SphericalHarmonicsPortal - Revision history

Patapom at 16:52, 9 December 2016

2016-12-09T16:52:48Z

Patapom at 16:51, 9 December 2016

2016-12-09T16:51:17Z

Patapom at 16:50, 9 December 2016

2016-12-09T16:50:20Z

Patapom at 16:49, 9 December 2016

2016-12-09T16:49:28Z

Patapom at 16:45, 9 December 2016

2016-12-09T16:45:12Z

Patapom at 21:09, 4 December 2016

2016-12-04T21:09:59Z

Patapom at 19:06, 4 December 2016

2016-12-04T19:06:55Z

Patapom at 18:33, 4 December 2016

2016-12-04T18:33:38Z

Patapom at 18:33, 4 December 2016

2016-12-04T18:33:22Z

Patapom at 18:25, 4 December 2016

2016-12-04T18:25:25Z

← Older revision		Revision as of 16:52, 9 December 2016
Line 830:		Line 830:


−	Here is a demonstration of the new result as a side-by-side comparison of a simple sphere illuminated by the Ennis environment: on the left I'm simply doing do AO*irradiance, on the right I'm using the new irradiance estimate accounting for AO.	+	Here is a demonstration of the new result as a side-by-side comparison of a simple sphere illuminated by the Ennis environment: on the left I'm simply doing AO*irradiance while on the right I'm using the new irradiance estimate accounting for AO.

	[[File:IrradianceAOSphereDemo.gif]]		[[File:IrradianceAOSphereDemo.gif]]

@@ Line 681: / Line 681: @@
 For the rest of the discussion, we'll let <math>t=\cos(\theta_{AO})</math>.<br />
-'''--- Order 0 ---'''
+<big>'''--- Order 0 ---'''</big><br />
 Carrying the integration of (9) for order 0 gives us:
 :<math>
@@ Line 695: / Line 696: @@
 :<math>\hat{A_0}'(\theta_{AO}) = \sqrt{4\pi} \, \left[\sqrt{\frac{\pi}{4}} \left(1 - t^2\right)\right] = \pi \left(1-t^2\right)</math>
-'''--- Order 1 ---'''
+<big>'''--- Order 1 ---'''</big><br />
 Carrying the integration of (9) for order 1 gives us:
 :<math>
@@ Line 709: / Line 711: @@
 :<math>\hat{A_1}'(\theta_{AO}) = \sqrt{\frac{4\pi}{3}} \, \left[\sqrt{\frac{\pi}{3}} \left(1 - t^3\right)\right] = \frac{2\pi}{3} \left(1-t^3\right)</math>
-'''--- Order 2 ---'''
+<big>'''--- Order 2 ---'''</big><br />
 Carrying the integration of (9) for order 2 gives us:
 :<math>

@@ Line 681: / Line 681: @@
 For the rest of the discussion, we'll let <math>t=\cos(\theta_{AO})</math>.<br />
-===== Order 0 =====
+'''--- Order 0 ---'''
 Carrying the integration of (9) for order 0 gives us:
 :<math>
@@ Line 695: / Line 695: @@
 :<math>\hat{A_0}'(\theta_{AO}) = \sqrt{4\pi} \, \left[\sqrt{\frac{\pi}{4}} \left(1 - t^2\right)\right] = \pi \left(1-t^2\right)</math>
-===== Order 1 =====
+'''--- Order 1 ---'''
 Carrying the integration of (9) for order 1 gives us:
 :<math>
@@ Line 709: / Line 709: @@
 :<math>\hat{A_1}'(\theta_{AO}) = \sqrt{\frac{4\pi}{3}} \, \left[\sqrt{\frac{\pi}{3}} \left(1 - t^3\right)\right] = \frac{2\pi}{3} \left(1-t^3\right)</math>
-===== Order 2 =====
+'''--- Order 2 ---'''
 Carrying the integration of (9) for order 2 gives us:
 :<math>

