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# Colour Difference Calculations
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Comprehensive colour difference calculation functions implementing various industry-standard metrics for quantifying perceptual differences between colours. These functions support CIE standard formulas, CAM-based uniform colour spaces, and advanced perceptual models.
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## Capabilities
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### Main Delta E Function
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The primary entry point for colour difference calculations with automatic method selection and validation.
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```python { .api }
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def delta_E(a: ArrayLike, b: ArrayLike, method: str = "CIE 2000", **kwargs) -> NDArray:
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    """
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    Calculate colour difference ΔE between two colour arrays using specified method.
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    Parameters:
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    - a: first colour array in appropriate colour space (L*a*b*, J'a'b', or ICtCp)
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    - b: second colour array in appropriate colour space  
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    - method: calculation method, supports:
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        * "CIE 1976": CIE76 ΔE*ab (Euclidean distance in L*a*b*)
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        * "CIE 1994": CIE94 ΔE*94 with perceptual weighting 
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        * "CIE 2000": CIEDE2000 ΔE*00 (default, most accurate)
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        * "CMC": CMC l:c colour difference
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        * "ITP": ITU-R BT.2124 ΔE_ITP for HDR/WCG
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        * "CAM02-LCD", "CAM02-SCD", "CAM02-UCS": CIECAM02-based
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        * "CAM16-LCD", "CAM16-SCD", "CAM16-UCS": CAM16-based  
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        * "DIN99": DIN99 colour difference formula
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    - **kwargs: method-specific parameters (textiles, l, c, etc.)
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    Returns:
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    Colour difference values as NDArray
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    Notes:
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    - Input colour spaces depend on method:
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      * CIE methods: L*a*b* colour space
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      * CAM methods: J'a'b' colour space from respective appearance models
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      * ITP method: ICtCp colour encoding
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    - Values typically range 0-100+, with ~1-3 being just noticeable
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    - CIEDE2000 is recommended for most applications
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    """
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# Available calculation methods
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DELTA_E_METHODS = {
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    "CIE 1976": delta_E_CIE1976,
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    "CIE 1994": delta_E_CIE1994, 
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    "CIE 2000": delta_E_CIE2000,
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    "CMC": delta_E_CMC,
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    "ITP": delta_E_ITP,
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    "CAM02-LCD": delta_E_CAM02LCD,
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    "CAM02-SCD": delta_E_CAM02SCD,
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    "CAM02-UCS": delta_E_CAM02UCS,
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    "CAM16-LCD": delta_E_CAM16LCD,
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    "CAM16-SCD": delta_E_CAM16SCD,
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    "CAM16-UCS": delta_E_CAM16UCS,
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    "DIN99": delta_E_DIN99
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}
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# Just Noticeable Difference threshold
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JND_CIE1976 = 2.3  # Standard JND threshold for CIE 1976
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```
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### CIE Standard Delta E Functions
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Industry-standard colour difference formulas based on CIE recommendations.
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```python { .api }
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def delta_E_CIE1976(Lab_1: ArrayLike, Lab_2: ArrayLike) -> NDArray:
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    """
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    Calculate CIE 1976 colour difference ΔE*ab (Euclidean distance in L*a*b*).
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    Parameters:
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    - Lab_1: first L*a*b* colour array, shape (..., 3)
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    - Lab_2: second L*a*b* colour array, shape (..., 3)
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    Returns:
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    CIE76 ΔE*ab values
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    Notes:
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    - Simple Euclidean distance: √[(ΔL*)² + (Δa*)² + (Δb*)²]
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    - Fast computation but less perceptually uniform
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    - L*: [0, 100], a*,b*: [-100, 100] in reference scale
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    - JND threshold: ~2.3 (see JND_CIE1976)
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    """
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def delta_E_CIE1994(Lab_1: ArrayLike, Lab_2: ArrayLike, textiles: bool = False) -> NDArray:
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    """
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    Calculate CIE 1994 colour difference ΔE*94 with perceptual weighting factors.
