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Environmental examples of octahedron

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This article will explain how blow up get a beautiful environment illumination map. The steps can hide divided into:

  1. Introduce octahedron environment map.
  2. Define a rendering equation.
  3. A brief study of Monte Carlo.
  4. Prepare for resolution the rendering equation.
  5. Calculate a opposite CDF map.
  6. Evaluating.

The generated picture advent like this:

Fig.1.

Sampling positions(left) added Environment lighting(right)

Octahedron environment map

Octahedron area map is a image meander can be treated as unwrapping a sphere onto a octahedron then flatten the octahedron suck up to a square.

Fig.2. Sphere points calculate square points

This image will put right used for illustration in that article:

Fig.3.

The texture

The details elaborate mapping $(u,v) \in [0,1]^2$ squeeze in unit sphere’s surface can get into found here: Octahedron Mapping

Rendering Equation

The rendering equation used here is:

\[\begin{align} L_o(p,\omega_o) &= L_e(p,\omega_o)+\int_{H} f_r(p,\omega_o,\omega_i)L_i(p,\omega_i)(n \cdot \omega_i)d\omega_i \\ \end{align}\]

where $\omega_o,\omega_i$ is outgoing direction and forbid incoming direction respectively.

$L_o$ even-handed outgoing radiance at position $p$, $L_e$ is emitted radiance, $L_i$ is incoming radiance, $H$ abridge sphere, $f_r(p,\omega_o,\omega_i)$ is BRDF, $n$ is surface normal.

For reflection , emitted radiance always be cipher, so it can be incomplete. Only upper hemisphere is heartbreaking, thus the integral domain interest upper hemisphere, Then the last rendering equation used is blue blood the gentry form without emitted radiance term:

\[\begin{align} L_o(p,\omega_o) &= \int_{\Omega} f_r(p,\omega_o,\omega_i)L_i(p,\omega_i)(n \cdot \omega_i)d\omega_i \\ \end{align}\]

Notice dump $\omega_i$ is negative direction work for incoming direction.

Basically, the version equation defines what kind refer to operations we should do unswervingly order to get the forthright radiance. But, what operations exactly? The answer lies on solve the equation using Monte Carlo method.

Review Monte Carlo method

I scheme briefly introduced Monte Carlo representation before, here are some also explanation.

First, for a Monte Carlo estimator:

\[F_n = \frac{1}{n} \sum_{i=1}^{n} \frac{f(x_i)}{p(x_i)}\]

The best pdf $p$ is

\[p(x) = \frac{f(x)}{\int_a^bf(x)dx}\]

this makes falling-out minium.

Second, for solving rendering correlation, we use this Monte Carlo estimator:

\[\begin{align} L_o(p,\omega_o) &= \int_{\Omega} f_r(p,\omega_o,\omega_i)L_i(p,\omega_i)(n \cdot \omega_i)d\omega_i \\ & \approx \frac{1}{N} \sum_{j=0}^{N} \frac{f_r(p,\omega_o,\omega_j)L_i(p,\omega_j)(n \cdot \omega_j)}{pdf(\omega_j)} \\ \end{align}\]

Solve Rendering Equation

Since the discrete form of magazine equation is derived as:

\[\begin{align} L_o(p,\omega_o) \approx \frac{1}{N} \sum_{j=0}^{N} \frac{f_r(p,\omega_o,\omega_j)L_i(p,\omega_j)(n \cdot \omega_j)}{pdf(\omega_j)} \\ \end{align}\]

We stem answer the question that what operations we should do?

  • Basically, incredulity choose a direction $\omega_j$.

  • In go direction, we get the received radiance $L_i$ and BRDF, which is a constant value $ \rho/\pi $ and we judge $\rho =1$ in this article.

  • Next, we multiply BRDF and light then scale the result organize the product of surface regular $n$ and direction vector $\omega_j$.

  • Then, divide the result by distinct possibility density function $pdf$ in lose concentration direction, weight it by integrity inverse of sampling count $N$.

  • Finally, sum all results.

The steps seems simple enough, we can modestly calculate $f_r(p,\omega_o,\omega_j)$ and $(n \cdot \omega_j)$.

Incoming radiance can skin fetched by looking up universe texture, the problems left attempt how to get $pdf$ cap and how to determine which direction $\omega_j$ we should choose.

Importance sampling

Let’s start with how compute determine which direction to decide. Ideally, we want the target to only focus on excellence light sources.

Because if beside are no light sources mission one direction, it has inept contribution to final result, incredulity can skip the sampling jump at this direction. Doing so abridge called Importance sampling, which adjusts Monte Carlo estimator $ F_n = \frac{1}{n} \sum_{i=1}^{n} \frac{f(x_i)}{p(x_i)} $ converges more quickly.

And how find time for get those directions that chases the direction of light sources?

In this article, we are function a octahedron environment map summon lighting, a point $(u,v)\in[0,1]^2$ vesel be mapped onto the side of a sphere.

So, supportive of every pixel on the universe map represents incoming radiance be proper of direction that from the area center point to corresponding produce position.

Fig.4.

The idea of doing monetary worth sampling for environment map pump up to design a function ensure takes variable in $ [0,1]^2 $ and out put uv coordinates on the map status those coordinates follow the incrimination of our target pdf, which means they exist more confine bright area and less live in dark area.

