Clouds - Revision history

Patapom at 14:13, 2 April 2011

2011-04-02T14:13:07Z

Patapom at 13:54, 1 April 2011

2011-04-01T13:54:47Z

Patapom at 13:31, 1 April 2011

2011-04-01T13:31:20Z

Patapom at 13:26, 1 April 2011

2011-04-01T13:26:48Z

Patapom at 13:15, 1 April 2011

2011-04-01T13:15:34Z

Patapom at 12:19, 1 April 2011

2011-04-01T12:19:30Z

Patapom at 12:17, 1 April 2011

2011-04-01T12:17:12Z

Patapom at 11:09, 1 April 2011

2011-04-01T11:09:03Z

Patapom at 20:29, 31 March 2011

2011-03-31T20:29:55Z

Show changes

Patapom at 18:09, 31 March 2011

2011-03-31T18:09:45Z

@@ Line 788: / Line 788: @@
 Here is a snapshot of the result :
 [[Image:Clouds.png|800px|thumb|center]]
 And the video is available here [http://www.youtube.com/watch?v=I1_f0z3Ht-k http://www.youtube.com/watch?v=I1_f0z3Ht-k]

@@ Line 649: / Line 649: @@
 To sum up :
-  <math>Li(P,\vec\omega) = \mathbf{p^\prime}(\widehat{\vec\omega \vec\omega_{sun}}) e^{-\tau(\vec\omega_{sun})} L_{sun}(\vec\omega_{sun})</math>
+  <math>Li(P,\vec\omega) = \mathbf{p^\prime}(\widehat{\vec\omega \vec\omega_{sun}}) e^{-\tau(P,\vec\omega_{sun})} L_{sun}(\vec\omega_{sun})</math>
   where :
-  <math>\tau(\vec\omega_{sun}) = \boldsymbol{\sigma} \boldsymbol{\rho}(x)</math>
+  <math>\tau(x,\vec\omega_{sun}) = \boldsymbol{\sigma} \boldsymbol{\rho}(x)</math>
   <math>\boldsymbol{\sigma}</math> is the global cloud extinction factor
   <math>\boldsymbol{\rho}(x)</math> is the accumulated density at <math>x</math> we talked about earlier

@@ Line 224: / Line 224: @@
   Step 0:
    <math>L(0) = BackgroundRadiance</math>
   Step 1:
    <math>L(1) = \sigma_s(1) \Delta_x L_i(1) + e^{-\sigma_t(1)\Delta_x} L(0)</math>
   Step 2:
    <math>\begin{align}L(2)
@@ Line 233: / Line 235: @@
   \end{align}
 </math>
   Step 3:
    <math>\begin{align}L(3)
@@ Line 240: / Line 243: @@
 \end{align}
 </math>
 Each step brings us forward toward the camera, where we obtain the final radiance reaching the sensor.
-From the reccurence above we see that, if we perform N steps of ray-marching :
+From the recurrence relation above we see that, if we perform N steps of ray-marching :
   <math>Extinction(N) = \prod_{i=1}^N e^{-\sigma_t(i)\Delta_x}</math>
   <math>InScattering(N) = \sum_{i=1}^N \mathbf{K}(i) \sigma_s(i) \Delta_x L_i(i)</math>