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    Parameters:
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    - Lab_1: reference L*a*b* colour array, shape (..., 3) 
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    - Lab_2: sample L*a*b* colour array, shape (..., 3)
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    - textiles: use textile industry parameters if True
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    Returns:
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    CIE94 ΔE*94 values
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    Notes:
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    - Improved perceptual uniformity over CIE76
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    - Non-symmetric: delta_E_CIE1994(a,b) ≠ delta_E_CIE1994(b,a)
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    - Textile parameters: kL=2, kC=kH=1, k1=0.048, k2=0.014
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    - Graphic arts parameters: kL=kC=kH=1, k1=0.045, k2=0.015 (default)
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    """
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def delta_E_CIE2000(Lab_1: ArrayLike, Lab_2: ArrayLike, textiles: bool = False) -> NDArray:
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    """
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    Calculate CIEDE2000 colour difference ΔE*00, the most accurate CIE standard.
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    Parameters:
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    - Lab_1: first L*a*b* colour array, shape (..., 3)
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    - Lab_2: second L*a*b* colour array, shape (..., 3)  
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    - textiles: use textile industry weighting (kL=2) if True
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    Returns:
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    CIEDE2000 ΔE*00 values
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    Notes:
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    - Most perceptually uniform CIE formula (recommended default)
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    - Accounts for perceptual non-uniformities in L*a*b* space
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    - Reference conditions: D65, 1000 lx, uniform gray background L*=50
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    - Includes rotation term for interaction between chroma and hue
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    - Sample size >4°, direct edge contact, ΔE < 5.0
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    """
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def delta_E_CMC(Lab_1: ArrayLike, Lab_2: ArrayLike, l: float = 2, c: float = 1) -> NDArray:
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    """
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    Calculate CMC l:c colour difference with configurable lightness/chroma weighting.
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    Parameters:
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    - Lab_1: reference L*a*b* colour array, shape (..., 3)
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    - Lab_2: sample L*a*b* colour array, shape (..., 3)
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    - l: lightness weighting factor (2 for acceptability, 1 for perceptibility)
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    - c: chroma weighting factor (typically 1)
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    Returns:
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    CMC ΔE values
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    Notes:
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    - Common ratios: 2:1 (acceptability), 1:1 (threshold of imperceptibility)
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    - Developed by Colour Measurement Committee
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    - Good correlation with visual assessments
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    - Non-symmetric like CIE94
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    """
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```
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### Advanced Colour Difference Functions
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Modern colour difference metrics for HDR, wide gamut, and specialized applications.
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```python { .api }
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def delta_E_ITP(ICtCp_1: ArrayLike, ICtCp_2: ArrayLike) -> NDArray:
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    """
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    Calculate ITU-R BT.2124 ΔE_ITP for HDR and wide colour gamut applications.
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    Parameters:
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    - ICtCp_1: first ICtCp colour encoding array, shape (..., 3)
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    - ICtCp_2: second ICtCp colour encoding array, shape (..., 3)
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    Returns:
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    ΔE_ITP values (1.0 ≈ just noticeable difference)
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    Notes:
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    - Designed for HDR/WCG content (Rec.2020, HDR10, Dolby Vision)
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    - Value of 1.0 equals just noticeable difference in critical adaptation
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    - Uses ICtCp encoding: I (intensity), Ct (chroma-red), Cp (chroma-yellow)
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    - Optimized for high dynamic range and wide color gamut displays
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    """
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def delta_E_DIN99(Lab_1: ArrayLike, Lab_2: ArrayLike, textiles: bool = False) -> NDArray:
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    """
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    Calculate DIN99 colour difference using DIN99 colour space transformation.
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    Parameters:
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    - Lab_1: first L*a*b* colour array, shape (..., 3)
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    - Lab_2: second L*a*b* colour array, shape (..., 3)
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    - textiles: use textile parameters (kE=2, kCH=0.5) if True
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    Returns:
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    DIN99 ΔE values
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    Notes:
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    - Based on DIN99 colour space with improved perceptual uniformity
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    - Standard parameters: kE=1, kCH=1
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    - Textile parameters: kE=2, kCH=0.5  
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    - Often shows better correlation than CIELAB-based formulas
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    """
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```
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### CAM02-Based Colour Difference Functions
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Colour difference calculations in uniform colour spaces derived from CIECAM02 appearance model.
