1 INTRODUCTION

Presently, modeling of light propagation is widely used in realistic computer graphics and for designing new materials and optical systems [1]. It is also used in architectural, automobile and aircraft design. If wave effects may be neglected in a problem, then stochastic ray tracing methods are a good choice. This group of methods mainly includes the simulation of radiation transport using the Metropolis method [2] and stochastic ray tracing [3]. The classical forward ray tracing beginning from the light source is inefficient for image generation, and for this reason it is replaced by bidirectional modifications of this method [4–6]. Among them, we consider the so-called bidirectional stochastic ray tracing with photon maps (BDPM – Bidirectional Photon Mapping) [5, 7]. A drawback of all stochastic methods is that they produce noisy results. Therefore, the noise reduction problem is always important, and it is considered in many works, e.g., see [8–10].

The level of noise in BDPM mainly depends on the random scattering of the forward and backward rays, on the choice of the vertex for their merging (or, in other words, on the vertex of the camera ray trajectory at which photon maps are used for estimating the luminance), and, finally, on the number of forward and backward rays traced in one iteration step. The majority of studies is devoted to the first two issues (e.g., [9–12]), and the number of rays got less attention. However, this is an important factor, and it often happens that the number of forward rays is already redundant and its further increase only increases the computation time but does not reduce the noise. In other cases, it may happen that the number of forward rays is indeed critical, while the number of traced backward rays is redundant.

This situation is illustrated in Table 1, where the dependence of variance (RMS) after a fixed computation time on the number of forward rays (from the light source) NF and backward rays (from the camera through the pixel) NB is shown. The scene description can be found in the section Results. In this example, there is almost no dependence on the number of backward rays NB (indicated in columns), while there is a minimum of noise level at NF ≈ 3000 depending on the number of rays emitted from the source. Here we see that the general claim that the more rays the lower the noise does not hold.

Table 1. Mean RMS over the region of the image marked by a red box in Fig. 2 after 1000 seconds of computation for various numbers of rays NF and NB
Full size table

It is usually difficult to predict which fraction of the forward and backward rays is optimal: however, a good choice can speed up the computations by several fold.

In this paper, we consider the optimization problem. In [8], a general rule determining noise in the BDPM was derived. It states that the variance of the pixel luminance is the sum of three components scaled by the reciprocal of the number of rays. These three components are independent of the number of rays and, therefore, they can be calculated once and then used for predicting the dependence of the noise level on the number of rays traced in one iteration step. In this way, we are going to predict the optimal number of rays. In other words, if we know these three components, then we are able to predict the level of noise for any number of rays.

Even though the mathematical expression is trivial, these values are difficult to calculate in the process of ray tracing. In this paper, we describe a method for their effective calculation, and it is shown how they can be used for choosing the optimal number of rays.

2 NOISE IN THE BIDIRECTIONAL RAY TRACING WITH PHOTON MAPS

In the BDPM, the computations are typically iterative. At each iteration step, NF light paths and NB(p) paths from the camera are traced for each pixel p. Next, we check each pair, and if the light path goes sufficiently close to a vertex on the path from the camera, we join them to obtain the complete path from the camera to the light source. Next, we calculate the contribution of this joined path to the pixel luminance and increase the pixel luminance by this magnitude. The accumulated sum divided by NFNB(p)NI (where NI is the number of iterations) converges to the luminance expectation. This mean value is independent of NF and NB(p), but its variance (noise) can vary significantly. For example, if there are too many light ray trajectories and the ray trajectories issued from the camera are few, then tracing the redundant light rays only wastes time without improving the image quality; therefore, it is reasonable to reduce the number of the redundant rays.

The optimal number of rays (which is common for all light paths in each scene and individual for each pixel for the paths from the camera) is the number that minimizes the noise after a fixed computation time. To solve this minimization problem, we need to know how the variance of the value obtained in one iteration step depends on the number of rays. This dependence is described by a simple algebraic formula [8]; however, its terms are not easy to find numerically.

