*2.1. Fourier Signature Descriptor*

The Fourier signature descriptor was firstly used by Menegatti et al. [19] to create an image-based memory for robot navigation. Payá et al. [21] studied the computational cost and the error in localization by using Fourier Signature (FS) and proposed a Monte Carlo approach to solve the localization problem in indoor environments.

This description method is based on the use of the Discrete Fourier Transform (DFT). After calculating the FS of a panoramic image, a new complex matrix is obtained *IM*(*u*, *v*). It can be decomposed into two real matrices, one containing the magnitudes and the other the arguments. The steps to obtain a global appearance descriptor from a panoramic image through the Fourier Signature (FS) are: First, departing from the intensity matrix of the original image, the DFT of each row is calculated. The result is a complex matrix with the same size as the original image (*IM*(*u*, *v*) ∈ C*Nx*×*Ny* ). Second, only the *k*<sup>1</sup> first columns of this matrix are retained since the main information is in the low frequency components. Third, the resultant matrix (*IM*(*u*, *<sup>v</sup>*) <sup>∈</sup> <sup>C</sup>*Nx*×*k*<sup>1</sup> ) is decomposed into the magnitudes and arguments matrices. The matrix of magnitudes (*A*(*u*, *<sup>y</sup>*) <sup>∈</sup> <sup>R</sup>*Nx*×*k*<sup>1</sup> ) is invariant against changes of the robot orientation in the movement plane if the image is panoramic. In the last step, the global appearance descriptor is obtained by arranging the *k*<sup>1</sup> columns of the magnitudes matrix in one single column (#»*<sup>d</sup>* <sup>∈</sup> <sup>R</sup>*Nx*·*k*1<sup>×</sup>1).
