Photon mapping with visible kernel domains

Abstract

Despite the strong efforts made in the last three decades, lighting simulation systems still remain prone to various types of imprecisions. This paper specifically tackles the problem of biases due to density estimation used in photon mapping approaches. We study the fundamental aspects of density estimation and exhibit the need for handling visibility in the early stage of the kernel domain definition. We show that properly managing visibility in the density estimation process allows to reduce or to remove biases all at once. In practice, we have implemented a 3D product kernel based on a polyhedral domain, with both point-to-point and point-to-surface visibility computation. Our experimental results illustrate the enhancements produced at every stage of density estimation, for direct photon maps visualization and progressive photon mapping.

Appendices

Appendices

A Insight into density estimation

This section discusses several aspects of density estimation and the importance of consistency with kernel density estimators [11]. In the univariate case, given n independent identically distributed observations $x_1, x_2,\ldots ,x_n$ of a given function f, the kernel density estimator $\langle f_h(x) \rangle$ of f(x) is:

$$\begin{aligned} \langle f_h(x) \rangle =\frac{1}{nh}\sum _{i=1}^n\;K_1\left( \frac{x-x_i}{h}\right) ,\quad x\in \mathbb {R},\;h>0, \end{aligned}$$

(10)

where $K_1$ is a 1D kernel function (with mean 0 and integral equal to 1) and h is the kernel bandwidth. In the following, we are mainly interested by kernels with a compact domain, i.e., kernels $K_1$, such that

$$\begin{aligned} K_1(x)=0,\, \forall x\not \in [-1,+1]. \end{aligned}$$

The multivariate kernel density estimator of f is:

$$\begin{aligned} \langle f_{\mathbf {H}}(\mathbf {x}) \rangle =\frac{1}{n\left| \mathbf {H}\right| }\sum _{i=1}^n\;K_d\left( \mathbf {H}^{-1}\left( \mathbf {x}-\mathbf {x}_i\right) \right) ,\quad \mathbf {x}\in \mathbb {R}^d, \end{aligned}$$

(11)

where $\mathbf {H}$ is a $(d\times d)$ nonsingular matrix generalizing the bandwidth and $K_d$ is a multivariate kernel function.

A popular representation for multivariate kernels consists in using product kernels, corresponding to product of the same univariate kernel function:

$$\begin{aligned} K_d(\mathbf {y})=\prod _{j=1}^d\;K_1(y_j),\quad \forall \mathbf {y}= [y_1,\ldots ,y_d], \end{aligned}$$

(12)

defined on a polyhedral domain.

Kernels are generally chosen with the following properties:

$$\begin{aligned}&\forall \mathbf {y}\in \mathbb {R}^d,\;K_d(\mathbf {y}) \ge 0\quad \text {(positive)}, \end{aligned}$$

(13)

$$\begin{aligned}&\int _{\mathbb {R}^d}K_d(\mathbf {y})\;\mathrm {d}\mathbf {y} = 1\quad \text {(normalized)}, \end{aligned}$$

(14)

$$\begin{aligned}&\int _{\mathbb {R}^d}\mathbf {y} K_d(\mathbf {y})\;\mathrm {d}\mathbf {y} = 0\quad \text {(symmetric)}, \end{aligned}$$

(15)

$$\begin{aligned}&\mu _2(K_d)\mathbf {I}_d =\int _{\mathbb {R}^d}\mathbf {y}\mathbf {y}^T K_d(\mathbf {y})\;\mathrm {d}\mathbf {y} \quad \text {(bounded)}, \end{aligned}$$

(16)

where $\mu _2(K_d)=\int y_i^2K_d(\mathbf {y})\mathrm {d}\mathbf {y}$ is independent of $i\in [1\ldots d]$, $\mu _2(K_d)<\infty$, and $\mathbf {I}_d$ is the $d\times d$ identity matrix.

Equations 13 and 14 ensure that the kernel can also be seen as a normalized probability density function; Eq. 15 states that $K_d$ is a symmetric function, which is generally a good property for a convolution, except near support boundaries; Eq. 16 is only useful to express the bias as a function over a finite value $\mu _2(K_d)$.

