1. An apparatus for recognizing an object by using an object image with a model, the apparatus comprising: an image input unit configured to input the object image including the object to be recognized; a feature point extraction unit configured to extract plural feature points of the object from the object image; a triangulation unit configured to divide the object image into regions of plural triangles having vertices which are the plural feature points of the object or are vertices of a convex hull containing all the plural feature points of the object; a feature point selection unit configured to select plural combinations of the feature points of the object, the each combination including three feature points, of the object, each corresponding to the each vertex of the each triangle; a basis calculation unit configured to obtain, from each of the plural selected combinations of the three feature points of the object, two reference vectors for these feature points, and calculate each basis spanned by these two reference vectors, the each basis including plural basis vectors, the each basis vector including components that provide a coordinate system representing the positions of feature points of the object and positional relation among the feature points of the object; a table memory unit configured to store a hash table data which associates a partial pattern of a model with an index parameter, the index parameter indicating a registration position of a partial pattern of the model in the hash table data, the index parameter including a parameter invariant to a geometric transformation; an index calculating unit configured to calculate the index parameter of the partial pattern of the object based on the each basis; an index retrieval unit configured to obtain the partial pattern of the model from the hash table data based on the index parameter of the partial pattern of the object; and a pattern similarity calculation unit to calculate a similarity between the obtained partial pattern of the model and the partial pattern of the object.