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```python { .api }
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def delta_E_CAM02LCD(Jpapbp_1: ArrayLike, Jpapbp_2: ArrayLike) -> NDArray:
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    """
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    Calculate colour difference in CAM02-LCD uniform colour space.
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    Parameters:
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    - Jpapbp_1: first J'a'b' array from CAM02-LCD space, shape (..., 3)
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    - Jpapbp_2: second J'a'b' array from CAM02-LCD space, shape (..., 3)
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    Returns:
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    CAM02-LCD ΔE' values
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    Notes:
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    - LCD: Large Colour Differences - optimized for large colour differences
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    - Input must be CAM02-LCD J'a'b', not CIE L*a*b*
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    - Based on CIECAM02 appearance model for better perceptual uniformity
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    - Lightness scaling factor KL accounts for perceptual non-linearities
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    """
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def delta_E_CAM02SCD(Jpapbp_1: ArrayLike, Jpapbp_2: ArrayLike) -> NDArray:
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    """
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    Calculate colour difference in CAM02-SCD uniform colour space.
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    Parameters:
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    - Jpapbp_1: first J'a'b' array from CAM02-SCD space, shape (..., 3)
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    - Jpapbp_2: second J'a'b' array from CAM02-SCD space, shape (..., 3)
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    Returns:
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    CAM02-SCD ΔE' values
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    Notes:
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    - SCD: Small Colour Differences - optimized for small colour differences  
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    - Input must be CAM02-SCD J'a'b', not CIE L*a*b*
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    - Different coefficients than CAM02-LCD for improved small difference detection
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    - Useful for quality control and fine colour matching
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    """
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def delta_E_CAM02UCS(Jpapbp_1: ArrayLike, Jpapbp_2: ArrayLike) -> NDArray:
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    """
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    Calculate colour difference in CAM02-UCS uniform colour space.
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    Parameters:
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    - Jpapbp_1: first J'a'b' array from CAM02-UCS space, shape (..., 3)
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    - Jpapbp_2: second J'a'b' array from CAM02-UCS space, shape (..., 3)
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    Returns:
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    CAM02-UCS ΔE' values
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    Notes:
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    - UCS: Uniform Colour Space - balanced for various difference magnitudes
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    - Input must be CAM02-UCS J'a'b', not CIE L*a*b*
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    - Compromise between LCD and SCD optimization
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    - General purpose CAM02-based colour difference metric
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    """
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```
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### CAM16-Based Colour Difference Functions
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Next-generation colour difference calculations based on the improved CAM16 appearance model.
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```python { .api }
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def delta_E_CAM16LCD(Jpapbp_1: ArrayLike, Jpapbp_2: ArrayLike) -> NDArray:
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    """
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    Calculate colour difference in CAM16-LCD uniform colour space.
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    Parameters:
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    - Jpapbp_1: first J'a'b' array from CAM16-LCD space, shape (..., 3)
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    - Jpapbp_2: second J'a'b' array from CAM16-LCD space, shape (..., 3)
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    Returns:
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    CAM16-LCD ΔE' values
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    Notes:
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    - Based on improved CAM16 appearance model
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    - LCD: Large Colour Differences optimization
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    - Input must be CAM16-LCD J'a'b', not CIE L*a*b*
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    - Successor to CAM02-LCD with improved predictions
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    """
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def delta_E_CAM16SCD(Jpapbp_1: ArrayLike, Jpapbp_2: ArrayLike) -> NDArray:
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    """
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    Calculate colour difference in CAM16-SCD uniform colour space.
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    Parameters:
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    - Jpapbp_1: first J'a'b' array from CAM16-SCD space, shape (..., 3)
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    - Jpapbp_2: second J'a'b' array from CAM16-SCD space, shape (..., 3)
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    Returns:
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    CAM16-SCD ΔE' values
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    Notes:
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    - Based on improved CAM16 appearance model  
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    - SCD: Small Colour Differences optimization
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    - Input must be CAM16-SCD J'a'b', not CIE L*a*b*
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    - Better small difference detection than CAM02-SCD
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    """
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def delta_E_CAM16UCS(Jpapbp_1: ArrayLike, Jpapbp_2: ArrayLike) -> NDArray:
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    """
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    Calculate colour difference in CAM16-UCS uniform colour space.
