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
• Principal Investigator Dr. Donald Walter
• Team Mentor Dr. Kuzman Adzievski.

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Outline

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

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What are wavelets?

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

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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.

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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
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Types of Wavelets

• Haar
• Daubechies
• Coiflets

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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.

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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.

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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.

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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)
  

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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

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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

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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.

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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.

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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.

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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.

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Results
Image with random noise
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Denoised Images 1
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Denoised Images 2
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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.