## 1 Introduction
Frequency domain watermarking schemes are generally preferred over the spatial domain watermarking approaches due to their inherent resistance against image processing attacks [8]. Also, as an embedding capacity extension is achieved, the frequency domain can provide more space while still achieving full imperceptibility [4]. However, the use of frequency domain techniques would incorporate some noise in the watermarked image which may be perceivable [9]. This has been countered by some researchers who came up with the idea of using optimization methods to select the best coefficients to embed watermarks, like particle swarm optimization (PSO) [10; 11] and genetic algorithms (GAs) [12; 13]. An alternative solution to increase the invisibility of watermarks is based on multiresolution decomposition techniques, such as dual-tree complex wavelet transform (DTCWT) [14; 15]. In recent years, GAs have been successfully applied, in particular for optimization problems in watermarking. GA-based wavelet-based image watermarking schemes are given in [9]. The embedding energy of a DWT-based watermarking scheme has also been optimized by means of an artificial neural network. A fuzzy clustering technique was used in [16] to determine the optimal embedding locations for a DCT-based watermarking scheme. These AI techniques enhance the performance of watermarking schemes and their parameters (capacity, strength, or location). In 2017, [PERSON] et al. [17] proposed an image watermarking scheme in the contourlet transform domain, in connection with a probabilistic neural network. They distilled the watermark extraction to a great degree of robustness. [PERSON] and [PERSON] [18] proposed an approach using an autoencoder to train two distinct neural networks (embedder and extractor) with a digital image using DCT coefficients. In [19], [PERSON] et al. proposed a generalized approach of using deep neural networks in digital watermarking. They presented a method that obtained great robustness under common image processing attacks, such as filtering, compression, and geometric transformations. Since the selection of transformation coefficients for embedding affects any watermarking method, researchers have exploited evolutionary algorithms to optimize the selection criteria. Researchers have used genetic algorithms widely in the watermarking of images in the last few years, and it has become an important research branch for many researchers [12; 20; 21; 22]. Moreover, PSO has been proven to give promising accuracies in terms of robustness and impeccability. As such, many studies have exploited this fact in the literature, with particular highlights being outstanding approaches such as the work carried out in [11; 23; 24]. Other interesting approaches can be found in the literature, incorporating nature-inspired optimization algorithms such as artificial bee colony optimization [25; 26], the bat algorithm [27], and gray wolf optimization [28]. Despite the advances in digital watermarking techniques, significant challenges remain in balancing imperceptibility and robustness. Conventional methods like DWT, DCT, or DFT usually leave visible artifacts on images or fail in preserving the integrity of the watermark under common malicious attacks (e.g., cropping, compression, and scaling). Although optimization algorithms have been applied in watermarking, their integration with newer wavelet-based transforms, like DTCWT, has not been fully explored. This study will address these gaps by exploring the following research questions:  
* How can the use of DTCWT and PSO be incorporated to improve the imperceptibility of watermarked images while retaining high robustness? * How well does the proposed method compare with traditional techniques like DWT and DCT in terms of PSNR (peak signal-to-noise ratio) and NCC (normalized cross-correlation) under standard image-processing manipulations such as compression, cropping, and scaling?