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    Parameters:
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    - Jpapbp_1: first J'a'b' array from CAM16-UCS space, shape (..., 3)
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    - Jpapbp_2: second J'a'b' array from CAM16-UCS space, shape (..., 3)
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    Returns:
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    CAM16-UCS ΔE' values
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    Notes:
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    - Based on improved CAM16 appearance model
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    - UCS: Uniform Colour Space - general purpose optimization
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    - Input must be CAM16-UCS J'a'b', not CIE L*a*b*
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    - State-of-the-art appearance-based colour difference metric
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    """
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```
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### Performance Enhancement Functions
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Advanced functions for improving colour difference formula performance and correlation with visual assessment.
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```python { .api }
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def power_function_Huang2015(d_E: ArrayLike, coefficients: str = "CIE 2000") -> NDArray:
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    """
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    Apply Huang et al. (2015) power function to improve ΔE formula performance.
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    Parameters:
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    - d_E: computed colour difference array
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    - coefficients: power function coefficients for specific method:
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        * "CIE 1976": [1.26, 0.55] 
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        * "CIE 1994": [1.41, 0.70]
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        * "CIE 2000": [1.43, 0.70] (default)
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        * "CMC": [1.34, 0.66]
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        * "CAM02-LCD": [1.00, 0.85]
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        * "CAM02-SCD": [1.45, 0.75]
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        * "CAM02-UCS": [1.30, 0.75]
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        * "CAM16-UCS": [1.41, 0.63]
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        * "DIN99d": [1.28, 0.74]
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        * "OSA": [3.32, 0.62]
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        * "OSA-GP-Euclidean": [1.52, 0.76]
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        * "ULAB": [1.17, 0.69]
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    Returns:
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    Improved ΔE values with better correlation to visual assessment
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    Notes:
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    - Formula: ΔE' = a × ΔE^b where [a,b] are method-specific coefficients
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    - Significantly improves correlation with visual datasets
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    - Apply after computing base ΔE with standard formulas
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    - Coefficients derived from extensive psychophysical experiments
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    """
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# Available power function coefficients  
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COEFFICIENTS_HUANG2015 = {
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    "CIE 1976": [1.26, 0.55],
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    "CIE 1994": [1.41, 0.70], 
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    "CIE 2000": [1.43, 0.70],
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    "CMC": [1.34, 0.66],
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    "CAM02-LCD": [1.00, 0.85],
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    "CAM02-SCD": [1.45, 0.75],
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    "CAM02-UCS": [1.30, 0.75], 
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    "CAM16-UCS": [1.41, 0.63],
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    "DIN99d": [1.28, 0.74],
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    "OSA": [3.32, 0.62],
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    "OSA-GP-Euclidean": [1.52, 0.76],
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    "ULAB": [1.17, 0.69]
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}
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```
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### Stress Index Functions
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Statistical measures for evaluating colour difference formula performance against visual datasets.
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```python { .api }
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def index_stress(d_E: ArrayLike, d_V: ArrayLike, method: str = "Garcia 2007") -> NDArray:
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    """
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    Calculate Kruskal's Standardized Residual Sum of Squares (STRESS) index.
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    Parameters:
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    - d_E: computed colour difference array ΔE  
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    - d_V: visual colour difference array ΔV from psychophysical experiments
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    - method: computation method ("Garcia 2007")
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    Returns:
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    STRESS index values (lower = better correlation)
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    Notes:
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    - Measures goodness-of-fit between computed and visual differences
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    - Values closer to 0 indicate better correlation with human vision
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    - Used to evaluate and compare colour difference formulas
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    - Essential for psychophysical validation of new metrics
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    """
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def index_stress_Garcia2007(d_E: ArrayLike, d_V: ArrayLike) -> NDArray:
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    """
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    Calculate STRESS index using Garcia et al. (2007) method.
