Purple fringing, a persistent artifact caused by Longitudinal Chromatic Aberration (LCA) in camera lenses, has long degraded the clarity and realism of digital imaging. Traditional solutions rely on complex and expensive apochromatic (APO) lens hardware and the extraction of handcrafted features, ignoring the data-driven approach. To fill this gap, we introduce DCA-LUT, the first deep learning framework for purple fringing removal. Inspired by the physical root of the problem-the spatial misalignment of RGB color channels due to lens dispersion, we introduce a novel Chromatic-Aware Coordinate Transformation (CA-CT) module, learning an image-adaptive color space to decouple and isolate fringing into a dedicated dimension. This targeted separation allows the network to learn a precise "purple fringe channel," which then guides the accurate restoration of the luminance channel. The final color correction is performed by a learned 5D Look-Up Table (5D LUT), enabling efficient and powerful non-linear color mapping. To enable robust training and fair evaluation, we constructed a large-scale synthetic purple fringing dataset (PF-Synth). Extensive experiments in synthetic and real-world datasets demonstrate that our method achieves state-of-the-art performance in purple fringing removal.

Purple flare, a diffuse chromatic aberration artifact commonly found around highlight areas, severely degrades the tone transition and color of the image. Existing traditional methods are based on hand-crafted features, which lack flexibility and rely entirely on fixed priors, while the scarcity of paired training data critically hampers deep learning. To address this issue, we propose a novel network built upon decoupled HSV Look-Up Tables (LUTs). The method aims to simplify color correction by adjusting the Hue (H), Saturation (S), and Value (V) components independently. This approach resolves the inherent color coupling problems in traditional methods. Our model adopts a two-stage architecture: First, a Chroma-Aware Spectral Tokenizer (CAST) converts the input image from RGB space to HSV space and independently encodes the Hue (H) and Value (V) channels into a set of semantic tokens describing the Purple flare status; second, the HSV-LUT module takes these tokens as input and dynamically generates independent correction curves (1D-LUTs) for the three channels H, S, and V. To effectively train and validate our model, we built the first large-scale purple flare dataset with diverse scenes. We also proposed new metrics and a loss function specifically designed for this task. Extensive experiments demonstrate that our model not only significantly outperforms existing methods in visual effects but also achieves state-of-the-art performance on all quantitative metrics.

Phase retrieval in quaternion domains presents unique computational challenges due to the non-commutative nature of quaternion algebra. Although several methods have been developed to address this issue, there remains significant potential for improvement. In this paper, we develop a novel quaternion-based alternating direction method of multipliers framework for quaternion phase retrieval from magnitude-only measurements based on the generalized Hamilton-real calculus, and provide a rigorous convergence analysis. Through extensive numerical simulations, we demonstrate that the proposed quaternion-based alternating direction method of multipliers framework outperforms existing methods, such as quaternion Wirtinger flow, quaternion truncated Wirtinger flow, and quaternion truncated amplitude flow, thereby establishing a new benchmark for quaternion phase retrieval problems. The theoretical framework and empirical results demonstrate that quaternion-based alternating direction method of multipliers offers a novel and effective approach for solving quaternion phase retrieval problems.

Price's theorem in statistical signal processing relates the expectation of a nonlinear function of normally distributed random variables to their covariances. However, such a key theorem is yet to be established and explored in quaternion statistics. To this end, we introduce Price's theorem for quaternion variables using the generalized Hamilton-real (GHR) calculus. This is achieved by first employing the chain rule of GHR calculus to derive two crucial quaternion matrix derivatives of functions with respect to the product of quaternion matrix variables. Next, we leverage quaternion second-order statistics to establish the relationship between the derivative of a function with respect to the augmented quaternion covariance matrix and its real counterpart. Based on the above results and Price's theorem in real signal processing, we finally propose a novel formulation of Price's theorem for quaternion random variables. This finding not only enriches the theory of quaternion statistical signals processing but also extends its applicability.