Filament Painting Science

🔬 The Science of Filament Painting & Optical TD Blending

Unlike 2D paper printing (which deposits ink droplets that chemically absorb light) or digital screens (which emit additive RGB light), 3D filament painting is an optical transmission phenomenon.

When light strikes the surface of a multi-color 3D relief, it penetrates translucent plastic layers, undergoes wavelength-dependent absorption and scattering, reflects off lower opaque layers, and emerges back to the viewer's eye.

TailWag Studio calculates this behavior in real time using the Beer-Lambert Law, Kubelka-Munk scattering theory, and CIE LAB perceptual color spaces ($\Delta E_{00}$).

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📏 1. What is Transmission Distance ($TD$)?

Transmission Distance ($TD$) is the fundamental optical metric of 3D printing filament.

Note

Definition: Transmission Distance is the thickness of printed filament (in millimeters) required for light to drop to $1/e$ ($\approx 36.8%$) of its incident intensity, or practically, the thickness where a translucent color completely blocks the visibility of a contrasting color beneath it.

• Low $TD$ (0.1 mm – 0.6 mm): Extremely opaque filament (e.g. Matte Black, Deep Navy, Dark Brown). These filaments block underlying light almost immediately within 1 to 3 layers.
• Medium $TD$ (1.0 mm – 3.0 mm): Semi-translucent colors (e.g. Red, Orange, Green, Blue). Light passes through 5 to 15 layers ($0.4\text{ mm} - 1.2\text{ mm}$), creating smooth tonal transitions as thickness increases.
• High $TD$ (4.0 mm – 8.0+ mm): High translucency filaments (e.g. White, Jade White, Yellow, Clear). Light penetrates deeply, allowing dark underlayers to show through as subtle tints.

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💡 2. The Optical Physics Models

Beer-Lambert Absorption Law

Light attenuation through a translucent layer of thickness $z$ and absorption coefficient $\alpha$:

$$I(z) = I_0 \cdot e^{-\alpha \cdot z}$$

Where:

• $I_0$ = Incident light intensity
• $z$ = Physical layer thickness (sum of layer heights, e.g. $N \times 0.08\text{ mm}$)
• $\alpha = \frac{1}{TD}$ = Wavelength-specific absorption coefficient of the filament material

When stacking multiple filament layers ($F_1, F_2, \dots, F_k$) each with thickness $z_k$ and color transmission spectra $T_k(\lambda)$:

$$T_{\text{total}}(\lambda) = \prod_{k=1}^m \exp\left(-\frac{z_k}{TD_k} \cdot [1 - T_k(\lambda)]\right)$$

Kubelka-Munk Scattering Model

For diffuse scattering materials like PLA with titanium dioxide ($\text{TiO}_2$) white pigment or carbon black, Beer-Lambert alone is insufficient because light scatters forward and backward between layers.

Kubelka-Munk calculates absorption ($K$) and scattering ($S$):

$$\frac{K}{S} = \frac{(1 - R_\infty)^2}{2 R_\infty}$$

Where $R_\infty$ is the reflectance of an infinitely thick sample. TailWag Studio combines forward Beer-Lambert transmission with Kubelka-Munk reflectance matrices to predict exact RGB values on the screen.

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👁️ 3. Perceptual Color Matching: CIEDE2000 ($\Delta E_{00}$)

Human eyes do not perceive color differences linearly in RGB space. A unit change in green is perceived very differently from a unit change in blue or yellow.

TailWag Studio converts all filament spectra and source pixels into *CIE $L^a^b^$ color space and evaluates matching accuracy using the CIEDE2000 ($\Delta E_{00}$) formula:

$$\Delta E_{00} = \sqrt{\left(\frac{\Delta L'}{k_L S_L}\right)^2 + \left(\frac{\Delta C'}{k_C S_C}\right)^2 + \left(\frac{\Delta H'}{k_H S_H}\right)^2 + R_T \left(\frac{\Delta C'}{k_C S_C}\right)\left(\frac{\Delta H'}{k_H S_H}\right)}$$

Interpreting $\Delta E$ Values

$\Delta E_{00}$ Range	Perceptual Difference	Quality Assessment in TailWag
$0.0 - 1.0$	Imperceptible to the human eye	Flawless match
$1.0 - 2.5$	Noticeable only by trained eyes on close inspection	Excellent fidelity
$2.5 - 5.0$	Noticeable color shift, but acceptable in print art	Good artistic reproduction
$> 5.0$	Distinctly different hue or tone	Requires additional filament or manual tuning

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🖼️ 4. Front-Lit (Filament Painting) vs. Back-Lit (Lithophanes)

TailWag Studio supports both optical modalities with a single toggle:

+-------------------------------------------------------+
|  FRONT-LIT (Filament Painting)                        |
|  Light Source ===> Surface ==> [Reflected to Eye]     |
|                    Black Base Layer (Absorbs All)     |
+-------------------------------------------------------+
|  BACK-LIT (Traditional Lithophane)                    |
|  Backlight Source ===> [Transmitted Through] ===> Eye |
|                        Varying Thickness Modulates    |
+-------------------------------------------------------+

1. Front-Lit Filament Painting:
   • Best for wall art, framed portraits, plaques, and decorative tiles.
   • Requires a black or dark base layer to act as an optical black hole, absorbing all un-reflected light.
   • Viewed in normal room lighting.
2. Back-Lit Lithophane:
   • Best for nightlights, lamp shades, and window ornaments.
   • Thicker regions appear dark; thinner regions appear bright when illuminated from behind.