The success of any transform coding technique depends on how well the basis functions represent the signal features. The discrete wavelet transform (DWT) performs a multiresolution analysis of a signal; this enables an efficient representation of smooth and detailed signal regions. Furthermore, computationally efficient algorithms exist for computing the DWT. For these reasons, recent image compression standards such as JPEG2000 use the wavelet transform. It is well known that orthogonality and symmetry are desirable transform properties in image compression applications. It is also known that the scalar wavelet transform does not possess both properties simultaneously. Multiwavelets overcome this limitation; the multiwavelet transform allows orthogonality and symmetry to co-exist. However recently reported image compression results indicate that the scalar wavelets still outperform the multiwavelets in terms of peak signal-to-noise ratio (PSNR). In a multiwavelet transform, the balancing order of the multiwavelet is indicative of its energy compaction efficiency (usually a higher balancing order implies lower mean-squared-error, MSE, in the compressed image). But a high balancing order alone does not ensure good image compression performance. Filter bank characteristics such as shift-variance, magnitude response, symmetry and phase response are important factors that also influence the MSE and perceived image quality. This thesis analyzes the impact of these multiwavelet characteristics on image compression performance. Our analysis allows us to explain---for the first time---reasons for the small performance gap between the scalar wavelets and multiwavelets. We study the characteristics of five balanced multiwavelets (and 2 unbalanced multiwavelets) and compare their image compression performance for grayscale images with the popular (9,7)-tap and (22,14)-tap biorthogonal scalar wavelets. We use the well-known SPIHT quantizer in our compression scheme and utilize PSNR and subjective quality measures to assess performance. We also study the effect of incorporating a human visual system (HVS)-based transform model in our multiwavelet compression scheme. Our results indicate those multiwavelet properties that are most important to image compression. Moreover, the PSNR and subjective quality results depict similar performance for the best scalar wavelets and multiwavelets. Our analysis also shows that the HVS-based multiwavelet transform coder considerably improves perceived image quality at low bit rates. / Master of Science

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Recent research in transform-based image compression has focused on the wavelet transform due to its superior performance over other transforms. Performance is often measured solely in terms of peak signal-to-noise ratio (PSNR) and compression algorithms are optimized for this quantitative metric. The performance in terms of subjective quality is typically not evaluated. Moreover, the sensitivities of the human visual system (HVS) are often not incorporated into compression schemes. This paper develops new wavelet models of the HVS and illustrates their performance for various scalar wavelet and multiwavelet transforms. The performance is measured quantitatively (PSNR) and qualitatively using our new perceptual testing procedure. Our new HVS model is comprised of two components: CSF masking and asymmetric compression. CSF masking weights the wavelet coefficients according to the contrast sensitivity function (CSF)---a model of humans' sensitivity to spatial frequency. This mask gives the most perceptible information the highest priority in the quantizer. The second component of our HVS model is called asymmetric compression. It is well known that humans are more sensitive to luminance stimuli than they are to chrominance stimuli; asymmetric compression quantizes the chrominance spaces more severely than the luminance component. The results of extensive trials indicate that our HVS model improves both quantitative and qualitative performance. These trials included 14 observers, 4 grayscale images and 10 color images (both natural and synthetic). For grayscale images, although our HVS scheme lowers PSNR, it improves subjective quality. For color images, our HVS model improves both PSNR and subjective quality. A benchmark for our HVS method is the latest version of the international image compression standard---JPEG2000. In terms of subjective quality, our scheme is superior to JPEG2000 for all images; it also outperforms JPEG2000 by 1 to 3 dB in PSNR. / Master of Science

Methods for digital image compression have been the subject of much study over the past decade. Advances in wavelet transforms and quantization methods have produced algorithms capable of surpassing the existing image compression standards like the Joint Photographic Experts Group (JPEG) algorithm. For best performance in image compression, wavelet transforms require filters that combine a number of desirable properties, such as orthogonality and symmetry. However, the design possibilities for wavelets are limited because they cannot simultaneously possess all of these desirable properties. The relatively new field of multiwavelets shows promise in removing some of the limitations of wavelets. Multiwavelets offer more design options and hence can combine all desirable transform features. The few previously published results of multiwavelet-based image compression have mostly fallen short of the performance enjoyed by the current wavelet algorithms. This thesis presents new multiwavelet transform methods and measurements that verify the potential benefits of multiwavelets. Using a zerotree quantization scheme modified to better match the unique decomposition properties of multiwavelets, it is shown that the latest multiwavelet filters can give performance equal to, or in many cases superior to, the current wavelet filters. The performance of multiwavelet packets is also explored for the first time and is shown to be competitive to that of wavelet packets in some cases. The wavelet and multiwavelet filter banks are tested on a much wider range of images than in the usual literature, providing a better analysis of the benefits and drawbacks of each. NOTE: (03/2007) An updated copy of this ETD was added after there were patron reports of problems with the file. / Master of Science

In Chapters 1 and 2, we introduce the definition of interpolating refinable function vectors in dimension one and high dimensions, characterize such interpolating refinable function vectors in terms of their masks, and derive their sum rule structure explicitly. We study biorthogonal refinable function vectors from interpolating refinable function vectors. We also study the symmetry property of an interpolating refinable function vector and characterize a symmetric interpolating refinable function vector in any dimension with respect to certain symmetry group in terms of its mask. Examples of interpolating refinable function vectors with some desirable properties, such as orthogonality, symmetry, compact support, and so on, are constructed according to our characterization results. In Chapters 3 and 4, we turn to the study of general matrix extension problems with symmetry for the construction of orthogonal and biorthogonal multiwavelets. We give characterization theorems and develop step-by-step algorithms for matrix extension with symmetry. To illustrate our results, we apply our algorithms to several examples of interpolating refinable function vectors with orthogonality or biorthogonality obtained in Chapter 1. In Chapter 5, we discuss some possible future research topics on the subjects of matrix extension with symmetry in high dimensions and frequency-based non-stationary tight wavelet frames with directionality. We demonstrate that one can construct a frequency-based tight wavelet frame with symmetry and show that directional analysis can be easily achieved under the framework of tight wavelet frames. Potential applications and research directions of such tight wavelet frames with directionality are discussed. / Applied Mathematics

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