Filmr Knowledge Base

Welcome to the technical documentation for Filmr, a high-fidelity, physics-based film simulation engine.

This book details the mathematical models and physical principles used in the simulation, including:

• Spectral Sensitivity
• Reciprocity Failure
• Chemical Color Coupling
• Grain Synthesis

Photochemistry Core (Quantum Scale)

When photons strike a silver halide crystal (AgX), a photolytic reaction is triggered:

1.1 Photoelectron Excitation

$$ \text{AgX} + h\nu \xrightarrow{k_{\text{photo}}} \text{Ag}^+ + \text{e}^- + \text{X}^0 $$

Where:

• $h\nu$ is the photon energy ($\nu$ is frequency)
• $k_{\text{photo}}$ is the photochemical reaction rate constant, following the quantum efficiency model: $$ k_{\text{photo}} = \eta \cdot \sigma_{\text{abs}} \cdot \Phi $$
  • $\eta$: Quantum efficiency (≈0.6-0.9)
  • $\sigma_{\text{abs}}$: Crystal absorption cross-section
  • $\Phi$: Photon flux (photons/(cm²·s))

1.2 Latent Image Center Formation

Released electrons are trapped by silver ions, forming silver atom clusters:

$$ \text{Ag}^+ + \text{e}^- \xrightarrow{k_{\text{trap}}} \text{Ag}^0 $$

Critical Condition: A stable latent image center is formed when a single crystal surface accumulates 5-10 silver atoms:

$$ n_{\text{Ag}} \geq n_{\text{critical}} \approx 5-10 $$

This process can be modeled using a Poisson distribution:

$$ P(n_{\text{Ag}} \geq n_{\text{crit}}) = 1 - \sum_{k=0}^{n_{\text{crit}}-1} \frac{\lambda^k e^{-\lambda}}{k!} $$

Where $\lambda = \eta \cdot N_{\text{photons}}$ is the expected number of electrons.

Exposure and Density Mapping (Macro Scale)

2.1 Exposure Definition

$$ E = I \cdot t $$

• $I$: Irradiance (lux)
• $t$: Exposure time (s)
• $E$: Exposure (lux·s)

2.2 Optical Density

The ability of silver grains to block light after development is quantified by Optical Density $D$:

$$ D = \log_{10}\left(\frac{I_0}{I_t}\right) = -\log_{10}(T) $$

• $I_0$: Incident light intensity
• $I_t$: Transmitted light intensity
• $T = I_t/I_0$: Transmittance

Characteristic Curve (Hurter-Driffield Curve)

This is the core mathematical model of film imaging, describing the S-shaped relationship between $D$ and $\log_{10}E$:

3.1 Piecewise Linear Model (Engineering Simplification)

$$ D(\log E) = \begin{cases} D_{\text{min}} + \frac{\log E - \log E_{\text{toe}}}{\gamma_{\text{toe}}} & \text{Toe (Underexposed)} \ D_{\text{min}} + \gamma \cdot (\log E - \log E_0) & \text{Linear Region (Normal Exposure)} \ D_{\text{max}} - \frac{\log E_{\text{shoulder}} - \log E}{\gamma_{\text{shoulder}}} & \text{Shoulder (Overexposed)} \end{cases} $$

Key Parameters:

• $\gamma$: Contrast coefficient (slope of the linear region) $$ \gamma = \frac{\Delta D}{\Delta \log E} = \frac{D_2 - D_1}{\log E_2 - \log E_1} $$
• Dynamic Range: $\text{DR} = \log_{10}(E_{\text{max}}/E_{\text{min}}) \approx 2.0$ (corresponding to 100:1 brightness ratio)

3.2 Precise Error Function Model (Scientific Grade)

Analytical solution provided by Kodak technical documents:

$$ D(E) = \frac{D_0}{2}\left[1 + \text{erf}\left(X + \ln\frac{E - E_0}{E_g - E_0}\right)\right] $$

Where the error function is:

$$ \text{erf}(y) = \frac{1}{\sqrt{\pi}} \int_{-\infty}^{y} e^{-z^2} dz $$

Parameter meanings:

