Probabilistic and Statistical Image Processing Concepts

1
Probabilistic and Statistical Image Processing
Concepts

• Sample mean of the set of pixel gray values in an
  N x M image A
• Sample variance of an image
• Standard deviation

2
Probabilistic and Statistical Image Processing
Concepts

• To convert an image A with mean and
  variance to a new image B with mean
• and variance , define B by
• This transformation will change a set of images
  with different means and variances to a set where
  the mean and variance are the same for each image
  in the set
• Possible use in watermarking/stegomarking several
  images simultaneously

3
Probabilistic and Statistical Image Processing
Concepts

• See Pattern recognition and image analysis, by
  Bose, Johnsonbaugh, and Jost, Prentice Hall,
  1996, ISBN 0-13-236415-8 for more
  statistical-based concepts applied to image
  processing.

4
Image Processing Concepts

• Linear transforms of images
• Other filters
• General image analysis schemes

5
Linear Transforms

• Recall from linear algebra any linear transform
  can be represented by a matrix
• Any image can be represented by a vector (simply
  vectorize image by putting all gray values, in
  raster-scan order, in an NM x 1 column vector)
• Thus, any linear transform to an image can be
  represented by a matrix vector multiplication

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Linear Transforms

• Additionally, any matrix can be investigated for
  different representations, such as a product of
  sparser matrices, a product of matrices each of
  which are banded, etc. using standard linear
  algebraic techniques and approaches
• A huge quantity of research has been done to
  optimize matrices for
• Implementation on various computer architectures
  and with various compilers for fast computation
• Implementation on hardware devices for optimal
  computation
• Implementation on custom-designed hardware
  devices
• Since so much is known about linear transforms
  and their implementations, and linear transforms
  are the first type of model that many scientists
  and engineers turn to when modeling phenomenon,
  linear transforms are used extensively in many
  scientific algorithms

7
Transform Domain Techniques for Steganography and
Watermarking

• Discrete Fourier Transform
• Discrete Cosine Transform
• Discrete Wavelet Transform
• Mellin-Fourier Transform
• Fresnel Transform
• Lapped Orthogonal Transform
• Related
• Singular Value Decomposition
• Minimax Eigenvector Decomposition

8
Transform Domain Techniques for Steganography and
Watermarking

• Discrete Fourier Transform and DWT combined
• A DWT-DFT Composite Watermarking Scheme Robust to
  Both Affine Transform and JPEG CompressionKang,
  Xiangui (Dept. of Electronics Engineering, Sun
  Yat-Sen University) Huang, Jiwu Shi, Yun Q.
  Lin, Yan Source IEEE Transactions on Circuits
  and Systems for Video Technology, v 13, n 8,
  August, 2003, p 776-786
• Fresnel Transform
• Digital image watermarking by Fresnel transform
  and its robustnessKang, Seok (Hokkaido Univ)
  Aoki, Yoshinao Source IEEE International
  Conference on Image Processing, v 2, 1999, p
  221-225.
• Lapped Orthogonal Transform
• Secure robust digital watermarking using the
  lapped orthogonal transform
• Pereira, Shelby (Univ of Geneva) O Ruanaidh,
  Joseph J.K. Pun, Thierry Source Proceedings of
  SPIE - The International Society for Optical
  Engineering, v 3657, Jan 25-27, 1999, p
  21-30Hough transform
• Hough Transform
• A rotation scaling and cropping invariant second
  generation watermarking scheme based on hough
  transformZhen, Ji (Faculty of Information
  Engineering) Zhang, Jihong Xiao, Weiwei Source
  Chinese Journal of Electronics, v 12, n 1,
  January, 2003, p 126-131

9
Transform Domain Techniques for Steganography and
Watermarking

• Mellin-Fourier transform
• Rotation, scale, and translation resilient
  watermarking for imagesLin, C.-Y. Wu, M.
  Bloom, J.A. Cox, I.J. Miller, M.L. Lui,
  Y.M.Image Processing, IEEE Transactions on
  , Volume 10 , Issue 5, May 2001 Pages767
  782
• Discrete Fourier transform
• Capacity estimates for data hiding in compressed
  imagesRamkumar, M. Akansu, A.N.Image
  Processing, IEEE Transactions on , Volume 10
  , Issue 8, Aug. 2001 Pages1252 - 1263

