# Development of Height Indicators using Omnidirectional Images and Global Appearance Descriptors
## 2 Omnidirectional Imaging and Global Appearance Descriptors
### Global Appearance Descriptors
#### 2.2.1 Fourier Signature  
The Fourier Signature (FS) was firstly described by [PERSON] et al. [37], who used it to carry out mapping and localization with a robot whose movement is restricted to the ground plane. It consists in the representation of a panoramic image calculating the one-dimensional DFT of each row. Therefore,  
Figure 1: Projection model of the catadioptric vision system. The image projected onto the image plane (plane of projection) is the omnidirectional view.  
each row of the original image \(\{r_{m}\}=\{r_{m,0}\), \(r_{m,1}\),..., \(r_{m,N-1}\}\), \(m=0,\ldots,M-1\) can be transformed into the sequence of complex numbers \(\{R_{m}\}=\{R_{m,0}\), \(R_{m,1}\),..., \(R_{m,N-1}\}\).  
When the DFT of every row of the image \(im(x_{i},y_{i})\in\mathbb{R}^{M\times N}\) is calculated, a new matrix \(IM(u,y_{i})\in\mathbb{C}^{M\times N}\) is obtained, being \(u\) the frequency variable (cycles/pixel). The components of this matrix are complex numbers thus it can be decomposed into a magnitudes matrix \(A(u,y_{i})\in\mathbb{R}^{M\times N}\) and an arguments matrix \(\Theta(u,y_{i})\in\mathbb{R}^{M\times N}\). Taking the properties of the DFT into account [37], only a subset of \(k\) columns can be retained to represent the image: \(A(u,y_{i})\in\mathbb{R}^{M\times k}\) and \(\Theta(u,y_{i})\in\mathbb{R}^{M\times k}\), \(k\leq N\).  
The FS presents another interesting property when used to describe a panoramic image. If the image comes from an omnidirectional vision sensor mounted vertically on the robot, then, the modules matrix \(A(u,y_{i})\) is invariant against rotations around the vertical axis. Let's consider two panoramic images captured from the same position on the ground plane but having the robot different orientations with respect to the vertical axis, with relative orientation \(\phi\), as shown in Figure 3. If the row \(m\) of the first image is represented as the sequence \(\{r_{m}\}=\{r_{m,n}\}\), \(n=0,\ldots,N-1\) then the same row in the second (rotated) image is \(\{r_{m,n-q}\}\), where \(q\) is the shift between images, measured in pixels, which is proportional to the relative rotation between images \(q=N\cdot\phi/360\), where \(\phi\) is measured in deg. Visually this shift appears as a circular shift of the columns of the image (Figure 3).  
The rotational invariance can be expressed by the DFT shift theorem as:  
\[\mathcal{F}[\{r_{m,n-q}\}]=R_{m,l}e^{-j\frac{2\pi l}{N}},\quad l=0,\ldots N-1, \tag{1}\]  
where \(\mathcal{F}[\{r_{m,n-q}\}]\) is the one-dimensional DFT of the shifted sequence, and \(R_{m,l}\) are the components of the one-dimensional DFT of the non-shifted sequence (row \(\{r_{m}\}\)).  
Taking this theorem into account, when the movement of the robot is contained in the ground plane, the magnitudes matrix can be used to estimate the position (since it is invariant to rotation), and the arguments matrix to estimate the relative orientation.  
Figure 2: (**a**) Sample omnidirectional image and (**b**) orthographic; (**c**) unit sphere and (**d**) cylindrical projections.