Multiresolution analysis (MRA) has fairly recently become important, and even essential, to image processing and signal analysis, and is thus having a growing impact on image and signal related areas. As one of the most famous family members of the MRA, the wavelet transform (WT) has demonstrated itself in numerous successful applications in various fields, and become one of the most powerful tools in the fields of image processing and signal analysis. Due to the fact that only the scale information is supplied in WT, the applications using the wavelet transform may be limited when the absolutely-referenced frequency and phase information are required. The Stockwell transform (ST) is a recently proposed multiresolution transform that supplies the absolutely-referenced frequency and phase information. However, the ST redundantly doubles the dimension of the original data set. Because of this redundancy, use of the ST is computationally expensive and even infeasible on some large size data sets. Thus, I propose the use of the discrete orthonormal Stockwell transform (DOST), a non-redundant version of ST. This thesis will continue to implement the theoretical research on the DOST and elaborate on some of our successful applications using the DOST. We uncover the fast calculation mechanism of the DOST using an equivalent matrix form that we discovered. We also highlight applications of the DOST in image compression and image restoration, and analyze the global and local translation properties. The local nature of the DOST suggests that it could be used in many other local applications.

Multiresolution analysis (MRA) has fairly recently become important, and even essential, to image processing and signal analysis, and is thus having a growing impact on image and signal related areas. As one of the most famous family members of the MRA, the wavelet transform (WT) has demonstrated itself in numerous successful applications in various fields, and become one of the most powerful tools in the fields of image processing and signal analysis. Due to the fact that only the scale information is supplied in WT, the applications using the wavelet transform may be limited when the absolutely-referenced frequency and phase information are required. The Stockwell transform (ST) is a recently proposed multiresolution transform that supplies the absolutely-referenced frequency and phase information. However, the ST redundantly doubles the dimension of the original data set. Because of this redundancy, use of the ST is computationally expensive and even infeasible on some large size data sets. Thus, I propose the use of the discrete orthonormal Stockwell transform (DOST), a non-redundant version of ST. This thesis will continue to implement the theoretical research on the DOST and elaborate on some of our successful applications using the DOST. We uncover the fast calculation mechanism of the DOST using an equivalent matrix form that we discovered. We also highlight applications of the DOST in image compression and image restoration, and analyze the global and local translation properties. The local nature of the DOST suggests that it could be used in many other local applications.