Clamp metallic Fresnel after specular_weight scaling via F0 and F82

[jamportz]

Fixes #314.

Summary

In v1.2 (#238) the metallic Fresnel is $\mathrm{clamp}(\xi_s \mathbf{F}_{82}(\mu), 0, 1)$, where $\xi_s$ = specular_weight. The MaterialX graph can't express this clamp, because the Fresnel is evaluated inside generalized_schlick_bsdf. So the reference graph just sets the lobe weight to $\xi_s$, and for $\xi_s > 1$ it reflects more light than it receives (see the furnace test below).

As discussed in the meeting, this PR instead clamps the two colors that parametrize the F82-tint curve:

$$\mathbf{F}_{\mathrm{metal}}(\mu) = \min(\xi_s, 1) \, \max\bigl(\mathbf{F}_{82}(\mu), 0\bigr)$$

where $\mathbf{F}_{82}(\mu)$ is the existing F82-tint curve, built from the colors

$$\mathbf{F}_0 = \min(k \, \mathtt{base\_weight} \, \mathtt{base\_color}, 1) \ , \quad \mathbf{C}_s = \min(k \, \mathtt{specular\_color}, 1) \ , \quad k = \max(\xi_s, 1) \ .$$

• $\xi_s \le 1$: identical to before ($k = 1$). The whole metallic lobe is scaled by $\xi_s$, so there is no look change in the normal range. The $\xi_s = 1$ renders below are pixel-identical to the old graph.
• $\xi_s > 1$: $\mathbf{F}_0$ and the F82 edge tint $\mathbf{C}_s$ are both boosted by $\xi_s$, each clamped to $[0, 1]$. The result is always a valid F82-tint curve, so it is bounded by 1 with no upper clamp. As $\xi_s \to \infty$ it tends to a perfect white mirror, the same limit as the previous clamp.
• Expressible in MaterialX with standard nodes, via the weight, color0 and color82 inputs of generalized_schlick_bsdf.

A footnote records an equivalent form for generalized-Schlick implementations: $\mathbf{F}_0 = \min(\xi_s \mathtt{base\_weight} \mathtt{base\_color}, 1)$ and $\mathbf{F}_{90} = \min(\xi_s, 1)$, with no lobe weight. This works because $\mathbf{C}_s$ is relative to the Schlick curve, so scaling the whole curve by $\xi_s \le 1$ is the same as scaling $\mathbf{F}_0$ and $\mathbf{F}_{90}$.

Changes

• Spec (index.html): $\mathbf{F}_{\mathrm{metal}}$ is now stated in terms of the existing $\mathbf{F}_{82}$ curve, with $\mathbf{F}_0$ and $\mathbf{C}_s$ defined as the boosted, clamped colors. Added the $\mathbf{F}_{90}$ footnote and updated the specular_weight description in the metal parameter table.
• MaterialX (reference/open_pbr_surface.mtlx): added metal_weight $= \min(\xi_s, 1)$, metal_boost $= \max(\xi_s, 1)$, metal_F0 and metal_F82, and wired them into all four metal generalized_schlick_bsdf nodes (base, thin-film, haze, haze + thin-film).
  • The coat-darkening albedo estimate Emetal is now metal_weight * metal_F0 $= \min(\xi_s \mathtt{base\_weight} \mathtt{base\_color}, 1)$. Note: this also brings base_weight into Emetal; previously it used base_color alone.
• CHANGELOG.md: added an entry.

Furnace test

A smooth ($\mathtt{specular\_roughness} = 0$) metal sphere in a uniform white environment of radiance 1. Each pixel is then exactly $\mathbf{F}_{\mathrm{metal}}(\mu)$, so anything brighter than the background creates energy. The images are shown one stop down, so the background is mid-grey. "max" is the largest pixel value in the image; the background is 1, so a physical result has max = 1. All renders use the MaterialX graph compiled to OSL and rendered in Arnold.