@@ Line 681: / Line 681: @@
 For the rest of the discussion, we'll let <math>t=\cos(\theta_{AO})</math>.<br />
-=== Order 0 ===
+===== Order 0 =====
 Carrying the integration of (9) for order 0 gives us:
 :<math>
@@ Line 695: / Line 695: @@
 :<math>\hat{A_0}'(\theta_{AO}) = \sqrt{4\pi} \, \left[\sqrt{\frac{\pi}{4}} \left(1 - t^2\right)\right] = \pi \left(1-t^2\right)</math>
-=== Order 1 ===
+===== Order 1 =====
 Carrying the integration of (9) for order 1 gives us:
 :<math>
@@ Line 709: / Line 709: @@
 :<math>\hat{A_1}'(\theta_{AO}) = \sqrt{\frac{4\pi}{3}} \, \left[\sqrt{\frac{\pi}{3}} \left(1 - t^3\right)\right] = \frac{2\pi}{3} \left(1-t^3\right)</math>
-=== Order 2 ===
+===== Order 2 =====
 Carrying the integration of (9) for order 2 gives us:
 :<math>

← Older revision		Revision as of 16:45, 9 December 2016
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	}		}

		+
		+	Here is a demonstration of the new result as a side-by-side comparison of a simple sphere illuminated by the Ennis environment: on the left I'm simply doing do AO*irradiance, on the right I'm using the new irradiance estimate accounting for AO.
		+
		+	[[File:IrradianceAOSphereDemo.gif]]
		+

@@ Line 729: / Line 729: @@
 You can see here the effect of Ambient Occlusion on diffuse lambertian reflectance coefficients.<br />
 The black curve is the numerical integration of the coefficients up to order 20 (order is represented horizontally).<br />
-The Red, Green and Black dots are the values analytically computed for orders 0, 1 and 2 using the above integrals:<br />
+The Red, Green and Blue dots are the values analytically computed for orders 0, 1 and 2 using the above integrals:<br />
 [[File:IrradianceAO.gif]]

