Odd terms in the SH irradiance expansion

Here’s one last observation about SH irradiance which is perhaps more a curiosity than anything else. Or at least we couldn’t work out how to make it useful!

Consider the integral for computing the L1 vector of coefficients \vec{\boldsymbol{R}}_1:

    \[ \vec{\boldsymbol{R}}_1=\frac{1}{4\pi} \oint \vec{\boldsymbol{\omega}}\, R(\vec{\boldsymbol{\omega}}) \, d\Omega \]

This looks somewhat similar to the definition of irradiance, and if we dot the whole expression by the normal vector we get:

    \[ \vec{\boldsymbol{n}} \! \cdot \! \vec{\boldsymbol{R}}_1=\frac{1}{4\pi}\oint\vec{\boldsymbol{n}}\!\cdot\!\vec{\boldsymbol{\omega}}\, R(\vec{\boldsymbol{\omega}}) \, d\Omega \]

    \[ =\frac{1}{4}\left( \frac{1}{\pi} \int_{H+}\vec{\boldsymbol{n}}\!\cdot\!\vec{\boldsymbol{\omega}}\, R(\vec{\boldsymbol{\omega}}) \, d\Omega +\frac{1}{\pi}\int_{H-}\vec{\boldsymbol{n}}\!\cdot\!\vec{\boldsymbol{\omega}}\, R(\vec{\boldsymbol{\omega}}) \, d\Omega \right) \]

which yields

    \[ \vec{\boldsymbol{n}}\!\cdot\!\vec{\boldsymbol{R}}_1=\frac{1}{4} \left( I(\vec{\boldsymbol{n}}) - I(-\vec{\boldsymbol{n}}) \right) \]

This is actually quite surprising — what it says is that the L1 band captures the antipodal difference in irradiance perfectly. If you know the exact irradiance in a direction, and you know the L1 vector, then you can compute the exact irradiance in the opposite direction. Or, to put it another way, the only odd terms in the irradiance expansion are the linear terms in the L1 band.

Perhaps this is just a curiosity, but it feels like it might give us a useful clue for improving higher-order SH irradiance models.

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