Chromatic-Aware Coordinate Transformation

• CA-CT is a suite of mathematically grounded color space mappings that embeds chromatic structure and perceptual non-uniformity into coordinate representations.
• It employs complex arithmetic to perform circular filtering and interpolation, naturally addressing hue periodicity and minimizing interpolation artifacts.
• Data-driven and geometric CA-CT variants enhance artifact suppression and observer compensation, leading to improved color fidelity in image processing.

Chromatic-Aware Coordinate Transformation (CA-CT) designates a diverse suite of mathematically grounded color space mappings that embed chromatic structure and perceptual non-uniformity directly into coordinate representations. CA-CTs are engineered to enable perceptually meaningful filtering, interpolation, adaptation, artifact suppression, or observer compensation by structuring coordinates so that metrics, arithmetic, or learned operations directly respect chromatic periodicity, hue sensitivity, color adaptation, and discrimination geometry.

1. Canonical CA-CT Construction in Complex Color Spaces

A principal CA-CT instance is established by mapping the perceptual CIELAB space (L, a, b) or its polar CIEHLC variant (L, C, h) onto a hybrid real-plus-complex coordinate system $(L, z)$ with $z = C e^{i h} \in \mathbb{C}$, isolating lightness $L$ as a real axis and chromaticity as a single complex coordinate. The forward mapping is:

• Given a CIELAB triplet, compute $C = \sqrt{a^2 + b^2}$ and $h = \mathrm{atan2}(b, a)$,
• Encode chromaticity $z = C e^{i h}$,
• Thus, each color becomes $(L, z) \in \mathbb{R} \times \mathbb{C}$.

Inverse mapping recovers polar chromaticity as $C = |z|$, $h = \arg(z)$ and projects back to $(a, b)$ via $a = C\cos h$, $b = C\sin h$.

This mapping is justified because the CIEHLC chromatic plane is circular in $h$: Euclidean operations in $\mathbb{C}$ naturally handle both hue periodicity and chromatic “distance,” and linear combinations in $\mathbb{C}$ remain within the convex hull of the chromatic data, obviating hue wraparound and interpolation artifacts. Filtering and barycentric interpolation become algebraically coherent and respect the shortest circular path between hues, i.e., minimal angular traversals (Akleman et al., 2023).

2. Circular Chromatic Filtering and Interpolation

The CA-CT formalism enables two fundamental operations that mitigate classical defects of Euclidean averaging in chromatic spaces:

• Circular Average Filtering (CAF): For input samples $(L_k, z_k)$ and weights $w_k$ with $\sum w_k=1$,
  • $z_\mathrm{filtered} = \sum_k w_k z_k$,
  • $L_\mathrm{filtered} = \sum_k w_k L_k$,
  • Output chromaticity as $C_\mathrm{filtered} = |z_\mathrm{filtered}|$, $h_\mathrm{filtered} = \arg(z_\mathrm{filtered})$.
• Circular Linear Interpolation (CLERP): Given $(L_1, z_1)$, $(L_2, z_2)$ and $t\in[0,1]$, define $\Delta h=\arg(z_2\,\overline{z_1})$, $h(t)=(1-t)h_1 + t(h_1+\Delta h)$, $C(t)=(1-t)C_1 + tC_2$, $L(t)=(1-t)L_1 + tL_2$, and interpolate as $z(t) = C(t) e^{i h(t)}$. This traces an Archimedean spiral in the chromatic plane—optimal in both chroma and hue—which ensures monotonic, vivid color transitions without low-saturation artifact zones or hue-wrapping (Akleman et al., 2023).

Visually, CA-CT-based filtering and interpolation eliminate hue-wrap and gray-passage artifacts typical to linear methods in CIELAB or RGB, as confirmed in empirical image examples showing improved perceptual uniformity.

3. Data-Driven Chromatic-Aware Transformations in Deep Learning

CA-CT methods extend beyond analytic mappings to data-driven transforms, notably in chromatic artifact suppression. Recent work introduces a learned CA-CT module for purple fringing removal (Lu et al., 15 Nov 2025). Here, the CA-CT is parameterized as an image-adaptive $3\times 3$ linear transformation $M$ mapping the raw RGB vector at each pixel to a Chromatic Aberration Space (CAS):

$I_\mathrm{CAS}(p) = M \cdot I_\mathrm{RGB}(p)$

Rows of $M$ define $C_\mathrm{lum}$ (luminance), $C_\mathrm{fringe}$ (purple fringing channel), and $C_\mathrm{ortho}$ (residual color). Orthogonality regularization on $M$ ensures axis interpretability and stability:

$$L_\mathrm{align} = \frac{1}{B} \sum_{b=1}^B \sum_{i\ne j}(r_{b,i} \cdot r_{b,j})^2$$

This adaptive CA-CT enables precise isolation of fringing artifacts for targeted suppression, outperforming static color transformations and heuristic methods. Quantitative metrics show that the learned $C_\mathrm{lum}$ removes fringe without cross-color bias, with signal metrics (PSNR, ECAS) improved over naive channel selection. The module integrates seamlessly with learned LUTs for artifact removal and overall chromatic restoration (Lu et al., 15 Nov 2025).