Here and below, we always deal with values for one specific pixel p; for this reason, we will omit the pixel indication and will write NB instead of NB(p).

It was shown in [8] that the luminance variance at a given pixel calculated during one iteration step is

$$\begin{gathered} V = \frac{1}{{{{N}_{F}}{{N}_{B}}}}(\langle \langle {{C}^{2}}\rangle \rangle - {{\langle \langle C\rangle \rangle }^{2}}) \\ + \frac{{1 - N_{F}^{{ - 1}}}}{{{{N}_{B}}}}({{\langle \langle C\rangle _{F}^{2}\rangle }_{B}} - {{\langle \langle C\rangle \rangle }^{2}}) \\ + \frac{{1 - N_{B}^{{ - 1}}}}{{{{N}_{F}}}}({{\langle \langle C\rangle _{B}^{2}\rangle }_{F}} - {{\langle \langle C\rangle \rangle }^{2}}) \\ \end{gathered} $$

where C(X(F), X(B)) is the contribution of the merged path from the source X(F) and the path from the camera X(B) to the pixel luminance, its mean 〈〈C〉〉 clearly coincides with the limiting (exact) pixel luminance L, and 〈⋅〉F and 〈⋅〉B denote averaging over the ensemble of the paths issued from the light source and the camera, respectively.

Note that this functional form is independent of whether or not the multiple importance sampling [13] is used and is even independent of whether vertex merging or vertex connection by an additional trajectory segment [9] is used; however, the quantities 〈〈C2〉〉, \({{\langle \langle C\rangle _{F}^{2}\rangle }_{B}}\), and \({{\langle \langle C\rangle _{B}^{2}\rangle }_{F}}\) depend on the computation strategy. In any case, they are independent of the number of rays, but they can depend on the radius of the integration sphere (i.e., on the radius of vertex merging).

3 CALCULATION OF THE COEFFICIENTS IN NOISE FORMULA

The calculation of 〈〈C2〉〉 is trivial (as in the calculation of the sample variance): by adding C to the accumulated pixel luminance, we also add C2 to 〈〈C2〉〉 accumulated in this pixel. That is, at each iteration step we calculate

$$C \equiv \frac{1}{{{{N}_{B}}{{N}_{F}}}}\sum\limits_{j = 1}^{{{N}_{B}}} {\sum\limits_{i = 1}^{{{N}_{F}}} {{{C}^{2}}(X_{i}^{{(F)}},X_{j}^{{(B)}})} } ,$$

and the mean of this value Cavg over the iterations converges to the desired value: Cavg → 〈〈C2〉〉. However, this method is certainly inapplicable for \({{\langle \langle C\rangle _{F}^{2}\rangle }_{B}}\) and \({{\langle \langle C\rangle _{B}^{2}\rangle }_{F}}\).

Indeed, assume that NF is so large that, for each ray issued from the camera, the sum \(\frac{1}{{{{N}_{F}}}}\sum\nolimits_{i = 1}^{{{N}_{F}}} {C(X_{i}^{{(F)}}} \), \(X_{j}^{{(B)}})\) gives a good approximation to 〈C〉F(\(X_{j}^{{(B)}}\)). At the same time, the number of rays passing through the same pixel NB is small; therefore the averaging \(\langle C\rangle _{F}^{2}(X_{j}^{{(B)}})\) over this small number of rays is insufficient. However, here we can add averaging over iterations because at each iteration step we have a random independent set of rays issued from the camera, and this yields a good estimate of \({{\langle \langle C\rangle _{F}^{2}\rangle }_{B}}\).

Certainly, this is just a thought experiment, and this method cannot be practically used because the number of rays issued from the light source is not large enough for providing a sufficiently accurate estimate of 〈C〉F(\(X_{j}^{{(B)}}\)) within one iteration step. Averaging over iterations in this case is extremely difficult because this quantity should be calculated for each trace \(X_{j}^{{(B)}}\) issued from the camera, and it occurs only once. It is clear that we actually will hit a neighborhood of this path regularly, so that if we will calculate and store the grid function \(X_{j}^{{(B)}}\) (on a sufficiently fine grid), the aim will be achieved. However, this is very inefficient since the space dimension, which equals the maximum length of the trace issued from the camera, is fairly large.

