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Singular Value Decomposition of images from scanned photographic plates

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

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Singular Value Decomposition of images from scanned photographic plates

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  1. 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 D0-02-275 of the Bulgarian National Science Fund, Bulgaria Rozhen 2010, 1 - 4 June

  2. 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, 1 - 4 June

  3. REVIEW Singular value decomposition (SVD) [1] is applied to a mid infrared ISOCAM spectral map of NGC 7023. • As a first result, this decomposition provides a mathematical analysis of the map in terms of a linear combination of elementary spectra. • 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. • Detection of faint stars • Noise removing • Continuum subtraction of spectral lines for radio-astronomical images • 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, 1 - 4 June

  4. Structure of SVD matrices decomposition , orthonormal matrices - U, V diagonal matrix - singular values σp 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 spacesof the matrix Rozhen 2010, 1 - 4 June

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  6. Example of weight image decomposition scanned photographic plate M45-556p.fits in the region of the Pleiades stellar cluster singular values Rozhen 2010, 1 - 4 June

  7. IMAGE SINGULAR VALUES ) ASI067 000556 (M45-556p.fits) in the region of the Pleiades stellar cluster (size 1122x1122) Singular values SPP BAM010M (nz194.fits) (size 9898x9897) Singular values Rozhen 2010, 1 - 4 June

  8. IMAGE SINGULAR VALUES • SPP ROZ200 001655 (size 18898 x 18240) singular values ROZ050 006419 (6419.fits) in the region of the Pleiades stellar cluster (size 9906x10060) singular values Rozhen 2010, 1 - 4 June

  9. Example of SVD k low - rank approximations scanned image of SPP BAM010M (nz194.fits) image size (9898x9897)usually k << rank (Image) Rozhen 2010, 1 - 4 June

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  11. Example of SVD k low - rank approximationsscanned image of ASI067 000556 (M45-556.fits) in the region of the Pleiades stellar clusterimage size (1122x1122) Rozhen 2010, 1 - 4 June

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  13. Image quality - Compression Ratio • Image quality measure used compressed ratio using • The first K - columns of U and V • They singular values Rozhen 2010, 1 - 4 June

  14. 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, 1 - 4 June

  15. Image rank - CR • Minimum number rank for readingclear 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, 1 - 4 June

  16. Conclusions • As rank k increases, the images quality increases but the same does the amount of memory needed to store the images ! • With large CR>97% we can see image details • 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 • The low - rank image approximation is faster from Wiener filtering. • SVD is numerically robust and stable algorithm • We can see image without fully reading image file – only up to 50 columns (row)! • For only 9 – 12 approximation reading notes of plate. • Therefore we can construct image database using SVD • For different k – different image approximation: a) Of the small low-rank approximation can select the Pleiades, galaxy, biggerplanet b) Of the larger low-rank approximation can select faint stars Rozhen 2010, 1 - 4 June

  17. Thank you for your attention ! • QUESTIONS ? • REMARKS ? • SUGGESTIONS ? Rozhen 2010, 1 - 4 June

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