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Compression and Denoising of Astronomical Images Using Wavelets

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Compression and Denoising of Astronomical Images Using Wavelets By: Kerry Baldeosingh, Paula Harrell, Trimaine Mc Fadden. South Carolina State University – PowerPoint PPT presentation

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Title: Compression and Denoising of Astronomical Images Using Wavelets


1
Compression and Denoising of Astronomical Images
Using Wavelets
  • By
  • Kerry Baldeosingh,
  • Paula Harrell,
  • Trimaine Mc Fadden.
  • South Carolina State University
  • Principal Investigator Dr. Donald Walter
  • Team Mentor Dr. Kuzman Adzievski.

2
Outline
  • Wavelets and some of their applications.
  • Haar and Daubechies wavelets.
  • Compression and Denoising.
  • Summary and results of image denoising.

3
What are wavelets?
  • Wavelets are functions that are generated from
    one single function, known as the mother wavelet.

4
Applications of Wavelets
  • Wavelets are used in many fields physics,
    astronomy, mathematics, biomedicine, computer
    graphics.
  • In our paper we use wavelets for compression and
    denoising of astronomical images.

5
Discrete signals and images
  • A discrete signal is a function f with values at
    discrete instances and usually it is expressed in
    the form
  • f ( f1, f2, f3, f4, f5, f6, fN-1, fN ).
  • A discrete image, f, is an array of M rows and N
    columns and is expressed in the form

f
6
Types of Wavelets
  • Haar
  • Daubechies
  • Coiflets

7
Haar Wavelets
  • The Haar wavelet is the simplest type of wavelet.
  • They are related to a mathematical operation
    called the Haar transform which serves as the
    model for other wavelet transforms.

8
Haar Transform
  • A 1D, 1-level Haar transform is performed on a
    signal,
  • f ( f1, f2, f3, f4, fN-1, fN ), is
  • f ?( a1 d1 )
  • where a1 (f1 f2 )/ v(2) , (f3 f4 )/ v(2),
  • and d1 (f1 f2 )/ v(2), (f3 f4 )/
    v(2),
  • a1 is called the trend or running average.
  • d1 is called the fluctuation or running
    difference.
  • This process can be repeated until there ceases
    to be an even number of averages.
  • Performing an inverse transform only to the
    trend sub signal would allow an approximation of
    the original signal.

9
Daubechies Wavelets and transforms
  • There are various types of Daubechies transforms.
    We use the Daub4 transform which is slightly
    more complex than the Haar transform.

10
Daub4 transform
  • This Daub4 transform involves using constant
    values a1, a2, a3 and a4 and ß1, ß2, ß3 and ß4
    which are found from solving a set of equations.
  • If a signal f ( f1, f2, f3, f4, f5, f6, fN-1,
    fN ), then a 1D, 1-level transform will be
  • f ?( a1 d1 )
  • where
  • a1 (a1f1 a2f2 a3f3 a4f4 ), (a1f3 a2f4
    a3f5 a4f6 ), (a1fn-1 a2fn a3f1 a4f2 )
  • d1 (ß 1f1 ß 2f2 ß 3f3 ß 4f4 ), (ß 1f3
    ß 2f4 ß 3f5 ß 4f6 ), (ß 1fn-1 ß 2fn ß
    3f1 ß 4f2 )
  • A 1D 2-level transform would be f ?(a2 d2 d1)

11
2D Wavelet transform
  • A 2D wavelet transform of a discrete image can be
    performed only when the image has an even number
    of rows and columns.
  • A 1-level wavelet transform of an image involves
  • A 1D, 1-level, wavelet transform on each row of
    the image, producing a new image.
  • Then on the new image, a 1D, 1-level wavelet
    transform is performed on each of its columns.
  • If f is an N x M, 2D array of an image, then
    under a 1-level wavelet transform f can be
    symbolized as

12
2D Wavelet transform
  • From an image f
  • a1 trend rows then trend columns.
  • h1 trend rows then fluctuation columns.
  • d1 fluctuation rows then fluctuations columns.
  • v1 fluctuation rows then trend columns.
  • A 2D 2-level transform is calculated by computing
    a 1-level transform of the trend subimage a1

13
Compression
  • Compression relies on converting data into a
    smaller format that allows the transmission of
    fewer bits.
  • There are two types of compression
  • Lossless
  • No image data is lost as a result of compression.
  • No errors in result.
  • Compression ratios of up to 21 can be obtained.
  • Lossy
  • Some image data is lost as a result of
    compression.
  • Results have small inaccuracies.
  • Compression ratios of up to 1001 can be
    obtained.

14
Method of compression
  • The basic steps of compression are as follows
  • Perform a wavelet transform of the signal.
  • Set equal to 0 all values of the wavelets
    transform which are insignificant, i.e., which
    lie below some threshold value.
  • Transmit only the significant, non-zero values of
    the transform obtained from Step. 2. This should
    be a much smaller data set than the original
    signal.
  • At the receiving end, perform the inverse wavelet
    transform of the data transmitted in Step. 3,
    assigning zero values to the insignificant values
    which were not transmitted. This decompression
    step produces an approximation of the original
    signal.

15
Denoising Gaussian noise
  • Find the mean, µ, and the standard deviation, s,
    of the image.
  • The image is then transformed once.
  • Using the value of the standard deviation, s, a
    thresholding value, T, can be established where T
    4.5 s.
  • Approximately 99 of the noise can be removed.
  • Success of this method depends on how well the
    transform compresses the signal into a few high
    magnitude values that stand out.

16
Soft and Hard thresholdiing
  • Hard thresholding
  • Soft thresholding
  • In hard thresholding, the transform is not
    continuous and thus those values near the
    threshold are greatly exaggerated. In soft
    thresholding the function does not abruptly
    change thus an inverse transform from soft
    thresholded values produce a better quality
    denoised image.

17
Results
Image with random noise
18
Denoised Images 1
19
Denoised Images 2
20
References and useful tools
  • Walker S. James. Wavelets and their Scientific
    Applications, A Primer. CRC Press LLC, Boca
    Raton, Florida, 1999.
  • http//www.crcpress.com/edp/download. FAWAV
    Software for Wavelet Analysis
  • Mathematica Software Package, Wolfram Research
    Inc.
  • Dr. Kuzman Adzievski. In-class Lectures and
    Recitations.
  • http//hubble.stsci.edu/gallery/showcase/stars/s1.
    html. Globular Cluster M80
  • Acknowledgement I would like to thank Dr. Daniel
    Smith for the helping me loading the JPEG images
    using the Mathematica software.
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