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Since wavelet analysis has become a powerful tool for signal and image processing, the multiresolution approach provides a solution for many practical applications. In X-ray computerized tomography algorithms for multiresolution 2D parallel beam, 20 fan-beam, and 3D cone-beam (Feldkamptype) reconstruction using tensor or quincunx wavelets were introduced in [1], [2], [3]. These reconstruction formulas are based on the strong relationship between the continuous wavelet transform and the Radon transform as mentioned in [4]. In this contribution we use a different approach to achieve an algorithm for reconstructing an object at different resolutions: The approximate inverse, introduced in [5], is a method for solving first kind operator equations AI = 9 in a stable way. Instead of determining the exact solution I, an inversion operator for Ie is calculated, where Ie is associated to I via the inner product (I, e) using a mollifier e. Applying the approximate inverse to computerized tomography yields a reconstruction algorithm of filtered backprojection type [6]. If we choose the mollifier as a wavelet, Ie represents the wavelet coefficients of f [7]. In this contribution we show a derivation of a nonseparable multiresolution reconstruction formula using the approximate inverse. For the sake of simplicity we introduce a nonseparable multiresolution inversion formula only for the 20 parallel scanning geometry. The resulting inversion formula using the approximate inverse is equal to the formula in [1]. In case of narrow cone-beam angles we make use of the Feldkamp algorithm [8]. We replace the standard ramp filter by the proposed ramp-wavelet filter, where the 2D multiresolution acts slice by slice. In addition we apply an image processing step to the reconstructed approximation coefficients to determine the potential defect areas. To reduce time and data the detail coefficients are only reconstructed inside of these regions of interest.Section II provides the basics for representing a signal using the wavelet expansion and is followed by a short introduction to the theory of Approximate Inverse in Section III. The progressive reconstruction method is presented in Section IV and its application to fuse reconstruction and image processing is outlined in Section V. Section VI contains the results of the proposed reconstruction method. Finally a conclusion is given in Section VII.

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