Fix wrong conversion between power and radiance of emitters #108505

New Issue

Weizhen Huang · 2023-06-01T11:29:36+02:00

Weizhen Huang commented

2023-06-01 11:29:36 +02:00

There are factors for converting power in watts to radiance in watts per steradian per square meter (eval_fac in Cycles). However, some scalings are incorrect. Adjusting the scaling preserves the energy of the light source and improves compatibility with other renderers.

Area Light

Assuming Lambertian emitter, the outgoing radiance integrated over the hemisphere and the projected area gives the total emitted power (radiant flux)

\Phi_e=\int_\mathcal{H^2}L_e\mathrm{d}\omega\mathrm{d}A^\bot=\int_\mathcal{H^2}L_e\cos\theta\mathrm{d}\omega\mathrm{d}A=\pi L_e\int\mathrm{d}A=\pi A L_e.

Therefore, eval_fac should be M_1_PI_F * invarea instead of 0.25f * invarea.
The factor normalize_spread in light.cpp ensures that the area light emits the same amount of power regardless of the spread angle.

Point Light

The physical meaning of a point light with radius is unclear. In Blender it appears to be something similar to sphere light although not exactly due to the visible solid angle. If treated as a sphere light, the total emitted power is given by

\Phi=\pi L_e\int\mathrm{d}A=4\pi^2 R^2L_e,

with R being the radius of the point light. In Cycles, spot.inv_area = 1.0f / (M_PI_F * radius * radius) (see light.cpp). Therefore, eval_fac = 0.25f * M_1_PI_F * invarea, as currently in the implementation.

HOWEVER, if we follow the current sampling procedure exactly, a point light with radius looks like a disk light from every direction. In this case, we can compute the radiant flux through a sphere of radius D around the point light center (Gauss's Law). The irradiance at arbitrary point on the sphere is

E=\int L_e(\omega_i\cdot n)\mathrm{d}\omega_i=\int_0^{2\pi}\int_0^{R}\frac{L_e\cos\theta_i\cos\theta_o}{d^2}r\mathrm{d}r\mathrm{d}\phi,

using the Jacobians \mathrm{d}\omega=\frac{\cos\theta_o}{d^2}\mathrm{d}A and \mathrm{d}A=r\mathrm{d}r\mathrm{d}\phi, and d=\sqrt{r^2+D^2} is the distance between a point on the light source and the point on the sphere. Substituting \cos\theta_i=\cos\theta_o=\frac{D}{d} in the above equation, we obtain

E=\pi L_e\frac{R^2}{D^2+R^2}.

Therefore, the total emitted radiant flux is

\Phi_e=\oiint_\mathcal{S^2} E\mathrm{d}S=4\pi D^2E =4\pi^2\frac{R^2D^2}{D^2+R^2}L_e.

The result depends on the receiver distance and is therefore not physically correct. The additional factor \frac{D^2}{D^2+R^2} plotted against \frac{D}{R} is

Therefore, it approximates a sphere light when approaching infinity.
The plan is to change point light to (double-sided) sphere light (#108506). In EEVEE a sphere light can be treated as a disk light that spans the same solid angle from the shading point.

Spot Light

Spot lights use the same sampling procedure as point lights, but emit light only in a cone: the function spot_light_attenuation() in spot.h acts as masking function (if blend is ignored), so the power specified in the UI corresponds to TWICE the power emitted by a spot light with \mathrm{half\_spread} = \frac{\pi}{2}. This behaviour is different than with area lights, where a normalization factor is used so that the intensity changes when adjusting the spread angle. According to Frostbite:

Such a coupling makes spotlight manipulation harder for artists and causes trouble when optimizing: artists cannot reduce the cone attenuation to affect fewer pixels without changing its intensity.

We could follow their approach to multiply the current eval_fac by 4, so that the power corresponds to the amount emitted by a spot light with \mathrm{half\_spread} = \frac{\pi}{3}, or just keep it as is so it gives the same intensity as a point light, or add an option for normalization.

Sun Light

For sun light, user specify an "intensity", which is the irradiance received by a surface perpendicular to the sun direction, the unit being Watt per square meter. Using the irradiance formula derived for point lights and treating the sun light as a disk light 1 unit away, we have

E=\int_0^{2\pi}\int_0^{R}\frac{L_eD^2}{d^4}r\mathrm{d}r\mathrm{d}\phi=\int_0^{2\pi}\int_0^{\tan\alpha}\frac{L_er}{(r^2+1)^2}\mathrm{d}r\mathrm{d}\phi,

where \alpha is half the angular diameter. Currently \mathrm{eval\_fac} = \frac{1}{\pi\tan^2\alpha\cos^3\phi}; using \cos\phi = \frac{1}{\sqrt{r^2+1}}, the irradiance is computed as

E=\frac{1}{\pi\tan^2\alpha}\int_0^{2\pi}\int_0^{\tan\alpha}\frac{Ir}{\sqrt{r^2+1}}\mathrm{d}r\mathrm{d}\phi=\frac{2(\sec\alpha-1)}{\tan^2\alpha}I.

Plotting the additional factor against the angular diameter:

for small angular diameter the value is close to 1, so the scaling for sun light seems fine.

