Recently, the notion of color has played a relevant role in a large number of computer vision applications. Color information provides features that are invariant to scale, translation, and rotation changes, which are suitable for image segmentation [1], image classification ^[2][3][2,3], or image retrieval ^[4][5][4,5]. One of the most critical tasks in computer vision applications is image denoising, which involves recovering an image from a degraded noisy version. Various types of noise can affect digital color images, particularly impulse noise [6].

Impulse noise in digital images is a random variation in the intensity of pixels caused by short-duration pulses of high energy. This type of noise can significantly degrade the quality of images and poses various challenges in real-world applications. For example, the impulse noise in dashboard camera footage under low-light conditions [7] can lead to the misinterpretation of videos, making it challenging to accurately identify vehicles involved in incidents. Additionally, impulse noise is a kind of ordinary noise in medical imaging (X-rays, MRIs, and CT scans) [8] that can result in the misinterpretation or disappearance of critical details important in diagnosis.

Impulse noise commonly occurs during the acquisition or transmission of an image caused by imperfections on the device lens, malfunctioning camera photosensors, the aging of the storage material, errors during the compression process, and the electronic instability of the image signal [9]. Addressing impulse noise in real-world applications often involves the use of various image processing techniques to restore the integrity of images. Impulse noise affects color digital images in such a way that the perturbed pixels differ significantly from their local neighborhood in the image domain.

Nature-inspired optimization algorithms have been widely applied in the image processing literature to address various challenges, including optimizing image quality evaluation [10], feature selection [11], and image reconstruction [12]. Of particular note is genetic programming [13], an evolutionary computing technique based on the principle of natural selection, which offers a flexible and adaptive methodology for addressing image processing problems. Like other evolutionary algorithms, the main idea is to transform a population of individuals (programs) by applying natural genetic operations such as reproduction, mutation, and selection [14]. The adaptability and robustness of genetic programming make it well suited for addressing the challenges associated with impulse noise removal, particularly for color digital images.

A large number of filters for impulse noise removal from color images have been proposed in the literature, e.g., [15]. Two types of filters, robust and adaptive, stand out in scenarios where images are degraded by complex noise patterns or variations in lighting and contrast. Robust filters deal with noisy pixels as a violation of the spatial coherence of the image intensities. The most well-known robust filter is the vector median filter (VMF) [16], a non-linear method that is still a reference in the field. The VMF orders the color input vectors based on their relative magnitude differences in a predefined sliding window. However, robust filters apply a correction procedure to every image pixel, even if the pixels are not noisy. Therefore, filtered images commonly present too much smoothing and extensive blurring [9]. Other methods exploit the sparsity of the image in some transform domains, formulating noise removal as an optimization problem. For example, the generalized synthesis and analysis prior algorithm (GSAPA) proposed in [17] uses the split-Bregman technique to break down the optimization problem with multiple regularization parameters into relatively easy-to-solve subproblems. In [18], an extension of the mean-shift technique was introduced to effectively reduce Gaussian and impulsive noise in color digital images; utilizing a novel similarity measure between pixels and a patch at the block’s center, the technique demonstrates efficiency in restoring heavily disturbed images.

On the other hand, adaptive filters assume that only some image pixels are perturbed by noise. These filters adapt to various image characteristics and noise statistics using different approaches. One of these approaches is the fuzzy peer-group concept, which associates a set of neighboring pixels whose distance to a central pixel does not exceed a threshold founded on fuzzy logic (see, for example, [19]). Under this approach, the fuzzy modified peer-group filter (FMPGF) [20] stands out due to its efficiency. Other approaches use fuzzy logic tools to measure magnitude differences, providing an adaptive framework to distinguish between noise and data. For example, the adaptive fuzzy filter proposed in [21] uses a noise-detection mechanism to select a small portion of input image pixels and convolves them with a set of weighted kernels to create a layer of convolved feature vectors. The feature vectors are then fed into a fuzzy inference system, where fuzzy membership degrees and a reduced set of fuzzy rules play a crucial role in categorizing pixels as either noise-free, associated with edges, or noisy. Another fuzzy inference rule-based filter for impulse noise is the noise adaptive fuzzy switching median (NAFSM) filter [22]. The NAFSM filter uses two-stage processing specialized for salt-and-pepper noise detection and removal, preserving strong edges through spatial averaging. A recent contribution in a similar vein is the adaptive fuzzy filter based on histogram estimation (AFHE) [23]. The AFHE filter dynamically adjusts the window size based on local noise densities using fuzzy-based criteria. Additionally, it incorporates an iterative procedure based on the Gaussian mean for further processing. Another recent development presented in [24] introduces a spatio-temporal filter designed to eliminate impulse noise from videos through the application of fuzzy logic. The distinctive feature of this filter is its ability to leverage both spatial and temporal correlations and color components in a sequential process, allowing it to effectively handle high impulse noise levels, even up to 70%. Other adaptive approaches are based on more straightforward decisions, such as the decision-based algorithm for the removal of impulse noise (DBAIN) [25]. The DBAIN filter processes a noisy image by first detecting salt-and-pepper noise (as its major decision) and then replacing the noisy pixel values with the median of their neighboring pixels. Some comprehensive studies of adaptive filters for impulse noise removal from color images are presented in ^[15][26][15,26].

