Light Transport in Tissue | |
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Variance reduction techniques are used to reduce the number of photons necessary to achieve the desired accuracy for a Monte Carlo calculation. These techniques have a long history and the important ones were first implemented by Kahn [35,36,38,37]. More recently, a summary of current variance reduction techniques is given by Hendricks and Booth [25]. A simple proof that the variance is smaller when implicit capture is used (described below) is given by Sobol' [57]. The flowchart in Figure 2.3 illustrates how variance reduction techniques fit into a Monte Carlo program.
The technique of implicit capture as described by Witt was used to reduce the variance in the Monte Carlo model [68]. This technique assigns a weight to each photon as it enters tissue. After each propagation step, the particle's weight is reduced by the probability of absorption. The usual method completely absorbs a photon according to the probability of absorption. Thus the implicit capture technique provides some absorption information at each photon step, rather than just at times when the photon is completely absorbed.
As an example, consider the use of the implicit capture technique with the variable stepsize method. In this case, the photon's weight is reduced by a factor of (1-a) which represents the fraction of the photon absorbed at each propagation step. This ensures that photons are not killed after a significant amount of computation time has been spent to transport them long distances. This is the basis for the improvement or, as Hendricks and Booth put it, ``All variance reduction schemes work by putting a large number of particles of low weight in regions of interest and allowing only a small number of particles with high weight in unimportant regions of phase space.''
The implicit capture technique is equivalent to propagating many photons (a packet) along each path through the tissue. The size of the packet is described by a weight coefficient. After each photon step a fraction (1-a) of the photons travelling along the path is absorbed, and the weight coefficient is reduced accordingly. The packet of photons is propagated until the weight coefficient drops below a specified tolerance.
How should the photon (packet) be terminated? The weight will never reach zero, and continuing to propagate a photon with a minuscule weight adds little information to the problem solution. Absorbing or discarding all the remaining weight, after the weight falls below a minimum, skews the absorption profile or violates energy conservation. A technique called roulette is used to terminate a photon once its weight drops below a specified minimum. The roulette technique gives such a photon (with weight w) one chance in m of surviving with a weight mw or else its weight is reduced to zero. The photon is thereby killed in an unbiased fashion, without sacrificing energy conservation and without continuing propagation until its weight has reached zero.
The converse technique, called splitting, might be used to improve statistics in another situation. When a photon passes into the a more ``interesting'' region, a photon with weight w may be split into m different photons each with weight w/m. This conserves energy and improves the statistics in the region of greater interest. When a photon passes into a region of lesser interest, then, roulette is used to reduce the number of photons in that region.