There are factors for converting power in watts to radiance in watts per steradian per square meter (`eval_fac` in Cycles). However, some scalings are incorrect. Adjusting the scaling preserves the energy of the light source and improves compatibility with other renderers. ## Area Light Assuming Lambertian emitter, the outgoing radiance integrated over the hemisphere and the projected area gives the total emitted power (radiant flux) \[\Phi_e=\int_\mathcal{H^2}L_e\mathrm{d}\omega\mathrm{d}A^\bot=\int_\mathcal{H^2}L_e\cos\theta\mathrm{d}\omega\mathrm{d}A=\pi L_e\int\mathrm{d}A=\pi A L_e.\] Therefore, `eval_fac` should be `M_1_PI_F * invarea` instead of `0.25f * invarea`. The factor `normalize_spread` in `light.cpp` ensures that the area light emits the same amount of power regardless of the spread angle. ## Point Light The physical meaning of a point light with radius is unclear. In Blender it appears to be something similar to sphere light although not exactly due to the visible solid angle. If treated as a sphere light, the total emitted power is given by \[\Phi=\pi L_e\int\mathrm{d}A=4\pi^2 R^2L_e,\] with \(R\) being the radius of the point light. In Cycles, `spot.inv_area = 1.0f / (M_PI_F * radius * radius)` (see `light.cpp`). Therefore, `eval_fac = 0.25f * M_1_PI_F * invarea`, as currently in the implementation. HOWEVER, if we follow the current sampling procedure exactly, a point light with radius looks like a disk light from every direction. In this case, we can compute the radiant flux through a sphere of radius \(D\) around the point light center (Gauss's Law). The irradiance at arbitrary point on the sphere is \[E=\int L_e(\omega_i\cdot n)\mathrm{d}\omega_i=\int_0^{2\pi}\int_0^{R}\frac{L_e\cos\theta_i\cos\theta_o}{d^2}r\mathrm{d}r\mathrm{d}\phi,\] using the Jacobians \(\mathrm{d}\omega=\frac{\cos\theta_o}{d^2}\mathrm{d}A\) and \(\mathrm{d}A=r\mathrm{d}r\mathrm{d}\phi\), and \(d=\sqrt{r^2+D^2}\) is the distance between a point on the light source and the point on the sphere. Substituting \(\cos\theta_i=\cos\theta_o=\frac{D}{d}\) in the above equation, we obtain \[E=\pi L_e\frac{R^2}{D^2+R^2}.\] Therefore, the total emitted radiant flux is \[\Phi_e=\oiint_\mathcal{S^2} E\mathrm{d}S=4\pi D^2E =4\pi^2\frac{R^2D^2}{D^2+R^2}L_e.\] The result depends on the receiver distance and is therefore not physically correct. The additional factor \(\frac{D^2}{D^2+R^2}\) plotted against \(\frac{D}{R}\) is ![point_light_factor.png](/attachments/831cbf01-d3d8-491b-830c-0f861a33e803) Therefore, it approximates a sphere light when approaching infinity. The plan is to change point light to (double-sided) sphere light (#108506). In EEVEE a sphere light can be treated as a disk light that spans the same solid angle from the shading point. ## Spot Light Spot lights use the same sampling procedure as point lights, but emit light only in a cone: the function `spot_light_attenuation()` in `spot.h` acts as masking function (if `blend` is ignored), so the power specified in the UI corresponds to TWICE the power emitted by a spot light with \(\mathrm{half\_spread} = \frac{\pi}{2}\). This behaviour is different than with area lights, where a normalization factor is used so that the intensity changes when adjusting the spread angle. According to [Frostbite](https://seblagarde.files.wordpress.com/2015/07/course_notes_moving_frostbite_to_pbr_v32.pdf): > Such a coupling makes spotlight manipulation harder for artists and causes trouble when optimizing: artists cannot reduce the cone attenuation to affect fewer pixels without changing its intensity. We could follow their approach to multiply the current `eval_fac` by 4, so that the power corresponds to the amount emitted by a spot light with \(\mathrm{half\_spread} = \frac{\pi}{3}\), or just keep it as is so it gives the same intensity as a point light, or add an option for normalization. ## Sun Light For sun light, user specify an "intensity", which is the irradiance received by a surface perpendicular to the sun direction, the unit being Watt per square meter. Using the irradiance formula derived for point lights and treating the sun light as a disk light 1 unit away, we have \[E=\int_0^{2\pi}\int_0^{R}\frac{L_eD^2}{d^4}r\mathrm{d}r\mathrm{d}\phi=\int_0^{2\pi}\int_0^{\tan\alpha}\frac{L_er}{(r^2+1)^2}\mathrm{d}r\mathrm{d}\phi,\] where \(\alpha\) is half the angular diameter. Currently \(\mathrm{eval\_fac} = \frac{1}{\pi\tan^2\alpha\cos^3\phi}\); using \(\cos\phi = \frac{1}{\sqrt{r^2+1}}\), the irradiance is computed as \[E=\frac{1}{\pi\tan^2\alpha}\int_0^{2\pi}\int_0^{\tan\alpha}\frac{Ir}{\sqrt{r^2+1}}\mathrm{d}r\mathrm{d}\phi=\frac{2(\sec\alpha-1)}{\tan^2\alpha}I.\] Plotting the additional factor against the angular diameter: ![sun_light_plot.png](/attachments/263f3efb-b082-446c-83b5-418e44ec0ad3) for small angular diameter the value is close to 1, so the scaling for sun light seems fine.

point_light_factor.png

17 KiB

sun_light_plot.png

26 KiB

👍 3

Weizhen Huang added this to the 4.0 milestone 2023-06-01 11:29:36 +02:00

Weizhen Huang added the

Module

Render & Cycles