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Terahertz data compression using wavelets

10,000 3,000

Topic Description

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Summary
Terahertz pulsed imaging (TPI) is a newly developed imaging technique with a wide range of possible
applications. This project set out to investigate compression methods on the data produced by TPI,
with a focus on using wavelets because they have recently emerged as powerful signal compression
tools.
Software was produced using Matlab to investigate different lossy compression methods
using both the wavelet transform and the wavelet packet transform. It also compared quantization of
the transform coefficients to a simple truncation approach in order to decide on an optimum
compression level where the original signal’s optical properties were not significantly altered. To
complete the compression, lossless Run-length encoding was compared to a value and position
entropy method, after which the final compression ratios of every method combination were
compared.
The first half of this paper begins by introducing Fourier analysis, the main branch of
mathematics that led to the discovery of wavelets, and details why there was a need to develop a
transform which solved its time-frequency resolution problem. This is followed by a general
description of compression methods and an explanation of how wavelets can be used to achieve
compression.
The second half of the paper details the design of the software produced to evaluate the
chosen compression methods. A great deal of emphasis in the project is placed on discovering how
different wavelet mother functions affect the compression quality rather than just using one wavelet.
A full set of results from running the implemented software are then displayed graphically and
analysed. It is shown that the best compression ratios are achieved by using the wavelet transform
with a Coiflet 2 wavelet, before truncation and value with position entropy. In fact a compression
ratio of 14.6:1 was achieved using this method on resin sample terahertz data when removing 95% of
the wavelet transform coefficients. Even at this level of compression, the refractive index and
absorption profiles of the reconstructed signals were not significantly altered. The results of this
research are particularly important for medical TPI use as it provides a very real reduction in
transmission and storage of scans.
Using the findings of this project, a pair of functions were written to perform compression
and decompression of Terahertz data. A Compact Disc accompanies this paper with these functions,
and all the functions created during the project on it. It also has the sample data used, and a selection
of results.

Contents
1 INTRODUCTION……………………………………………………………………………………………………………1
1.1 TERAHERTZ SIGNALS…………………………………………………………………………………………………….1
1.1.1 TERAHERTZ PULSED IMAGING……………………………………………………………………………………….2
1.2 PROJECT AIMS ………………………………………………………………………………………………………………2
1.2.1 MINIMUM REQUIREMENTS…………………………………………………………………………………………….3
1.2.2 POTENTIAL ENHANCEMENTS …………………………………………………………………………………………3
1.3 PROJECT PLAN………………………………………………………………………………………………………………3
1.3.1 REQUIRED RESEARCH …………………………………………………………………………………………………..3
1.3.2 METHODOLOGY……………………………………………………………………………………………………………4
1.3.3 SCHEDULE …………………………………………………………………………………………………………………..4
2 BACKGROUND……………………………………………………………………………………………………………..5
2.1 EXISTING RESEARCH AND USE OF WAVELETS ………………………………………………………………….5
2.2 COMPRESSION……………………………………………………………………………………………………………….5
2.2.1 LINEAR TRANSFORMER…………………………………………………………………………………………………6
2.2.2 QUANTIZER …………………………………………………………………………………………………………………6
2.2.3 ENTROPY ENCODER ……………………………………………………………………………………………………..7
2.3 TRANSFORMS ………………………………………………………………………………………………………………..8
2.3.1 DISCRETE FOURIER TRANSFORM ……………………………………………………………………………………8
2.3.2 SHORT TERM FOURIER TRANSFORM……………………………………………………………………………..10
2.3.3 CONTINUOUS WAVELET TRANSFORM …………………………………………………………………………..11
2.3.4 DISCRETE WAVELET TRANSFORM………………………………………………………………………………..13
2.3.5 WAVELET PACKET TRANSFORM…………………………………………………………………………………..14
2.3.6 THE WAVELETS ………………………………………………………………………………………………………….15
2.4 WHY WAVELETS? ………………………………………………………………………………………………………..16
3 DESIGN AND IMPLEMENTATION……………………………………………………………………………..18
3.1 SOFTWARE CHOICE……………………………………………………………………………………………………..18
3.1.1 MATLAB ……………………………………………………………………………………………………………………18
3.1.2 WAVELAB………………………………………………………………………………………………………………….18
3.2 TERAHERTZ SAMPLE DATA ………………………………………………………………………………………….19
IV
3.3 METHOD OVERVIEW……………………………………………………………………………………………………20
3.3.1 EXPERIMENT VARIABLES …………………………………………………………………………………………….20
3.4 IMPLEMENTATION OF THE TRANSFORM COMPRESSION ………………………………………………….21
3.4.1 WAVELET TRANSFORM COMPRESSION ………………………………………………………………………….22
3.4.2 WAVELET PACKET TRANSFORM COMPRESSION ……………………………………………………………..23
3.5 OPTICAL PROPERTIES FOR EVALUATION ………………………………………………………………………23
3.5.1 CALCULATION OF REFRACTIVE INDEX ………………………………………………………………………….24
3.5.2 CALCULATION OF ABSORPTION COEFFICIENTS ………………………………………………………………25
3.6 OBTAINING THE TRANSFORM COMPRESSION RESULTS……………………………………………………27
3.6.1 TRUNCATION……………………………………………………………………………………………………………..27
3.6.2 QUANTIZATION ………………………………………………………………………………………………………….30
3.7 IMPLEMENTING THE ENTROPY……………………………………………………………………………………..30
3.7.1 TRUNCATION ENTROPY……………………………………………………………………………………………….31
3.7.2 QUANTIZATION ENTROPY…………………………………………………………………………………………….32
4 COMPRESSION RESULTS AND EVALUATION…………………………………………………………34
4.1 TRANSFORM RESULTS WITH TRUNCATION …………………………………………………………………….34
4.1.1 REFRACTIVE INDEX…………………………………………………………………………………………………….34
4.1.2 ABSORPTION COEFFICIENTS…………………………………………………………………………………………34
4.2 QUANTIZATION RESULTS ……………………………………………………………………………………………..37
4.2.1 REFRACTIVE INDEX…………………………………………………………………………………………………….37
4.2.2 ABSORPTION COEFFICIENTS…………………………………………………………………………………………38
4.3 ENTROPY RESULTS………………………………………………………………………………………………………40
4.3.1 TRUNCATED SIGNALS………………………………………………………………………………………………….40
4.3.2 QUANTIZED SIGNALS…………………………………………………………………………………………………..41
5 CONCLUSION………………………………………………………………………………………………………………43
5.1.1 RECOMMENDATIONS FOR FURTHER RESEARCH ………………………………………………………………44
BIBLIOGRAPHY ……………

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