Cone-beam CT Reconstruction Based On Hessian Schatten Penalty

Computed Tomography(CT) can obtain the internal structure information without causing damage to the object. Physician can diagnose the size and location of lesions according to the obtained structure information of human organs or tissues in radiotherapy. The conventional CT uses a narrow beam of X-rays and it is necessary to scan several times to get the whole projection data. The projection is one-dimensional and we can get the two-dimensional image through certain reconstruction method, and stack the successive two-dimensional image data to get the three-dimensional image. Cone-beam CT uses a wide beam of X-rays. We can get all the needed projection data after a 360 degree rotation. Compared with the conventional CT, cone-beam CT can significantly improve the utilization rate of X-ray. Meanwhile, the utilization of the surface detector can significantly improve the acquisition speed of the projection data. Moreover, cone-beam CT can reduce the ionizing radiation dose. Consequently, the damage caused to people's normal cells can been dramatically reduced. All the advantages of cone-beam CT make it get a wide application in clinic applications.The total variation(TV) penalty has shown state-of-the-art performance in suppressing noises and preserving edges. However, it produces the well-known staircase effect. In this study, to avoid the staircase effect, we proposed to use a new penalty—Hessian Schatten penalty—that involves higher order derivatives. We utilized the Schatten norms of the Hessian matrix, computed at every pixel of the image, as the penalty term to form the objective function. We utilized mainly the Hessian Schatten nuclear norm penalty, Hessian Schatten Frobenius norm penalty and Hessian Schatten spectral norm penalty in this paper. By the derivation of a primal-dual formulation, our Hessian Schatten penalty method can solve the corresponding constrained optimization problem under any choice of Schatten norm, and overcome the non-smooth nature of the problem. The fast iterative shrinkage-thresholding algorithm(FISTA) was employed to solve the optimization of the objective function in this study. This method provides state-of-the-art convergence rate by combining two successive iterates.This paper conducts computer simulations with MATLAB 2012 and Visual Studio 2012. We reconstructed the underlying image from the projection data using FDK method, Total Variation(TV) method and the Hessian Schatten method, and compared their performance with given performance indexes. These indexes we utilized are peak signal to noise ratio(PSNR), improvement signal to noise ratio(ISNR), contrast to noise ratio(CNR) and structural similarity index(SSIM). The experiments on two digital phantoms and two physical phantoms demonstrated the outstanding ability of the Hessian Schatten penalty over the TV-based penalty in suppressing the staircase effect.