Karthik Gurumoorthy Ajit Rajwade Arunava Banerjee Anand Rangarajan Department of CISE University of Florida 1.

Karthik GurumoorthyAjit Rajwade

Arunava BanerjeeAnand Rangarajan

Department of CISEUniversity of Florida

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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.2

Vector

Conventional learning methods in vision like PCA,

ICA, etc.

21 MM

Image 211 MM

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Our approach following Rangarajan [EMMCVPR-2001] &

Ye [JMLR-2004]

Treated as a21 MM

Image 21 MM

Matrix

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Image of size divided into N patches of size

each treated as a Matrix.

21 MM

Image

21 qq

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21 MM

2211 , MqMq

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=P U S V

TUSVP

U and V:Ortho-normal matrices

S: Diagonal Matrix of singular values

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useful for compression (e.g.: SSVD [Ranade et al-IVC 2007]).

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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.

Tiiii VSUP

Produce ortho-normal basis-pairs, common for all N patches.

Since storing the basis pairs is not expensive.

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NK

NK

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iP aU iaS aV

Taiaai VSUP

iaS•Non-diagonal•Non-sparse

aiTaia VPUS

What sparse matrix will optimally reconstruct from ?

Optimally = least error:

Sparse = matrix has at most some non-zero elements.

2|||| FTaiaa VSUP

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),( aa VUiPiaS

iaS T

We have a simple, provably optimal greedy method to compute such a

1. Compute the matrix . 2. In matrix , nullify all except the largest

elements to produce .

aiTaia VPUW

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iaS

iaW T

iaS

A set of N image patches .

Learning K << N ortho-normal basis

pairs )},{( aa VU

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)1(, NiPi

2

1 1

||||}),),,({( FTaiaai

N

i

K

aiaiaiaaa VSUPMMSVUE

aIVVUU aTaa

Ta , aiTSia ,,|||| 0

MembershipsProjection Matrices

Input: N image patches of size .

Output: K pairs of ortho-normal bases

called as dictionary.

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21 qq

)},{( ii VU

Divide each test image into patches of size

Fix per-pixel average error (say e), similar to the “quality” user-parameter in JPEG.

21 qq

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.

.

.

.

.

.

1U 1V

2U 2V

KU KV

iP

.

.

.

111 VPUS iT

i

222 VPUS iT

i

KiTKiK VPUS

eqq

VSUP FTaiaai

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2||||

222 VPUS iT

i

)(log10 10 ePSNR

RPP = number of bits per pixel

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0.5 bits 0.92 bits 1.36 bits

1.78 bits 3.023 bits

Size of original database is 3.46 MB.Size of dictionary of 50 ortho-normal

basis pairs is 56 KB=0.05MB.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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)(log10 10 ePSNR

RPP = number of bits per pixel

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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.21

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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