Typical single-pixel imaging techniques for edge detection are mostly based on first-order differential edge detection operators. In this paper, we present a novel edge detection scheme combining Fourier single-pixel imaging with a second-order Laplacian of Gaussian (LoG) operator. This method utilizes the convolution results of an LoG operator and Fourier basis patterns as the modulated patterns to extract the edge detail of an unknown object without imaging it. The simulation and experimental results demonstrate that our scheme can ensure finer edge detail, especially under a noisy environment, and save half the processing time when compared with a traditional first-order Sobel operator.