This paper addresses the challenges of image reconstruction in computed tomography (CT) due to under-sampling and insufficient data, particularly in divergent-beam CT. The authors develop and investigate an iterative image reconstruction algorithm based on minimizing the total variation (TV) of the image. The TV algorithm is designed to handle sparse and insufficient data problems, such as few-view projections, limited angular ranges, and gaps in projection data caused by bad detector bins. The algorithm is validated through numerical demonstrations using fan-beam CT simulations, showing its effectiveness in reconstructing accurate images from sparse or insufficient data. The TV algorithm is also shown to be robust under non-ideal conditions, such as data inconsistency due to noise, and can be generalized to cone-beam CT and other tomographic imaging modalities. comparisons with standard algorithms like EM and ART highlight the ill-posedness of the imaging problems considered, demonstrating the superior performance of the TV algorithm in handling sparse and insufficient data.This paper addresses the challenges of image reconstruction in computed tomography (CT) due to under-sampling and insufficient data, particularly in divergent-beam CT. The authors develop and investigate an iterative image reconstruction algorithm based on minimizing the total variation (TV) of the image. The TV algorithm is designed to handle sparse and insufficient data problems, such as few-view projections, limited angular ranges, and gaps in projection data caused by bad detector bins. The algorithm is validated through numerical demonstrations using fan-beam CT simulations, showing its effectiveness in reconstructing accurate images from sparse or insufficient data. The TV algorithm is also shown to be robust under non-ideal conditions, such as data inconsistency due to noise, and can be generalized to cone-beam CT and other tomographic imaging modalities. comparisons with standard algorithms like EM and ART highlight the ill-posedness of the imaging problems considered, demonstrating the superior performance of the TV algorithm in handling sparse and insufficient data.