• $D_0$: Maximum saturation density
• $E_0$: Threshold exposure
• $E_g$: Sensitivity reference point
• $X$: Development condition correction term

Development Kinetics (Chemical Amplification)

Development amplifies latent image centers by $10^6$-$10^8$ times. Its rate equation:

4.1 First-Order Kinetics Model

$$ \frac{d[\text{Ag}]}{dt} = k_{\text{dev}} \cdot [\text{Dev}] \cdot [\text{Ag}^+] \cdot N_{\text{latent}} $$

• $[\text{Dev}]$: Developer concentration
• $N_{\text{latent}}$: Number of latent image centers (proportional to exposure)

4.2 Time-Temperature Compensation (Arrhenius Equation)

$$ k_{\text{dev}} = A \cdot e^{-E_a/(RT)} $$

• $E_a$: Activation energy (≈50-70 kJ/mol)
• $R$: Gas constant
• $T$: Absolute temperature (K)

Film Characteristics

The "flavor" differences between films from different manufacturers are essentially reflected in measurable parameters determined by chemical formulas and crystal structures. Below are real parameters from authoritative technical data sheets.

1. KODAK TRI-X 400 (Classic News Film)

Source: Kodak Official Technical Data Sheet F-4017

Flavor Characteristics: Wide exposure latitude (±2 stops still usable), rugged "breathing" grain, good shadow detail retention.

2. FUJIFILM Velvia 50 (Landscape Slide Film)

Source: Fujifilm Technical Data Sheet

Flavor Characteristics: High color saturation (especially red, orange), strong contrast, deeper blue sky rendering, fine grain but narrow exposure latitude (±0.5 stops).

3. ILFORD HP5 Plus (General Purpose B&W)

Source: ILFORD Official Data Sheet

Flavor Characteristics: Excellent push performance (usable at EI 3200), soft grain structure, smooth highlight roll-off, rich shadow gradations.

4. Mathematical Expression of Parameter Differences

These parameters directly determine the shape of the H-D Curve:

$$ D(E) = D_{\text{min}} + \frac{D_{\text{max}}-D_{\text{min}}}{1 + 10^{\gamma(\log E_0 - \log E)}} $$

• Tri-X: Lower $E_0$, long toe, large latitude
• Velvia: Extremely high $D_{\text{max}}$, large $\gamma$, steep shoulder
• HP5: Lower $D_{\text{min}}$, $E_0$ shifts right when pushed

Spectral Sensitivity Differences: $$ S(\lambda){\text{Velvia}} > S(\lambda){\text{Tri-X}} > S(\lambda)_{\text{HP5}} \quad (\lambda > 600\text{nm}) $$

5. Authoritative Verification Suggestions

1. Official Data Sheets: Kodak Alaris (kodakprofessional.com), Fujifilm, ILFORD (ilfordphoto.com)
2. Third-party Testing: Imatest software MTF analysis, X-Rite densitometer measurement of Dmax/Dmin
3. Measurement Methods: Verify Gamma using Stouffer step wedge + standard development process

These parameters are direct quantifications of film chemical formulas and crystal technologies, which are more reliable than subjective "flavor" descriptions.

Reference Data

All data comes from manufacturer official technical data sheets and ISO standard tests. Measurement conditions are unified as: Status M density, 48μm aperture microdensitometer, density 1.0 above Dmin.

1. Fujifilm Color Reversal Film (E-6 Process)

2. Fujifilm Color Negative Film (C-41 Process)

3. Kodak B&W Negative Film

4. ILFORD B&W Negative Film

Note: Ilford does not publish RMS values; data from independent testing (analog.cafe).

5. Kodak Color Negative Film

Note: Kodak switched to PGI (Print Grain Index) after 2007. Lower is better. PGI 25 ≈ RMS 8.

6. Classic Discontinued Films (Historical Data)

7. Data Authenticity Assurance

Measurement Standard Traceability:

• ISO Sensitivity: ISO 5800:1987, measured at +1, 0, -1 stops in specified developer.
• RMS Granularity: ISO 6328:2000, 48μm aperture, measured at density 1.0, 12x magnification.
• Resolution: ISO 6328, USAF 1951 chart, tested at 1000:1 and 1.6:1 contrast.
• Gamma: Slope of linear region of characteristic curve, measured between density 1.0-2.0.

