The classic Monte-Carlo ray tracing is a powerful method which allows to simulate virtually all effects in ray optics, but it may be inadmissibly slow for many cases, such as calculation of images seen by a lens or pin-hole camera. In this cases another stochastic method is more efficient such as the bi-directional ray Monte-Carlo tracing with photon maps (BDPM). The level of noise i.e. the r.m.s. (“root mean square”) of pixel luminance calculated in one iteration of the method, depends on various parameters of the method, such as the number of light and camera paths, radius of integration sphere etc. so it is desirable to be able to predict this dependence to choose optimal parameters of the method. It was shown that this r.m.s is a sum of 3 functions scaled by reverse number of camera and light rays. These functions themselves are independent of the number of rays, so knowing them one can predict the noise for any number of rays and thus find the optimal one. These functions are a sort of correlations and their calculation from ray tracing is not a trivial problem. In this paper we describe a practical method of calculation and demonstrate the usage of its results for the choice of ray number.
Currently simulation of light propagation is widely used not only in optical engineering
but also in design of new materials. It is intensively applied in architectural,
automotive and aircraft design tasks. If wave effects can be neglected, a kind of stochastic
ray tracing is a good choice. This realm mainly includes Metropolis Light Transport
and Monte-Carlo ray tracing (MCRT) method [
The level of noise in this method mainly depends on the method of random
scattering of the forward and backward rays, on choosing the vertex to merge (or, in other
words, in which vertex of the camera path to use photon maps for estimation of the
illumination) and, at last, on the number of forward and backward rays traced during
one iteration. Most publications are devoted to the first two means, for example, [
In this paper we address the latter problem. In [
Although their mathematical definition is trivial, these three values are not that easy to calculate numerically by ray tracing. In this paper we describe a method of their efficient calculation and demonstrate their usage to choose the optimal number of rays. 2
Usually BDPM proceeds iteration by iteration. In each iteration we trace NF light (“forward”) paths and, for each pixel p, NB(p) camera (“backward”) paths. Then each light path is checked against each camera path and if there are vertices they run close, they are “merged” into a join path (which connects camera and light source). We then calculate contribution of this join path to the luminance of pixel and increment pixel luminance by this value. The accumulated sum, divided by NF NB(p)NI where NI is the number of iterations, converges to the expectation. This average is independent from NF and NB(p), but the noise (variance) is. Say, if there is already too many light paths while few camera paths, tracing these excess light rays only takes time without gain, so it is advantageous to reduce their number.
The optimal number of rays (total per scene for light paths and individually for each
pixel for camera paths) is that which results in the least noise after the fixed calculation
time. To solve this minimization problem we need to know how the variance of the
value produced in one iteration depends on that number of rays. This dependence is
described by the 3-term law [
Here and below we shall always speak about the values for one given pixel p and so drop this notation, writing NB instead of NB(p) etc.
As shown in [
Calculation of Noise Components for BDPM 3 where C(X(F ); X(B)) is the contribution to pixel luminance from “merging” the light path X(F ) and camera path X(B), its average hhCii is obviously the limiting pixel luminance L, and h iF , h iB denote the averages over the ensemble of light and camera paths, respectively.
Notice this law holds regardless of whether we use Multiple Importance Sampling
[
Calculation of hhC2ii is easy: whenever we add C to the pixel luminance, we also add C2 to the “error image”. Then at each iteration we calculate
C
1 XNB XNF C2(Xi(F ); Xj(B))
NBNF j=1 i=1 and averaging it over iterations C just gives what we need: C ! hhC2ii. But with hhCi2F iB and hhCi2BiF we face difficulty.
For example, let us suppose there is very many light paths per iteration. Then for each camera ray the sum over them gives a good approximation to hCiF . The average over camera paths is bad because normally we have a few (10, max. 100) camera paths through one pixel per iteration. But if we additionally average over iterations, we get a good approximation to hhCi2F iB. This, however, does not work if there are not that many light paths (to be precise, not that many intersections). In this case the average over light paths from one iteration deviates seriously from hCiF . Meanwhile averaging the squared value over iterations may give averaged square, not square of average because averaging over iterations is hh ii. What to do then?
