# Geometric Parameter Self-Calibration Based on Projection Feature Matching for X-Ray Nanotomography
## 1 Introduction  
Nanotomography is a technique that uses the attenuation-based information from X-rays after penetrating an object to achieve high-resolution three-dimensional imaging of the object's internal structure [1]. It has the advantages of nondestructive imaging, high-resolution accuracy, and fast imaging speed. It is widely used in fields such as nondestructive testing [2], material analysis [3], and 3D printing [4]. However, the mismatch between the actual spatial structure parameters of the system and the ideal model parameters leads to the generation of geometric artifacts. The blurring and ghosting in 3D reconstruction images seriously affect image information acquisition. The calibration of geometric parameters presents the foundation for obtaining high-quality 3D-reconstructed images. Therefore, the calibration of geometric parameters is essential. Researchers have recently proposed and developed many methods for calibrating the geometric parameters of CT systems. They can be divided into the phantom-based geometric parameter calibration method, the geometric parameter self-calibration method based on reconstructed image features, and the geometric parameter self-calibration method based on projected image features. The basic principle underlying the phantom-based method is the use of standard templates with known geometric shapes. The system's geometric parameters are then solved using the phantom projection trajectory calculation parameters [5; 6; 7; 8]. This method has the advantages of high correction accuracy and fast solution speed. However, this method requires the fabrication and placement of a high-precision calibration phantom, consequently making its application difficult in nanotomography which uses a single-imaging field of only a few millimeters. The geometric parameter self-calibration method based on reconstructing image features uses the reconstructed image feature information of the detected object to construct a criterion to calibrate geometric parameters. In 2007, [PERSON] et al. [9] proposed a cross-correlation-based software correction method to correct the blurred reconstructed image caused by image misalignment. In 2008, [PERSON] et al. [10] proposed to reconstruct the image entropy information to minimize the correction of geometric artifacts. In 2010, [PERSON] et al. [11] proposed reconstruction of the image sharpness minimization to achieve geometric parameter calibration. When the image texture is complex, the correction effect of entropy and sharpness methods conspicuously decreases. In 2013, [PERSON] et al. [12] applied the idea of interval division to divide the solution parameters into multiple cells, and iteratively narrowed the interval by comparing the reconstructed image quality corresponding to the endpoint values of the interval, thus realizing the solution of the geometric parameters of a CT system. In 2017, [PERSON] et al. [13] proposed a classification network based on deep learning to classify reconstructed image slices into geometric artifacts and non-geometric artifacts to achieve artifact correction. In 2021, [PERSON] et al. [14] proposed a deep neural network containing six-layer convolution and six-layer deconvolution to achieve the correction of geometric artifacts in a data-driven manner. The deep-learning-based method provides a new idea for geometric artifact correction. However, the technique requires extensive data for network training. The method based on the reconstruction of image features requires multiple iterative reconstruction, and the correction efficiency is low. The geometric calibration method based on the projected image feature directly uses object projection information to realize the calibration of geometric parameters. In 2008, [PERSON] et al. [15] extended the redundancy of the sinusoidal curve projected by fan-beam CT in the central plane to cone-beam CT projection data, using the redundancy of projected data to construct cost functions to realize geometric parameter calibration. In 2009, [PERSON] et al.