## 4 Fingerprinting-Based Positioning with Spatial Side Information
### Pattern-Matching Methods
#### 4.4.4 Pattern-Matching Method 4-MultFingIn-ClassOut
However, the fingerprints from the fixed and moving tiles in the overlapping parts will bear resemblance to some extent, such that these can be recognized by the 2D CNN as a translation of part of a fixed tile with a certain tile-class probability. Figure 11: Example of tile-class labels assigned to the fixed tiles (numbers inside circles) and the border of a superimposed random moving tile. For a given input of fingerprints, here that corresponding to the moving tile, the 2D CNN returns as its output tile-class probabilities (associated with each fixed tile). Here, we exemplify an ideal outcome where the tile-class probabilities are proportional to the areas of the fixed tiles covered by the overlapping part of the moving tile (percentages shown at the bottom of each fixed tile). A 2D CNN takes as its input a tile in the form of a tensor of fingerprints \(\mathcal{F}\), interpreted as a 2D image with \(N_{F}\) channels, produced by the 2D sensor array. The input tensor has a size \(N_{SS}\times N_{SS}\times N_{F}\) given by the size of the 2D sensor array and the number of channels (equivalent to the number of fingerprint sources, \(N_{F}\)). In the training phase, the CNN is trained with fixed tiles as a classification model to map the fingerprints \(\mathcal{F}\) to a fixed tile-class. The CNN outputs a tile-class likelihood probability vector, \(\boldsymbol{p}_{\mathcal{F}}\), of size \(N_{T}\), indicating the likelihood of matching each tile-class for the \(N_{T}\) tiles available in the positioning area in a target scenario. The tile-class likelihood probability vector \(\boldsymbol{p}_{\mathcal{F}}\) is the output of the NN from a normalized dense layer with a softmax activation function. In the testing phase, the CNN takes as its input the tensor of fingerprints \(\mathcal{F}\) of the size \(N_{SS}\times N_{SS}\times N_{F}\) from a moving tile and produces as its output the tile-class likelihood probability vector \(\boldsymbol{p}_{\mathcal{F}}\). A known mapping between the tile-classes and the center positions of the respective fixed tiles is used to calculate the estimated position coordinates, \(\hat{\boldsymbol{x}}_{\mathcal{F}}\) and \(\hat{y}_{\mathcal{F}}\), of the moving tile (2D sensor array). The final position coordinate estimate, \(\hat{\boldsymbol{x}}_{\mathcal{F}}\) and \(\hat{y}_{\mathcal{F}}\), for a given input tensor with fingerprints \(\mathcal{F}\) is calculated by aggregating the tile-class probabilities assigned as weights to the position coordinates of the corresponding fixed tiles:  
\[\hat{\boldsymbol{x}}_{\mathcal{F}}=\sum_{j\in\Theta}\boldsymbol{p}_{\mathcal{ F}_{j}}x_{T}^{(j)}, \tag{4}\]  
\[\hat{y}_{\mathcal{F}}=\sum_{j\in\Theta}\boldsymbol{p}_{\mathcal{F}_{j}}y_{T}^{( j)}, \tag{5}\]  
where \(\Theta\) is the set of fixed tiles, with cardinality \(N_{T}\) and elements \(j\in\Theta\), \(x_{T}^{(j)}\) and \(y_{T}^{(j)}\) are the coordinates of the center of the tile \(j\), and \(\boldsymbol{p}_{\mathcal{F}_{j}}\) is the \(j\)th element of the tile-class likelihood probability vector \(\boldsymbol{p}_{\mathcal{F}}\), associated with the tile \(j\), where  
\[\sum_{j\in\Theta}\boldsymbol{p}_{\mathcal{F}_{j}}=1. \tag{6}\]  
It is assumed that there is a suitable data structure, e.g., a look-up table, with the mapping between the tile-class \(j\) and its center coordinates \(x_{T}^{(j)}\) and \(y_{T}^{(j)}\). Other ways to exploit the tile-class probabilities output by the softmax layer could be devised. However, our goal is to investigate the possible gains with the proposed approach, so other ways to exploit the tile-class probabilities are left for future work. * Translation Invariance Property It is unclear to us whether the so-called translation invariance property of CNNs is present and contributing to boosting the recognition of parts of a moving tile, which are translated parts of the overlapped fixed tiles. This translation invariance property is also discussed in terms of shift invariance [209].