Wavelet Analysis Of Quaternion Valued Function And Application Of Wavelet Theory

In this thesis, we study quaternion-valued multi-resolution analysis and the the-ory of wavelets. By using the theory of multi-resolution in vector-valued functionspace, we show that quaternion-valued multi-resolution analysis. And we give theconstruction about quaternion-valued scaling functions and wavelets. By definingquaternion valued inner product, we establish the wavelet theory on L~2(R~2, H; dx).Moreover, we discuss the continuous wavelet X-ray transform and its discrete for-mula on L~2(R~n), and give an inverse formula of continuous wavelet X-ray transform.Discrete transform are described which make use of wavelet orthonormal bases onL~2(R~n), and more generally, of biorthogonal system of wavelet Riesz bases and ofwavelet frame.