Time and frequency domain based methods

2.5.3.1 Discrete wavelet transform

The discrete wavelet transform is used for compression of A-scans due to its time-frequency representation of information in datasets [24]. A-scans are represented with wavelet coefficients after pre- processing using discrete wavelet transform [63]. The basic idea of using discrete wavelet transform based compression (DWT) for A-scans is their shape-similarity to the pulse shape of coded exci- tation. By this fact it can be expected, that the number of wavelet-

coefficients may be small against the number of samples. The

wavelet-coefficients are saved instead of samples of A-scans and the resulting compression may be high.

A-scans are convoluted with a family of basis functions, i.e. mother and father wavelets, to achieve the wavelet coefficients. The father wavelet also called scaling function can be derived from the mother wavelet. The mother and the father wavelets are orthogonal func- tions. The time information of the ultrasound pulses corresponds to the time position of the convoluted basis functions. In order to get the frequency information, the basis functions are convoluted in different scales with the A-scans. The original A-scans can be reconstructed perfectly with the uncompressed wavelet coefficients using the inverse discrete wavelet transform [25, 26].

It is not easy to find a basis function that can be used as the mother wavelet. The simplest mother wavelet is Haar, whose definition is shown with following function:

WHaar(x) =      1 _{0 ≤ x < 1/2,} −1 1/2 < x ≤ 1, 0 otherwise.

The standard mother wavelets [64] for DWT are collected to con- struct the wavelet filter bank. The contents of the filter bank are the coefficients of mother and father wavelets. The selection of the

standard mother wavelets influences the performances of the dis- crete wavelet transform in data compression.

With DWT the wavelet coefficients are achieved with the selected mother wavelets at a large range of scales. The achieved wavelet coefficients are quantized and saved as compressed datasets for USCT. The performance of different standard mother wavelets are tested experimentally.

2.5.3.2 Multi-fractal analysis

A-scans are considered to have the fractal features, since the noise in ultrasound signal may show singular behavior. I.e. some points of signal in time cannot be approximated with a Taylor series using a finite number of terms [65]. The fractals in A-scans are analyzed with the DWT method, since the properties of fractals are related to the wavelet coefficients [66]. The method introducted in [67] uses the multifractal analysis by processing wavelet coefficients to re- duce noise in signals. This method (called MultiFractal in this work) is used to reduce the noise in A-scans and increase the performance of data compression in USCT.

The main process of the MultiFractal method used in this work is as follows: In the first step the wavelet coefficients are registered with DWT to represent the A-scans. Then these wavelet coefficients are damped with factors which are power law related to the cor- responding frequencies of the wavelet coefficients. High frequency components are reduced strongly. Finally the processed wavelet co- efficients are quantized and compressed.

The wavelet coefficients corresponding to the high frequencies are considered to have more information about noise. Thus these wavelet coefficients are decreased more strongly.

The damping factor can be selected by users and has a large value for the original dataset with a high noise level. The performance of MultiFractal with the selected damping factor has to be evaluated with USCT datasets in experiments due to the unknown characte- ristics of noise.

2.5.3.3 Continuous wavelet transform

The continuous wavelet (CWT) is used mostly for research of sig- nal properties instead of implementation of data compression. The local characteristics of signals are represented with the achieved coefficients. In this work CWT is combined with a statistic peak de- tection method [68] and used for the compression of USCT datasets (WavePDT). The name WavePDT method is an abbreviation of the continuous wavelet based peak detection method.

The wavelet coefficientsFwfor a functionf (x) with a mother wavelet

W are calculated as follows: Fw(a, b) = 1 √_a Z ∞ −∞ f (x)W∗ t − b a  dx

where _{∗ stands for the operation of complex conjugate. a and b are}

scale and translational parameters respectively.

In the WavePDT method the A-scans are represented with the wavelet coefficients by using the CWT. The mother wavelet used in CWT is the Morlet mother wavelet defined in [69]. Wavelet coefficients of real experimental signals are measured to construct a decision tree. This decision tree is based on the values of wavelet coefficients of the available ultrasound pulse. The wavelet coefficients of A-scans, which corresponds to an ultrasound pulse in decision tree, are se- lected. The selected wavelet coefficients represent the information of the ultrasound pulses in A-scans.

The information included in the selected wavelet coefficients is rep- resented by the time-of-arrival, center frequency, phase, bandwidth and amplitude of detected ultrasound pulses [69]. These parame- ters are saved as the compressed data for A-scans.

CWT in WavePDT method uses mother wavelets which are designed under different conditions as for the discrete wavelet transform. The generally used mother wavelets for CWT are Morlet, Meyer and Mexican Hat [64]. The statistic peak detection method in WavePDT method uses an iteration process, until every ultrasound pulse in A-scans is detected. Details can be found in [68].

The WavePDT method uses the advantages of CWT and a peak de- tection method with a decision tree to represent the information in A-scans. The number of wavelet coefficients is smaller than the number of samples to achieve a high compression ratio. However

the iteration process results in a high computational complexity for implementation of these compression methods.