Density estimation is intrinsically biased. The bias is defined as:

$$\begin{aligned} b\left( \langle f_{\mathbf {H}}(\mathbf {x}) \rangle \right) = E\left\{ \langle f_{\mathbf {H}}(\mathbf {x}) \rangle \right\} - f(\mathbf {x}), \end{aligned}$$

where $E\lbrace \langle f_{\mathbf {H}}(\mathbf {x}) \rangle \rbrace$ is the expected value of function $\langle f_{\mathbf {H}}(\mathbf {x}) \rangle$. When the estimator is biased, the expected value does not tend to the actual function:

$$\begin{aligned} b\left( \langle f_{\mathbf {H}}(\mathbf {x}) \rangle \right) \ne 0 \quad \Rightarrow \quad E\left\{ \langle f_{\mathbf {H}}(\mathbf {x}) \rangle \right\} \ne f(\mathbf {x}). \end{aligned}$$

In practice, users are mostly interested in density estimators that are consistent: $\forall \mathbf {x}\in \mathbb {R}^d$, $b\left( \langle f_{\mathbf {H}}(\mathbf {x}) \rangle \right) {\scriptscriptstyle ^{_\rightarrow }}0$, when all values of $\mathbf {H}$ tend to 0.

A good trade-off between bias and variance is difficult to find since increasing the bandwidth h allows to reduce variance but increases bias; conversely, decreasing h reduces bias but increases variance.

B Boundary bias in statistics

Boundary bias is well known in statistics. When the estimation domain is bounded, bias increases at the boundary, leading to inconsistent density estimation. This can be explained using a Taylor expansion (see [30] for the multivariate case, exhibiting the same behavior). In the univariate case, a kernel defined on a compact domain $[-1,+1]$ leads to the following bias expression:

$$\begin{aligned} b(\langle f_h(x) \rangle )&= f(x)\int _{-h}^{+h} K_1\left( \frac{z}{h}\right) \,\mathrm {d}z-f(x) \end{aligned}$$

(17)

$$\begin{aligned}&-hf'(x)\int _{-h}^{+h} zK_1\left( \frac{z}{h}\right) \,\mathrm {d}z \end{aligned}$$

(18)

$$\begin{aligned}&+\frac{h^2}{2}f''(x)\int _{-h}^{+h}z^2K_1\left( \frac{z}{h}\right) \,\mathrm {d}z+o(h^2). \end{aligned}$$

(19)

Using Eq. 14 cancels the first two terms, Eq. 15 removes the third one, and the condition corresponding to Eq. 16 allows to bound the last term and to conclude that bias is proportional to $h^2$. Finally, when h decreases down to 0, the bias also tends to 0: The estimator is consistent.

Let us now consider a function f defined over $[0,+\infty ]$. Density estimation is performed using Eq. 10 for $0<x<h$, and the previous bias approximation leads to:

$$\begin{aligned} b(\langle f_h(x) \rangle )&= f(x)\int _0^{+h} K_1\left( \frac{z}{h}\right) \,\mathrm {d}z-f(x)\\&\quad -hf'(x)\int _0^{+h} zK_1\left( \frac{z}{h}\right) \,\mathrm {d}z \\&\quad +\frac{h^2}{2}f''(x)\int _0^{+h}z^2K_1\left( \frac{z}{h}\right) \,\mathrm {d}z+o(h^2). \end{aligned}$$

In this case, Eqs. 13, 14, 15, 16 do not ensure a consistent estimator anymore. Hence, the bias becomes a function of f(x), which cannot be decreased with more observations.

In order to ensure a consistent estimation, Jones proposes to normalize the kernel at each estimation point [14]; in other words, a normalization factor is calculated for each estimated point x:

$$\begin{aligned} \mathbb {K}_1(x) = \int _{-h}^{+x} K_1\left( \frac{x-z}{h}\right) \,\mathrm {d}z, \end{aligned}$$

(20)

and density estimation becomes:

$$\begin{aligned} \langle f_h(x) \rangle =\frac{1}{nh\mathbb {K}_1(x)}\sum _{i=1}^n\;K_1\left( \frac{x-x_i}{h}\right) . \end{aligned}$$

(21)

Consequently, density bias is in $\mathcal O(h)$ and thus consistent.

The generalization in any dimension d of this normalization process requires to determine the normalization factor $\mathbb {K}_d(\mathbf {x})$ resulting from the kernel integration over a domain $\mathcal {D}=\{ \mathcal {D}_1 \times \mathcal {D}_2 \times \cdots \times \mathcal {D}_d \}$:

$$\begin{aligned} \mathbb {K}_d(\mathbf {x})&= \int _{\mathcal {D}} K_d( u_1,\ldots ,u_d ) \;\mathrm {d}u_1\ldots \mathrm {d}u_d \nonumber \\&= \int _{\mathcal {D}_d}\ldots \int _{\mathcal {D}_1} K_1(u_1)\;\mathrm {d}u_1\ldots \mathrm {d}u_d, \end{aligned}$$

(22)

where each sub-domain $\mathcal {D}_i$ is orthogonal to the others, which makes it separable and thus easier to estimate.