Application of the wavelet based Radon transform

The theory of the Radon transform forms the foundation for problems of reconstruction from projections. For example, in computerized tomography (CT) the raw data can be identified with the Radon transform of the image. The desired image is found by applying the inverse Radon transform to the projection data. In cases where it is desired to image a local region that is small in comparison to the entire image there is a problem due to the nature of the global properties of the inverse Radon transform in two dimensions. From a practical point of view this means we must have projection data for regions that are not in the region of interest (ROI) in order to stabilize the inversion process that yields the ROI. Introduction of the wavelet transform as an intermediate part of the inversion leads to an important improvement in this procedure. It is possible to devise algorithms such that significantly less radiation exposure is required without causing a noticeable degradation of the image in the ROI. The key is to make use of wavelets with several vanishing moments and to do appropriate sparse sampling away from the ROI. A review of Radon transform inversion is discussed for three major inversion algorithms, and a brief summary of wavelets is given. The current situation on wavelet based Radon transform inversion is reviewed along with potential applications to CT, limited angle CT, and single photon emission computed tomography (SPECT).