# Gait Energy Response Functions for Gait Recognition against Various Clothing and Carrying Status
## 3 Gait Recognition Using GERF
### Representation of SD-GERF  
Because the Global GERF trains a single common response function for the whole GEI, it cannot consider differences in the importance of static and dynamic components among the body parts (e.g., the static components are more important for the head, while the dynamic components are more important for the leg). We therefore introduce spatial dependency into the GERF framework. The most straightforward way to do this is to define different GERFs for individual spatial positions, i.e., for every pixel, and to optimize an SD-GERF that is a concatenated vector of the multiple GERFs. This strategy, however, significantly increases the number of variables (i.e., the dimension of the SD-GREF vector in proportion to the image size) and hence suffers from generalization errors as well as an increase in space and time complexity. We therefore consider reducing the dimension of the SD-GERF vector. First, as the body parts are divided by vertical positions in most part-based approaches to gait recognition [28; 29], we also consider only vertical spatial dependency while neglecting horizontal spatial dependency. Second, because GERFs of adjacent vertical positions should be similar, in other words, abrupt changes of the GERFs among adjacent vertical positions are unlikely to occur, we set the SD-GERFs of a smaller number of vertical positions, using an interval coarser than a single pixel. We then represent in-between GERFs by linear interpolation. Third, while the Global GERF is defined by \((I_{\text{max}}+1)\) components, i.e., at every intensity level, we represent a GERF with a smaller number of intensity levels at a coarser intensity interval, and then represent in-between response values by linear interpolation as well. In summary, we introduce a set of control points distributed over vertical positions and intensity levels at certain intervals as shown in Figure 3 where response values are defined and estimate intermediate response values by bilinear interpolation from the adjacent control points. Note that not only pixels on the control points but also those in-between control points are used for training and testing. Next, we introduce a mathematical definition of the SD-GERF based on the previously described concept. A control point for the \(i\)-th intensity \(I_{i}\) and the \(j\)-th row (vertical position) \(y_{j}\) is defined as  
\[\mathbf{r}_{i,j}=\left[I_{i},y_{j}\right]^{T}, \tag{6}\]  
where \(I_{i}=iQ_{I},i\in\{0,1,...,N_{I}\}\) and \(y_{j}=jQ_{y},j\in\{0,1,...,N_{y}\}\), and \(Q_{I}\) and \(Q_{y}\) are intervals for the intensity and the row directions, respectively, and \((N_{I}+1)\) and \((N_{y}+1)\) are the number of control points for the intensity and row directions, respectively (see Figure 3). We then denote a response value \(f_{i,j}\) be the transformed gait energy at the corresponding control point \(\mathbf{r}_{i,j}\) of the \(i\)-th intensity value \(I_{i}\) and the \(j\)-th row \(y_{j}\), where both the original value \(I_{i}\) and response value \(f_{i,j}\) mean intensities (gait energies). In other words, given an intensity value \(I_{i}\) on the \(j\)-th row, it is transformed into the response value \(f_{i,j}\). Similarly to the Global GERF, we represent the SD-GERF by concatenating the response values from all the control points to form a column vector \(\mathbf{f}=\left[f_{0,0},...,f_{N_{I},0},\ldots,f_{0,N_{y}},...,f_{N_{I},N_{y}} \right]^{T}\in\mathbb{R}^{N_{d}}\), \(N_{d}=(N_{I}+1)(N_{y}+1)\). We subsequently introduce the bilinear interpolation of response values for intermediate intensities and rows from the adjacent control points. Given a pixel with intensity \(I\) at the position \((x,y)\), we compute an intensity index \(i=\lfloor I/Q_{I}\rfloor\) and row index \(j=\lfloor y/Q_{y}\rfloor\) of an adjacent control point, where \(\lfloor\cdot\rfloor\) is a floor function.