Application Of Wavelet In The Reconstruction Of Local Tomography

Computed Tomography technique(CT) has become a new nondes –tructive detection method in recent years, which is of good quality and high resolution. It can be processed and analyzed digitally, so it is app -lied widely in aviation, shipping, automobile, public security, medical and other areas.In the preface of the thesis, we summarize the theory of CT, and recall its developing history. In the second chapter, we introduce the basic theory of CT. The projection data is the intergral of linear attenuation on the object. Estimating the interior distribution of density is the essential of image reconstruction, which mathematics foundation is Radon transform and Radon theorem that are the theory core of FBP. Because of the magnification of differential operator in Radon theorem we always implement FBP by Fourier slice theorem. Fourier slice theorem point out that the Fourier transform of the Radon transform at some angle is the Fourier transform of function along a line passing through the origin. The flow chart of FBP is given.In chapter three, we present an important theory tool—wavelet analysis. First of all, we give the definition of wavelet and several simple examples, and compare the characters of a few common wavelets. Secondly, we introduce continuous wavelets transform, describe the conception of multi-resolution space and list the notable arithmetic of decomposition and reconstruction—Mallat arithmetic. Finally, we describe separate tensor product wavelet, specially give the notion of vanishing moment of scale function. If content , then we call the function has number of vanishing moments.In the fourth chapter, we introduce the algorithms of wavelet-base Multiresolution local tomography. Above all, we present the math description in two dimensions. Suppose is a continuous function, in , there is a disk which radius is , and some intergrals of lines through the origin, let Region of Interest is a disk which radius is (), we must reconstruct ROI by some intergrals. Then we give the reason that why we can 't reconstruct ROI using FBP. It follows In the above equation is a derivative operator, is Hilbert convolution operator:,so The derivative part is local operator, but Hilbert transform is not, so FBP is of no effect.Then we present the algorithm that can be used to obtain the wavelet coefficients of a function on from its Radon transform data.Given a real-valued, square integrable function on that satisfies continuous wavelets transform condition, let be given on , the wavelet transform of function can be reconstructed from its 1-D projection by where ,In the discrete case the above formula becomeswhere . The right-hand side can be evaluated in two steps, the filtering step, In fact, according to convolution theory in Fourier domain, and the backprojection where is replaced by .If the wavelet basis is separable, the approximation and detail coefficients are given by (3.2) and (3.5) in chapter three. These coefficients can be obtained from the projection data, replacing by ,,,.The approximation coefficients are obtained by The filtering part in the Fourier is given by In a similar way This means that the wavelet and scaling coefficients of the image can be obtained by filtered backprojection method while the ramp filter is replaced by which are called the scaling and wavelet ramp filters. The reconstruction part uses the conventional multiresolution reconstruction filterbank.And then we make clear the local character of the algorithm. In the backprojection formula where , If have compact support and many vanishing moments, they will still be rapid decay in limit extent after Hilbert transform. More specifically, the following holds.Lemma: Suppose that outside the interval and satisfies ,then for , If and corresponding...