@@ Line 257: / Line 257: @@
 To reconstruct the band-limited version of the original signal from a bunch of SH coefficients, you simply need to apply the inverse transform:
-<math>f(\theta,\phi) \; \simeq \; \displaystyle\sum_{l=0}^{N} \displaystyle\sum_{m=-l}^{l} C_l^m Y_l^m(\theta,\phi)\sin(\theta) \text{ where } N \text{ is the SH order}</math>
+<math>f(\theta,\phi) \; \simeq \; \displaystyle\sum_{l=0}^{N} \displaystyle\sum_{m=-l}^{l} C_l^m Y_l^m(\theta,\phi) \text{ where } N \text{ is the SH order}</math>
@@ Line 335: / Line 335: @@
 For example:<br />
 * If the transfer function simply is the BRDF*cosine term as mentioned above, then your object will simply be ''directly'' lit by the environment as is normally the case with standard lighting
-* But the interesting part is that, since the transfer function could be anything, it may as well be the BRDF*cosine term '''plus''' what the object perceives of its environment: you would then obtain the direct lighting '''plus''' a first bounce of indirect lighting and even shadowing!
+* But the interesting part is that, since the transfer function could be anything, it may as well be the BRDF*cosine term '''plus''' what the object perceives of its environment: you would then obtain the direct lighting '''plus''' the indirect lighting!
 This is illustrated by the image on the right where we increase the realism of the scene's lighting. What's even better is that we can completely change the surrounding lighting environment on the fly at any time for absolutely no cost at all!<br />
-You can see an example of that technique in my [http://www.patapom.com/topics/WebGL/cathedral/intro.html StJean cathedral project] (requires WebGL!).<br />
+You can see an example of that technique applied in my [http://www.patapom.com/topics/WebGL/cathedral/intro.html St Jean cathedral project] (requires WebGL!).<br />
 In this example, I stored 2 bands (order 1 = 4 coefficients) of the pre-computed radiance transfer coefficients for every vertex of the model.<br />
 '''NOTE:''' Although it seems cool to do that, it's pretty heavy for real game scenes that now use tens of millions of vertices, it's also pretty long to pre-compute since you must ray-trace the scene from every vertex and finally, it's limited to a static environment...
@@ Line 361: / Line 361: @@
 <br />
-Then:
+Then:<br />
 :<math>
 C_i \; = \; \displaystyle\sum_{j=0}^{N^2} A_i \; \mathbf{M_{ij}}
@@ Line 497: / Line 496: @@
   	r[6] -= C4*t;
   	// entry count=13
-  	// '''multiply count=120'''
+  	// '''multiplications count=120'''
   	// '''addition count=74'''
+  }
@@ Line 574: / Line 573: @@
 We can now write the SH version of the diffuse reflection integral (1) from the previous section:<br />
-:<math>E(\theta_n, \phi_n) \; = \; \displaystyle\sum_{l,m} L_l^m \, A_l \, \times [-R^l_{m,0}(\theta_n) \sin(m \phi_n)] \qquad (7)</math>
+:<math>E(\theta_n, \phi_n) \; = \; \displaystyle\sum_{l,m} L_l^m \, A_l \, \times [R^l_{m,m'}(\theta_n) \sin(-m \phi_n)] \qquad (7)</math>
-I skipped the description of the full rotation of the incoming lighting but it now comes into play as the expression <math>-R^l_{m,0}(\theta_n) \sin(m \phi_n)</math> where <math>R^l_{m,0}(\theta_n)</math> represents a <math>(2l+1)\times(2l+1)</math> matrix that tells us how a SH transforms under rotation about the Y axis, ''i.e.'', how to rewrite a rotated SH as a linear combination of all the SH of the same order.<br />
+I skipped the description of the full rotation of the incoming lighting but it now comes into play as the expression <math>R^l_{m,m'}(\theta_n) \sin(-m \phi_n)</math> where <math>R^l_{m,m'}(\theta_n)</math> represents a <math>(2l+1)\times(2l+1)</math> matrix that tells us how a SH transforms under rotation about the Y axis, ''i.e.'', how to rewrite a rotated SH as a linear combination of all the SH of the same order.<br />
 This matrix is further simplified by the fact that the transfer function <math>A = \cos(\theta_n)</math> has no azimuthal dependence so terms with <math>m'\neq0</math> will vanish.
 It can be shown that:
-:<math>-R^l_{m,0}(\theta_n) sin(m \phi_n) \, = \, \sqrt{\frac{4\pi}{2l+1}} Y_l^m(\theta_n, \phi_n)</math>
+:<math>R^l_{m,0}(\theta_n) sin(-m \phi_n) \, = \, \sqrt{\frac{4\pi}{2l+1}} Y_l^m(\theta_n, \phi_n)</math>

← Older revision		Revision as of 18:33, 4 December 2016
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−	In the early 2000's, people like Ravi Ramamoorthi and Peter-Pike Sloan introduced a new powerful and amazing tool to the Computer Graphics ~~industry~~: Spherical Harmonics (or SH).<br />	+	In the early 2000's, people like Ravi Ramamoorthi and Peter-Pike Sloan introduced a new powerful and amazing tool to the Computer Graphics society: Spherical Harmonics (or SH).<br />
	Although Spherical Harmonics have always been around for quite some time, for example in the excellent [https://cseweb.ucsd.edu/~ravir/6998/papers/siggraph-rebuild.pdf "Predicting Reflectance Functions from Complex Surfaces"] 1992 paper from Westin et al. (and well before that in quantum physics!), they only piqued the public interest since their introduction as an efficient way of creating realistic and interactive indirect lighting rendering via Pre-computed Radiance Transfer (PRT).		Although Spherical Harmonics have always been around for quite some time, for example in the excellent [https://cseweb.ucsd.edu/~ravir/6998/papers/siggraph-rebuild.pdf "Predicting Reflectance Functions from Complex Surfaces"] 1992 paper from Westin et al. (and well before that in quantum physics!), they only piqued the public interest since their introduction as an efficient way of creating realistic and interactive indirect lighting rendering via Pre-computed Radiance Transfer (PRT).

← Older revision		Revision as of 18:25, 4 December 2016
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	In any case, we notice that it might be interesting once again to restrict the angle of the cone we use to estimate the irradiance:		In any case, we notice that it might be interesting once again to restrict the angle of the cone we use to estimate the irradiance:

−	[[File:ThreeConeEstimates.png]]	+	[[File:ThreeConeEstimates.png\|800px]]