@@ Line 213: / Line 213: @@
 We can rewrite the equation above in terms of discrete ray-marching steps :
-  <math>L(N) = \sigma_s(N) \Delta_x L_i(N) + e^{-\sigma_t(N)\Delta_x} L(N-1)</math>
+  <math>L(i) = \sigma_s(i) \Delta_x L_i(i) + e^{-\sigma_t(i)\Delta_x} L(i-1)</math>
   Where :
-  N is the index of the ray-marching step.
+  <math>i</math> is the index of the ray-marching step.
   <math>L(0)</math> is the background radiance we start with. It's basically the color of your pre-rendered scene.
@@ Line 222: / Line 222: @@
 Let's write a few steps of ray-marching using the above notation :
-  Step 0:  <math>L(0) = BackgroundRadiance</math>
+  Step 0:
-  Step 1:  <math>L(1) = \sigma_s(1) \Delta_x L_i(1) + e^{-\sigma_t(1)\Delta_x} L(0)</math>
+  <math>L(0) = BackgroundRadiance</math>
-  Step 2:  <math>L(2) = \sigma_s(2) \Delta_x L_i(2) + e^{-\sigma_t(2)\Delta_x} L(1) = \sigma_s(2) \Delta_x L_i(2) + e^{-\sigma_t(2)\Delta_x} \big[ \sigma_s(1) \Delta_x L_i(1) + e^{-\sigma_t(1)\Delta_x} L(0) \big]</math>
+  Step 1:
-  Step 3:  <math>L(3) = \sigma_s(3) \Delta_x L_i(3) + e^{-\sigma_t(3)\Delta_x} L(2) = \sigma_s(3) \Delta_x L_i(3) + e^{-\sigma_t(3)\Delta_x} \big[ \sigma_s(2) \Delta_x L_i(2) + e^{-\sigma_t(2)\Delta_x} \big[ \sigma_s(1) \Delta_x L_i(1) + e^{-\sigma_t(1)\Delta_x} L(0) \big] \big]</math>
+  <math>L(1) = \sigma_s(1) \Delta_x L_i(1) + e^{-\sigma_t(1)\Delta_x} L(0)</math>
 Each step brings us forward toward the camera, where we obtain the final radiance reaching the sensor.

← Older revision		Revision as of 13:15, 1 April 2011
Line 183:		Line 183:

	<math>L_i(x^\prime,\vec\omega)</math> is what I call the '''incoming irradiance''' which is the contribution of radiances from all directions about <math>x^\prime</math> that get somehow reflected in the view direction <math>\vec\omega</math>, the probability of such reflection is given by the phase function that we will describe later.		<math>L_i(x^\prime,\vec\omega)</math> is what I call the '''incoming irradiance''' which is the contribution of radiances from all directions about <math>x^\prime</math> that get somehow reflected in the view direction <math>\vec\omega</math>, the probability of such reflection is given by the phase function that we will describe later.
		+
		+	This is the infamous '''radiosity''' that was for a long time approximated by a single ambient term, by lack of methods or computing power to solve this complicated equation.

← Older revision		Revision as of 12:17, 1 April 2011
Line 210:		Line 210:
	We can rewrite the equation above in terms of discrete ray-marching steps :		We can rewrite the equation above in terms of discrete ray-marching steps :

−	<math>~~L_{~~N} = \sigma_s _N \Delta_x L_i _N + e^{-\sigma_t _N\Delta_x} ~~L_{~~N-1}</math>	+	<math>L(N) = \sigma_s(N) \Delta_x L_i(N) + e^{-\sigma_t(N)\Delta_x} L(N-1)</math>

		+	Where N is the index of the ray-marching step.
		+
		+	<math>L^{0}</math> is the radiance we start with. It's either 0 (cold emptiness of space) or the Sun's radiance if the ray is looking straight into the Sun.
		+
		+	Let's write a few steps of ray-marching using the above notation :
		+
		+	Step 0: <math>L(0) = BackgroundRadiance</math>
		+	Step 1: <math>L(1) = \sigma_s(1) \Delta_x L_i(1) + e^{-\sigma_t(1)\Delta_x} L(0)</math>
		+	Step 2: <math>L(2) = \sigma_s(2) \Delta_x L_i(2) + e^{-\sigma_t(2)\Delta_x} L(1) = \sigma_s(2) \Delta_x L_i(2) + e^{-\sigma_t(2)\Delta_x} \big[ \sigma_s(1) \Delta_x L_i(1) + e^{-\sigma_t(1)\Delta_x} L(0) \big]</math>
		+	Step 3: <math>L(3) = \sigma_s(3) \Delta_x L_i(3) + e^{-\sigma_t(3)\Delta_x} L(2) = \sigma_s(3) \Delta_x L_i(3) + e^{-\sigma_t(3)\Delta_x} \big[ \sigma_s(2) \Delta_x L_i(2) + e^{-\sigma_t(2)\Delta_x} \big[ \sigma_s(1) \Delta_x L_i(1) + e^{-\sigma_t(1)\Delta_x} L(0) \big] \big]</math>
		+
		+	Each step brings us forward toward the camera, where we obtain the final radiance reaching the sensor.
		+
		+	From the reccurence above we see that, if we perform N steps of ray-marching :
		+	<math>Extinction(N) = \prod_{i=1}^N e^{-\sigma_t(i)\Delta_x}</math>
		+	<math>InScattering(N) = \sum_{i=1}^N \mathbf{K}(i) \sigma_s(i) \Delta_x L_i(i)</math>
		+
		+	where :
		+	<math>Extinction(0) = 1</math>
		+	<math>InScattering(0) = BackgroundRadiance</math>
		+	<math>\mathbf{K}(i) =
		+	\begin{cases}
		+	{\prod_{j=i+1}^N e^{-\sigma_t(j)\Delta_x}} & i < N \\
		+	1 & i = N
		+	\end{cases}
		+	</math>
		+
		+	And finally, after N steps :
		+	<math>\begin{align}L(\vec\omega) & = InScattering(N) + Extinction(N) * InScattering(0) \\
		+	& = InScattering(N) + Extinction(N) * BackgroundRadiance \\
		+	\end{align}</math>
		+
		+	where :
		+	<math>BackgroundRadiance</math> is the color of your pre-rendered scene

