Glass Bunny - Showcase

glassbunny

The source cuda code can be found here, which you should noted is that the stb_image.h, stb_image_write.h and the stanford-bunny.obj must be placed at the same file level.

1. Rendering Equation

1.1 Radiance and Solid Angle

Radiance $L (x, ω)$ : the energy density at point $x$ along direction $ω$ , and the most fundamental quantity in rendering.
Hemispherical integration domain: the hemisphere above the surface normal $n$ , denoted $H^{2} (n)$ .
Cosine term: $cos θ = n \cdot ω_{i}$ , representing projected-area foreshortening due to the angle between incident direction and normal.

1.2 Surface Rendering Equation (Kajiya)

A common form for opaque surfaces is:

L_{o} (x, ω_{o}) = L_{e} (x, ω_{o}) + \int_{H^{2} (n)} f_{s} (x, ω_{i}, ω_{o}) L_{i} (x, ω_{i}) ∣ n \cdot ω_{i} ∣ d ω_{i}

Where:

$L_{o}$ : outgoing radiance
$L_{e}$ : emitted radiance
$f_{s}$ : BSDF (a unified form covering BRDF/BTDF)
$L_{i}$ : incident radiance (recursively contributed by other scene points)

The challenge for glass is that $f_{s}$ contains both reflection and transmission, and ideal glass is delta-distributed, so it cannot be handled directly with ordinary continuous PDFs.

2. Interface Physics of Liuli (Dielectric Glass)

Glass is usually modeled as a non-conductive dielectric. At the interface, three events may occur:

Fresnel reflection
Refraction (Snell)
Possible total internal reflection (TIR)

2.1 Snell’s Law

η_{i} sin θ_{i} = η_{t} sin θ_{t}

$η_{i}$ : index of refraction on incident side (air is about 1.0)
$η_{t}$ : index of refraction on transmitted side (glass is commonly 1.45~1.55)

2.2 Fresnel Term

Exact dielectric Fresnel (parallel/perpendicular polarization):

F = \frac{1}{2} [(\frac{η _{t} cos θ _{i} - η _{i} cos θ _{t}}{η _{t} cos θ _{i} + η _{i} cos θ _{t}})^{2} + (\frac{η _{i} cos θ _{i} - η _{t} cos θ _{t}}{η _{i} cos θ _{i} + η _{t} cos θ _{t}})^{2}]

In practice, Schlick’s approximation is widely used:

F (cos θ) \approx F_{0} + (1 - F_{0}) (1 - cos θ)^{5}, F_{0} = (\frac{η _{i} - η _{t}}{η _{i} + η _{t}})^{2}

2.3 Total Internal Reflection

When traveling from a higher IOR medium to a lower one, if the refraction angle has no real solution, then $F = 1$ and only reflection remains.

3. Glass “Color” Comes from Volumetric Absorption: Beer-Lambert

Ideally, non-absorbing glass is effectively colorless. Colored liuli comes from the medium absorption coefficient $σ_{a}$ (per RGB channel):

T (d) = exp (- σ_{a} d)

$d$ : geometric distance traveled by light inside glass
$T (d)$ : transmittance (per channel)

In path tracing, whenever “the previous segment is inside glass,” multiply throughput by $exp (- σ_{a} \cdot t_{hi t})$ .

This is why $σ_{a}$ controls glass tint strength in many implementations.

4. Liuli BSDF: From Ideal to Rough

4.1 Ideal Smooth Glass (Specular Delta)

An ideal interface has only two discrete directions:

Specular reflection direction $ω_{r}$
Refraction direction $ω_{t}$

It can be written in delta form (schematically):

f (ω_{i}, ω_{o}) = F δ (ω_{i} - ω_{r}) + (1 - F) δ (ω_{i} - ω_{t}) \cdot J

$J$ is the measure-conversion term for transmission (related to $η$ ). In many practical implementations, this is absorbed via event sampling probability + throughput update.