However strange it may seem, the solution is simple. Assume for convenience that the number of rays issued from the camera and from the light source is independent of the iteration step. To calculate \({{\langle \langle C\rangle _{F}^{2}\rangle }_{B}}\), we divide the whole set NF of rays issued from the source into two parts (not necessarily equal) consisting of \({{N}_{{{{F}_{1}}}}}\) and \({{N}_{{{{F}_{2}}}}}\) rays, respectively, and calculate at each iteration step

$$\begin{gathered} B \equiv \frac{1}{{{{N}_{B}}}}\sum\limits_{j = 1}^{{{N}_{B}}} {\left( {\left( {\frac{1}{{{{N}_{{{{F}_{1}}}}}}}\sum\limits_{i \in hal{{f}_{1}}}^{} {C(X_{i}^{{(F)}},X_{j}^{{(B)}})} } \right)} \right.} \\ \left. { \times \left( {\frac{1}{{{{N}_{{{{F}_{2}}}}}}}\sum\limits_{i \in hal{{f}_{2}}}^{} {C(X_{i}^{{(F)}},X_{j}^{{(B)}})} } \right)} \right). \\ \end{gathered} $$

The mean Bavg of this quantity over the iterations is nothing more nor less than the mean over the ensemble of NF rays issued from the source and NB rays issued from the camera. These two means are independent and, therefore, they commute. Let us first average B over the ensemble of rays issued from the source: Bavg = 〈〈B〉F〉B. Since the rays in the first and second parts of this ensemble are obviously independent, the mean of the product of the two internal sums is the product of the means over these halves:

$${{\langle B\rangle }_{F}} \equiv \frac{1}{{{{N}_{B}}}}\sum\limits_{j = 1}^{{{N}_{B}}} {({{{\langle C\rangle }}_{{{{F}_{1}}}}}(X_{j}^{{(B)}}){{{\langle C\rangle }}_{{{{F}_{2}}}}}(X_{j}^{{(B)}}))} .$$

Furthermore, since the statistical characteristics of the rays in the first and second parts are certainly identical, the means over the halves of the ensemble and over the entire ensemble are also identical:

$${{\langle B\rangle }_{F}} \equiv \frac{1}{{{{N}_{B}}}}\sum\limits_{j = 1}^{{{N}_{B}}} {\langle C\rangle _{F}^{2}(X_{j}^{B})} $$

The remaining averaging over the ensemble of the rays issued from the camera yields

$${{B}_{{{\text{avg}}}}} = {{\langle {{\langle B\rangle }_{F}}\rangle }_{B}} = {{\langle \langle C\rangle _{F}^{2}\rangle }_{B}}$$

Thus, the second coefficient in the noise formula can be calculated to an arbitrary accuracy by the sequence of iterations even if each iteration contains a small number of rays.

The computation of \({{\langle \langle C\rangle _{B}^{2}\rangle }_{F}}\) is organized similarly by dividing into two parts the set of rays issued from the camera and passing through the given pixel and by calculating at each iteration step the quantity

$$\begin{gathered} F \equiv \frac{1}{{{{N}_{F}}}}\sum\limits_{i = 1}^{{{N}_{F}}} {\left( {\left( {\frac{1}{{{{N}_{{{{B}_{1}}}}}}}\sum\limits_{j \in hal{{f}_{1}}}^{} {C(X_{i}^{{(F)}},X_{j}^{{(B)}})} } \right)} \right.} \\ \left. { \times \left( {\frac{1}{{{{N}_{{{{B}_{2}}}}}}}\sum\limits_{j \in hal{{f}_{2}}}^{} {C(X_{i}^{{(F)}},X_{j}^{{(B)}})} } \right)} \right) \\ \end{gathered} $$

Its mean Favg over the iterations converges to \({{\langle \langle C\rangle _{B}^{2}\rangle }_{F}}\).

