Motivated by the important archaeological application of exploring cultural heritage objects, in this paper we study the challenging problem of automatically segmenting curve structures that are very weakly stamped or carved on an object surface in the form of a highly noisy depth map. Different from most classical low-level image segmentation methods that are known to be very sensitive to the noise and occlusions, we propose a new supervised learning algorithm based on Convolutional Neural Network (CNN) to implicitly learn and utilize more curve geometry and pattern information for addressing this challenging problem. More specifically, we first propose a Fully Convolutional Network (FCN) to estimate the skeleton of curve structures and at each skeleton pixel, a scale value is estimated to reflect the local curve width. Then we propose a dense prediction network to refine the estimated curve skeletons. Based on the estimated scale values, we finally develop an adaptive thresholding algorithm to achieve the final segmentation of curve structures. In the experiment, we validate the performance of the proposed method on a dataset of depth images scanned from unearthed pottery shards dating to the Woodland period of Southeastern North America.