D’oh! Thanks Thomas. That would have taken me ages to debug.

For future references, the paper formula assumes near = 1.

Should divide by (near^2) unless I’m overlooking something again.

Hello Phi,

The pixel area is

area of shadowmap / number of pixels,

i.e., you must scale your pixel area by the total area of the shadow map, which can be calculated from the field of view and the distance to the virtual plane.

=================Python Code=======================

import numpy
n_x = 512.0
n_y = 512.0
h_fov = numpy.radians(75.0)
alpha = n_y / n_x
near = 1.0
shadowmap_width = 2.0 * near*numpy.tan(h_fov/2.0)
shadowmap_height = alpha * shadowmap_width
area_pixel = shadowmap_width * shadowmap_height / (n_x * n_y)

forward = [0.0, 0.0, -near]
r = [0.5,
0.5,
-near]

norm_r = numpy.linalg.norm(r)
r /= norm_r

print(4.0 * alpha * (numpy.tan(h_fov / 2.0) ** 2) * (numpy.dot(r, forward) ** 3) / (n_x * n_y))
print(area_pixel * numpy.dot(r, forward) / (norm_r ** 2))

Cool stuff. When I was reading the paper, I think the final equation of the projected solid angle w.r.t. the pixel is wrong (right-most one)? I wrote some code to check the values and can’t seem to make them agree.

I checked “RSMFS.glsl” and my derivation almost matches it: how did the cosine(theta_p) disappear?

cos(theta_p) = dot(normalized_forward_vector, r_p) / norm(r_p)

This would explain the pow-1.5 factor.

Thank you,
Phi

=================Python Code=======================
n_x = 512
n_y = 512
h_fov = numpy.radians(75)
alpha = n_y / n_x
area_pixel = 1 / (n_x * n_y)
near = 1

forward = [0, 0, -near]
r = [0.5,
0.5,
-near]
norm_r = numpy.linalg.norm(r)
r /= norm_r

print(4 * alpha * (numpy.tan(h_fov / 2) ** 2) * (numpy.dot(r, forward) ** 3) * area_pixel)
print(area_pixel * numpy.dot(r, forward) / (norm_r ** 2))

where Y_00(theta,phi) = 1 / sqrt(4pi) and Y_10(theta,phi) = sqrt(3/4pi) * cos(theta).
The usual complex conjugation of the Y_lm’s can be omitted for m=0. c_1,1 and c_1,-1 are zero.

The injection step is to replace the incoming refracted light by a virtual
point light which angular dependence is assumed to be proportional to the
clamped cosine function centred along the refracted direction. If you
expand this function in spherical harmonics, the coefficients are obtained
by applying a rotation matrix (corresponding to the refracted direction) to
the c_00 and c_10 coefficients above.

Shouldnt it be: vec2(0.282094791, 0.488602511);
The coefficients are 1/(2*sqrt(PI) and sqrt(3)/(2*sqrt(PI)). I assume there might be some compensation where you might have managed to offset the value back to the original values. I presumed that the division by PI would have compensated for the values for the 0 coefficient, but that would also mean the attentuation factor in equation (1) from your paper might have been negated.