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Lec 16 Ray Tracing 4(Monte Carlo Path Tracing)

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Monte Carlo Integration

Why: we want to solve an integral, but it can be too difficult to solve \(\int_a^b f(x)dx\) analytically.

What & how: estimate the integral of a function by averaging random samples of the function's value.

Monte Carlo estimator:

\[X_i\sim p(x)\quad\int_a^b f(x)dx \]
\[\int f(x)dx=\frac{1}{N}\sum_{i=1}^N\frac{f(X_i)}{p(X_i)}\]
  • The more samples, the less variance.
  • Sample on X, integration on X.

Path Tracing

Problems of Whitted-Style:

  • Glossy texure (Utah teapot)
  • Color bleeding (Cornell box)

A simple example: direct illumination

\[ L_o(\mathrm{p},\omega_o)=\int_{\Omega^+}L_i(\mathrm{p},\omega_i)f_r(\mathrm{p},\omega_i,\omega_o)(n\cdot\omega_i)\mathrm{d}\omega_i \]
  • \(f(x)\): \(L_i(\mathrm{p},\omega_i)f_r(\mathrm{p},\omega_i,\omega_o)(n\cdot\omega_i)\)
  • \(pdf(\omega)\): \(1/2\pi\)

Monte Carlo integration:

\[ L_o(\mathrm{p},\omega_o)\approx\frac{1}{N}\sum_{i=1}^N\frac{L_i(\mathrm{p},\omega_i)f_r(\mathrm{p},\omega_i,\omega_o)(n\cdot\omega_i)}{pdf(\omega_i)} \]

Code:

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shade (p, wo)
    Randomly choose N directions wi~pdf
    Lo = 0.0
    For each wi
        Trace a ray r(p, wi)
        If ray r hit the light
            Lo += (1 / N) * L_i * f_r * cosine / pdf(wi)
    Return Lo

Introduce global illumination:

The light reflect from Q to P, equals to the direct illumination ar Q observed at P.

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shade (p, wo)
    Randomly choose N directions wi~pdf
    Lo = 0.0
    For each wi
        Trace a ray r(p, wi)
        If ray r hit the light
            Lo += (1 / N) * L_i * f_r * cosine / pdf(wi)
        Else If ray r hit an object at q
            Lo += (1 / N) * shade(q, -wi) * f_r * cosine / pdf(wi)
    Return Lo

Problems 1: Explosion of rays as bounces go up
Rays will not explode iff N = 1.

This is "path tracing".

Code:

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shade (p, wo)
    Randomly choose ONE directions wi~pdf(w)
    Lo = 0.0
    For each wi
        Trace a ray r(p, wi)
        If ray r hit the light
            Return L_i * f_r * cosine / pdf(wi)
        Else If ray r hit an object at q
            Return shade(q, -wi) * f_r * cosine / pdf(wi)

Ray Generation

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ray_genaration(camPos, pixel)
    Uniformly choose N sample positions within the pixel
    pixel_radiance = 0.0
    For each sample in the pixel
        Shoot a ray t(camPos, cam_to_sample)
        If ray r hit the scene at p
            pixel_radiance += 1 / N * shade(p, sample_to_cam)
    Return pixel_radiance

Problem 1: The recursion won't stop.
Solution: Russian Roulette (RR)

Suppose we manually set a probability P (0<P<1).
With probability P, shoot a ray and return the shading result divided by P is Lo / P.
With probability 1-P, don't shoot a ray adn you'll get 0.

In this way, you can still expect to get Lo:
E = P _ (Lo / P) + (1 - P) _ 0 = Lo

Code:

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(Add)
    Manually specify a probability P_RR
    Randomly select ksi in a uniform dist. in [0, 1]
    If (ksi > P_RR) retunr 0.0

    Return ... / P_RR

Problem 2: Inefficient. When SPP (samplees per pixel) is low, get noisy results

Only a few rays hit the light, so a lot of rays are "wasted".

Solution: sample on the light

Assume pdf = 1 / A. But the rendering equation intefrates on the solid angle.

To sample on the light and integrate on the light, need the relationship between \(d\omega\) and \(dA\).

sample the light

\[d\omega=\frac{dA\cos\theta'}{|| x'-x ||^2}\]

Rewrite the rendering equation:

\[ \begin{align*} L_o(\mathrm{p},\omega_o)&=\int_{\Omega^+}L_i(\mathrm{p},\omega_i)f_r(\mathrm{p},\omega_i,\omega_o)\cos\theta\,\mathrm{d}\omega_i \\ &=\int_A L_i(\mathrm{p},\omega_i)f_r(\mathrm{p},\omega_i,\omega_o)\frac{\cos\theta\cos\theta'}{|| x'-x ||^2}\mathrm{d}A \end{align*} \]

Code:

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shade(p, wo)
    # Contribution from the light source.
    Uniformlu sanple the light at x' (pdf_light = 1 / A)
    L_dir = L_i * f_r * cos theta * cos theta' / |x' - p|^2 / pdf_light

    # Contribution from other reflectors
    L_indir = 0.0
    Test Russian Roulette with probability P_RR
    Uniformly sample the hemisphere toward wi (pdf_hemi = 1 / 2pi)
    Trace a ray r(p, wi)
    If ray r hir a non-emitting object ar q
        L_indir = shade(q, -wi) * f_r * cos theta / pdf_hemi / P_RR

    Return L_dir + L_indir

Final problem: need to test if the ray os not blocked in the middle.

Hard to handle point light source.

Things haven't covered:

  • Uniformly sampling the hemisphere

  • How? And in gengral. how to sample any function?

  • Monte Carlo intefration allows arbitatry pdfs

  • What's the best choice? (importance sampling)

  • Do random numbers matter?

  • Yes (low discrepancy sequences)

  • Sample the hemimsphere and the light

  • Can I combine them? Yes (multipe immp. sampling)

  • The radiance of a pixel is the average of radiance on all paths passing through it

  • Why? (pixel reconstruction filter)

  • Is the radiance of a pixel the color of a pixel?

  • No. (gamma correction, curves, color space)