**2. Related work**

CNNs are becoming more capable for image analysis application, due to the increasing number of layers and complex connections. However, adding more layers to increases performance produce more intricate models and higher computational cost. Eventually, deeper networks may start losing accuracy. On the other hand, MRA keeps all the information contained and distributed into its coefficients through the different resolution levels. Each one contains specific features; higher resolutions show more details and lower resolutions shows only the strongest features. In recent years, more studies have applied MRA to the CNN. Some of these studies are presented in **Table 1**, where the first two columns indicate the publication year and the application.

All the studies shown are applications embedded within the CNN to replace the pooling layers. The last three columns describe the kind of wavelet solution implemented. For example, the third column shows that Discrete Wavelet Transform (DWT), Lifting Scheme Wavelet Transform (LSWT) and Wavelet Package Transform (WPT) are the only wavelet transforms utilized in the analyzed studies. At the same time, the last three columns complete each model information by adding the wavelet family selected, the number of wavelet coefficients applied and the levels of decomposition. The wavelets used are families like Daubechies or Coiflet. The 2D wavelet transforms produce 4 coefficients and all analyzed models require the approximation coefficients. However, not all studies utilize the details coefficients.


**Table 1.** *Wavelets Transforms used to improve CNN pooling layers.*

In Ma et al. [24], the detail coefficients are used partially. In the last column indicates the number of levels of decomposition applied in each model. These studies introduce new frameworks with interesting results, like Bastidas-Rodriguez et al. [25], this model is embedded in a larger CNN to detect scenes and classify textures or compress images [24]. Other approaches also detect scenes and classify handwriting digits [26]. But, replacing pooling layers is the most common uses of wavelets within CNN like in Refs. [9–12, 23], as well as activation functions as reported in Refs. [27–31]. It is used for most of the previous applications, as well as for image restoration, lung tumors detection, among many more.

The aim of the study, initially inspired by Williams and Li [9], is to use the Wavelet Transform Coefficients randomly in a formal model as a pooling layer, to emulate the Max-pooling effect but independently from the signal's shape. Achieving a measurable increment of its accuracy using a benchmark dataset.
