Rendering Pipeline

Complete technical reference for filmr's physics-based film simulation. Every formula is verified against source code.

Pipeline Overview

┌─────────────────────────────────────────────────────────┐
│  Input: sRGB u8 image                                   │
├─────────────────────────────────────────────────────────┤
│  Stage 0: Linearize          sRGB → Linear f32          │
│  Stage 1: Light Leak + Halation                         │
│  Stage 2: Full-Spectrum Develop                          │
│    ├ Pass 1: Per-pixel spectral propagation              │
│    ├ Pass 2: Scattering diffusion                        │
│    ├ Pass 2.5: White balance + warmth                    │
│    └ Pass 3: H-D curves + color matrix + inhibition      │
│  Stage 3: MTF blur                                       │
│  Stage 4: Grain                                          │
│  Stage 5: Output conversion   Density → sRGB u8         │
├─────────────────────────────────────────────────────────┤
│  Output: sRGB u8 image                                  │
└─────────────────────────────────────────────────────────┘

Stage 0: Linearize

Digital images are stored in sRGB gamma encoding. The first step converts each pixel to linear light, which represents actual physical irradiance.

\[ \text{linear} = \begin{cases} \dfrac{v}{12.92} & v \le 0.04045 \\[6pt] \left(\dfrac{v + 0.055}{1.055}\right)^{2.4} & v > 0.04045 \end{cases} \]

A 256-entry lookup table is precomputed for speed.

Stage 1: Light Leak + Halation

Halation

In real film, light passes through the emulsion layers, reflects off the film base, and scatters back into the emulsion. This creates a warm glow around bright highlights.

1. Compute luminance: \( L = 0.2126R + 0.7152G + 0.0722B \)
2. Extract highlights above threshold \( T \)
3. Gaussian blur with \( \sigma = W \times \text{halation_sigma} \)
4. Blend back: \( \text{pixel} \mathrel{+}= \text{blurred} \times \text{tint} \times \text{strength} \)

The tint is typically warm orange [1.0, 0.6, 0.3], matching the reddish glow seen in real film scans.

Stage 2: Full-Spectrum Develop

This is the core of the simulation. Light is propagated wavelength-by-wavelength through the film's physical layer structure — modelling how real photons interact with silver halide crystals, dye layers, and the film base.

The Film Layer Stack

A colour negative film is a sandwich of thin layers:

Each layer has:

• Refractive index \( n \) (~1.50–1.65)
• Spectral absorption \( \alpha(\lambda) \) — 81 values from 380–780 nm
• Scattering coefficient \( s \) — Mie/Rayleigh scattering in emulsion
• Thickness \( d \) in micrometers

Precomputation

Before processing pixels, the engine precomputes per-layer coefficients for all 81 wavelength bins:

Fresnel reflection at each interface (how much light bounces back when entering a new medium):

\[ R = \left(\frac{n_1 - n_2}{n_1 + n_2}\right)^2, \quad t = 1 - R \]

Beer-Lambert transmission (how much light survives passing through a layer):

\[ \tau_i = \exp\bigl(-(\alpha_i + s) \cdot d\bigr) \]

Absorption deposit (fraction of light captured by an emulsion layer):

\[ \delta_i = (1 - \tau_i) \times \frac{\alpha_i}{\alpha_i + s} \]

This separates absorption (which exposes the film) from scattering (which just removes light from the beam).

Pass 1: Per-Pixel Spectral Propagation

Spectral Reconstruction

Each pixel's RGB values are expanded into a full 81-wavelength spectrum using precomputed sRGB spectral sensitivities and the D65 daylight illuminant:

\[ P(\lambda_i) = R \cdot m_{R,i} + G \cdot m_{G,i} + B \cdot m_{B,i} \]

where \( m_{ch,i} = S_{ch}(\lambda_i) \times D_{65}(\lambda_i) \).

Forward Pass (Light Entering the Film)

For each layer, top to bottom:

1. Fresnel: \( P_i \mathrel{\times}= t \) — some light reflects at the interface
2. Absorb (emulsion only): \( E_{ch,i} \mathrel{+}= P_i \times \delta_i \) — film captures energy
3. Attenuate: \( P_i \mathrel{\times}= \tau_i \) — remaining light continues downward

What happens to different colours:

Backward Pass (Base Reflection)

Light that reaches the base partially reflects back:

\[ P_{\text{back},i} = P_{\text{residual},i} \times \left(\frac{n_{\text{base}} - 1}{n_{\text{base}} + 1}\right)^2 \]

This reflected light travels back up through all layers, depositing additional exposure in each emulsion. This is the physical origin of halation — the red emulsion (closest to the base) receives the most reflected light.

Integration

The per-wavelength exposure is integrated into three scalar values using the trapezoidal rule:

\[ E_{ch} = \left[\sum_{i=1}^{79} e_i + \frac{1}{2}(e_0 + e_{80})\right] \times 5\text{ nm} \]

Finally, exposure is scaled by a normalization factor and the user's exposure time setting.