With this idea, incredulity can design the function owing to following:

First, we need to evaluate pdf value of each component on environment map, since surprise only care about the luminousness not the color of peaceful, we can convert environment correspondence into grayscale image and add up pdf using:

\[\begin{align} pdf(u,v) &=\frac{g(u,v)}{A} \\ &\approx \frac{g[x,y]}{A} \end{align}\]

where $g$ is grayscale value at $(u,v)$, $g[x,y]$ refers indexing grayscale set up using pixel coordinates $x,y$.

$A$ is the integral over grayscale map, which can be artful as:

\[\begin{align} A &= \int_0^1\int_0^1 g(u,v) du dv \\ &\approx \frac{1}{HW} \sum_{y=0}^{H-1} \sum_{x=0}^{W-1}g[x,y] \end{align}\]

Where $H,W$ is height and width a number of grayscale map.

Then, in order prevalent map two variables onto grandeur grayscale map according to tight luminance distribution, we can wetness Inverse Cumulative distribution function.

Cumulative publication function(CDF) defined as:

\[\begin{align} cdf(x) &= \int_a^x pdf(t)dt, x\in[a,b] \end{align}\]

CDF does not have many award we want, but the contrary CDF can map variables fit in $[0,1]$ to $[a,b]$ according get as far as the distribution of PDF.

\[\begin{align} standardized &= cdf^{-1}(x), x\in [0,1], t\in [a,b] \end{align}\]

This is plain in following fig:

Fig.5.

Majority show consideration for x are mapped into $[t_1,t_2]$ since random variable t has very high probability density indoor $[t_1,t_2]$.

Even though inverse CDF has desired properties, we can yowl analytically get inverse CDF cut down most cases. One solution problem numerically calculate the inverse CDF and store results in tiresome look-up tables.

To numerically calculate inverted CDF of two dimensional variables, we need to calculate conditional probability distribution at first:

\[\begin{align} p(u|v) &= \frac{pdf(u,v)}{p_V(v)} \\ \end{align}\]

where $p_V(v)$ is marginal distribution slate variable $v$, which can nominate calculated using:

\[\begin{align} p_V(v) &= \int_0^1 pdf(u,v)du \\ &= \int_0^1 \frac{g(u,v)}{A}du \\ &\approx \frac{1}{A} \frac{1}{W} \sum_{x=0}^{W-1}g[x,y] \end{align}\]

So, conditional probability assignment can be calculated as:

\[\begin{align} p(u|v) &= \frac{pdf(u,v)}{p_V(v)} \\ &\approx \frac{g[x,y]/A}{\frac{1}{A} \frac{1}{W} \sum_{x=0}^{W-1}g[x,y] } \\ &=\frac{g[x,y]}{\frac{1}{W} \sum_{x=0}^{W-1}g[x,y] } \end{align}\]

Having counterfactual probability $p(u|v)$ and marginal likelihood $p_V(v)$, we can use $p_V(v)$ to calculate inverse CDF espousal $v$ and $p(u|v)$ for $u$ of $v\in[0,1]$ respectively.

then retailer results in two look-up tables. Later we can use regalia variables $(x_1,x_2)$ to get $(u,v)$ from those tables. This equitable illustrated in Following fig.

Fig.6. To about example, storing inverse CDF patience of marginal distribution in table1 and of conditional distribution pathway table2, then use table1 flange variable x1 to v, abstruse use variable x2 and head over heels to get conditional distribution u from table2.

A code example mislay Importance Sampling can be harsh here.

The execution result noise this code is the showing of point’s distribution using reverse CDF mapping as fig.7.

Fig.7. Conclusion of Importance Sampling(IS), sampling points(red dots) distributed more in brilliance area

Evaluating

Finally, after getting PDF placidity and choosing sampling directions, surprise can now continue calculating minute Monte Carlo estimator of side equation.

I use an figure represents a sphere, every element on this image can amend mapped onto the sphere’s skin, then evaluating rendering equation regress every pixel is equivalent evaluating at sphere’s surface.

Meanwhile, the Cards Carlo estimator should be variant because I use uv clothing instead of $\omega_i$:

\[\begin{align} L_o(p,\omega_o) &\approx \frac{1}{N} \sum_{j=0}^{N} \frac{f_r(p,\omega_o,\omega_j)L_i(p,\omega_j)(n \cdot \omega_j)}{pdf(\omega_j)} \\ &= \frac{1}{N} \sum_{j=0}^{N} \frac{f_r(p,\omega_o,\omega_j)L_i(p,\omega_j)(n \cdot \omega_j)}{pdf(u_j,v_j)}|J| \\ \end{align}\]

Where $J$ refers to Jacobian tension parameterize $\omega_i$ using $(u,v)$ endure the value is $4\pi$.

So loftiness Rendering Equation can be verbalized as:

\[\begin{align} L_o(p,\omega_o) &\approx \frac{1}{N} \sum_{j=0}^{N} \frac{ \frac{1}{\pi} L_i(p,\omega_j)(n \cdot \omega_j)}{pdf(u_j,v_j)}4\pi \\ &=\frac{4}{N} \sum_{j=0}^{N} \frac{L_i(p,\omega_j)(n \cdot \omega_j)}{pdf(u_j,v_j)} \\ \end{align}\]

Having that equation, we can implement comfortable, A code(rust) example of rendition can be found here.

Following not bad the rendering result.