Note that the set of rays issued from the camera may be partitioned into unequal parts. However, this is senseless because the sums over these two parts appear in the formulas symmetrically, and there is no reason to improve one of the parts at the expense of the other. In practice, it is simpler to use the partition into even and odd rays.

Now, knowing the RMS means and the limiting pixel luminance L, we can calculate the variance of the contribution of one iteration to this luminance as

$$\begin{gathered} V = \frac{1}{{{{N}_{F}}{{N}_{B}}}}({{C}_{{{\text{avg}}}}} - {{L}^{2}}) + \frac{{1 - N_{F}^{{ - 1}}}}{{{{N}_{B}}}}({{B}_{{{\text{avg}}}}} - {{L}^{2}}) \\ + \frac{{1 - N_{B}^{{ - 1}}}}{{{{N}_{F}}}}({{F}_{{{\text{avg}}}}} - {{L}^{2}}); \\ \end{gathered} $$
(1)

after NI iterations, the variance will be \(\frac{V}{{{{N}_{I}}}}\).

Note that Cavg, Bavg, and Favg, as well as L, depend on the pixel but are independent of the number of rays. However, as has been mentioned above, Cavg, Bavg, and Favg can depend on the radius of the integration sphere. This is a very important property because it makes it possible to predict the noise for an arbitrary number of rays. Therefore, we may make one preliminary calculation for a certain, maybe even a very poor choice of the number of rays, find Cavg, Bavg, Favg, and L, and then calculate the optimal number of rays (in the general case, NB will be specific for each pixel), which ensures the minimal noise.

4 FEATURES OF DIRECT LIGHTING

In the BDPM method, the direct illuminance can be taken into account in two different ways. In the first method, all the rays issued from the source—both direct (not scattered)) and indirect (scattered) are processed uniformly. That is, the direct illuminance is also taken from the photon maps. In the second method, the direct illuminance is calculated straightforwardly: each vertex of the path from the camera is connected to the light source, its luminance is calculated along this segment, and sum with the luminance estimate is taken from the photon maps [14] (formula (2) for w0 = 0 and w1 = 1). As a rule, the second method is more effective because in it the direct luminance component does not contain the noise of the photon maps.

It is clear that in both cases the noise is described by formulas of Section 3. Indeed, if all the luminance components are taken from the photon maps as in the first case, then the number of nonzero terms C(\(X_{i}^{{(F)}}\), \(X_{j}^{{(B)}}\)) is much less than the maximal number NFNBNp (where Np is the number of pixels in the image) because the probability of trace “intersection” is low. For this reason, the computation of the sums in C, B, and F will be efficient.

In the second method, the situation is different; indeed, in this case each path from the camera makes certain contribution due to direct illuminance. Now, all C(\(X_{i}^{{(F)}}\), \(X_{j}^{{(B)}}\)) are nonzero, even if \(X_{i}^{{(F)}}\) and \(X_{j}^{{(B)}}\) do not intersect. The computation of the sums in C, B, and F becomes two expensive. Therefore, it is reasonable to slightly transform the second method to simplify computations.

More precisely, to avoid this effect, we should separate the contribution of the direct illuminance in the sums because it is independent of the ray issued from the light source and may, therefore, be taken out of the summation sign:

$$C(X_{i}^{{(F)}},X_{j}^{{(B)}}) = {{C}^{{(I)}}}(X_{i}^{{(F)}},X_{j}^{{(B)}}) + {{C}^{{(0)}}}(X_{j}^{{(B)}})$$

where C(I)(\(X_{i}^{{(F)}}\), \(X_{j}^{{(B)}}\)) is the contribution of the intersection of the camera and indirect rays issued from the source and C(0)(\(X_{j}^{{(B)}}\)) is the contribution of the direct illuminance obtained by connecting the node on the camera ray with the light source.

