Rozhen 2010, June Singular Value Decomposition of images from scanned photographic plates Vasil Kolev Institute of Computer and Communications Systems Bulgarian Academy of Sciences Milcho Tsvetkov, Katya Tsvetkova, Ana Borisova Institute of Astronomy, Bulgarian Academy of Sciences This work has been supported by the research project D of the Bulgarian National Science Fund, Bulgaria

Rozhen 2010, June Advantages of SVD There are several reasons: The fact that the decomposition is achieved by unitary matrix makes it an ideal vehicle for discussing the geometry of n –space SVD it is stable, small perturbation in A correspondent to small perturbation in and conversely Decomposition provides low rank approximation to A There exist efficient, stable algorithms to compute the SVD

Rozhen 2010, June REVIEW Singular value decomposition (SVD) [1] is applied to a mid infrared ISOCAM spectral map of NGC As a first result, this decomposition provides a mathematical analysis of the map in terms of a linear combination of elementary spectra. 2. After further processing, it is shown that the spectrum observed on each pixel can be described as the physical superposition of four components. Separation of data to image and noise subspaces using SVD [2]. Subspace techniques have previously being used in image compression as well as image comparison. has not been used in (radio) astronomical image processing. 1. Detection of faint stars 2. Noise removing 3. Continuum subtraction of spectral lines for radio-astronomical images 4. Automatic image classification [1] Boissel P, Joblin C., and Pernot P - Singular value decomposition: A tool to separate elementary contributions in ISOCAM spectral maps”,vol.373, A&A, pp.L15-L18, 2001 [2] Yatawatta S., Subspace Techniques for Radio-Astronomical Data Enhancement, Astrophysics, 2008

Rozhen 2010, June Structure of SVD matrices decomposition orthonormal matrices - U, V diagonal matrix - singular values σ p, Columns of U is called left singular vectors Columns of V is called right singular vectors The SVD gives us important information about - the rank of the matrix, - the column and row spaces of the matrix

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Example of weight image decomposition scanned photographic plate M45-556p.fits in the region of the Pleiades stellar cluster singular values

Rozhen 2010, June IMAGE SINGULAR VALUES ) Singular values ASI (M45-556p.fits) in the region of the Pleiades stellar cluster ( size 1122x1122 ) SPP BAM010M (nz194.fits) (size 9898x9897) Singular values

Rozhen 2010, June IMAGE SINGULAR VALUES SPP ROZ ( size x ) ROZ (6419.fits) in the region of the Pleiades stellar cluster ( size 9906x10060 ) singular values

Rozhen 2010, June Example of SVD k low - rank approximations scanned image of SPP BAM010M (nz194.fits) image size (9898x9897) usually k << rank (Image)

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Example of SVD k low - rank approximations scanned image of ASI (M fits) in the region of the Pleiades stellar cluster image size (1122x1122)

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Image quality - Compression Ratio Image quality measure used compressed ratio using The first K - columns of U and V They singular values

Rozhen 2010, June Memory usage – image rank (k) 5.35% with k=30 (1122x1122) 1.60% with k=50 (9898x9897) 1.01% with k=50 (9906x10060)

Rozhen 2010, June Image rank - CR Minimum number rank for reading clear notes of plates : - rank 12 with CR=97.86%, image size (1122x1122) - rank 9 with CR=98.82%, image size (9906x10060) - - rank 9 with CR=99.83%, image size (9898x9897)

Rozhen 2010, June Conclusions 1. As rank k increases, the images quality increases but the same does the amount of memory needed to store the images ! 2. With large CR>97% we can see image details 3. This approach provides a natural way to compress the image data, since here singular values represent the relative contribution of the image with respect to the noise in each low-rank approximation 4. The low - rank image approximation is faster from Wiener filtering. 5. SVD is numerically robust and stable algorithm 6. We can see image without fully reading image file – only up to 50 columns (row)! 7. For only 9 – 12 approximation reading notes of plate. 8. Therefore we can construct image database using SVD 9. For different k – different image approximation: a) Of the small low-rank approximation can select the Pleiades, galaxy, bigger planet b) Of the larger low-rank approximation can select faint stars

Rozhen 2010, June Thank you for your attention ! QUESTIONS ? REMARKS ? SUGGESTIONS ?