How to represent a point using spherical harmonics?

Short answer: You can't really. Representing a point-light requires an infinite number of coefficients.

Long answer: You can approximate it using a circular shape. The paper "Algorithms for Spherical Harmonic-Lighting" By Ian G. Lisle and S.-L. Tracy Huang gives a method for calculating the coefficients for this directly.

You'd also have to know the distance to the light.

Short answer: You can't really. Representing a point-light requires an infinite number of coefficients. Long answer: You can approximate it using a circular shape. The paper "Algorithms for Spherical Harmonic-Lighting" By Ian G. Lisle and S.-L. Tracy Huang gives a method for calculating the coefficients for this directly.

Hey ginkgo! Well, yes - but isn't that exactly the idea behind spherical harmonics - approximating a function using a polynomial of a finite degree (e.g. degree 2 already gives an error rate less than 1%).

I have found several methods for computing the coefficients, amongst others, the original project page for irradiance computation:
Link, which provides the function prefilter.c to compute coefficients.

But the input to this function is an environment map (an image), for which the lighting is computed.

You'd also have to know the distance to the light.

Well, as I mentioned, let's take the direction vector [0 0 1], with an infinite point light source.


Further suggestions?

Short answer: You can't really. Representing a point-light requires an infinite number of coefficients. Long answer: You can approximate it using a circular shape. The paper "Algorithms for Spherical Harmonic-Lighting" By Ian G. Lisle and S.-L. Tracy Huang gives a method for calculating the coefficients for this directly.

Hey ginkgo! Well, yes - but isn't that exactly the idea behind spherical harmonics - approximating a function using a polynomial of a finite degree (e.g. degree 2 already gives an error rate less than 1%).
I have found several methods for computing the coefficients, amongst others, the original project page for irradiance computation: Link, which provides the function prefilter.c to compute coefficients.
But the input to this function is an environment map (an image), for which the lighting is computed.

You'd also have to know the distance to the light.

Well, as I mentioned, let's take the direction vector [0 0 1], with an infinite point light source.

Further suggestions?

Well, as I mentioned, let's take the direction vector [0 0 1], with an infinite point light source.

Ah ok. The confusion was because an infinite/directional and a point/omni light are different things.
The details for directional lights are contained in the Stupid SH Tricks paper in the "Analytic Models" section.

[quote name='opengl_beginner' timestamp='1353232912' post='5001996']
Well, as I mentioned, let's take the direction vector [0 0 1], with an infinite point light source.

Ah ok. The confusion was because an infinite/directional and a point/omni light are different things.
The details for directional lights are contained in the Stupid SH Tricks paper in the "Analytic Models" section.
[/quote]


Hey Hodgman!


Thanks a lot for the reference. I had seen the tutorial before, but never managed to notice the relevant chapter!

Directional lights are trivial to compute, you simply evaluate the SH basis functions in the given direction and scale appropriately (see Normalization section.) Spherical Light sources can be efficiently evaluated using zonal harmonics. Below is a diagram showing an example scene, we want to compute the incident radiance, a spherical function, at the receiver point P. Given a spherical light source with
center C, radius r, what is the radiance arriving at a point P d units away? The sin of the half-angle subtended by the light source is r/d, so you just need to compute a light source that subtends an appropriate part of the sphere. The ZH coefficients can be computed in closed form as a function of this angle: ? = integral ( integral (y_l, \theta, 0, 2Pi ) ,\theta, 0, a ) where a is the half-angle d subtended. See Appendix A3 ZH Coefficients for Spherical Light Source for the expressions through order 6.
[/quote]

I assume a = r/d, where d is infinite and thus a = 0.
I've looked into the Appendix for L = 1, ...., 6:

L=0: ?sqrt(?)(?1+cos(a))
L =1: 1/2 sqrt(3) sqrt(?) sin(a)2
L=2: ?1/2 * sqrt(5) * sqrt(?) * cos(a) (?1 + cos(a)) (cos(a) + 1)
L=3 ?1/8 *sqrt(7)* sqrt(?) (?1 + cos(a)) (cos(a) + 1) (5 cos(a)^2 ? 1)
L=4 ?3/8*sqrt(?)*cos(a) (?1 + cos(a)) (cos(a) + 1) (7 cos(a)2 ? 3)
L=5 ? 1/16* sqrt(11)*sqrt(?)* (?1 + cos(a)) (cos(a) + 1) (21 cos(a)4 ? 14 cos(a)2 + 1)

[/quote]


So, clearly, all coefficients will be 0 due to the term (-1 + cos(a)), which is 0 for a = 0.


Where is the flaw in my derivation?

