*2.1. Dimensionality Reduction Using Autoencoders*

To develop data driven models for localization and estimation of loads from depth measurements while providing realtime performance, an autoencoder [18] framework is proposed to be utilized. This is because the full field measurements that are acquired from the depth sensors are inherently rich, but can be very large in size, thus working with them becomes computationally expensive and can hinder realtime performance. Autoencoders can effectively reduce the large number of features obtained from depth sensors while retaining the critical information, thus encoding the original input at a much smaller dimension. Furthermore, to ensure that maximum informative data is obtained, Kullback-Leibler divergence (*KLDiv*) [19] was used to avoid obtaining binary encoded data by enforcing the mean and standard deviation of the encoded data to be some desired values. In this work, logarithmic normalization was utilized to minimize this large range of data due to the possible presence of a large gap between the values of the input depth measurements. The overall algorithm for the utilized autoencoder is given, as follows:

$$Y = \Gamma(<\log(X), W\_1 > +B\_1) \tag{1}$$

$$Z = \Gamma(<\text{'}, W\_2> + B\_2) \tag{2}$$

$$KL\_{Div} = a\_d \log \frac{\alpha\_d}{\alpha} + (1 - \alpha\_d) \log \frac{1 - \alpha\_d}{1 - \alpha} \tag{3}$$

$$CF\_{AE} = \frac{1}{N} \sum\_{i=1}^{N} \left(X\_i - Z\_i\right)^2 + \beta KL\_{Div} \tag{4}$$

where *X* is the input depth vector, *Y* is the output of the encoder, *Z* is the output of the decoder, Γ is the activation function, *αd*, and *α* are the desired and actual mean and/or standard deviation of the encoded data, respectively, *W*<sup>1</sup> and *W*<sup>2</sup> are the weight matrices, *B*<sup>1</sup> and *B*<sup>2</sup> are the bias vectors, *CFAE* is the autoencoder cost function to be minimized, and < ·, · > is the dot product.

After the critical information is extracted from the input depth features and is encoded at a smaller dimension using the proposed autoencoder, two different supervised ANNs for realtime localization and estimation of loads can then be utilized, as illustrated in Figure 1.

**Figure 1.** Proposed realtime load localization and estimation framework for SHM.