Now, we write C as

$$C = \frac{1}{{{{N}_{B}}{{N}_{F}}}}\sum\limits_{j = 1}^{{{N}_{B}}} {\sum\limits_{i = 1}^{{{N}_{F}}} {{{{({{C}^{{(I)}}}(X_{i}^{{(F)}},X_{j}^{{(B)}}) + {{C}^{{(0)}}}(X_{j}^{{(B)}}))}}^{2}}} } $$
$$\begin{gathered} = \frac{1}{{{{N}_{B}}{{N}_{F}}}}\sum\limits_{j = 1}^{{{N}_{B}}} {\sum\limits_{i = 1}^{{{N}_{F}}} {({{{({{C}^{{(I)}}}(X_{i}^{{(F)}},X_{j}^{{(B)}}))}}^{2}}} } \\ + \,2{{C}^{{(I)}}}(X_{i}^{{(F)}},X_{j}^{{(B)}}){{C}^{{(0)}}}(X_{j}^{{(B)}})) \\ + \frac{1}{{{{N}_{B}}}}\sum\limits_{j = 1}^{{{N}_{B}}} {{{{({{C}^{{(0)}}}(X_{j}^{{(B)}}))}}^{2}}} \\ \end{gathered} $$

Here, the sums contain the same number of nonzero terms as in the absence of direct illuminance (i.e., for C(0) = 0).

Next, we transform the expression for B as

$$\begin{gathered} B = \frac{1}{{{{N}_{B}}}}\sum\limits_{j = 1}^{{{N}_{B}}} {\left( {\left( {\frac{{{{\Sigma }_{{i \in hal{{f}_{1}}}}}{{C}^{{(I)}}}(X_{i}^{{(F)}},X_{j}^{{(B)}})}}{{{{N}_{{{{F}_{1}}}}}}} + {{C}^{{(0)}}}(X_{j}^{{(B)}})} \right)} \right.} \\ \left. { \times \left( {\frac{{{{\Sigma }_{{i \in hal{{f}_{2}}}}}{{C}^{{(I)}}}(X_{i}^{{(F)}},X_{j}^{{(B)}})}}{{{{N}_{{{{F}_{2}}}}}}} + {{C}^{{(0)}}}(X_{j}^{{(B)}})} \right)} \right) \\ = \frac{1}{{{{N}_{B}}}}\sum\limits_{j = 1}^{{{N}_{B}}} {{{B}_{1}}(X_{j}^{{(B)}}){{B}_{2}}(X_{j}^{{(B)}})} \\ \end{gathered} $$

where

$${{B}_{k}}(X_{j}^{{(B)}}) \equiv {{C}^{{(0)}}}(X_{j}^{{(B)}}) + \frac{1}{{{{N}_{{{{F}_{k}}}}}}}\sum\limits_{i \in hal{{f}_{k}}}^{} {{{C}^{{(I)}}}(X_{i}^{{(F)}},X_{j}^{{(B)}})} $$

Similarly,

$$\begin{gathered} F = \frac{1}{{{{N}_{F}}}}\sum\limits_{i = 1}^{{{N}_{F}}} {((F_{1}^{{(I)}}(X_{i}^{{(F)}}) + F_{1}^{{(0)}})(F_{2}^{{(I)}}(X_{i}^{{(F)}}) + F_{2}^{{(0)}}))} \\ = \frac{1}{{{{N}_{F}}}}\sum\limits_{i = 1}^{{{N}_{F}}} {((F_{1}^{{(I)}}(X_{i}^{{(F)}})F_{2}^{{(I)}}(X_{i}^{{(F)}})} \\ + \,F_{1}^{{(I)}}(X_{i}^{{(F)}})F_{2}^{{(0)}} + F_{2}^{{(I)}}(X_{i}^{{(F)}})F_{1}^{{(0)}})) + F_{1}^{{(0)}}F_{2}^{{(0)}} \\ \end{gathered} $$

where

$$\begin{gathered} F_{k}^{{(I)}}(X_{i}^{{(F)}}) \equiv \frac{1}{{{{N}_{{{{B}_{k}}}}}}}\sum\limits_{i \in hal{{f}_{k}}}^{} {{{C}^{{(I)}}}(X_{i}^{{(F)}},X_{j}^{{(B)}})F_{k}^{{(0)}}} \\ \equiv \frac{1}{{{{N}_{{{{B}_{k}}}}}}}\sum\limits_{i \in hal{{f}_{k}}}^{} {{{C}^{{(0)}}}(X_{j}^{{(B)}})} \\ \end{gathered} $$

Now, all the sums contain the same number of nonzero terms as in the absence of direct illuminance (i.e., for C(0) = 0); therefore, the computations are fairly efficient. The computation time is approximately the same as in the calculation of the image luminance itself.