Data Source Verification:

• Kodak: imaging.kodakalaris.com Official Data Sheets (F-4017, F-4016 series).
• Ilford: ilfordphoto.com 2021 Technical Specification White Papers.
• Fujifilm: fujifilm.com 2020 "Professional Film Data Guide".
• Independent Verification: analog.cafe re-tests using X-Rite 361T densitometer.

Update Note:

• Portra 400 produced after 2024 uses improved emulsion, PGI improved to 32 (finer grain).

Spectral Intuition and Color Shift

1. Key Premise Clarification

Film Simulation ≠ Color Preservation Algorithm

Film simulation typically allows for overall tonal changes (such as warm tones, cool tones, curve compression), but does NOT allow:

• Unexpected channel crosstalk (e.g., Red being asymmetrically pulled)
• Neutral colors becoming non-neutral
• Uncontrollable hue shifts related to luminance

So the goal is not "Output = Input", but rather:

Verify if there are abnormal color shifts within the "known acceptable range of variation"

2. Core Idea: Use "Diagnostic Images"

Natural photos cannot be used for algorithm verification because you can never prove whether the color cast comes from the algorithm or the content itself.

You need Synthetic / Diagnostic Images.

3. Category 1: Neutral Axis Verification (Most Important)

1️⃣ Input: Ideal Neutral Grayscale

Construct an image:

• R = G = B
• From 0 → 255 (or linear 0 → 1)
• Include different luminance sections (Shadows / Mid-gray / Highlights)

Expected Behavior

• After film simulation:
  • R′ ≈ G′ ≈ B′
  • Overall brightening/darkening/curve changes are allowed
  • Systematic deviation of Δ(R′−G′) with luminance is NOT allowed

Calculable Metrics

Metric A: Neutral Color Deviation

Δ_neutral = mean(|R' - G'| + |G' - B'| + |B' - R'|)

Metric B: Luminance-Correlated Deviation

corr(L_in, (R'-G'))
corr(L_in, (G'-B'))

If color cast changes with luminance → Typical Algorithmic Error (Commonly seen when incorrect curves are applied in RGB space instead of luminance space)

4. Category 2: Pure Color Channel Integrity Test

2️⃣ Input: Pure Colors and Single Channel Gradients

Construct:

• (R=1, G=0, B=0)
• (0,1,0)
• (0,0,1)
• And gradients for each channel 0→1

Expected Behavior

• Film allows:
  • Pure Red → Shifts to Orange / Magenta (This is style)
• But it must be explainable and symmetric

Calculable Metrics

Metric C: Channel Leakage Matrix

You can approximate a 3×3 matrix:

[ R' ]   [ a b c ] [ R ]
[ G' ] ≈ [ d e f ] [ G ]
[ B' ]   [ g h i ] [ B ]

Validation Framework

To pass industrial-grade film simulation certification, a measurement standard from molecular scale to perceptual scale must be established. Below is the complete 7-layer verification system, with quantifiable thresholds and Rust-readable test protocols for each layer:

Layer 0: Spectral Fidelity

0.1 Emulsion Spectral Sensitivity $S(\lambda)$

Standard: ISO 5800:2001 / ISO 6846 Measurement Object: Quantum efficiency of Blue/Green/Red sensitive layers to monochromatic light Data Format: 380–780 nm, 5 nm step, 81-dimensional vector Pass Thresholds:

• Peak wavelength error $< \pm 5$ nm
• Full Width at Half Maximum (FWHM) error $< \pm 15$ nm
• Interlayer crosstalk (Blue/Green sensitivity ratio at 480 nm) $< 15%$

Reference Data Source:

#![allow(unused)]
fn main() {
// Kodak Portra 400 Official Sensitivity (Pre-normalization)
const S_BLUE: [f32; 81] = [0.01, 0.03, ..., 0.95, 0.02]; // 380-780nm, 5nm step
const S_GREEN: [f32; 81] = [...];
const S_RED: [f32; 81] = [...];
}

