# American Institute of Mathematical Sciences

January  2018, 12(1): 239-259. doi: 10.3934/ipi.2018010

## A scaled gradient method for digital tomographic image reconstruction

 1 Department of Mathematics, Shanghai University, Shanghai 200444, China 2 Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA

* Corresponding author: James G. Nagy

Received  March 2017 Revised  July 2017 Published  December 2017

Fund Project: The first author is supported by grant no. 15ZR1416300 from the Shanghai Municipal Natural Science Foundation, the third author is supported by grant no. DMS-1522760 from the US National Science Foundation

Digital tomographic image reconstruction uses multiple x-ray projections obtained along a range of different incident angles to reconstruct a 3D representation of an object. For example, computed tomography (CT) generally refers to the situation when a full set of angles are used (e.g., 360 degrees) while tomosynthesis refers to the case when only a limited (e.g., 30 degrees) angular range is used. In either case, most existing reconstruction algorithms assume that the x-ray source is monoenergetic. This results in a simplified linear forward model, which is easy to solve but can result in artifacts in the reconstructed images. It has been shown that these artifacts can be reduced by using a more accurate polyenergetic assumption for the x-ray source, but the polyenergetic model requires solving a large-scale nonlinear inverse problem. In addition to reducing artifacts, a full polyenergetic model can be used to extract additional information about the materials of the object; that is, to provide a mechanism for quantitative imaging. In this paper, we develop an approach to solve the nonlinear image reconstruction problem by incorporating total variation (TV) regularization. The corresponding optimization problem is then solved by using a scaled gradient descent method. The proposed algorithm is based on KKT conditions and Nesterov's acceleration strategy. Experimental results on reconstructed polyenergetic image data illustrate the effectiveness of this proposed approach.

Citation: Jianjun Zhang, Yunyi Hu, James G. Nagy. A scaled gradient method for digital tomographic image reconstruction. Inverse Problems & Imaging, 2018, 12 (1) : 239-259. doi: 10.3934/ipi.2018010
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##### References:
From left to right and upper to bottom: original first material, reconstructed first material, original second material, reconstructed second material, sinogram image and RErr with iteration
From left to right and upper to bottom: original first material, reconstructed first material, original second material, reconstructed second material, sinogram image and RErr with iteration
From left to right and upper to bottom: original first material, reconstructed first material, original second material, reconstructed second material, sinogram image and RErr with iteration
From left to right and upper to bottom: original first material, reconstructed first material, original second material, reconstructed second material, sinogram image and RErr with iteration
From left to right and upper to bottom: sum along each row for first material, sum along each column for first material, sum along each row for second material, sum along each column for second material, relative error of reduced resolution image along rows and columns for first material, relative error of reduced resolution image along rows and columns for second material
From left to right and upper to bottom: sum along each row for first material, sum along each column for first material, sum along each row for second material, sum along each column for second material, relative error of reduced resolution image along rows and columns for first material, relative error of reduced resolution image along rows and columns for second material
From left to right and upper to bottom: original first material, reconstructed first material, original second material, reconstructed second material, sinogram image and RErr with iteration
From left to right and upper to bottom: original first material, reconstructed first material, original second material, reconstructed second material, sinogram image and RErr with iteration
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