Why is reflectance calculated with math.pi/irradiance?

[patrickcgray]

One line 265 of image.py reflectance is calculated from radiance via self.__reflectance_image = self.radiance() * math.pi / irradiance. I'm curious why math.pi is used instead of just dividing radiance by irradiance from the DLS/panel? Is this because the irradiance is a full hemisphere? I don't quite understand the physical/geometric basis of this conversion.

[poynting]

Yes, it is a geometric factor due to the conversion between radiance and irradiance when using a panel and a radiance sensor. See for example equation 1 in this paper, copied below.

[patrickcgray]

Okay thanks! But what about when the irradiance values come from the DLS? Would it need to also be pi/irradiance? Or is the horizontal_irradiance that comes from the DLS somehow analogous to the panel measurement and needs the same treatment?

[fdarvas]

The DLS is calibrated to measure irradiance - unlike the camera pixels, which measure light coming from a narrow cone, the DLS integrates light from all directions, i.e. the whole hemisphere. If we assume a cosine response and constant radiance (for simplicity sake) and integrate the radiance over the hemisphere for the DLS we have DLS_output = Radiance Integral_0_2 pi dphi Integral_0_pi/2 cos(psi) sin(psi) dpsi = pi Radiance. So the DLS already measures Irradiance (the factor pi is "built" into these measurements), while the camera measures radiance .

[patrickcgray]

Okay and because pi is baked into the measurement, in order to cancel it out we need to use the rad*pi / irradiance rather than just rad/irradiance to correctly calculate reflectance from the DLS measurements?

[fdarvas]

Correct.