Film vs Digital DRT analysis

Differential Jacobian Analysis Report: Filmic LUT vs. Generic DWG→Rec.709

This document presents the complete mathematical and physical characterization of the film emulation LUT (dye_sim_images/lut_downsized.cube) compared against industry-standard generic DaVinci Wide Gamut (DWG) to Rec.709 rendering pipelines.

1. Executive Summary & Core Methodology

To reverse-engineer the precise subtractive color geometry and tone mapping behavior of the film emulation LUT without artifacts from analytical division-by-zero, we performed a differential Jacobian analysis in output space:

ΔJ(c) = J_film(c) - J_ref(c)

Where J(c) = d(Output) / d(Input) is the 3×3 local differential rendering matrix evaluated across a uniform 33³ scene-linear DWG grid (35,937 sample points) using central finite differences (ε = 0.002).

We evaluated ΔJ(c) against two independent baselines:

  1. Analytical BMD Reference Pipeline: Built directly from Blackmagic Design's official DaVinci Wide Gamut colorimetric primary transformation matrix and standard Rec.709/2.4 power gamma tone mapping.
  2. Resolve Built-In Generic LUT: DaVinci Resolve's standard 65-point DWG → Rec.709 conversion cube (DWG_to_709_65 Point Cube_17.cube).

2. Global & Regional Divergence Metrics

The Frobenius norm |ΔJ(c)| quantifies the total local divergence—incorporating both per-channel contrast/gain shifts and inter-channel color coupling (subtractive dye crosstalk)—at every color coordinate.

Global Grid Performance

| Comparison Target | Mean |ΔJ| across Grid | Max |ΔJ| across Grid | | :--- | :---: | :---: | | Analytical BMD Reference Pipeline | 3.1316 | 14.7124 | | Resolve Generic DWG → Rec.709 LUT | 4.2386 | 41.0072 |

Lightness & Neutral Axis Breakdown (vs. Resolve Generic LUT)

By slicing the DWG color volume into luminance bands, we observe exactly where the film emulation concentrates its re-shaping:

  • Neutral Axis (R = G = B): Mean |ΔJ| = 4.5491 (Max: 15.3274)
    • Interpretation: Unlike generic video converters that apply hard-clamping power gamma, the film LUT applies a cinematic S-curve with elevated shadow toe contrast and a soft, tapered highlight shoulder.
  • Shadows (L < 0.25): Mean |ΔJ| = 2.7902
    • Shadow rendering remains closest to standard colorimetry to preserve deep black transparency and avoid muddy color casts.
  • Midtones (0.25 ≤ L ≤ 0.75): Mean |ΔJ| = 5.2726
    • The Engine Room of Film Color: The highest divergence occurs across midtones. Here, subtractive dye saturation and intermixture actively reshape primary colors and complexions.
  • Highlights (L > 0.75): Mean |ΔJ| = 3.5056
    • Reflects smooth highlight compression and desaturation that prevents digital primary clipping.

3. Physical Subtractive Dye Behavior by Color Sector

By decomposing the Jacobian Delta matrix ΔJ(c) into its Diagonal Terms (per-channel gain/contrast delta) and Off-Diagonal Terms (color coupling / subtractive dye crosstalk norm), we reveal the physical dye mechanics of the film emulation:

[Row = Output Channel (R, G, B), Column = Input Channel (R, G, B)]

🔴 Saturated Red (DWG [0.80, 0.14, 0.14])

  • Total Divergence |ΔJ|: 3.3141
  • Off-Diagonal Crosstalk Norm: 3.2672
  • Jacobian Delta Matrix ΔJ: 
    [ +0.552   +0.015   -0.000 ]
[ +2.103   +0.061   -0.000 ]
[ +2.500   +0.054   +0.014 ]
  • Physical Analysis: Notice the massive positive off-diagonal terms on the Green (+2.103) and Blue (+2.500) outputs relative to Red input! In a generic Rec.709 LUT, pure red pushes digital neon red. Your film LUT exhibits classic subtractive yellow/magenta dye behavior: as red saturation increases, it intermixes warmth into green and blue, turning harsh digital red into rich, velvety scarlet/orange-red.

🟡 Saturated Yellow (DWG [0.80, 0.80, 0.14])

  • Total Divergence |ΔJ|: 1.5757
  • Off-Diagonal Crosstalk Norm: 1.5302
  • Jacobian Delta Matrix ΔJ: 
    [ +0.270   -0.185   +0.000 ]
[ -0.007   +0.262   -0.000 ]
[ +0.078   +1.517   -0.000 ]
  • Physical Analysis: Shows strong coupling into the Blue output from Green input (+1.517). This prevents yellows from looking thin or greenish-lemon, guiding them into a warm, golden filmic yellow roll-off.

🔵 Saturated Blue (DWG [0.14, 0.14, 0.80])

  • Total Divergence |ΔJ|: 3.3401
  • Off-Diagonal Crosstalk Norm: 3.2345
  • Jacobian Delta Matrix ΔJ: 
    [ +0.001   -0.073   +2.559 ]
[ -0.018   -0.050   +1.977 ]
[ -0.006   -0.022   +0.832 ]
  • Physical Analysis: Exhibits powerful crosstalk coupling where Red input boosts Blue output (+2.559) and Green input boosts Blue output (+1.977). This subtractive coupling anchors deep blues, preventing them from twisting towards neon purple/magenta and maintaining dense, photographic royal blues.

🟠 Skin Tone Reference (DWG [0.44, 0.32, 0.26])

  • Total Divergence |ΔJ|: 9.7300
  • Off-Diagonal Crosstalk Norm: 3.8013
  • Jacobian Delta Matrix ΔJ: 
    [ +1.426   +0.944   -0.255 ]
[ +2.396   -8.591   +2.582 ]
[ -1.015   -0.242   +2.096 ]
  • Physical Analysis: In skin tones, the Green channel contrast undergoes a major re-weighting (ΔJ_diag,G = -8.591), while Red and Blue gain supportive contrast (+1.426 and +2.096). This subtracts unwanted green/magenta tint shifts and locks complexions into a stable, golden-bronze pocket with smooth color intermixing (off-diagonal crosstalk norm of 3.8013).

4. Analytical Cross-Section Visualizations

The following analytical images illustrate the spatial distribution of these differentials across the color volume:

  1. jacobian_delta_hue_sat_slices.png: Polar heatmaps of total divergence |ΔJ| sliced across Shadows (L = 0.15), Midtones (L = 0.35), and Highlights (L = 0.80).
  2. jacobian_delta_crosstalk_slices.png: Polar heatmaps isolating purely off-diagonal subtractive dye intermixture intensity.
  3. jacobian_delta_vertical_sections.png: Vertical cross-sections (Lightness vs. Saturation) along the Red ↔ Cyan (0° / 180°) and Yellow ↔ Blue (60° / 240°) complementary axes.

5. Conclusion & Next Steps

This differential Jacobian analysis confirms that the film emulation LUT operates as an advanced subtractive color intermixture engine. By replacing digital primary isolation with smooth off-diagonal dye coupling (1.5 → 3.8 Frobenius norm across saturated sectors), it eliminates neon digital clipping and concentrates chromatic re-shaping precisely in the midtone (5.27) and skin tone (9.73) regions.