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opengl_beginner
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In Topic: How to represent a point using spherical harmonics?
19 November 2012  03:53 AM
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
In Topic: How to represent a point using spherical harmonics?
18 November 2012  12:40 PM
Ah ok. The confusion was because an infinite/directional and a point/omni light are different things.
Well, as I mentioned, let's take the direction vector [0 0 1], with an infinite point light source.
The details for directional lights are contained in the Stupid SH Tricks paper in the "Analytic Models" section.
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 halfangle 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: 𝑧 = integral ( integral (y_l, \theta, 0, 2Pi ) ,\theta, 0, a ) where a is the halfangle d subtended. See Appendix A3 ZH Coefficients for Spherical Light Source for the expressions through order 6.
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(π)(−1+cos(a))
L =1: 1/2 sqrt(3) sqrt(π) sin(a)2
L=2: −1/2 * sqrt(5) * sqrt(π) * cos(a) (−1 + cos(a)) (cos(a) + 1)
L=3 −1/8 *sqrt(7)* sqrt(π) (−1 + cos(a)) (cos(a) + 1) (5 cos(a)^2 − 1)
L=4 −3/8*sqrt(π)*cos(a) (−1 + cos(a)) (cos(a) + 1) (7 cos(a)2 − 3)
L=5 − 1/16* sqrt(11)*sqrt(π)* (−1 + cos(a)) (cos(a) + 1) (21 cos(a)4 − 14 cos(a)2 + 1)
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?
In Topic: How to represent a point using spherical harmonics?
18 November 2012  04:01 AM
Short answer: You can't really. Representing a pointlight requires an infinite number of coefficients. Long answer: You can approximate it using a circular shape. The paper "Algorithms for Spherical HarmonicLighting" 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?
In Topic: How to represent a point using spherical harmonics?
18 November 2012  04:00 AM
Short answer: You can't really. Representing a pointlight requires an infinite number of coefficients. Long answer: You can approximate it using a circular shape. The paper "Algorithms for Spherical HarmonicLighting" 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?
In Topic: Spherical Harmonics  Sphere maps, cube maps and user interaction
30 October 2012  01:05 AM
This project shows what I would like to do: http://www.yasrt.org/shlighting/
These are the steps I have been thinking of:
1) Compute SH coefficients for different light probes (e.g. Grace Cathedrale)
2) Generate a sphere, which represents the SH illumination model
3) Create a 3D object (e.g. aring)
4) Fix albedo = 1
5) Rendering the ring under Grace Cathedrale illumination
6) Rotate SH model, and as a result, the scene should change in realtime
I've managed to carry out step 1) using the prefilter.c function given here.
Now  how do I create a sphere that represents the model (similar to aforementioned project)?
Thanks!