Recent investigations on human vision discover that the retinal image is a landscape or a geometric surface, consisting of features such as ridges and summits. However, most of existing popular local image descriptors in the literature, e.g., scale invariant feature transform (SIFT), histogram of oriented gradient (HOG), DAISY, local binary Patterns (LBP), and gradient location and orientation histogram, only employ the first-order gradient information related to the slope and the elasticity, i.e., length, area, and so on of a surface, and thereby partially characterize the geometric properties of a landscape. In this paper, we introduce a novel and powerful local image descriptor that extracts the histograms of second-order gradients (HSOGs) to capture the curvature related geometric properties of the neural landscape, i.e., cliffs, ridges, summits, valleys, basins, and so on. We conduct comprehensive experiments on three different applications, including the problem of local image matching, visual object categorization, and scene classification. The experimental results clearly evidence the discriminative power of HSOG as compared with its first-order gradient-based counterparts, e.g., SIFT, HOG, DAISY, and center-symmetric LBP, and the complementarity in terms of image representation, demonstrating the effectiveness of the proposed local descriptor.