### Multi-wavelength free path sampling using ratio tracking

Posted:

**Wed Feb 22, 2017 5:15 am**Hi,

I'm curious about whether using ratio tracking for estimating Monte Carlo constribution (f(x) / p(x), not transmittance) with delta tracking for heterogeneous media is unbiased.

We can get a MC contribution of free path sampling by delta tracking for a wavelength used for the procedure.

There is a demand to know MC contributions for other wavelengths simultaneously,

but we don't have the transmittance and PDF as separated values.

Therefore it's not possible to apply delta tracking to estimate MC contributions for multiple wavelengths.

On the other hand, estimating transmittance for a given segment in participating media for multiple wavelengths in unbiased fashion is possible by performing ratio tracking for every wavelengths.

I consider things like following:

perform delta tracking to sample free path for the wavelength wl_i and get the distance d.

PDF of the procedure is

p(d, wl_i) = sigma_e(d, wl_i) * exp(-int_0^d sigma_e(s, wl_i) ds)

Transmittance for another wavelength wl_j is

T(d, wl_j) = exp(-int_0^d sigma_e(s, wl_j) ds)

Therefore, MC contribution for the wavelength wl_j is

T(d, wl_j) / p(d, wl_i) = 1 / sigma_e(d, wl_i) * exp(-int_0^d (sigma_e(s, wl_j) - sigma_e(s, wl_i)) ds)

sigma_e(s, wl_j) - sigma_e(s, wl_i) can be a negative value, but ratio tracking is able to handle correctly.

Does this become unbiased estimate when I integrate this procedure into a light transport algorithm?

Thanks,

I'm curious about whether using ratio tracking for estimating Monte Carlo constribution (f(x) / p(x), not transmittance) with delta tracking for heterogeneous media is unbiased.

We can get a MC contribution of free path sampling by delta tracking for a wavelength used for the procedure.

There is a demand to know MC contributions for other wavelengths simultaneously,

but we don't have the transmittance and PDF as separated values.

Therefore it's not possible to apply delta tracking to estimate MC contributions for multiple wavelengths.

On the other hand, estimating transmittance for a given segment in participating media for multiple wavelengths in unbiased fashion is possible by performing ratio tracking for every wavelengths.

I consider things like following:

perform delta tracking to sample free path for the wavelength wl_i and get the distance d.

PDF of the procedure is

p(d, wl_i) = sigma_e(d, wl_i) * exp(-int_0^d sigma_e(s, wl_i) ds)

Transmittance for another wavelength wl_j is

T(d, wl_j) = exp(-int_0^d sigma_e(s, wl_j) ds)

Therefore, MC contribution for the wavelength wl_j is

T(d, wl_j) / p(d, wl_i) = 1 / sigma_e(d, wl_i) * exp(-int_0^d (sigma_e(s, wl_j) - sigma_e(s, wl_i)) ds)

sigma_e(s, wl_j) - sigma_e(s, wl_i) can be a negative value, but ratio tracking is able to handle correctly.

Does this become unbiased estimate when I integrate this procedure into a light transport algorithm?

Thanks,