0.2 Dye Spectral Absorption Cross-section $\varepsilon(\lambda)$

Standard: Status A Densitometry Calibration (ISO 5-3) Measurement Object: Absorption spectra of Yellow/Magenta/Cyan dyes at maximum density Pass Thresholds:

• Peak absorbance correlation with Kodak patent data $R^2 > 0.995$
• Dye overlap (Yellow dye absorbance at 550 nm $< 12%$ of peak)

Layer 1: Exposure Response

1.1 H-D Curve (Hurter-Driffield)

Standard: ISO 6 (B&W) / ISO 5800 (Color) Measurement Object: $\log_{10}(\text{Exposure})$ vs Density $D$ Key Parameters: | Parameter | Symbol | Acceptable Range | Measurement Method | |-----------|--------|------------------|-------------------| | Fog Density | $D_{\min}$ | $0.15 \pm 0.03$ | Unexposed base | | Max Density | $D_{\max}$ | $> 2.8$ | Overexposed by 5 stops | | Contrast | $\gamma$ | $0.55 \pm 0.05$ | Slope of linear region | | Latitude | $L$ | $> 2.8$ stops | $D_{\min}+0.1$ to $D_{\max}-0.1$ |

Rust Verification:

#![allow(unused)]
fn main() {
assert!(hd_curve.gamma >= 0.50 && hd_curve.gamma <= 0.60);
assert!(hd_curve.d_max >= 2.8);
}

1.2 Reciprocity Failure

Standard: ISO 839 Measurement Object: Exposure time $t$ from 1/1000 s to 10 s, keeping $H = I \cdot t$ constant Pass Thresholds:

• Density drift $< 0.15$ (1/1000 s to 1 s)
• Color shift $\Delta E_{00} < 3.0$ (Layer failure desynchronization at long exposures)

Layer 2: Chemical Coupling (Dye Coupling)

2.1 Coupling Efficiency $\beta$

Standard: Kodak internal process specs (reverse engineerable) Measurement Object: Dye density generated per unit silver density Formula: $D_{\text{dye}} = \beta \cdot (D_{\text{silver}} - D_{\min})$ Pass Thresholds:

• Yellow layer $\beta_Y = 1.8 \pm 0.1$
• Magenta layer $\beta_M = 2.0 \pm 0.1$ (Higher extinction coefficient for Magenta dye)
• Cyan layer $\beta_C = 1.9 \pm 0.1$

2.2 Interlayer Interimage Effects (IIE)

Standard: ISO 4090 Measurement Object: Inhibition/Enhancement of lower layer development by upper layer exposure Test Chart: Red/Green/Blue monochromatic wedges + Neutral gray wedge side-by-side Pass Thresholds:

• Development inhibition rate $< 8%$ (High density in upper layer causes density drop in lower layer)
• Edge Effect MTF 50% frequency $> 40$ lp/mm

Layer 3: Optical Output

3.1 Status A Density (Print Viewing)

Standard: ISO 5-3 Measurement Object: RGB density measured through Wratten 106/92/88 filters Pass Thresholds:

• Neutral Gray $R=G=B \pm 0.05$ (Status A)
• Orange Mask Base Color $D_R - D_B = 0.70 \pm 0.05$ (Typical Color Negative Mask)

3.2 Spectral Transmittance $T(\lambda)$

Standard: ISO 5-1 Measurement Object: $T(\lambda) = 10^{-A(\lambda)}$, where $A(\lambda) = \sum C_i \varepsilon_i(\lambda)$ Pass Thresholds:

• RMSE with real film transmittance spectra $< 0.02$ (400-700 nm)

Layer 4: Colorimetric Accuracy

Metrics and Diagnosis

Dynamic Range and Highlight Roll-off

A common observation in film simulation is that digital dynamic range numbers win, but film highlight behavior wins on "character". This is not a "resolution" issue, but a dimensionality superiority of chemical response.

1. Dynamic Range: The Definition Trap

Key: CMOS hard clips at 14 bit (Full Well Capacity), whereas film dye continues to stack after $D_{\max}=2.8$—it just transitions from linear growth to logarithmic decay. Scanners might clip, but optical enlargers can retrieve this, giving film its "highlight retention" ability.