10
Transform Domain Techniques for Steganography and
Watermarking

• Singular Value Decomposition
• An SVD-based watermarking scheme for protecting
  rightful ownershipRuizhen Liu Tieniu
  TanMultimedia, IEEE Transactions on , Volume
  4, Issue 1 , March 2002 Pages121 128
• Minimax Algebra
• An Interlaced Minimax Eigenvector Decomposition
  Algorithm for Steganography, Davidson, J.L., and
  Kuan, D., in progress, 2004.

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Discrete Cosine Transform

• The discrete cosine transform is a linear
  transform
• Unlike the discrete Fourier transform, DCT has
  only real values in its computation
• Recall any linear transform can be represented
  by a matrix
• The matrix for the DCT is unitary (real and
  orthogonal)
• Used in JPEG compression

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Discrete Cosine Transform

• The forward equation, for NxN image A, is

• The inverse equation, for NxN image B, is

• Here

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Basis functions of DCT (8x8)
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DCT Basis Function Evaluation

• These basis functions are for NM8
• There are 88 64 basis functions or basis
  images
• To determine the actual values of the each of the
  64 basis images, we select a u and a v, both
  between 0 and 7. This indexes which of the 64
  matrices (on the previous slide) you will be
  calculating the 8 x 8 matrix of values for.
• Then, you create the 8 x 8 matrix using the
  kernel equation and the 64 pairs (i,j), where i
  and j each range from 0 to 7.

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DCT Basis Function Evaluation

• Example select u 1, v 3, N8
• General form for the kernel or basis matrix K

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DCT Basis Function Evaluation

• Evaluating,
• Once you have the 88 64 values for the kernel
  matrix K, you then need to apply it to the image

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Applying DCT to Image Data

• Once in the basis image form, you simply overlay
  the basis image directly onto the input image you
  want to transform, multiply the corresponding
  basis image values and input image values, sum
  them up, multiply with the appropriate constants
  in front, and output the value into the output
  image array

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Applying DCT to Image Data

• Now, sum up all 64 entries, multiply by
  constants, and output

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Applying DCT to Image Data

• We do this calculation for all other entries in
  the output image B,
• The image values in the output image B are called
  the transform coefficients for the DCT of A

20
Applying DCT to Image Data

• The transform actually multiplies each basis
  image pointwise with the input image, and reduces
  that to a single value
• This process can be viewed as a sort of
  matching of the input image with each of the
  basis images the more the magnitude of the
  individual image values match up with the
  corresponding basis image values, the higher the
  product of those two values will be, and
  consequently the higher the single value will be
  when all the 64 values are added up
• Sidenote this is the idea of a matched filter
  in signal and image processing

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Some Comments about the DCT

• If you look at the basis images, you can tell the
  general type of frequency information contained
  in each transform coefficient
• For example, the basis image has values
• for each i,j. When the constants are put in,

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Some Comments about the DCT

• we get
• which makes the transform coefficient b(0,0) a
  sort of average of the input image values
• Sidenote This is similar to the discrete Fourier
  transform (0,0) transform coefficient, giving a
  DC value

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Some Comments about the DCT

• The other basis images have similar frequency
  information the top row of basis images
  measure vertical frequencies from low (0,1) to
  high (0,7)
• The first column of basis images measure
  horizontal frequencies from low to high
• The middle basis images are combinations of these
• The basis images in the top left corner measure
  lowest frequencies
• The basis image in the bottom right corner
  measure the highest frequencies

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Some Comments about the DCT

• The transform coefficients in the top left corner
  are typically higher in magnitude than the
  transform coefficients in the bottom right
  corner, which measure the highest frequencies in
  the image
• This relative magnitude information has been
  exploited for compression purposes and for
  watermarking and stegomarking
• For compression, the higher magnitude transform
  coefficients are omitted and the image still can
  have very good fidelity
• For stegomarking and watermarking, putting the
  hidden bits into the high-magnitude DCT
  coefficients makes those coefficients recoverable
  after the image is compressed