1. A local feature descriptor extracting apparatus comprising: a memory that stores a program; and a processor configured to run the program stored in the memory that is configured to cause the processor to detect a plurality of feature points in an image and output feature point information that is information regarding each feature point, select a prescribed number of feature points in a descending order of scale from the plurality of detected feature points, based on the feature point information, acquire a local region corresponding to each selected feature point, divide each local region into a plurality of subregions, generate a feature vector of a plurality of dimensions for each subregion in each local region, and select a dimension from the feature vector for each subregion so that at least one gradient direction differs between neighboring subregions, based on a positional relationship between subregions in each local region and output an element of the selected dimension as a feature descriptor of the local region.