# Hybrid method: Algebraic/Inverse Radon Transform for region of interest reconstruction of computed tomography images

#### Abstract

Computed tomography is one of the most rapidly advancing method in modern medical imaging, because of is relatively low cost and outstanding quality. The main shortcoming of the method is the relatively high dose of radiation used. Consequently, considerable efforts have been made to obtain good image reconstructions from fewer projections or from using lower intensity beams (hence, noisier data). It has been known for decades that a scan with high intensity beams and large number of projections (typically, 180-360 for a planar image) will lead to a very good image reconstruction via an Inverse Radon Transform (IRT), but the method is not accurate when less than 30 projections are employed. On the other hand, algebraic methods are very efficient for small images; unfortunately, for an N x N image, the number of coefficients of the matrix corresponding to the linear system of equations is of the order of N4. Hence, even for a relatively small images with N=256, the matrix have around 4 billion coefficients. For almost all practical applications of the computed tomography, algebraic reconstruction are prohibitively time consuming. ^ The aim of this novel method is to combine the advantages of both IRT and ART to obtain a good image for a small region of the total image. Starting with the total projections of the large image, a Filtered Back Projections (FBP) algorithm is used to create an intermediate image; a small Region of Interest (ROI) is selected from the large image and the partial projections belonging to the ROI alone are calculated. These partial projections are subsequently used with an algebraic method to reconstruct the ROI. Assuming a region of interest (e.g., any abnormality suspected to be clinically important) is 10 times smaller than the whole image, the speed of the algebraic method is increased by a factor of roughly 106 (e.g., about one second instead of 10 days); even more important, for three dimensional images, the factor is 109 (one second instead of 30 years). ^ Furthermore, for a small ROI, the system of equations to be solved algebraically become overdetermined even for a small number of available projections, while for the total image the system is severely under determined. ^ The influence of noise and of the reduced number of projections on the quality of the approximate projections of the ROI image is also investigated. ^ A new parameter δ is introduced and its defined as the difference of signal levels between the real image and the FBP reconstruction. The optimal value of the new parameter could be determined without using the original image. The quality of the reconstruction of the ROI depends on its size, with a better quality for larger sizes. Also, it was found that the method is robust even in the presence of noise.^