Corresponding sampling:

Sample reflection with probability $F$
Sample transmission with probability $1 - F$
Under TIR, force reflection

4.2 Rough Glass

Real liuli surfaces have microstructure, so microfacet models are used:

Normal distribution function $D (m)$ (commonly GGX/Beckmann)
Geometric masking-shadowing term $G$
Fresnel term $F$

Rough glass includes:

Microfacet reflection BRDF
Microfacet transmission BTDF

Characteristics:

Highlights/caustics become more diffuse instead of razor sharp
Noise convergence strongly depends on the sampling strategy (BSDF-only sampling is often still noisy)

5. Decomposing Lighting Integral: Direct + Indirect

Split the rendering equation into:

L_{o} = L_{e} + L_{d i rec t} + L_{in d i rec t}

$L_{d i rec t}$ : direct contribution from visible light sources at current point (can use NEE)
$L_{in d i rec t}$ : recursive bounce contribution

For glass scenes, relying only on “randomly bouncing into light” is very noisy, especially with small area lights and caustic paths. So typically:

Perform NEE on diffuse / non-delta vertices
Use MIS to combine BSDF sampling and light sampling

6. Path Tracing Estimator (Monte Carlo)

Let path throughput be $β$ . Update at each hit $x_{k}$ :

β \leftarrow β \cdot \frac{f _{s} ( ω _{i} , ω _{o} ) ∣ n \cdot ω _{i} ∣}{p ( ω _{i} )}

If inside glass, also apply absorption:

β \leftarrow β \cdot exp (- σ_{a} d)

Final pixel estimate:

\hat{L} = \frac{1}{N} s = 1 \sum N k \sum β_{k} L_{e mi t / e n v}

7. NEE (Next Event Estimation)

At current point $x$ , sample light point $y$ directly:

L_{d i rec t} \approx \frac{f _{s} ( x , ω _{i} , ω _{o} ) L _{e} ( y , - ω _{i} ) G ( x , y ) V ( x , y )}{p _{l i g h t} ( y )}

Where:

$G (x, y) = \frac{∣ n _{x} \cdot ω _{i} ∣ ∣ n _{y} \cdot ( - ω _{i} ) ∣}{∥ x - y ∥ ^{2}}$
$V (x, y)$ is visibility (shadow ray)
If an area light is sampled in area measure first, convert area PDF to solid-angle PDF:

p_{ω} = p_{A} \frac{∥ x - y ∥ ^{2}}{∣ n _{y} \cdot ( - ω _{i} ) ∣}

8. MIS (Multiple Importance Sampling)

When the same integral can be estimated by two strategies (light sampling / BSDF sampling), MIS reduces variance.

Common Power Heuristic:

w_{a} = \frac{p _{a}^{2}}{p _{a}^{2} + p _{b}^{2}}, w_{b} = \frac{p _{b}^{2}}{p _{a}^{2} + p _{b}^{2}}

Then:

Multiply light-sampling contribution by $w_{l i g h t}$
Multiply BSDF-hit-light contribution by $w_{b s df}$

This is more stable when lights are small, BSDFs are sharp, or both are complex.

9. Russian Roulette (Unbiased Path Termination)

Deep paths are expensive, so after bounce >= 3 a survival test is usually applied:

p_{s u r v i v e} = min (max (β), p_{ma x})

If terminated: break
If survived: $β / = p_{s u r v i v e}$

This keeps the estimator unbiased while significantly reducing average path length.

10. Engineering-Level Denoising

Glass + area light often produces high-frequency bright speckle noise (fireflies). Common combinations:

Physics-level variance reduction

NEE + MIS
Better BSDF/light sampling
Higher spp while camera is still

Statistical denoising

Firefly clamp (limit extreme samples)
Joint bilateral / temporal accumulation / A-Trous

Perceptual post-process

Perform filtering in HDR domain before tonemapping to avoid color shift

Note: clamping introduces slight bias, but greatly improves usability and is common in interactive preview.

11. Practical Parameter Ranges (Liuli)

$i or$ : 1.45~1.55 (typical glass)
$σ_{a}$ : 0~2 (depends on model scale)
$roughness$ : 0.02~0.15 (common micro-frosted liuli range)
$max_bounces$ : 8~12
$spp_per_frame$ : higher when camera is still, lower while moving

If model scale changes significantly, recalibrate $σ_{a}$ first, otherwise color can become too strong or too weak.