1. A feature point selecting system comprising: a recognition task executing unit that executes a recognition task using an importance of each of a plurality of feature point candidates on a three-dimensional shape model for a plurality of evaluation images which are generated from the three-dimensional shape model and which are used to evaluate a recognition error in the recognition task; a recognition error evaluating unit that evaluates a recognition error related to all evaluation images from a difference between a recognition result of the recognition task executing unit and correct data of the recognition task for each evaluation image; a feature point importance determining unit that determines the importance of each feature point candidate by setting a cost function which is a function for the importance of each feature point candidate and which is represented as a function obtained by adding a restriction condition that an importance of an unimportant feature point candidate becomes close to zero, to the recognition error related to the all evaluation images, and calculating the importance of each feature point candidate which minimizes a value of the cost function; and a feature point selecting unit that selects a feature point which needs to be used in the recognition task from the feature point candidates on the three-dimensional shape model based on the importance of each feature point candidate, wherein, with the recognition task executing unit, the recognition error evaluating unit and the feature point importance determining unit, until the value of the cost function which is set based on the importance of each feature point candidate determined by the feature point importance determining unit converges, repeatedly, the recognition task executing unit executes the recognition task, the recognition error evaluating unit evaluates the recognition error related to the all evaluation images and the feature point importance determining unit determines the importance of the feature point candidates.