For the erroneous detection of feature points and multi-adjustment of the thresholds caused by global thresholds in the extraction of feature points from scattered point clouds, an adaptive feature extraction method based on Morse theory is proposed. Firstly, the potential feature points are marked by assigning weights that are computed by covariance analysis. Secondly, the mean angle along the principle direction between the point and its neighboring points is defined as the local feature descriptor, in order to compute the discrete gradient of the potential feature points. Finally, the Voronoi diagram of each local feature neighborhood is established,the linear interpolation method is utilized to compute the discrete gradient of the vertices of the Thiessen polygonal, and the gradient extreme points are marked as feature points according to the local discrete gradient vector domain. In order to improve the robustness and the anti-noise performance, the discrete gradient computation is performed using multi-scale where the neighborhood scale is used as the scale parameter, and then the features are extracted with multi-scale analysis. The experimental results demonstrate that the proposed method is simple, robust and does not depend on the sharpness of the features; furthermore, it extracts both sharp and blunt features. The results are satisfactory at different levels of noises from 0.03 dB to 0.05 dB, even though some of the features may be missing under 0.05 dB noise level.