## 1 Introduction  
Tunnel displacement and deformation monitoring are critical factors for engineers to evaluate the tunnel health and safety. Some indices such as the cross section's chord [1], convergence and dislocations [2], tunnel axis's settlement [2], and tunnel model's clearance [3] are important measures to evaluate the stability of tunnels. At the initial stage, some traditional methods are used to monitor the tunnel's health. The instruments such as tape extensometer [4] and total station [5] are extensively used for high precision deformation monitoring. Although they have a very high precision, these traditional ground surveying measurements are time-consuming, labor-intensive and costly. In some cases, they fail to get the deformation because sparsely measured points are not dense enough to evaluate the overall tunnel. Apart from the above geodetic surveys, an alternative method based on digital photogrammetry has been proposed because more measured points along a profile can be extracted efficiently. However, many of the acquired images easily suffer from occlusions, low contrast and unfavorable perspectives, thereby making automatic image matching and understanding harder. For example, by combining stereo images and laser-lit spots, [PERSON] et al. [5; 6] propose a profile-image method to measure cross sections. Although this method captures the densified points along a profile more than traditional geodetic surveys, it is restricted by insufficient lighting conditions in an actual tunnel. We recommend the readers to the recent survey [7] to get the review of the state-of-the-art methods and applications in tunnel inspections based on photogrammetric and computer vision techniques. Recently, the Terrestrial Laser Scanning (TLS) technique has been developed to efficiently and accurately collect three-dimensional point clouds from the reflected objects. Since the advantages of TLS technique include being highly accurate, efficient, automatic and has a lack of contact, the TLS technique has been proved to be an effective way to monitor the deformation of engineering structures, particularly in the fields of tunnel management [8; 9] and deformation analysis [10; 11; 12]. Based on TLS point clouds, tunnel deformation monitoring techniques can be generally categorized into profile-based methods and model-based methods. The profile-based method tries to analyze the tunnel's deformation by using a series of thin profiles directly derived from TLS tunnel point clouds. Numerous published papers along this line [1; 13; 14; 15] demonstrate the accuracy and efficiency of this type of method. For example, [PERSON] et al. [13] propose an automated and efficient method to estimate the tunnel centerline and cross-sectional plane. In their method, the centerline is first estimated by skeletonizing a binary image produced by projecting three-dimensional point clouds onto a horizontal plane. Based on the tunnel boundary points and the centerline, the cross-sectional planes are estimated and further adjusted, and the final cross sections are generated by projecting the nearby points to the adjusted planes. Although the authors improve the efficiency of surveying and data processing, they cannot extract the continuous profiles because the parametric equation of tunnel centerline is not provided. In addition, this method is sensitive to the non-lining points, i.e., pipes and equipment attached to the lining. To overcome the drawback regarding the above centerline, the methods in [1; 3] use multiple models, i.e., straight line, transition curve and curve to commonly fit the different segments of tunnel centerline, and then adopt the global least-squares adjustment to maintain the consistency between adjacent fitting models. To cope with missing data, [PERSON] et al. [1] recover the blank areas of tunnel profiles with a surficial interpolation algorithm.