Pass 2: Scattering Diffusion

Scattering within emulsion layers causes light to spread laterally, slightly blurring the image:

\[ \sigma_{\text{px}} = \frac{\sum_l d_l \cdot s_l}{36000} \times W \]

In practice this is sub-pixel for most film stocks and has negligible visual effect.

Pass 2.5: White Balance

Optionally equalizes the average colour of the image:

\[ \text{gain}{ch} = 1 + \left(\frac{\bar{L}}{\bar{C}{ch}} - 1\right) \times \text{strength} \]

Warmth shifts the red/blue balance: \( R \mathrel{\times}= (1 + w \cdot 0.1) \), \( B \mathrel{\times}= (1 - w \cdot 0.1) \).

Pass 3: H-D Curves + Color Matrix + Inhibition

The Characteristic Curve

The Hurter-Driffield (H-D) curve is the fundamental relationship between exposure and density in photographic film. It's the S-shaped curve that gives film its distinctive tonal response.

\[ D = D_{\min} + (D_{\max} - D_{\min}) \cdot \frac{1}{1 + e^{-k \cdot x}} \]

where:

\[ x = \log_{10} E - \log_{10} E_0, \quad k = \frac{4\gamma}{D_{\max} - D_{\min}} \]

• Toe (low exposure): gentle roll-off, shadow detail
• Linear region: proportional response, midtones
• Shoulder (high exposure): compression, highlight roll-off

Shoulder Softening

At very high densities, silver halide crystals saturate:

\[ D > D_s: \quad D' = D - \frac{(D - D_s)^2}{D_s + (D - D_s)} \]

Color Matrix

Models inter-layer dye coupling. Each channel's net density is mixed:

\[ \begin{pmatrix} D_R' \\ D_G' \\ D_B' \end{pmatrix} = \mathbf{M} \begin{pmatrix} D_R - D_{\min,R} \\ D_G - D_{\min,G} \\ D_B - D_{\min,B} \end{pmatrix} + \begin{pmatrix} D_{\min,R} \\ D_{\min,G} \\ D_{\min,B} \end{pmatrix} \]

Interlayer Inhibition

During development, chemical byproducts from one layer suppress development in adjacent layers (DIR coupler effect). This enhances colour separation.

The inhibition operates on density deviation from the mean, so neutral grays are unaffected:

\[ \bar{D} = \frac{D_R + D_G + D_B}{3}, \quad \Delta D_i = D_i - \bar{D} \]

\[ D_i^{,\text{final}} = D_i + \sum_j \text{inh}_{i,j} \cdot \Delta D_j \]

Off-diagonal values are negative (suppression), increasing the spread between channels.

Stage 3: MTF Blur

Every film has a finite resolving power. This is modelled as a Gaussian blur:

\[ \sigma = \frac{0.5}{\text{resolution_lp_mm}} \times \frac{W}{36} \]

where \( W \) is image width in pixels and 36 mm is the standard 35mm film width.

Stage 4: Grain

Film grain arises from the random distribution of silver halide crystals. The noise variance depends on density:

\[ \sigma^2 = \bigl(\alpha \cdot D^{1.5} + \sigma_{\text{read}}^2 + \sigma_{\text{shot}}^2\bigr) \times \bigl(1 + r \cdot \sin(\pi D)\bigr) \]

• \( \alpha \cdot D^{1.5} \): grain increases with density (more developed crystals)
• \( \sigma_{\text{read}}^2 \): base fog noise
• \( \sigma_{\text{shot}}^2 = s \cdot e^{-2D} \): photon shot noise (strongest in shadows)
• Roughness \( r \): frequency modulation in midtones

Two grain layers are blended: fine crystals everywhere, plus coarse clumps that appear in highlights (silver halide aggregation at high density).

Stage 5: Output Conversion

Converts the density image to a viewable photograph.

Density → Transmission

\[ T = 10^{-(D - D_{\min})} \]

High density = low transmission = dark on the negative.

Dye Self-Absorption

At high densities, Beer's Law deviates slightly:

\[ D > 1.5: \quad T' = T \times \text{clamp}\bigl(1 + 0.02(D - 1.5),; 0.97,; 1.03\bigr) \]

Print Simulation

The negative is "printed" onto paper with its own gamma:

\[ n = \frac{T_{\max} - T}{T_{\max} - T_{\min}}, \quad \text{output} = n^{\gamma_{\text{paper}}} \]

sRGB Encoding

\[ \text{sRGB} = \begin{cases} 12.92 \cdot v & v \le 0.0031308 \\[4pt] 1.055 \cdot v^{1/2.4} - 0.055 & v > 0.0031308 \end{cases} \]

Spectral Data

All spectral computations use standard reference data:

• CIE 1931 2° Standard Observer — colour matching functions \( \bar{x}, \bar{y}, \bar{z} \)
• CIE Standard Illuminant D65 — daylight spectrum
• sRGB primaries — derived from XYZ CMF × IEC 61966-2-1 matrix

Wavelength range: 380–780 nm, 5 nm steps, 81 bins.

Performance

Key optimization: layer coefficients (Fresnel, Beer-Lambert, deposit) are precomputed once per frame, eliminating ~1782 exp() calls per pixel.

Verification

72 tests ensure physical correctness: