【计算机图形学】体渲染专题 (二)

首先，老规矩：

未经允许禁止转载(防止某些人乱转，转着转着就到蛮牛之类的地方去了)

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【Computer Graphics】Photorealistic Rendering of Volume Effect

Heskey0 (Bilibili)

December 2021

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

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

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作者：Heskey0
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