【Computer Graphics】Offline Rendering

Heskey0 (Bilibili)

December 2021

Based On Mark Pauly's Thesis[1999] and 《PBRT》

Contents

Chapter 1 . Sampling Techniques In Path Tracing

1.1. Inverse CDF (Cumulative Density Function)

There are many techniques for generating random variates from a specified probability distribution such as the normal, exponential, or gamma distribution. However, one technique stands out because of its generality and simplicity: the inverse CDF sampling technique.

The algorithm is as follows:

1. Obtain or generate a draw (realization) \(u\) from the standard uniform distribution \(U∼Unif(0,1)\)
2. The draw \(x\) from the CDF \(F(x)\) is given by \(x=F^{-1}(u)\)

Example of inverse CDF method:

Let \(p(\theta)=sin\theta\) be the probability density function, \(F(\theta)=1-cos\theta\) the cumulative density function of \(\theta\)

1. generate a draw \(\xi\) from the standard uniform distribution \(U∼Unif(0,1)\)

2. the draw \(\theta\) from the PDF \(p(\theta)\) is given by \(\theta=F^{-1}(\xi)=arccos(1-\xi)\)

1.2. Uniformly sampling a hemisphere

a uniform distribution means that the density function is a constant, so we know that \(p(x)=c\)

\[\int_{S^2} p(\omega) d\omega=1\Longrightarrow c\int_{S^2}d\omega=1\Longrightarrow c=\frac1{2\pi} \]

hence \(p(\omega)=\frac1{2\pi}\), \(d\omega=sin\theta d\theta d\phi\), \(p(\theta,\phi)=\frac{sin\theta}{2\pi}\)

Consider sampling \(\theta\) first. To do so, we need \(\theta\)'s marginal density function \(p(\theta)\):

\[p(\theta)=\int_0^{2\pi}p(\theta,\phi)d\phi=\int_0^{2\pi}\frac{sin\theta}{2\pi}d\phi=sin\theta \]

Now, compute the conditional density for \(\phi\):

\[p(\phi|\theta)=\frac{p(\theta,\phi)}{p(\theta)}=\frac1{2\pi} \]

Notice that the density function for \(\phi\) itself is uniform, then use the inverse CDF sampling technique to sample each of these PDFs in turn

\[P(\theta)=\int_0^{\theta}sin\theta^{\prime}d\theta^{\prime}=1-cos\theta\Longrightarrow \theta=arccos\xi_1 \]

\[P(\phi|\theta)=\int_0^{\phi}\frac1{2\pi}d\phi^{\prime}=\frac{\phi}{2\pi}\Longrightarrow\phi=2\pi\xi_2 \]

Converting these back to Cartesian coordinates, we get the final sampling formula:

\[x=sin\theta cos\phi=cos(2\pi\xi_2)\sqrt{1-\xi_1^2} \]

\[y=sin\theta sin\phi=sin(2\pi\xi_2)\sqrt{1-\xi_1^2} \]

\[z=cos\theta=\xi_1 \]

1.3. Sample area light

def sample_area_light(hit_pos, pos_normal):
    # sampling inside the light area
    x = ti.random() * light_x_range + light_x_min_pos
    z = ti.random() * light_z_range + light_z_min_pos
    on_light_pos = ti.Vector([x, light_y_pos, z])
    return (on_light_pos - hit_pos).normalized()

1.4. Cosine-weighted Sampling

\[p(\omega)\propto cos\theta\Longrightarrow p(\omega)=c*cos\theta \]

\[\int_{S^2}p(\omega)d\omega=1\Longrightarrow c=\frac1{\pi}\Longrightarrow p(\omega)=\frac{cos\theta}{\pi}\Longrightarrow p(\theta,\phi)=\frac1{\pi}cos\theta sin\theta \]

We could use the inverse CDF sampling technique as before, but instead we can use a technique known as Malley’s method to generate these cosine-weighted points.

The algorithm is as follows:

1. sample a unit disk (Concentric Mapping)

   \[r=x;\phi=\frac yx\frac{\pi}4 \]

2. project up to the unit hemisphere

1.5. Multiple importance sampling

MIS allows us to combine \(m\) different sampling strategies to produce a single unbiased estimator by weighting each sampling strategy by its probability distribution function.

\[\langle I_j\rangle=\sum_{i=1}^m\frac1{n_i}\sum_{j=1}^{n_i}w_i(X_{i,j})\frac{f(X_{i,j})}{p_i(X_{i,j})} \]

where \(X_{i,j}\) are independent random variables drawn from some distribution function pi and \(w_i(X_{i,j})\) is some heuristic for weighting each sampling technique with respect to pdf.

balance heuristic:

\[w_s(x)=\frac{n_sp_s(x)}{\sum_in_ip_i(x)} \]

power heuristic:

\[w_s(x)=\frac{(n_sp_s(x))^{\beta}}{\sum_i(n_ip_i(x))^{\beta}} \]

Veach determined empirically that \(\beta=2\) is a good value