Hey ginkgo! Well, yes - but isn't that exactly the idea behind spherical harmonics - approximating a function using a polynomial of a finite degree (e.g. degree 2 already gives an error rate less than 1%).

Yes, you generally use spherical harmonics as a means of approximating some function defined about a sphere using a compact set of coefficients. The issue that ginkgo was alluding to has to do with the fact that spherical harmonics are essentially a frequency-space representation of a function, where lower coefficients correspond lower-frequency components of the function and the higher coefficients correspond to higher-frequency components. With your typical "punctual" light source (point light, directional light, etc.) the incoming radiance in terms of a sphere surrounding some point in space (such as the surface you're rendering) is essentially a dirac delta function. A delta function would require infinite coefficients to be represented in spherical harmonics, so it's basically impossible. You can get the best approximation for some SH order by directly projecting the direction of the delta onto the basis functions (which is mentioned in Stupid SH tricks), but if you were to display the results for 2nd-order SH you'd find that you basically end up with a big low-frequency blob oriented about the direction. This is why "area" lights that have some volume associated with them work better with SH, since they can be represented better with less coefficients. The same goes for any function defined about a sphere, for instance a BRDF or an NDF.

MJP's explanation is exactly right. It would be much easier to develop some intuition for this using Fourier series to approximate a delta function.

"Projecting the direction of the delta onto the basis functions" is the same thing as evaluating the basis functions at the point where the delta is centered, so in the case of Fourier series, you just get cos(x) + cos(2x) + cos(3x) + cos(4x) + ... as an approximation to the delta function centered at 0.

You can see it here expanded up to cos(10x): http://fooplot.com/plot/k49yaatz5z

I expect the equivalent situation in spherical harmonics will also have funny ripples that it would take many terms to make small. A delta function is just not something that can be well approximated with low-frequency functions.

I expect the equivalent situation in spherical harmonics will also have funny ripples that it would take many terms to make small.

Indeed. The visual artifacts caused by this sort of "ringing" can actually be very severe when you're working with HDR intensities.

Firstly, thanks all, for your remarks. Things are becoming much clearer now.
I would like to say the following.

1) Although, I do not quite understand why a point light source is a Dirac Delta function in the frequency space. I know that the span of the spatial domain is inversely proportional to the span in the frequency domain, i.e. if a function is spread in (x, y, z) coordinates, it will be concentrated in the frequency domain.
BUT - a point light source is concentrated in the spatial domain (it's just a line) and thus, I would expect it to be spread out in the frequency domain.
Why is this not the case?


2) Perhaps I should explain what the motivation behind the problem is. Very often we come across images which contain dark spots. By adding a point light source to the current scene illumination, I wanted to make these spots lighter, i.e. 'more visible'. Since all of you are advising against using a point light source, do you see any other variant of how to address this problem?

Thanks,
opengl_beginner

Point lights being Dirac delta functions has nothing to do with frequency space. The function that we are trying to approximate using spherical harmonics is the radiant intensity (measured in Watts per steradian). If light is coming concentrated from a single point, that function is actually a delta (some amount of light is coming from a single point, which means something like infinite radiant intensity at that particular point).

By the way, delta functions are not actually functions: They are measures. I felt dirty after having written the paragraph above.

1) Although, I do not quite understand why a point light source is a Dirac Delta function in the frequency space. I know that the span of the spatial domain is inversely proportional to the span in the frequency domain, i.e. if a function is spread in (x, y, z) coordinates, it will be concentrated in the frequency domain.
BUT - a point light source is concentrated in the spatial domain (it's just a line) and thus, I would expect it to be spread out in the frequency domain.
Why is this not the case?

A point light is a delta in the spatial domain, not the frequency domain. A point light is infinitely small, so in the frequency domain the response is infinitely large (hence why you need infinite SH coefficients).

2) Perhaps I should explain what the motivation behind the problem is. Very often we come across images which contain dark spots. By adding a point light source to the current scene illumination, I wanted to make these spots lighter, i.e. 'more visible'. Since all of you are advising against using a point light source, do you see any other variant of how to address this problem?

Well you can still somewhat approximate the contribution (irradiance) from a point light source if you only care about diffuse, and not specular. Just project it onto SH, and convolve with a cosine kernel to SH containing lambertian reflectance. Then you just "look up" into the SH at runtime in your shader by projecting your normal direction onto SH and calculating a dot product between the two SH vectors. You can even roll the cosine convolution into the runtime lookup, since it's just a multiply by 3 scalars. Games have done this in the past to approximate the contribution from "unimportant" light sources. Like I said before though you may ringing artifacts if you project lights that can have high intensities.

By the way, delta functions are not actually functions


double const diracDelta( std::function<double const(double const) > testFunction )
{
return testFunction(0.0);
}


:-D

*ducks*