Effective image compression requires a non-expansive discrete wavelet transform (DWT) be employed; consequently, image border extension is a critical issue. Ideally, the image border extension method should not introduce distortion under compression. It has been shown in literature that symmetric extension performs better than periodic extension. However, the non-expansive, symmetric extension using fast Fourier transform and circular convolution DWT methods require symmetric filters. This precludes orthogonal wavelets for image compression since they cannot simultaneously possess the desirable properties of orthogonality and symmetry. Thus, biorthogonal wavelets have been the de facto standard for image compression applications. The viability of symmetric extension with biorthogonal wavelets is the primary reason cited for their superior performance. Recent matrix-based techniques for computing a non-expansive DWT have suggested the possibility of implementing symmetric extension with orthogonal wavelets. For the first time, this thesis analyzes and compares orthogonal and biorthogonal wavelets with symmetric extension. Our results indicate a significant performance improvement for orthogonal wavelets when they employ symmetric extension. Furthermore, our analysis also identifies that linear (or near-linear) phase filters are critical to compression performance---an issue that has not been recognized to date. We also demonstrate that biorthogonal and orthogonal wavelets generate similar compression performance when they have similar filter properties and both employ symmetric extension. The biorthogonal wavelets indicate a slight performance advantage for low frequency images; however, this advantage is significantly smaller than recently published results and is explained in terms of wavelet properties not previously considered. / Master of Science

In this thesis, a method to calculate scaling function coefficients for a new bi-orthogonal wavelet family derived directly from an impulse response waveform is presented. In literature, the Daubechies wavelets (DB wavelet) and the Morlet wavelet are the most commonly used wavelets for the dyadic wavelet transform (DWT) and the continuous wavelet transform (CWT), respectively. For a specific vibration signal processing application, a wavelet basis that is similar or is derived directly from the signal being studied proves to be superior to the commonly used wavelet basis. To assure a wavelet basis has a direct relationship to the signal being studied, a new formula is proposed to calculate coefficients which capture the characteristics of an impulse response waveform. The calculated coefficients are then used to develop a new bi-orthogonal wavelet family.