2. Three Stages of Chemical Highlight Roll-off

Stage 1: Linear Region (0–0.8 Density)

Silver halide grains respond linearly according to Beer-Lambert law: $D = \gamma \cdot \log_{10}(H)$. This looks like standard LOG encoding.

Stage 2: Shoulder Softening (0.8–1.8 Density)

Chemical Mechanisms:

1. Grain surface development centers saturate → Electron trapping efficiency $\eta$ drops, following Mott-Gurney Space Charge Limit: $J \propto V^2/d^3$
2. Developer oxidation product (QDI) decomposes at high pH → Coupling rate $k_{\text{coupling}}$ decays
3. Interlayer Inhibition: High density in upper layers releases Bromide ions (Br⁻) to lower layers, inhibiting their development (ISO 4090 IIE effect)

Mathematically: $$ D(H) = \gamma \log_{10}(H) \cdot \left(1 - \frac{H}{H_{\text{sat}}}\right)^{\alpha} $$ Where $\alpha \approx 0.65$ (Kodak Patent US4508812A). This is the core of highlight softening.

Stage 3: Toe Clipping (>1.8 Density)

Dye molecules stack to the Förster Resonance Energy Transfer distance (< 2 nm), quenching energy → Density growth approaches $D_{\max}$ asymptotically.

3. Why Digital Post-Processing Falls Short

3.1 Dimensionality Gap

CMOS highlights have only 16384 discrete levels (14 bit). Once clipped, data is gone. Film highlights still have 31-dimensional spectral changes, just with slower density growth. Post-processing can only interpolate, not extrapolate real spectra.

3.2 Irreversibility of Non-linear Coupling

Photoshop's "Highlight Compression" is $y = \log(x + \epsilon)$ or $y = x^{0.8}$, applied independently per channel. Film involves Yellow + Magenta + Cyan dye densities inhibiting each other. Mathematically, it's a ternary non-linear coupled system:

$$ \begin{cases} C_Y = f_Y(H_B, H_G, H_R) \ C_M = f_M(H_G, H_R, H_Y) \ C_C = f_C(H_R, H_Y, H_G) \end{cases} $$

Standard tools don't model "upper layer exposure poisoning lower layer".

3.3 Noise Texture Differences

CMOS highlight noise is Poisson + Readout Noise. Film is Granularity following Wiener Spectrum: $G(f) = \frac{a}{f^2 + b}$, which decreases as density increases. Digital noise addition is just "adding salt", lacking density-dependent grain size distribution.

4. Hard Simulation: Adding "Chemical Constraints"

4.1 Shoulder Softening Model

Apply a space charge limit after Exposure → Density mapping:

#![allow(unused)]
fn main() {
fn shoulder_softening(density: f32, shoulder_point: f32) -> f32 {
    if density > shoulder_point {
        let excess = density - shoulder_point;
        density - excess * excess / (shoulder_point + excess)
    } else {
        density
    }
}
}

Where shoulder_point = 0.8 (ISO 5800 standard point).

4.2 Interlayer Inhibition Matrix

#![allow(unused)]
fn main() {
// Interlayer effect matrix from ISO 4090 measurements
// Row i = Real Dye i / Theoretical Dye i
let interlayer: nalgebra::Matrix3<f32> = nalgebra::Matrix3::new(
    1.00, -0.08, -0.03, // Yellow inhibited by Magenta/Cyan
    -0.05, 1.00, -0.07, // Magenta inhibited by Yellow/Cyan
    -0.02, -0.06, 1.00, // Cyan inhibited by Yellow/Magenta
);

let dyes_real = interlayer * dyes_theoretical;
}

4.3 Dye Self-Absorption Correction

#![allow(unused)]
fn main() {
// Dye self-absorption causes spectral shift at high densities
fn dye_self_absorb(c: f32, t: &mut [f32]) {
    // c: dye concentration, t: transmittance spectrum
    for wl in 0..31 {
        // Beer's law deviation of 1-3% when density > 1.5
        let correction = 1.0 + (c - 1.5) * 0.02;
        t[wl] *= correction.clamp(0.97, 1.03);
    }
}
}