A SIFT algorithm in spherical coordinates for omnidirectional images is proposed. This algorithm can generate two types of local descriptors, Local Spherical Descriptors and Local Planar Descriptors. With the first ones, point matching between two omnidirectional images can be performed, and with the second ones, the same matching process can be done but between omnidirectional and planar images. Furthermore, a planar to spherical mapping is introduced and an algorithm for its estimation is given. This mapping allows to extract objects from an omnidirectional image given their SIFT descriptors in a planar image. Several experiments, confirming the promising and accurate performance of the system, are conducted.