit sequence D=[110001];
- The image 5*5, has 12 gray values

- Use as an example:
- Histogram is

-5

-4

0

-2

-1

1

2

3

-3

4

5

6

- Step1: expand image histogram
- From right side, h[4]=0, h[4] to h[5]

-5

-4

0

-2

-1

1

2

3

-3

4

5

6

- Step1: expand image histogram
- From right side, h[5]=0, h[5] to h[6]

-5

-4

0

-2

-1

1

2

3

-3

4

5

6

- Step1: expand image histogram
- From left side, h[-4]=0, h[-4] to h[-5]

-5

-4

0

-2

-1

1

2

3

-3

4

5

6

- Step1: expand image histogram
- From center h[3]=0, h[3] to h[4]

-5

-4

0

-2

-1

1

2

3

-3

4

5

6

- Step1: expand image histogram
- Histogram is
- Three histogram pairs: from right to left to center
- right [1,0], left [0,2], center [3,0]

-5

-4

0

-2

-1

1

2

3

-3

4

5

6

- Step2: Embedding Data
- from right to left to center D=[110001];
- right [1,0], capacity=1, embedded 1 using histogram pair method

-5

-4

0

-2

-1

1

2

3

-3

4

5

6

- Step2: Embedding Data
- from right to left to center D=[110001];
- left [0,2], capacity=2, embedded 10 using histogram pair method

-5

-4

0

-2

-1

1

2

3

-3

4

5

6

- Step2: Embedding Data
- from right to left to center D=[110001];
- Center [3,0], capacity=3, embedded 001 using histogram pair method

-5

-4

0

-2

-1

1

2

3

-3

4

5

6

- Step2: Embedding Data
- Histogram is

-5

-4

0

-2

-1

1

2

3

-3

4

5

6

- For application in “Lena” image, for certain payload, PSNR is good.

- Retrieval Algorithm is inverse to the embedding process;
- To retrieval data, the order is still from right to left to center, to check number of pixels on gray value (4,5), (-3,-4), (2,3) because those pairs are embedded data;
- Using the expansion function to get original cover image.

- Introduction
- Applications
- Methods
- Histogram Pair
- Optimum Histogram Pair

- Conclusion
- Simulation

- Comparison between two methods:

- Introduction
- Applications
- Methods
- Histogram Pair
- Optimum Histogram Pair

- Conclusion
- Simulation

- For Histogram Pair method, to hiding data sentence:
- “ECE 643 Digital Image Processing Course Project by Su Yu”
- In “Lena” image.

- Z. Ni, Y. Q. Shi, N. Ansari and W. Su, Reversible Data Hiding
- G. Xuan, C. Yang, Y. Zhen, Y. Q. Shi, and Z. Ni, Reversible Data Hiding Using Integer Wavelet Transform and CompandingTechnique
- 1. R. C. Gonzalez and R. E. Woods, < Digital Image Processing >, Prentice Hall, 3rd (2007) edition

Thank you!

Questions?