HEALpix and spherical harmonic sampling

 Posts: 7
 Joined: October 20 2004
 Affiliation: FYMA/PHYS/UCL
 Contact:
HEALpix and spherical harmonic sampling
Hi,
I have still a question about HEALpix grid in the particular context of spherical harmonic (SH) sampling and SH transform.
Are we sure that SH transform can be exactly realized to all orders exactly as it can be done on traditional equiangular grid (I mean, D theta = cste, D phi = cste)?
The problem is that, in polar caps, the number of points on each isolatitude ring decreases as its distance to the North pole, i.e. from 4*Nside to 4. However, spherical harmonics Y_l^m of order (l,m) (with m<=l) contain the term exp(i m phi) which needs at least m+1 points to be correctly sampled (without aliasing) on each isolatitude ring.
So, I'm not sure that Spherical Fourier transforms, i.e. spherical harmonic transform, can be realized exactly on HEALpix grid. In best cases, this can be just an approximation with a certain error.
What do you think ? Perhaps it is already well known but I haven't read such remark elsewhere. Do you know a reference which explains that effect ?
Best,
Laurent.
I have still a question about HEALpix grid in the particular context of spherical harmonic (SH) sampling and SH transform.
Are we sure that SH transform can be exactly realized to all orders exactly as it can be done on traditional equiangular grid (I mean, D theta = cste, D phi = cste)?
The problem is that, in polar caps, the number of points on each isolatitude ring decreases as its distance to the North pole, i.e. from 4*Nside to 4. However, spherical harmonics Y_l^m of order (l,m) (with m<=l) contain the term exp(i m phi) which needs at least m+1 points to be correctly sampled (without aliasing) on each isolatitude ring.
So, I'm not sure that Spherical Fourier transforms, i.e. spherical harmonic transform, can be realized exactly on HEALpix grid. In best cases, this can be just an approximation with a certain error.
What do you think ? Perhaps it is already well known but I haven't read such remark elsewhere. Do you know a reference which explains that effect ?
Best,
Laurent.

 Posts: 1485
 Joined: September 23 2004
 Affiliation: University of Sussex
 Contact:
Healpix sampling
Yes, it is approximate.
However near the poles the spherical harmonics go like
[tex]Y_{lm} \sim (1)^m \frac{e^{im\phi}}{m!}\sqrt{\frac{(2l+1)}{4\pi} \frac{(l+m)!}{(lm)!}} \left( \frac{\theta}{2}\right)^m[/tex]
so for large m and small [tex]\theta[/tex] the contributions go like [tex]\theta^m[/tex] which is tiny. So although HealPix does include all [tex]m\le l[/tex], near the polar caps you can to very good accuracy neglect highm contributions (much better than numerical precision).
This makes sense because [tex]e^{i\ell\phi}[/tex] has a spatial frequency much higher near the poles than at the equator, so you wouldn't expect a significant contribution near the poles unless m is small compared to [tex]\ell[/tex]. So a small number of pixels near the poles is not so bad.
However near the poles the spherical harmonics go like
[tex]Y_{lm} \sim (1)^m \frac{e^{im\phi}}{m!}\sqrt{\frac{(2l+1)}{4\pi} \frac{(l+m)!}{(lm)!}} \left( \frac{\theta}{2}\right)^m[/tex]
so for large m and small [tex]\theta[/tex] the contributions go like [tex]\theta^m[/tex] which is tiny. So although HealPix does include all [tex]m\le l[/tex], near the polar caps you can to very good accuracy neglect highm contributions (much better than numerical precision).
This makes sense because [tex]e^{i\ell\phi}[/tex] has a spatial frequency much higher near the poles than at the equator, so you wouldn't expect a significant contribution near the poles unless m is small compared to [tex]\ell[/tex]. So a small number of pixels near the poles is not so bad.

 Posts: 7
 Joined: October 20 2004
 Affiliation: FYMA/PHYS/UCL
 Contact:
OK. Thanks for the answer.
It will be good to control this error (don't know how) which is not present on equiangular grid (I mean for bandlimited signals). I think that, for instance, CMB analysis relies partially on these spherical harmonic transform and results in this field have to be errorcontrolled.
However, IMHO, approximation is not so harmless since for instance for [tex]i = N_{\rm side}/2 [/tex] (i.e. in the North cap and with [tex]i[/tex] the isolatitude ring label), spherical harmonics are so not well estimated by [tex]C e^{{\rm i} m\phi} \big(\frac{\theta}{2}\big)^m [/tex], leading to more important contributions and preventing good computations of SH transform on [tex]m\geq N_{\rm side}/2 [/tex] in case where [tex]l\geq N_{\rm side}/2 [/tex] (for high frequencies).
Best,
Laurent.
It will be good to control this error (don't know how) which is not present on equiangular grid (I mean for bandlimited signals). I think that, for instance, CMB analysis relies partially on these spherical harmonic transform and results in this field have to be errorcontrolled.
However, IMHO, approximation is not so harmless since for instance for [tex]i = N_{\rm side}/2 [/tex] (i.e. in the North cap and with [tex]i[/tex] the isolatitude ring label), spherical harmonics are so not well estimated by [tex]C e^{{\rm i} m\phi} \big(\frac{\theta}{2}\big)^m [/tex], leading to more important contributions and preventing good computations of SH transform on [tex]m\geq N_{\rm side}/2 [/tex] in case where [tex]l\geq N_{\rm side}/2 [/tex] (for high frequencies).
Best,
Laurent.

 Posts: 1485
 Joined: September 23 2004
 Affiliation: University of Sussex
 Contact:
From the differential equation for the [tex]P_{lm}[/tex] you can show that the turning point (where the character changes from oscillatory to decaying) is at [tex]\sin\theta \sim m/\ell[/tex]. Hence for [tex]m >> \ell \sin\theta[/tex] you are always well in the damped tail, and the contributions are small.
However it's true that the pixelization error is worse around the polar caps. I believe they are writing a paper on the error properties.
However it's true that the pixelization error is worse around the polar caps. I believe they are writing a paper on the error properties.