1. A three-dimensional-object detecting apparatus that has a single image-capture device provided at a predetermined position to capture an image of a predetermined image-capture region including a predetermined image-capture surface and that detects a three-dimensional object on the image-capture surface on a basis of the captured image of the image-capture region, the three-dimensional-object detecting apparatus comprising: a detection-image creating device configured to detect the three-dimensional object on the image-capture surface on a basis of the captured image and to create a detection image in which a silhouette of only the detected three-dimensional object is left by eliminating an unnecessary region other than the detected three-dimensional object; a density-map creating device configured to pre-store mask images, obtained by arranging virtual cuboids having a certain width, depth, and height at predetermined coordinate points in a coordinate plane set on the image-capture surface and projecting silhouettes of the arranged virtual cuboids onto an imaging plane of the image-capture device, for the corresponding coordinate points different from each other, to determine, as the three-dimensional-object's spatial densities at the corresponding coordinate points, area ratios with respect to respective portions where the three-dimensional-object silhouette in the detection image and the cuboid silhouettes in the mask images for the corresponding coordinate points overlap each other, and to arrange, in accordance with an arrangement sequence of the coordinate points, pixels for the respective coordinate points such that the pixels have pixel values corresponding to the determined spatial densities at the respective coordinate points, to thereby create a density map indicating distribution of the spatial densities on the image-capture surface; and a three-dimensional-object position detecting device configured to extract at least one high-density region in which the pixel value exceeds a threshold pixel value from the density map created by the density-map creating device and to detect, as the position of the three-dimensional object corresponding to the high-density region, coordinates of a representative point that is included in the extracted high-density region and that is closest to the image-capture device.