Theory

Operators in GI

Operators allow convenient representations of complex equations. So

$$L(x) = L_e(x) + \rho(x) \int_{\cal E} L(x') {\rm vis}(x,x')\f... ...{ \left\vert \left\vert x - x' \right\vert \right\vert ^2}dA.$$

can be expressed as

L(x) = L_e(x) + ML(x)

or

(I - M)L(x) = L_e(x).

What does this buy us? Ease of notation and manipulation and easier to play with the properties of the operator. These operators are linear, which makes life a lot simpler. Written like this, it is easy to see that we are interested in finding an linear operator that is the inverse of (I - M) and applying it to the emission functions.

Using the Neumann series expansion, the inverse can be written

$$\mbox{${(I - M)}^{-1}$ } = I + M+ M^2 + M^3 + \cdots$$

if the series converges (if $M^k \rightarrow 0$). We now need to show that the series converges for powers of M, and in order to do this, we need a way of knowing the size of the operator.

For an example of an operator which is invertible and whose Neumann series does not coverge, try

$$\left\vert\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right... ...vert\begin{array}{rr} 0 & -2 \\ -2 & 0 \end{array}\right\vert.$$

On a side note, the linearity of the (I - M) and it's inverse has a few nice implications. The property of linearity can be expressed as

$$(I - M)(\alpha L_1(x) + \beta L_2(x)) = \alpha(I - M)L_1(x) + \beta(I - M)L_2(x) = \alpha L_{e1}(x) + \beta L_{e2}(x)$$

This means that global illumination can be done independently for each light source and the resulting solutions or images can be scaled and added together to show the effect of a combination of lights with different intensities.

Function Norms

One way of measuring size or magnitude is to use a norm. A norm is a function that maps a scalar, vector, function, or operator to a non-negative scalar. Norms have several properties

$$\left\vert \left\vert x \right\vert \right\vert$$ >= (6)

$$\left\vert \left\vert x \right\vert \right\vert 0 \Leftrightarrow x = 0$$ = (7)

$$\left\vert \left\vert \alpha x \right\vert \right\vert \alpha \left\vert \left\vert x \right\vert \right\vert$$ = (8)

$$\left\vert \left\vert x + y \right\vert \right\vert \left\vert \left\vert x \right\vert \right\vert + \left\vert \left\vert y \right\vert \right\vert$$ <= (9)

For a function f, the commonly used norms are

L₁

${\left\vert \left\vert f \right\vert \right\vert}_{1} = \int \vert f(x)\vert$

L₂

${\left\vert \left\vert f \right\vert \right\vert}_{2} = (\int f(x)^2)\frac{1}{2}$

$L_{\infty}$

${\left\vert \left\vert f \right\vert \right\vert}_{\infty} = \max_x \vert f(x)\vert$

Using norms we can determine the relative magnitudes of different radiance functions, and more importantly, the relative magnitude of the difference between two radiance functions. This will allow us to determine the magnitude of the error introduced by some approximation. Error is always associated, either explicitly or implicitly, with respect to some norm.

Operator Norms

In order to show that $M^k \rightarrow 0$ we need to define what a norm means for an operator. We do this by finding out the maximum effect the operator can have on any of the functions it operates on.

$$\left\vert \left\vert N \right\vert \right\vert = \max_{x \ne... ...ight\vert }{ \left\vert \left\vert x \right\vert \right\vert }$$

since $\left\vert \left\vert \alpha x \right\vert \right\vert = \vert\alpha\vert \left\vert \left\vert x \right\vert \right\vert$ we can look only at the ball of radius one and rewrite the definition as

$$\left\vert \left\vert N \right\vert \right\vert = \max_{x : \... ...ht\vert = 1} \left\vert \left\vert Nx \right\vert \right\vert$$

In effect we apply the operator to the ball of all things of size 1 and look at the resulting blob. The norm of the operator is resulting size of the object that was moved furthest away from the origin.

Decomposition

The properties of M are hard to analyze because M does several different things. In order to get a better understanding of the operator, we will decompose it into two other operators, G and K. G will take care of the the global transfer of energy, and K takes care of the local reflection (see Figure 2.1.4), so M= KG. This follows the decomposition of Arvo[3]. (Another decomposition was described by Gershbein [11].)

$$(GL)(x,\omega) = L(x'(x,\omega),\omega)$$

and

$$(KL_i)(x,\omega) = \rho(x) \int_{H^{in}} L_i(x,\omega') \cos(\theta_{x}) d\omega.$$

  

Figure: The action of G and K at a single point x.

We want to show $\left\vert \left\vert M \right\vert \right\vert = \left\vert \left\vert KG \right\vert \right\vert < 1$. Based on Holder's inequality, ${\left\vert \left\vert KG \right\vert \right\vert}_{1} \leq {\left\vert \left\... ...rt \right\vert}_{\infty} {\left\vert \left\vert G \right\vert \right\vert}_{1}$. In an enclosed environment, G can be thought of as rearranging or shuffling L. This has absolutely no effect on the one norm of L, therefore ${\left\vert \left\vert G \right\vert \right\vert}_{1} = 1$. K effectively smooths out L_i and weights the result by $\rho(x)$. So for the L₁ and L_infty norms, $\left\vert \left\vert K \right\vert \right\vert \leq \max_{x} \rho(x) < 1$. For a full proof, see Arvo [2]. Therefore

$${\left\vert \left\vert KG \right\vert \right\vert}_{1} \leq ... ...vert G \right\vert \right\vert}_{1} \leq \max_{x} \rho(x) < 1.$$

So, since $M^k \rightarrow 0$, we have

$$\mbox{${(I - M)}^{-1}$ } = I + M+ M^2 + \cdots$$

More Reading

The Cohen and Wallace book has a page on Norms ([10] pg 133) Glassner's book [12]has a chapter (16) on integral equations that discusses operators as well, and Appendix A has a chapter on Linear Spaces. Jim Arvo has a paper on functional analysis [2] that has a more accurate discussion of norms than covered here. Most analysis books cover norms.