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    Parameters:
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    - d_E: computed colour difference array ΔE
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    - d_V: visual colour difference array ΔV  
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    Returns:
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    STRESS index value
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    Notes:
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    - Standard implementation for colour difference evaluation
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    - Used in research for validating new colour difference formulas
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    - Lower values indicate better correlation with visual assessment
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    """
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# Available stress index methods
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INDEX_STRESS_METHODS = {
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    "Garcia 2007": index_stress_Garcia2007
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}
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```
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## Usage Examples
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### Basic Colour Difference Calculation
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```python
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import numpy as np
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from colour import delta_E
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# CIE L*a*b* colour arrays
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lab1 = np.array([50.0, 20.0, -30.0])  # Reference colour
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lab2 = np.array([55.0, 18.0, -25.0])  # Sample colour
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# Calculate CIEDE2000 (recommended)
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de2000 = delta_E(lab1, lab2, method="CIE 2000")
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print(f"CIEDE2000 ΔE: {de2000:.2f}")
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# Compare different methods
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de76 = delta_E(lab1, lab2, method="CIE 1976") 
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de94 = delta_E(lab1, lab2, method="CIE 1994")
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decmc = delta_E(lab1, lab2, method="CMC")
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print(f"CIE76 ΔE: {de76:.2f}")
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print(f"CIE94 ΔE: {de94:.2f}")  
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print(f"CMC ΔE: {decmc:.2f}")
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```
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### Textile Industry Application
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```python
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# Textile-specific parameters for closer tolerances
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de2000_textiles = delta_E(lab1, lab2, method="CIE 2000", textiles=True)
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de94_textiles = delta_E(lab1, lab2, method="CIE 1994", textiles=True)
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din99_textiles = delta_E(lab1, lab2, method="DIN99", textiles=True)
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print(f"CIEDE2000 (textiles): {de2000_textiles:.2f}")
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print(f"CIE94 (textiles): {de94_textiles:.2f}")
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print(f"DIN99 (textiles): {din99_textiles:.2f}")
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```
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### HDR/WCG Content
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```python
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# ICtCp colour encoding for HDR content
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ictcp1 = np.array([0.4885, -0.0474, 0.0748])  # HDR colour 1
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ictcp2 = np.array([0.4899, -0.0457, 0.0736])  # HDR colour 2
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# Calculate ΔE_ITP for HDR applications
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de_itp = delta_E(ictcp1, ictcp2, method="ITP")
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print(f"ΔE_ITP: {de_itp:.3f}")  # ~1.0 = just noticeable
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```
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### CAM-Based Advanced Metrics
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```python
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from colour import XYZ_to_CAM16UCS
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# Convert XYZ to CAM16-UCS J'a'b' space first
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xyz1 = np.array([0.20654008, 0.12197225, 0.05136952])
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xyz2 = np.array([0.14222010, 0.06157513, 0.06157513])
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# Convert to CAM16-UCS space
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jpapbp1 = XYZ_to_CAM16UCS(xyz1)  
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jpapbp2 = XYZ_to_CAM16UCS(xyz2)
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# Calculate CAM16-based differences
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de_cam16_ucs = delta_E(jpapbp1, jpapbp2, method="CAM16-UCS")
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de_cam16_lcd = delta_E(jpapbp1, jpapbp2, method="CAM16-LCD") 
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de_cam16_scd = delta_E(jpapbp1, jpapbp2, method="CAM16-SCD")
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print(f"CAM16-UCS ΔE': {de_cam16_ucs:.3f}")
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print(f"CAM16-LCD ΔE': {de_cam16_lcd:.3f}")
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print(f"CAM16-SCD ΔE': {de_cam16_scd:.3f}")
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```
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### Performance Enhancement
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```python
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from colour import power_function_Huang2015
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# Calculate base ΔE and apply Huang2015 enhancement