5 RESULTS

As an example, we used the well-known benchmark scene Cornell Box. An isotropic point light source is located slightly below the ceiling center. All the surfaces are Lambertian with the albedo 0.5. The method BDPM was used without multiple importance sampling, and the ray issued from the camera was forcefully interrupted after the second diffuse intersection. The integration sphere radius was 1/120 of the scene size. The direct luminance was calculated by connecting the surface point to the light source rather than taken from photon maps.

The image obtained by the virtual camera is shown in Fig. 1.

Fig. 1.
figure 1

Virtual photo of the model Cornell Box scene.

Full size image

The three terms of the noise were calculated as described in Sections 3 and 4, and the total variance of the contribution to the luminance in one iteration was found by formula (1). For comparison, it was also calculated as sample variance over a series of iterations, which is possible because the contributions of different iterations are independent.

The noise amplitude, i.e., \(\sqrt V \), calculated using these two methods is shown in Fig. 2. It is seen that the results are almost undistinguishable.

Fig. 2.
figure 2

Noise \(\sqrt V \) for NB = 25 and NF = 1000—(a, above) benchmark value found as the sample variance over the sequence of iterations; (b, below) V is found by formula (1). The mean RMS over the region within the red box in the upper image is 1.2767 and 1.2769 in the lower image.

Full size image

To illustrate the contribution of all three terms in formula (1) to the total variance, they are show in Fig. 3 as the (R, G, B) components of the image. More precisely, the quantities \(\sqrt {\frac{1}{{{{N}_{F}}{{N}_{B}}}}({{C}_{{{\text{avg}}}}} - {{L}^{2}})} \), \(\sqrt {\frac{{1 - N_{F}^{{ - 1}}}}{{{{N}_{B}}}}({{B}_{{{\text{avg}}}}} - {{L}^{2}})} \), and \(\sqrt {\frac{{1 - N_{B}^{{ - 1}}}}{{{{N}_{F}}}}({{F}_{{{\text{avg}}}}} - {{L}^{2}})} \) are written to the R, G, and B channels of the image, respectively.

Fig. 3.
figure 3

Contributions of \(\sqrt {\frac{1}{{{{N}_{F}}{{N}_{B}}}}({{C}_{{{\text{avg}}}}} - {{L}^{2}})} \), \(\sqrt {\frac{{1 - N_{F}^{{ - 1}}}}{{{{N}_{B}}}}({{B}_{{{\text{avg}}}}} - {{L}^{2}})} \), and \(\sqrt {\frac{{1 - N_{B}^{{ - 1}}}}{{{{N}_{F}}}}({{F}_{{{\text{avg}}}}} - {{L}^{2}})} \) to the total noise written in the color channels R, G, and B of the image—(a, above) NB = 25, (b, below) NB = 100; in both cases NF = 1000. The mean values over the region within the red box in the upper image are (0.3812065, 1.21883, 0.01907) and (0.19602,0.610129,0.0204105) in the lower image; thus, the first two components decreased by a factor of \(\sqrt 4 \). The total RMS, i.e. \(\sqrt V \), averaged over the red box is 1.27689 in the upper image and 0.639361 in the lower image.

Full size image

It is seen that the image in Fig. 3 is mainly green, i.e., the second component is dominant. Therefore, to reduce the noise in this case, we should mainly reduce this dominant second term, i.e., increase NB because Cavg, Bavg, Favg, and L are independent of the number of rays. The increase of the number of rays NF issued from the light source has almost no effect and only slows down the computation. The result for NB increased by a factor of three is shown in Fig. 3b.

