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Maximum value of a spherical harmonic

Started by glowkeeper March 27, 2007
I have a function on a sphere represented in terms of a 3rd order
spherical harmonic (http://www.research.scea.com/gdc2003/spherical-
harmonic-lighting.pdf).  This is somewhat similar to how wavelet
compression, such as jpeg, represents a function over a 2d area in
frequency space.  I want to find the maximum value of this function
analytically without converting from frequency space.  An upper bound
can be easily found by summing the maximum value of all the basis
functions, but this will be a significant over estimate.  I don't
really need the exact value, so a more accurate estimate would be
interesting (as well as an exact solution).

On Mar 27, 5:26 am, "glowkeeper" <glowkee...@yahoo.co.uk> wrote:
> I have a function on a sphere represented in terms of a 3rd order > spherical harmonic (http://www.research.scea.com/gdc2003/spherical- > harmonic-lighting.pdf). This is somewhat similar to how wavelet > compression, such as jpeg, represents a function over a 2d area in > frequency space. I want to find the maximum value of this function > analytically without converting from frequency space. An upper bound > can be easily found by summing the maximum value of all the basis > functions, but this will be a significant over estimate. I don't > really need the exact value, so a more accurate estimate would be > interesting (as well as an exact solution).
Hello Glowkeeper, There may be some confusion on terminology. The 3rd order Spherical Harmonic (L=3,m=0) is simply sqrt(7/16pi)(5cos(theta)^3 - 3 cos(theta)) It has a global maximum at theta = 0, and the max is sqrt(7/4pi). I'm not sure about your comment pertaining summing the maximum values of the basis functions. Did you not know how to handle the two cos terms? There are of course 7 different standard 3rd order Spherical Harmonic functions, but I gave you the only real valued one. Or do you have an expansion in terms of all 7 3rd order functions? IHTH, Clay
On Mar 27, 3:58 pm, "Clay" <phys...@bellsouth.net> wrote:
> On Mar 27, 5:26 am, "glowkeeper" <glowkee...@yahoo.co.uk> wrote: > > > I have a function on a sphere represented in terms of a 3rd order > > spherical harmonic (http://www.research.scea.com/gdc2003/spherical- > > harmonic-lighting.pdf). This is somewhat similar to how wavelet > > compression, such as jpeg, represents a function over a 2d area in > > frequency space. I want to find the maximum value of this function > > analytically without converting from frequency space. An upper bound > > can be easily found by summing the maximum value of all the basis > > functions, but this will be a significant over estimate. I don't > > really need the exact value, so a more accurate estimate would be > > interesting (as well as an exact solution). > > Hello Glowkeeper, > > There may be some confusion on terminology. The 3rd order Spherical > Harmonic (L=3,m=0) is simply sqrt(7/16pi)(5cos(theta)^3 - 3 > cos(theta)) > > It has a global maximum at theta = 0, and the max is sqrt(7/4pi). > > I'm not sure about your comment pertaining summing the maximum values > of the basis functions. Did you not know how to handle the two cos > terms? > > There are of course 7 different standard 3rd order Spherical Harmonic > functions, but I gave you the only real valued one. Or do you have an > expansion in terms of all 7 3rd order functions? > > IHTH, > > Clay
Clay Yes I expect there is confusion in terminology. I'm no expert with Spherical Harmonics and I was taking my terminology from the paper I linked to. I am starting with an empirically defined function on a sphere, I am then compressing this by representing the function as a scaled sum of Spherical Harmonic basis functions. So as you say the function is in terms of all the 3rd order basis function, but also the 1st and 2nd order functions as well, each function being scaled by a constant value. I can see how you can find the maximum of any individual function, in just the way you describe, but I want to find the maximum of the scaled sum of all the functions. Clearly a much harder problem.