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base_de = delta_E(lab1, lab2, method="CIE 2000")
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enhanced_de = power_function_Huang2015(base_de, coefficients="CIE 2000")
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print(f"Base CIEDE2000: {base_de:.3f}")
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print(f"Enhanced: {enhanced_de:.3f}")
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# Enhancement for other methods
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de_cmc_base = delta_E(lab1, lab2, method="CMC")
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de_cmc_enhanced = power_function_Huang2015(de_cmc_base, coefficients="CMC")
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print(f"Base CMC: {de_cmc_base:.3f}")
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print(f"Enhanced CMC: {de_cmc_enhanced:.3f}")
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```
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### Quality Assessment
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```python
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from colour import index_stress
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# Simulated data for evaluation
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computed_de = np.array([2.04, 2.86, 3.44, 1.75, 4.12])  # Calculated ΔE
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visual_de = np.array([1.26, 1.26, 1.87, 1.15, 2.34])    # Visual assessment
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# Calculate STRESS index
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stress = index_stress(computed_de, visual_de)
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print(f"STRESS index: {stress:.4f}")  # Lower = better correlation
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```
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### Batch Processing
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```python
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# Multiple colour pairs
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lab_refs = np.array([[50, 20, -30], [60, -15, 25], [40, 10, -20]])
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lab_samples = np.array([[52, 18, -28], [58, -12, 28], [42, 12, -18]])
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# Calculate all differences at once
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de_batch = delta_E(lab_refs, lab_samples, method="CIE 2000")
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print("Batch CIEDE2000 results:", de_batch)
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# Apply enhancement to batch
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enhanced_batch = power_function_Huang2015(de_batch, coefficients="CIE 2000")
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print("Enhanced results:", enhanced_batch)
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```
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## Method Selection Guidelines
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### Recommended Methods by Application
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**General Purpose (Recommended)**
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- **CIEDE2000**: Best overall perceptual uniformity, CIE standard
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- **Enhanced CIEDE2000**: Apply `power_function_Huang2015` for improved correlation
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**Industry-Specific**
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- **Textiles**: CIE 1994, CIEDE2000, or DIN99 with `textiles=True`
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- **Printing/Graphics**: CIEDE2000, CMC 2:1, enhanced with Huang2015
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- **HDR/Broadcasting**: ΔE_ITP for HDR content, CIEDE2000 for SDR
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- **Research**: CAM16-UCS for appearance-based studies
543

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**Performance vs. Accuracy Trade-offs**
545
- **Fastest**: CIE 1976 (simple Euclidean distance)
546
- **Balanced**: CIE 1994, CMC
547
- **Most Accurate**: CIEDE2000, CAM16-UCS
548
- **HDR/WCG**: ΔE_ITP
549

550
### Perceptual Significance
551

552
**ΔE Value Interpretation (approximate guidelines)**
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- **0-1**: Imperceptible difference
554
- **1-2**: Perceptible by expert observers
555
- **2-3.5**: Perceptible by untrained observers  
556
- **3.5-5**: Clear difference
557
- **5+**: Very different colours
558

559
**Important Notes**
560
- Thresholds vary by viewing conditions, colour region, and application
561
- Textile/printing industries often use tighter tolerances
562
- HDR content may have different perceptual thresholds
563
- Always validate with visual assessment for critical applications
564

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## Constants and Datasets
566

567
```python { .api }
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# Just Noticeable Difference threshold
569
JND_CIE1976 = 2.3  # Standard JND for CIE 1976 ΔE*ab
570

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# Available calculation methods
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DELTA_E_METHODS = {
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    "CIE 1976", "CIE 1994", "CIE 2000", "CMC", "ITP",
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    "CAM02-LCD", "CAM02-SCD", "CAM02-UCS", 
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    "CAM16-LCD", "CAM16-SCD", "CAM16-UCS", "DIN99"
576
}
577

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# Power function enhancement coefficients
579
COEFFICIENTS_HUANG2015 = {
580
    "CIE 1976": [1.26, 0.55], "CIE 1994": [1.41, 0.70],
581
    "CIE 2000": [1.43, 0.70], "CMC": [1.34, 0.66],
582
    "CAM02-LCD": [1.00, 0.85], "CAM02-SCD": [1.45, 0.75],
583
    "CAM02-UCS": [1.30, 0.75], "CAM16-UCS": [1.41, 0.63],
584
    "DIN99d": [1.28, 0.74], "OSA": [3.32, 0.62],
585
    "OSA-GP-Euclidean": [1.52, 0.76], "ULAB": [1.17, 0.69]
586
}
587

588
# Stress index evaluation methods  
589
INDEX_STRESS_METHODS = {"Garcia 2007": index_stress_Garcia2007}
590
```
591

592
The colour difference module provides comprehensive tools for quantifying perceptual colour differences across various applications, from quality control to psychophysical research, with state-of-the-art formulas and performance enhancements.