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

Master of Science / Department of Mathematics / Nathan Albin / A new methodology is introduced for use in discrete periodic extension of non-periodic functions. The methodology is based on a band-limited step function, and utilizes the computational efficiency of FC-Gram (Fourier Continuation based on orthonormal Gram polynomial basis on the extension stage) extension database. The discrete periodic extension is a technique for augmenting a set of uniformly-spaced samples of a smooth function with auxiliary values in an extension region. If a suitable extension is constructed, the interpolating trigonometric polynomial found via an FFT(Fast Fourier Transform) will accurately approximate the original function in its original interval. The discrete periodic extension is a key construction in the FC-Gram algorithm which is successfully implemented in several recent efficient and high-order PDEs solvers. This thesis focuses on a new flexible discrete periodic extension procedure that performs at least as well as the FC-Gram method, but with somewhat simpler implementation and significantly decreased setup time.