Methods of Image Compression by PHL Transform Dziech, Andrzej Slusarczyk, Przemyslaw Tibken, Bernd Journal of Intelligent and Robotic Systems Volume: 39,

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

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.

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.

Haar Function

Define

and

for j a nonnegative integer and 120 jk

Haar Function – Cont.

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.

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.

PHL Transform The set of PHL functions is

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

PHL Transform The forward and inverse PHL transform can be

presented in matrix form as follows:

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

Image Compression Using PHL Transform

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

Test Images

1. Natural images (Lena, Bridge)

2. Scanned document (Text)

3. Computer generated images (Slope, Circles)

4. Compound image (Montage)

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

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.

Threshold Sampling – Cont.Natural images are also well compressed. For compression ratio up to 75%, reconstructedimage 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.

Threshold Sampling – PHL vs. DCT

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.

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.

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.

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.

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.

Scalar Quantization – Cont.

Entropy Coding

The scanning sequence is specified as above. The two-dimensional quantized table is converted into six one-dimensional sequences: 1–15, 16–28, 29–37, 38–46, 47–55, 56–64. If the remaining coefficients in formed sequences are all zero, there are rejected and an end-of-block symbol is inserted.

Entropy Coding – Huffman Alg.

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.