[Algorithms] Spherical Harmonic Filtering

[Alen Ladavac]

Hi all,

I was revisiting some of our old shader code (order 2, i.e. 9-coefficients) trying to fix ringing in some high contrast environments. I am looking at Stupid Spherical Harmonics (SH) Tricks by Peter-Pike Sloan, the section about Windowing.

The Lanczos and Han functions mentioned are of form:

sin(pi*x/w)/(x/w)

and

(1+cos(pi*x/w))/2

respectively, for a given filtering window of width "w". The rest of text, as far as I can see, seems to imply that w should be equal to the order of SH used. (w=6 for 6th order).

Initially, I would expecting filtering to be done by convolution in spatial domain, by projecting the filter function to the SH basis and then multiplying its y[l,0] coefficients into appropriate bands (y[l,m]) of the filtered function. However, all the other notions in there seem to point to this actually being done as multiplication in frequency domain. More specifically, as if the filtering function as presented here above _already is_ the frequency domain presentation. So it would just multiply of y(l,m) coefficients of the filtered function by values of the filter evaluated at integers.

This looks rather unusual to me, but I guess there is some reasoning behind all that.

As a side note, I got some ok-ish results with filtering in spatial domain (aka. bluring) using Han function as above (*) with value w=1.5 through convolution as described above. (Where the number w=1.5 was "tuned" by manual binary search with visual assessment of results to minimize visible ringing without introducing "too much" blur.) While on the other hand, using the multiplication directly in frequency domain looked way too blurry for w=3 and was ringing too much if I tried increasing the number (as with multiplication in the frequency domain wider "window" leads to less blur, while it opposite happens in the spatial domain.)

I would appreciate if anyone can shed a bit of light onto this.

Thanks a lot in advance,
Alen

(*) That form of the function does look strange for spatial domain but 1.5 is an arbitrary "hand-tuned" factor anyway.

[Jon Watte]

This is a little late, but better late than never :-)

Did you get a reasonable solution? Do you have any images to show?

The way I understand it, those functions are both spatial domain functions.
The first one is the "sinc()" function, and the second one is the "cosine
window" function from sampling theory.
Also, when you use the sinc function, you typically want to either cut off
at a zero crossing (multiple of pi) or further window the sinc() function,
typically with a cosine window.

Finally, to adjust the cosine window, you can raise it to some power for a
trade-off between "preserving detail" and "removing aliasing"; known as a
"raised cosine window function."

Sincerely,

jw





Sincerely,

Jon Watte


--
"I pledge allegiance to the flag of the United States of America, and to
the republic for which it stands, one nation indivisible, with liberty and
justice for all."
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Post by Alen Ladavac
Hi all,
I was revisiting some of our old shader code (order 2, i.e.
9-coefficients) trying to fix ringing in some high contrast environments. I
am looking at Stupid Spherical Harmonics (SH) Tricks by Peter-Pike Sloan,
the section about Windowing.
sin(pi*x/w)/(x/w)
and
(1+cos(pi*x/w))/2
respectively, for a given filtering window of width "w". The rest of text,
as far as I can see, seems to imply that w should be equal to the order of
SH used. (w=6 for 6th order).
Initially, I would expecting filtering to be done by convolution in
spatial domain, by projecting the filter function to the SH basis and then
multiplying its y[l,0] coefficients into appropriate bands (y[l,m]) of the
filtered function. However, all the other notions in there seem to point to
this actually being done as multiplication in frequency domain. More
specifically, as if the filtering function as presented here above _already
is_ the frequency domain presentation. So it would just multiply of y(l,m)
coefficients of the filtered function by values of the filter evaluated at
integers.
This looks rather unusual to me, but I guess there is some reasoning behind all that.
As a side note, I got some ok-ish results with filtering in spatial domain
(aka. bluring) using Han function as above (*) with value w=1.5 through
convolution as described above. (Where the number w=1.5 was "tuned" by
manual binary search with visual assessment of results to minimize visible
ringing without introducing "too much" blur.) While on the other hand,
using the multiplication directly in frequency domain looked way too blurry
for w=3 and was ringing too much if I tried increasing the number (as with
multiplication in the frequency domain wider "window" leads to less blur,
while it opposite happens in the spatial domain.)
I would appreciate if anyone can shed a bit of light onto this.
Thanks a lot in advance,
Alen
(*) That form of the function does look strange for spatial domain but 1.5
is an arbitrary "hand-tuned" factor anyway.
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