Rows: v1.2 graph is the current graph (unclamped); A boosts $\mathbf{F}_0$ only; B (this PR) boosts $\mathbf{F}_0$ and $\mathbf{C}_s$; C boosts $\mathbf{F}_0$ and the absolute 82° reflectivity (see "Alternatives considered").

Gold: $\mathtt{base\_color} = (1, 0.72, 0.315)$, $\mathtt{specular\_color} = (1, 0.973, 0.597)$

	$\xi_s = 1$	$\xi_s = 2$	$\xi_s = 4$	$\xi_s = 8$
v1.2 graph	max 1.00	max 2.00	max 4.00	max 8.00
A	max 1.00	max 1.00	max 1.00	max 1.00
B (this PR)	max 1.00	max 1.00	max 1.00	max 1.00
C	max 1.00	max 1.00	max 1.00	max 1.00

Tinted metal: $\mathtt{base\_color} = (0.5, 0.5, 0.5)$, $\mathtt{specular\_color} = (0.3, 0.55, 0.9)$

	$\xi_s = 1$	$\xi_s = 2$	$\xi_s = 4$	$\xi_s = 8$
v1.2 graph	max 1.00	max 1.60	max 3.18	max 6.35
A	max 1.00	max 1.00	max 1.00	max 1.00
B (this PR)	max 1.00	max 1.00	max 1.00	max 1.00
C	max 1.00	max 1.00	max 1.00	max 1.00

The old graph glows once $\xi_s > 1$: the sphere becomes brighter than the environment illuminating it, with pixel values up to about $\xi_s$. A, B and C never exceed the background. As $\xi_s$ grows they tend to a perfect white mirror, which is correctly invisible in a furnace. The differences between A, B and C are confined to a thin band at the silhouette (82° maps to the outer ~1% of the sphere's radius), so they are compared using the Fresnel curves under "Alternatives considered".

Shader ball (gold, roughness 0.2)

Same rows as above.

	$\xi_s = 1$	$\xi_s = 2$	$\xi_s = 4$	$\xi_s = 8$
v1.2 graph
A
B (this PR)
C

Alternatives considered

All three options below are identical for $\xi_s \le 1$; they differ only in the edge tint used for $\xi_s > 1$.

	Edge tint for $\xi_s > 1$	Behaviour
A	$\mathbf{C}_s = \mathtt{specular\_color}$ (boost $\mathbf{F}_0$ only)	Keeps the edge dip, but boosting never brightens the edges. Hits a ceiling below a white mirror for tinted metals. Darker than the previous behaviour, by up to 5–8% in albedo.
B (this PR)	$\mathbf{C}_s = \min(k \mathtt{specular\_color}, 1)$	Stays within ~1–3% albedo of the previous behaviour and has the same $\xi_s \to \infty$ limit. The edge tint washes out quickly: it has no effect once $\xi_s \mathtt{specular\_color} \ge 1$. It is somewhat brighter than a pure curve scaling around 82°, because the tint boost compounds with the boosted Schlick curve.
C	boost the absolute 82° reflectivity $\mathbf{C}_s \mathbf{F}_\mathrm{Schlick}(\bar\mu)$ by $k$, clamped	Most faithful to "scale the curve", and always lies between A and B. It needs extra graph nodes and is nearly identical to B for realistic metals.

Analytic Fresnel curves for single channels, compared with the old $\xi_s \mathbf{F}_{82}$ graph and the v1.2 spec clamp:

🤖 Generated with Claude Code

[fpsunflower]

LGTM!

[jstone-lucasfilm]

This looks great to me, @jamportz, and clamping before the BSDF evaluation should work well in MaterialX/OSL/MDL.

One minor, non-blocking suggestion for the specification text: the statement that the curve "tends to a perfect (white) mirror as specular_weight -> inf" holds only where the colors are non-zero. Since the boost is multiplicative, a zero channel in specular_color would stay at zero for any specular_weight, so the edge reflectance in that channel would remain pinned at zero.

Perhaps this could be qualified as "tends to a perfect (white) mirror as specular_weight -> inf, for non-zero colors"?