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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.

$\begin{align*}\left\{\begin{array}{ll}\left(a+b∈t_{\mathbb R^N}|(-Δ)^{\frac{s}{2}}u|^2\right)(-Δ)^su=\lambda u+μ|u|^{q-2}u+|u|^{2_{s}^*-2}u&\ \ \mbox{in}\ \ Ω, \\ u=0&\ \ \mbox{in}\ \mathbb R^N\backslashΩ, \end{array}\right.\end{align*}$ |

$N>2s$ |

$a, b, \lambda, μ>0$ |

$s∈(0, 1)$ |

$Ω$ |

$(-Δ)^s$ |

$2<q≤q\min\{4, 2_s^*\}$ |

$b$ |

$μ$ |

In this paper, we study a class of nonlinear Choquard type equations involving a general nonlinearity. By using the method of penalization argument, we show that there exists a family of solutions having multiple concentration regions which concentrate at the minimum points of the potential *V*. Moreover, the *monotonicity* of *f*(*s*)=*s* and the so-called *Ambrosetti*-*Rabinowitz* condition are not required.

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