Rozhen 2010, 1 - 4 June

Singular Value Decomposition of images

from scanned photographic plates

Vasil KolevInstitute of Computer and Communications Systems

Bulgarian Academy of Sciences

Milcho Tsvetkov, Katya Tsvetkova, Ana BorisovaInstitute 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

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

REVIEW

Singular value decomposition (SVD) [1] is applied to a mid infrared ISOCAMspectral map of NGC 7023. 1. 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 stars2. Noise removing3. 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 separateelementary 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

Structure of SVD matrices decomposition

orthonormal matrices - U, Vdiagonal matrix -

singular values σp

Σ

,

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

Rozhen 2010, 1 - 4 June

Rozhen 2010, 1 - 4 June

Example of weight image decomposition

scanned photographic plate M45-556p.fits in the region of the Pleiades stellar cluster

T555

T444

T333

T222

T111 vuvuvuvuvu 5A

Tj

n

1jjj

Tn

T2

T1

nn2211 vuσ

v

v

v

uσuσuσ

TVUA

singular values

Rozhen 2010, 1 - 4 June

IMAGE SINGULAR VALUES)

Singular values ASI067 000556 (M45-556p.fits)

in the region of the Pleiades stellar cluster

(size 1122x1122)

SPP BAM010M (nz194.fits)

(size 9898x9897) Singular values

Rozhen 2010, 1 - 4 June

IMAGE SINGULAR VALUES

SPP ROZ200 001655

(size 18898 x 18240)

ROZ050 006419 (6419.fits) in the region ofthe Pleiades stellar cluster

(size 9906x10060) singular values

singular values

Rozhen 2010, 1 - 4 June

Example of SVD k low - rank approximations

scanned image of SPP BAM010M (nz194.fits)

image size (9898x9897)

usually k << rank (Image)

rankimagek

Rozhen 2010, 1 - 4 June

Rozhen 2010, 1 - 4 June

Example of SVD k low - rank approximations

scanned image of ASI067 000556 (M45-556.fits) in the region of the

Pleiades stellar clusterimage size (1122x1122)

Rozhen 2010, 1 - 4 June

Rozhen 2010, 1 - 4 June

Image quality - Compression Ratio

Image quality measure used compressed ratio

using

The first K - columns of U and V They singular values

% ,100zeY(A)sizeX(A)si

1)sizeY(A)zeX(A)rank(A)(si1Rationn CompressioCR

Rozhen 2010, 1 - 4 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, 1 - 4 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, 1 - 4 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, 1 - 4 June

Thank you for your attention !

QUESTIONS ?

REMARKS ?

SUGGESTIONS ?