@@ Line 1: / Line 1: @@
-This page talks about the complex task of rendering clouds in real-time. It's more of a guide through my own experience in attempting to render clouds rather than an extensive review of all existing techniques. The target audience is the skillful amateurs of the demoscene and video games developers community rather than the expert scientists writing articles for the Siggraph, and I hope I didn't make too many mistakes in the various formulas and statements used herefater.
+This page talks about the complex task of rendering clouds in real-time. It's more of a guide through my own experience rather than an extensive review of all existing techniques. The target audience is the skillful amateurs of the demoscene and video games developers community rather than the expert scientists writing articles for the Siggraph, and I hope I didn't make too many mistakes in the various formulas and statements used herefater.
@@ Line 10: / Line 10: @@
 * Chapter 1 and 2 are an introduction to cloud shapes and complex lighting physics occurring within them.
-* Chapter 3 rather concentrates on some rendering algorithms I have tested.
+* Chapter 3 concentrates on some rendering algorithms I have tested.
@@ Line 44: / Line 44: @@
 Some are great for introducing regular randomness (plain noise), some others are good for clouds (sum 1/f(noise)) while even others are good for terrain modeling (sum 1/f(|noise|)) and are given funny names like "ridged multi fractal" or "fractal brownian motion".
-Once you have your basic noise function, it's all a matter of intelligently combining several noises at several amplitudes and frequencies.
+Once you have your basic noise function, it's all a matter of intelligently combining several noises at different amplitudes and frequencies.
@@ Line 66: / Line 66: @@
 These are the big guys (and my favorite [[File:s13.gif]]) !
-I remember being baffled by the foundation paper [http://nis-lab.is.s.u-tokyo.ac.jp/~nis/cdrom/sig00_cloud.pdf "A Simple, Efficient Method for Realistic Animation of Clouds"] by Dobashi et al. in 2000, where these guys managed to render impressively realistic clouds using particles while the clouds were modeled using a cellular automaton. This paper served as a base for [http://www.markmark.net/clouds/ Mark Harris] and his well known clouds.
+I remember being baffled by the foundation paper [http://nis-lab.is.s.u-tokyo.ac.jp/~nis/cdrom/sig00_cloud.pdf "A Simple, Efficient Method for Realistic Animation of Clouds"] by Dobashi et al. in 2000, where these guys managed to render impressively realistic clouds using particles, while the clouds were modeled using a cellular automaton. This paper served as a base for [http://www.markmark.net/clouds/ Mark Harris] and his well known clouds.
 [[Image:MarkHarris.png|400px|thumb|center]]
@@ Line 207: / Line 207: @@
 You can view the light ray-marching process as gaining some energy (i.e. in-scattering, an addition) then marching a little and losing some energy (i.e. extinction, a multiplication).