## 3 Background  
The cross-correlation formula and its variations have been widely used for accomplishing many signal processing tasks, for example, pattern recognition and digital watermarking, to name just a few. Although there are various forms of this equation, one of the most useful for computing the cross correlation (\(\bar{x_{k}}\)) between two signals \(x=[x_{1},x_{2},\ldots,x_{N}]\) and \(y=[y_{1},y_{2},\ldots,y_{N}]\) is as follows:  
\[\bar{x_{k}}=\frac{1}{N}\sum_{i=1}^{N}x_{i}y_{i}. \tag{1}\]  
Despite the advances in the signal processing field, the cross-correlation formula continues to be used by many authors as a means to measure the similarity between two signals due to its remarkable performance; however, like many models, it has its weaknesses.  
On the other hand, to overcome some of the weakness of that model, we deduced in our previous works [5,6] a fractional model of a correlation-like equation for watermarking derived using principles of FC. Recognizing the complexity of fractional calculus and to facilitate its implementation, we have provided comprehensive details on simplification and approximation techniques in Appendix A. This appendix offers a solid foundation, ensuring that readers can effectively navigate and apply the mathematical concepts involved.  
In this paper, Equation (2) will be referred to as the \"Fractional Cross-correlation\"; this model outperforms Equation (1) in matching signals under adverse conditions; thus, we are interested in extending the use of this model to other related areas. The fractional cross-correlation is defined as follows:  
\[\bar{x_{k}}=-\operatorname{Im}\left[\frac{3}{4}\frac{1}{N}\sqrt{\left(\sum_{ i=1}^{N}x_{i}y_{i}\right)^{2}-\frac{2}{3}\sigma^{2}N[2 NH-1]}-\epsilon\right], \tag{2}\]  
for signals \(x\) and \(y\), where:  
\[\epsilon=\frac{3}{4}\frac{1}{N}\sqrt{\frac{2}{3}\sigma^{2}N[2 NH-1]},\]  
and \(H=\ln(\sqrt{2\pi\sigma^{2}\epsilon})\) for a signal with variance \(\sigma^{2}\).  
As we stated earlier in this section, this FC model performs remarkably well in matching signals under severe conditions, and now we present an example to clarify this. Let us examine Figure 1a-c. In this figure, we illustrate the case of a generic watermark detector based on Equation (1) for detecting watermarks. Such a device must output an almost zero value when no watermark is embedded in a presumably watermarked image and produce a much higher value when it detects a watermark in the tested image, as shown in Figure 1b. Usually, a watermarking system relies on a threshold for deciding whether a watermark is present or not. The main issue here is that, depending on several factors, the response of the system can be higher than expected for the case of no watermarks actually embedded in the image, leading to false positives.  
Figure 1b exemplifies how Equation (1) is unable to resolve ownership on the watermarked image shown in Figure 1 since there are many false positives. Conversely, Figure 1c shows how the response of a system based on Equation (2) is able to overcome this particular scenario since there are no false positives. We also observed that the rate of the matching and the non matching responses is higher in the case of the fractional cross-correlation. This evidence motivated us to use this fractional model for other applications in the signal processing area; in the case of this paper, we focused on developing an image processing filter.