4. Geometric CA-CTs and Observer Compensation

A geometric approach to CA-CT is exemplified by Riemannian-metric based observer compensation (Oshima et al., 2015). Here, the color space of a given observer (normal or color-weak) is endowed with a spatially varying metric tensor $g_{ij}(x)$ derived from just-noticeable difference (jnd) ellipsoid measurements. The goal is to construct an isometry $F$ between $(C_\mathrm{normal}, g)$ and $(C_\mathrm{weak}, \tilde g)$, i.e.,

$$g_{ij}(x) = \frac{\partial F^p}{\partial x^i}\,\tilde g_{pq}(F(x))\,\frac{\partial F^q}{\partial x^j}$$

This is achieved using Riemann normal coordinates (RNC) around a reference point (white point D65): both observer spaces are mapped to Euclidean space via exponential and logarithmic maps, so $F(x) = \exp^{(w)}_{x_0}(\log^{(n)}_{x_0}(x))$ serves as a global, non-linear, chromatic-aware coordinate transformation. Implementation options range from 2D chromaticity slices to full 3D RNC compensation, with semantic-differential testing demonstrating improved perceptual correspondence between normal and compensated vision (Oshima et al., 2015).

5. Projective and Algebraic CA-CT: proLab and Split-Quaternions

Perceptually uniform and chromaticity-aware transforms are also realized via projective and algebraic structures:

• proLab: Defines a 3D projective CA-CT $\Phi: \mathrm{CIE\, XYZ}\to$ proLab with homogeneous coordinates $X_h=[X,Y,Z,1]^T$ and nonlinear $4\times 4$ matrix $P$. This warping equalizes perceptual-metric density (STRESS metric: proLab 0.209, CIELAB 0.259) and normalizes angular errors across hue via local JND estimates. proLab preserves linear manifolds (chromatic constancy planes, shading lines), supports homoscedastic noise transformation, and enables Euclidean analysis concordant with CIEDE2000. Trade-offs include slightly less uniformity than CAM16-UCS, but higher computational tractability for regression and subspace analysis (Konovalenko et al., 2020).
• Split-Quaternion CA-CT: The split-CAT approach encodes HCV (hue-chroma-value) colors as split-quaternions $q = V + C\cos H\,i + C\sin H\,j$ within the 3D Jordan subalgebra. Chromatic adaptation is effected via the sandwich mapping $q'(x) = p_e^{-1/2}q(x)p_e^{-1/2}$, where $p_e$ encodes the illuminant and the operation implements a Lorentz transformation in the corresponding Minkowski space. This algebraic structure rigorously models quantum-like observer effects and enables accurate and invertible chromatic adaptation. Empirical testing yields improved performance over standard von Kries diagonal models across multiple color-difference metrics (e.g., CIE 94, DIN99, CAM16 UCS), especially in reducing hue distortion (Berthier et al., 21 Apr 2025).

6. Applications and Advantages of CA-CTs

CA-CTs serve as foundational enablers for:

• Perceptually stable filtering, interpolation, and spline construction free from hue-wraparound and desaturation,
• Image-specific artifact suppression (e.g., purple fringing) via targeted channelization and adaptive transforms,
• Cross-observer image compensation with global geometric accuracy,
• Physics-informed chromatic adaptation through quantum-inspired, algebraically encoded observer models,
• Homoscedastic and uniform coordinate systems facilitating linear regression, clustering, and metric-based retrieval in color processing,
• Efficient integration into learned pipelines requiring artifact isolation and color restoration.

The diversity in mathematical structure—complex numbers, Riemann geometry, projective transforms, split-quaternions—demonstrates the flexibility and depth of CA-CTs in unifying perceptual, geometric, and computational requirements in color image processing. The implementations supported by perceptual and quantitative benchmarks reveal that CA-CTs offer concrete improvements in visual smoothness, artifact suppression, color fidelity, and task-specific accuracy, and are broadly adaptable across both classical and data-driven domains (Akleman et al., 2023, Lu et al., 15 Nov 2025, Oshima et al., 2015, Konovalenko et al., 2020, Berthier et al., 21 Apr 2025).