Image Compression by Learning Matrix Ortho-normal Bases

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Image Compression by Learning Matrix Ortho-normal
Bases

• Karthik Gurumoorthy
• Ajit Rajwade
• Arunava Banerjee
• Anand Rangarajan
• Department of CISE
• University of Florida

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Overview

• A new approach to lossy image compression based
  on machine learning.
• Key idea Learning of Matrix Ortho-normal Bases
  from training data to efficiently code images.
• Applied to compression of well-known face
  databases like ORL, Yale.
• Competitive with JPEG.

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Background Images as Vectors
Vector


Image
Conventional learning methods in vision like PCA,
ICA, etc.
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Background Images as Matrices
Treated as a

Image
Matrix
Our approach following Rangarajan EMMCVPR-2001
Ye JMLR-2004
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Image Patches


Image
Image of size divided into N
patches of size each treated as a
Matrix.
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SVD of a Patch
P
U
S
V

U and V Ortho-normal matrices
S Diagonal Matrix of singular values
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Exponentially decreasing Singular Values
useful for compression (e.g. SSVD Ranade et
al-IVC 2007).
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SVD for Ensemble of Patches?

• Consider a set of N image patches
• SVD of each patch gives
• Costly in terms of storage as we need to store N
  ortho-normal basis pairs.

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Common Orthonormal Basis-pairs?

• Produce ortho-normal
  basis-pairs, common for all N patches.
• Since storing the basis pairs
  is not expensive.

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Away from SVD

• Non-diagonal
• Non-sparse

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Away from SVD

• What sparse matrix will optimally
  reconstruct from ?
• Optimally least error
• Sparse matrix has at most some
  non-zero elements.

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Away from SVD

• We have a simple, provably optimal greedy method
  to compute such a
• Compute the matrix .
• In matrix , nullify all except the
  largest elements to produce .

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Learning algorithm

• A set of N image patches
  .
• Learning K ltlt N ortho-normal basis pairs

Projection Matrices
Memberships
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Summary of Training Algorithm

• Input N image patches of size .
• Output K pairs of ortho-normal bases
• called as dictionary.
  
  

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Testing phase

• Divide each test image into patches of size
• Fix per-pixel average error (say e), similar to
  the quality user-parameter in JPEG.

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Testing phase
. . .
. . .
. . .
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Results ROC Curve (ORL Database)
RPP number of bits per pixel
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Sample Reconstructions
0.92 bits
1.36 bits
0.5 bits
1.78 bits
3.023 bits
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Results ORL Database

• Size of original database is 3.46 MB.
• Size of dictionary of 50 ortho-normal basis pairs
  is 56 KB0.05MB.
• Size of database after compression and coding
  with our method with e 0.0001 is 1.3 MB.
• Total compression rate achieved is 61.

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Results ROC Curve (Yale Database)
RPP number of bits per pixel
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Conclusions

• New lossy image compression method using machine
  learning.
• Key idea 1 matrix based image representation.
• Key idea2 Learning small set of matrix
  ortho-normal basis pairs tuned to a database.
• Results competitive with JPEG standard.
• Future extensions video compression.

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References

• A. Rangarajan, Learning matrix space image
  representations, Energy Minimizing Methods in
  Computer Vision and Pattern Recognition, 2001.
• J. Ye, Generalized low rank approximation of
  matrices, Journal of Machine Learning Research
  ,2004.
• M. Aharon, M. Elad and A. Bruckstein, The K-SVD
  An algorithm for designing of overcomplete
  dictionaries for sparse representation. IEEE
  Transactions on Signal Processing, 2006.
• A. Ranade, S. Mahabalarao and S. Kale. A
  variation on SVD based image compression. Image
  and Vision Computing, 2007.

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Questions
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