A Technique for Selfembedding of Digital Images

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A Technique for Self-embedding of Digital Images
Leonard Popyack, Victor Skormin
Vladimir Gorodetski
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Summary

• 1. Introduction
• 2. Singular value decomposition of a digital
  image Mathematical foundation
• 3. Experimental Exploration of the CSVG image
• 4. Quantization and encoding of CSVD image
• 5. Self-embeddingidea, embedding and recovery
  procedures, examples, application
• 7. Conclusion

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Singular Value decomposition of a Digital Image
Mathematical Basis

• Let be the matrix of a
  digital image of size m? n. Representation

• is called its Singular Value Decomposition (SVD),
  X, Y left and right singular vectors and
  - its singular values.
  Representation

• is called Cut Singular Value Decomposition
  (CSVD) of the digital image A. Method of
  computing of the SVD are well known.

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Cut SVD Image Examples

• Contribution of each i-th layer into forming of
  the original image is proportional to the
  singular value , since singular vectors are
  normalized to 1.

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Simulation-based Exploration of the CSVD Image
Segmentation

• The formal criterion (the CIQM) for assessing the
  quality of the cut image compared to the original
  one

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Lessons learnt from the simulation

• 1.There exists a positive correlation between the
  value of the criterion ?(s) (the CIQM) and the
  quality of CSVD image. The value ?(s) can play
  the role of threshold of CSVD image quality and
  it must be chosen between 0.85 and 0.95.
  Regardless, it may be used as a preliminary
  assessment of the CSVD image quality.
• 2. Variation of the block size makes it possible
  to vary the bit rate of the compressed image. On
  the one hand, an increase of a block size
  increases the number of MSLs needed to
  provide the appropriate quality of a CSVD
  image. On the other hand, an increase of the
  block size makes it possible to achieve more
  percentage of image compression. A possible
  tradeoff is making use of segmentation into 10?
  10 blocks.

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Results of simulation-based exploration of the
CSVD images and criterion ?(s)
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Quantization of the CSVD image

• A quality of the recovered image depends on the
  precision of the recovery of singular vectors and
  takes the most memory. Quantization is based on
  factoring in the empirical probability
  distribution of the components of each singular
  vector as follows.
• Let s be the number of bits reserved for coding
  the value x. Then is the number of
  values that a singular vector component can take.
  Let us choose quantization due to the constraint

• i.e. quantization is designed in such a way that
  provides the equal values of probabilities of
  random events, i1,2,,N.

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Functions of probability distribution for the
segmentation into blocks 10?10 and 12 ? 12 for
the first (left) and for the second (right)
singular vectors

• Segmentation
• into 10?10 blocks

• Segmentation
• into 12?12 blocks

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Encoding of CSVD image

• Two main ideas are used in image encoding
  procedure
• 1. Approximation of the direction of the singular
  vector.
• 2. Root mean squire of the approximation error
  minimization.

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Encoding of CSVD image

• Criterion of optimization

• where

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Self-Embedding

• Self-Embedding is embedding an image into itself.
• Steps of self-embedding procedure
• 1. Convert Image to be embedded is into SVD
  compressed format.
• 1. Generate and save a secret key that determines
  transpositions of lines and columns of the image
  to be embedded. image .
• (The transpositions must a) exclude an overlap of
  blocks of container with the identical blocks of
  embedded image, b) meet constraints providing
  detectability of the corrupted blocks.)
• 2. Perform permutation of the blocks according
  to the secret key and form permutated matrix of
  the image to be embedded.
• 3. Insert code of last image into LSBs of
  container and assign password.

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Recovery Procedure

• Steps of the recovery procedure
• 1. Detect whether the received image was
  corrupted in transmission process.
• 2. Determine the blocks that were corrupted if
  any.
• 3. Determine location of the copies of blocks
  that were corrupted.
• 4. Replace the corrupted blocks with their
  self-embedded copies.
• Details of the algorithm of the corrupted image
  recovery contained self-embedded SVD compressed
  copy are described in the paper .

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Encoding of CSVD image

• CSVD of the image "Lena" preserving 2 MSL
  segmented into 12? 12 blocks and using 5 bit per
  component
• 1) quantized evenly and used flexible encoding of
  singular vectors (left), 2) quantized as it is
  proposed and using flexible encoding (center), 3)
  precise picture comprising two MSL.

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Communication with anti-tampering abilities An
Example
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Communication with anti-tampering abilities An
Example
Self-embedded TSVD image (uncorrupted)
Corrupted original image with embedded TSVD image
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Conclusion

• The paper focus is a technique for digital image
  self-embedding. The new results are as follows
• 1. A new SVD-based technique for image
  compression is developed and explored via
  simulation. The results were used as a formal
  basis for the development of a new image
  compressed format which makes possible to provide
  less than 2 bpp data rate while preserving needed
  image quality.
• 2. The developed compression technique is applied
  to the self-embedding task. An algorithm of the
  recovery of the corrupted image using
  self-embedding technique is developed. This
  technique makes it possible to detect the
  corrupted blocks and the possibility to recover
  them.
• The extended simulation-based exploration of
  the results was performed on the basis of the
  developed software tool using Visual C 6.0.