On Mar 28, 6:40 am, "glowkeeper" <glowkee...@yahoo.co.uk> wrote:
> On Mar 27, 3:58 pm, "Clay" <phys...@bellsouth.net> wrote: > > > > > > > On Mar 27, 5:26 am, "glowkeeper" <glowkee...@yahoo.co.uk> wrote: > > > > I have a function on a sphere represented in terms of a 3rd order > > > spherical harmonic (http://www.research.scea.com/gdc2003/spherical- > > > harmonic-lighting.pdf). This is somewhat similar to how wavelet > > > compression, such as jpeg, represents a function over a 2d area in > > > frequency space. I want to find the maximum value of this function > > > analytically without converting from frequency space. An upper bound > > > can be easily found by summing the maximum value of all the basis > > > functions, but this will be a significant over estimate. I don't > > > really need the exact value, so a more accurate estimate would be > > > interesting (as well as an exact solution). > > > Hello Glowkeeper, > > > There may be some confusion on terminology. The 3rd order Spherical > > Harmonic (L=3,m=0) is simply sqrt(7/16pi)(5cos(theta)^3 - 3 > > cos(theta)) > > > It has a global maximum at theta = 0, and the max is sqrt(7/4pi). > > > I'm not sure about your comment pertaining summing the maximum values > > of the basis functions. Did you not know how to handle the two cos > > terms? > > > There are of course 7 different standard 3rd order Spherical Harmonic > > functions, but I gave you the only real valued one. Or do you have an > > expansion in terms of all 7 3rd order functions? > > > IHTH, > > > Clay > > Clay > > Yes I expect there is confusion in terminology. I'm no expert with > Spherical Harmonics and I was taking my terminology from the paper I > linked to. I am starting with an empirically defined function on a > sphere, I am then compressing this by representing the function as a > scaled sum of Spherical Harmonic basis functions. So as you say the > function is in terms of all the 3rd order basis function, but also the > 1st and 2nd order functions as well, each function being scaled by a > constant value. > > I can see how you can find the maximum of any individual function, in > just the way you describe, but I want to find the maximum of the > scaled sum of all the functions. Clearly a much harder problem.- Hide quoted text - > > - Show quoted text -
Hello GlowKeeper, Do you know if your result is purely real? Try plotting your sum of Ylm(theta,phi) over [0<=theta<=2pi,0<=phi<=pi] and see where it is biggest. If your sum is purely real, then you can dispense with the phi part and just plot theta from 0 to 2pi and see where the max occurs. You can use a simple bisection or golden ratio method to easily find the max. IHTH, Clay
On Mar 28, 2:43 pm, "Clay" <phys...@bellsouth.net> wrote:
> On Mar 28, 6:40 am, "glowkeeper" <glowkee...@yahoo.co.uk> wrote: > > > > > > > On Mar 27, 3:58 pm, "Clay" <phys...@bellsouth.net> wrote: > > > > On Mar 27, 5:26 am, "glowkeeper" <glowkee...@yahoo.co.uk> wrote: > > > > > I have a function on a sphere represented in terms of a 3rd order > > > > spherical harmonic (http://www.research.scea.com/gdc2003/spherical- > > > > harmonic-lighting.pdf). This is somewhat similar to how wavelet > > > > compression, such as jpeg, represents a function over a 2d area in > > > > frequency space. I want to find the maximum value of this function > > > > analytically without converting from frequency space. An upper bound > > > > can be easily found by summing the maximum value of all the basis > > > > functions, but this will be a significant over estimate. I don't > > > > really need the exact value, so a more accurate estimate would be > > > > interesting (as well as an exact solution). > > > > Hello Glowkeeper, > > > > There may be some confusion on terminology. The 3rd order Spherical > > > Harmonic (L=3,m=0) is simply sqrt(7/16pi)(5cos(theta)^3 - 3 > > > cos(theta)) > > > > It has a global maximum at theta = 0, and the max is sqrt(7/4pi). > > > > I'm not sure about your comment pertaining summing the maximum values > > > of the basis functions. Did you not know how to handle the two cos > > > terms? > > > > There are of course 7 different standard 3rd order Spherical Harmonic > > > functions, but I gave you the only real valued one. Or do you have an > > > expansion in terms of all 7 3rd order functions? > > > > IHTH, > > > > Clay > > > Clay > > > Yes I expect there is confusion in terminology. I'm no expert with > > Spherical Harmonics and I was taking my terminology from the paper I > > linked to. I am starting with an empirically defined function on a > > sphere, I am then compressing this by representing the function as a > > scaled sum of Spherical Harmonic basis functions. So as you say the > > function is in terms of all the 3rd order basis function, but also the > > 1st and 2nd order functions as well, each function being scaled by a > > constant value. > > > I can see how you can find the maximum of any individual function, in > > just the way you describe, but I want to find the maximum of the > > scaled sum of all the functions. Clearly a much harder problem.- Hide quoted text - > > > - Show quoted text - > > Hello GlowKeeper, > > Do you know if your result is purely real? Try plotting your sum of > Ylm(theta,phi) over [0<=theta<=2pi,0<=phi<=pi] and see where it is > biggest. If your sum is purely real, then you can dispense with the > phi part and just plot theta from 0 to 2pi and see where the max > occurs. You can use a simple bisection or golden ratio method to > easily find the max. > > IHTH, > > Clay- Hide quoted text - > > - Show quoted text -
Good suggestion, thanks Clay!