Solution is very simple. Let the number of light and camera paths traced in one iteration be the same (for all iterations). To calculate hhCi2F iB, let us subdivide the set of light paths traced in one iteration into two halves containing NF1 and NF2 rays. Then, let at each iteration (1) (2) (3) Its average over iterations is just hhCi2F iB. Indeed, the average over iterations is hh iF iB = hh iBiF , because of linearity the order of averaging has no effect. For any camera path Xj(B) the two inner sums are independent because the two halves of light paths are statistically independent. The average of each of them is naturally hCiF (Xj(B)). Therefore averaging it over iterations
B = hhBiF iB = hhCiF hCiF iB = hhCi2F iB (4) gives just what we need.
Similarly, to calculate hhCi2BiF we subdivide the set of camera paths (through the pixel) into two halves, NB1 and NB2 , calculate at each iteration F
NF 00 1 X
1
X and average it over iterations.
Notice the two halves have not to be equal (i.e. contain a different number of rays). Usually it is convenient to relate odd rays to the first half and even rays to the second half.
Knowing these quadratic averages and the limiting image luminance L one can calculate the variance of pixel luminance as
Notice C; B; F as well as L depend on pixel but are independent of the number of rays. This latter property is very important because allows to predict noise level for any number of rays. One can perform calculation for some inoptimal number of rays and find C; B; F ; L and then find the number of rays (with NB generally depending on pixel) which minimizes the noise. 3
We traced the famous Cornell Box with an isotropic point light source slightly below
the center of the floor. All scene surfaces are gray Lambert with albedo 50%. Both
direct and indirect illumination was taken from photon maps. Integration sphere radius
was 1/120 of the scene size. “Diffuse depth” i.e. the maximal allowed number of diffuse
events for camera ray [
Pixel luminance calculated in one iteration is a random value which is independent from another iteration. Therefore the reference value of noise can be calculated directly from the sample variance of the series of iterations. Also, the above 3 noise terms had been calculated as described above and the total noise had been calculated from them according to (6).
The amplitude of noise calculated by these two ways are shown in Figure 2. One can see that these latter nearly coincide.
The contribution of the three noise components is shown in Figure 3.
Fig. 3. Noise components q NF1NB (C L2); q 1 NNBF 1 (B L2); q 1 NNFB 1 (F L2) put into the R, G and B color channel of the image. Left panel: NB = 100, right panel: NB = 300. In both cases NF = 300000. The average values of the three components over the red rectangle is (0:0293; 0:595; 0:229) in the left image and (0:0169; 0:344; 0:230) in the right one, so the first two components decreased just p3 times. Average r.m.s. over the red rectangle is 0.639 in the left image and 0.414 in the right one.
In Figure 3, q NF1NB (C L2), q 1 NNBF 1 (B L2) and q 1 NNFB 1 (F L2) are put into the R, G and B color channels. One can see that the image is mainly green, i.e. the second component dominates. Since C; B; F and L are independent of the number of rays, to reduce noise one needs to increase the number of camera rays per pixel NB while increase of the number of light rays is nearly useless. Results for NB increased to 300 are shown in the right panel of Figure 3. 4
We proposed a method for calculation the components C; B; F of the noise in BDPM and demonstrated how it can be used to choose the optimal number of rays traced in one iteration of the BDPM method.
The noise itself can be calculated directly from the sample variance of the series of values accumulated in successive iterations, but this does not let one decide which number of rays (camera or light) is critical and must be increased. Meanwhile knowing the three noise components C; B; F one can see which of them dominates and thus which number of rays (camera or light) must be increased. If time spent on tracing camera and light rays is known together with time spent on their merging, one can also calculate the optimal number of rays i.e. the one which gives the smallest noise during the same calculation time.