In practice, it is not so much the variance of the contribution of one iteration V that matters as the variance of the luminance after a given computation time T. During this time, NI = \(\frac{T}{\tau }\) iteration steps will be performed, and the variance of the resulting image will be VT = \(\frac{{\tau V}}{T}\). While V monotonically decreases with increasing number of rays, the time taken by one iteration step monotonically increases. These two “opposing factors” provide the optimum in the number of rays, as shown in Fig 4 and Table 2. They illustrate the behavior of the mean RMS (i.e. \(\sqrt {{{V}_{T}}} \)) over the red box for the benchmark scene under consideration, depending on the number of rays per one iteration step for T = 1000 s. Note that the number of rays issued from the camera was the same for all pixels (not only within the red box).

Fig. 4.
figure 4

The mean RMS over the region of the image marked by the red box in Fig. 2 after 1000 seconds of computation for various NB and NF (on the horizontal axis).

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Table 2. Statistical data for tracing 25 rays issued from the camera per pixel depending on the number of rays issued from the source during 1000 s for the benchmark scene
Full size table

It is clearly seen in Fig. 4 that the dependence on the number of rays issued from the camera (shown by plots of different colors) is very weak in this example. On the contrary, the dependence on the number of rays issued from the light source (plotted on the horizontal axis) has a minimum in a neighborhood of NF = 3100. The exact value of this minimum can slightly vary depending on the number of rays issued from the camera. However, this minimum is very flat; i.e., RMS is almost invariable in a wide range of NF variation, and its growth becomes noticeable only when the deviation from the optimum is by several orders of magnitude.

Note that the smooth curve in Fig. 4 is constructed on the basis of a lot of points obtained by calculating the RMS over one iteration by the exact formula (1), and the time taken by one iteration step is found by the approximate formula τ = αNF + βNB + γNBNF the coefficients in which are obtained by fitting to the data in Table 1. This approximation works very well.

More detailed information is shown in Table 2 for the case NB = 25. The time τ of one iteration step is composed of the time spent on tracing the rays issued from the camera τBMCRT, tracing the rays issued from the light source τFMCRT, and the time spent on merging these traces τmerge. In addition to the total noise \(\sqrt {{{V}_{T}}} \), Table 2 shows the variance over one iteration \(\sqrt V \), the number of iterations performed in the time T, the time spent on one iteration τ, and its components.

It is clearly seen from Table 2 that the variance in one iteration decreases with increasing number of rays issued from the light source (which could be expected),while the final noise increases. The analysis of the time components allows us to estimate their contribution to the total tracing time and understand why an increase in the number of rays ultimately leads to degradation in the quality of the final image.

6 CONCLUSIONS

A practical method for computing the coefficients 〈〈C2〉〉, \({{\langle \langle C\rangle _{F}^{2}\rangle }_{B}}\), and \({{\langle \langle C\rangle _{B}^{2}\rangle }_{F}}\) appearing in the formula for the BDPM noise on the basis of quantities obtained by ray tracing is proposed. A practical method for computing the coefficients for the case of the straightforward calculation of the direct luminance is also considered.

The final noise value can be found using the sample variance of a series of values accumulated in a sequence of iteration steps. This yields its value only for a specific number of rays; however, it does not allow us to predict how the variance will change for another number of rays. Meanwhile, it is easy to calculate, knowing the coefficients 〈〈C2〉〉, \({{\langle \langle C\rangle _{F}^{2}\rangle }_{B}}\), \({{\langle \langle C\rangle _{B}^{2}\rangle }_{F}}\). Therefore, given the results of only one computation, we are able to predict the variance (the contribution of one iteration step) for an arbitrary number of rays.

If the time spent on tracing the rays issued from the camera and the light source is known together with the time spent on their merging (as shown in Table 2), then the optimal number of rays can be calculated, which is the number of rays that ensures the minimum noise during the given computation time.