A constrained kernel regression model is proposed to solve the problem of one-dimensional (1D) pose estimation. Unlike the traditional kernel regression model, a circular constraint is applied to the output of the regression function, i.e. using 2D coordinates on a unit circle as output instead of 1D pose angles from 0 to 360°. The experimental results show that with this constraint, the performance of kernel regression on the 1D pose estimation can be improved significantly, and the constrained kernel regression model can run in real-time.Introduction: To automatically recognise the pose/viewpoint of an object is a challenging problem of visual object recognition in the area of computer vision, and pose estimation is also a key issue in many applications, including human-machine interaction, robotics, aerospace etc. Practice people usually recognise objects from several familiar viewpoints, e.g. we recognise people from the front or side view of the face, but not from the back side of the head. For most applications, e.g. a robot finding where the handle of a mug is, or a spacecraft flying around the target spacecraft for rendezvous and docking, onedimensional (1D) pose might be enough. Therefore, in this Letter, we focused on 1D pose estimation, i.e. a camera looking at the object on a viewing circle. There have been several previous works to solve such a 1D pose estimation problem recently, such as [1][2][3][4][5][6][7][8]. These approaches can be classified into two groups. One group, including [1][2][3][4][5], solves the problem in a discrete way, i.e. using several discrete (4, 8 or 16) view-based classifiers. However, the other group, including [6-8], solves pose estimation in a continuous way, practically based on regression frameworks [6,7]. Generally, pose estimation is fundamentally a continuous problem, so we followed up [6-8] estimate continuous 1D poses.Torki and Elgammal [6] learnt a kernel regression (KR) function from an embedded representation of local features in visual images. El-Gaaly et al.[7] used a multi-kernel learning approach to extend the regression framework in [6], called multi-KR in which two kernels were defined for visual inputs and depth inputs separately. Both of them kept the output of their regression function unconstrained, i.e. directly outputting the 1D pose angles from 0 to 360°. The two images with pose angles 1 and 359°s hould be quite similar in visual appearance, but their outputs of the traditional KR function, which are supposed to be 1 and 359°, are extremely different. According to [8], it is clear that the images of one object with 1D pose variation (or captured on a viewing circle) lie on a closed 1D manifold, and such a manifold is homeomorphic to a unit circle. Using the homeomorphic manifold (the unit circle) as a common representation can significantly improve the pose estimation results. However, the model in [8] is generative, and cannot run in realtime, as it needs to minimise the reconstruction error for inference. Hence, we introduce such a circular cons...