Machine learning-based techniques have also been applied to remove impulse noise from color images. For example, a support vector machine (SVM)-based filtering technique was proposed in [27]. Later, fuzzy c-means (FCM) clustering combined with a fuzzy-support vector machine (FSVM) was introduced in [28]. Additionally, multiple neuro-fuzzy filters were trained with an artificial bee colony algorithm and combined with a decision tree algorithm in [29] to denoise corrupted images. Also, neural network-based denoising methods are another popular approach. For instance, a convolutional neural network-based denoising method mainly composed of a sparse block, a feature enhancement block, an attention block, and a reconstruction block was described in [30]. Also, a convolutional autoencoder-based feature map domain was combined with low-rank models to improve denoising quality in [31]. Recently, a novel denoising network called DeQCANet was introduced in [32] for removing color random-valued impulse noise. This work implements a quaternion convolutional neural network, incorporating a novel quaternion map construction strategy to enhance color features across channels. The proposed denoising technique exhibits competitive performance compared to other well-established methods for denoising color images. Another recent development is the impulse detection convolutional neural network (IDCNN) introduced in [33], which utilizes a switching filtering technique consisting of a deep neural network architecture to detect noisy pixels, followed by a restoration stage of these pixels through the fast adaptive mean filter. In [34], a neural network architecture was introduced to recognize images affected by random-valued impulse noise, comprising a preprocessing stage and incorporating a pixel distortion detector, a cleaning mechanism, and a neural network complex for image recognition. An overview of deep learning techniques on image denoising can be found in [35]. While convolutional neural networks and deep learning-based approaches excel in performance, they are complex and opaque. Their intricate architectures and complex hierarchical representations in these techniques make it challenging to decipher the exact decision-making process, limiting the transparency and understanding of how the denoising outcomes are achieved. Achieving transparency is crucial for various reasons, especially in applications where interpretability is essential for user trust, accountability, and ethical considerations.

Alternative filtering techniques tackle the challenge of image denoising through an evolutionary computing approach. For instance, in [36], Toledo et al. introduced a denoising method employing a genetic algorithm. In their method, an input noisy image is represented as an individual, and the individuals of the initial population are represented as mutated versions of the image. The evolutionary process applied to the population allows for the continuous refinement and adaptation of the original noisy individual until an enhanced image is found. In a similar vein, in [37], researchers addressed image denoising by merging output images of robust and adaptive filters to integrate them as an evolving population for a genetic algorithm. Genetic algorithms, in this context, are frequently employed to fine-tune specific filter parameters, essentially framing them as an optimization task. On the other hand, genetic programming has emerged as a highly promising field for the creation of adaptive filters aimed at noise reduction; however, no filters use genetic programming to remove impulse noise from color digital images. Nevertheless, there are a few works that have addressed impulse noise removal for grayscale images, the most relevant being ^[38][39][40][38,39,40].

Petrovic and Crnojevic [38] used genetic programming to produce two binary classification models for identifying noisy pixels in grayscale images following a cascade approach. While the first model identifies the majority of noisy pixels located within homogeneous regions of the image, the second is trained to specialize in identifying complex cases of noisy pixels (e.g., those with amplitudes close to their local neighborhood). The classification models were trained using test images contaminated with a mixture of two impulse noise models: salt and pepper and uniform. Once the noisy pixels are detected, they are corrected using a conventional α-trimmed mean method as part of a two-stage filter. On the other hand, Majid et al. [39] used genetic programming to estimate the optimal value of every noisy pixel in a grayscale image. The estimation combines the useful information of local clean pixels within a small neighborhood with arithmetic operators. In their approach, Majid et al. used the directional derivative to detect pixels perturbed by salt-and-pepper and uniform noise. Their filter was tested using standard images perturbed with a noise density from 10 to 90%. Their experimental results showed that, compared to other approaches, their filter can restore noisy images while preserving edges and fine details, especially in the presence of high impulse noise density. Khmag et al. [40] presented a filter based on a two-step switching scheme to remove both salt-and-pepper and additive white Gaussian noises. Their filter uses a patch-based approach, which decomposes a noisy image into a group of patches and then applies a clustering process to gather all patches with the same features and textures. They use genetic programming to generate an adaptive local filter to denoise each singular class of patches with different textural forms. The restoration process of these filters is performed by applying second-generation wavelet thresholding to the clustered patches. Other applications of genetic programming in image processing can be found in [41]. It is important to emphasize that although many of the techniques developed for monochrome images can be directly applied to color image denoising, the independent processing of color image channels is commonly inappropriate, leading to the generation of strong artifacts. To address this challenge, developing methodologies that effectively exploit the inherent correlation among color channels becomes imperative, ensuring more accurate and artifact-free denoising results in the realm of color image processing.