Methods of Image Compression by PHL Transform

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Methods of Image Compression by PHL Transform

• Dziech, Andrzej Slusarczyk, Przemyslaw Tibken,
  Bernd
• Journal of Intelligent and Robotic Systems
• Volume 39, Issue 4, April 2004. pp. 447-458.
• Presented by Xiao Zou

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Abstract

• An image data compression scheme based on
  Periodic Haar Piecewise-Linear (PHL) transform
  and quantization tables is proposed.
• Evaluating the effectiveness of the compression
  for different classes of images.
• Comparing the compression quality using PHL and
  DCT transforms.

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Basic Idea

• Using Periodic Haar Piecewise-Linear (PHL)
  Transform (integrating Haar function)
• For some applications, PHL transform is better
  than DCT transform
• PHL transform has very fast algorithm for
  computation.

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Haar Function
Define
and
for j a nonnegative integer and
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Haar Function Cont.
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Haar Function Cont.
? A function f(x) can be written as a series
expansion by
? The functions ?j k and ? are all orthogonal
in 0, 1 , with
0
? Can be used to define Wavelets.
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Periodic Haar Piecewise-Linear Transform

• The set of Periodic Haar Piecewise-Linear (PHL)
  functions is obtained by integrating the
  well-known set of Haar functions.

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PHL Transform

• The set of PHL functions is linearly independent
  but not orthogonal. Figure 1 shows the set of PHL
  functions for N 8.

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PHL Transform

• The forward and inverse PHL transform can be
  presented in matrix form as follows

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PHL Transform

• Computational algorithms of PHL transform are
  very fast and easy for implementation.
• The forward PHL transform algorithm requires (2N
  -3) additions, (N -2) binary shifts and (N - 2)
  normalizations
• The inverse PHL transform requires (3N/4)
  additions, (N-3) multiplications and (N - 2)
  normalizations

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Image Compression Using PHL Transform

• The PHL transform decomposes input image on
  subimages being sequential approximations of
  input data. The hierarchical representation is
  created.

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Test Images

1. Natural images (Lena, Bridge)
2. Scanned document (Text)
3. Computer generated images (Slope, Circles)
4. Compound image (Montage)

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Threshold Sampling

• To evaluate compression ability of PHL transform,
  selected thresholds in 2D PHL spectral domain are
  applied.
• Each sample whose magnitude is greater than the
  threshold level is selected and the rest are set
  to zero.
• An inverse 2D transformation is then performed to
  obtain a reconstructed image.
• Plots of the Peak Signal-to-Noise Ratio (PSNR)
  versus compression ratio for the test images are
  shown

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Threshold Sampling Cont.
PHL transform has very good decorrelation
properties especially for computer generated
images. For Slope image PSNR equals 55.5 dB for
compression ratio of 80 and falls to 47.1 dB for
95. The others computer images Circles and
Text can be perfectly reconstructed for 93.3
and 61.05 of rejected coefficients.
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Threshold Sampling Cont.
Natural images are also well compressed. For
compression ratio up to 75, reconstructed image
quality measured by PSNR is better for PHL
transform than for DCT transform. However
detail analysis of reconstructed images shows
some distortions. As it is seen the reconstructed
images are well visible and for compression ratio
around 90 PSNR falls below 30 dB.
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Threshold Sampling PHL vs. DCT
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Threshold Sampling Block Coding
Block coding has no significant influence on
compression quality. For the same compression
ratio differences in PSNR are below 1 dB. To
achieve higher quality larger size of blocks
should be used.
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Threshold Sampling Block Coding
The effect of block coding becomes visible at
high compression ratio. This effect can be
reduced by using frames, i.e., blocks with
overlapped boundaries.
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Zonal Sampling
Good image transforms have ability to pack
decorrelated coefficients within the smallest
zone of spectrum. This property is especially
important for efficient spectrum coding.
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Scalar Quantization

• Using quantization table to quantize PHL spectral
  coefficients.
• Spectral coefficients with the same localization
  are divided by the quantization table and then
  rounded to the nearest integer number.

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Scalar Quantization
Quantization table from Figure 12(b) has been
designed to preserve best image quality. It can
be optimized for selected applications and higher
compression ratios can be achieved. Using
presented algorithm, quantization table of any
size can be created.
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Scalar Quantization Cont.
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Entropy Coding
The scanning sequence is specified as above. The
two-dimensional quantized table is converted into
six one-dimensional sequences 115, 1628,
2937, 3846, 4755, 5664. If the remaining
coefficients in formed sequences are all zero,
there are rejected and an end-of-block symbol is
inserted.
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Entropy Coding Huffman Alg.
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Conclusion

• PHL transform constitutes an alternative approach
  in reference to the transforms based on harmonic
  functions.
• PHL transform is very fast and easy for
  implementation computational algorithm that is
  much faster than that of DCT.
• Comparing the results of compression, it is seen
  that for computer-generated images the
  compression properties of PHL transform are
  better than of DCT transform.
• Performed analysis shows that PHL transform is
  suitable for compression of compound images,
  e.g., computer presentations, scanned documents
  with images and computer graphics.