Linearly Transformed Cosines¶

引擎相关代码：Engine/Shaders/Private/RectLight.ush

In [1]:

import sys
import numpy as np
import matplotlib.pyplot as plt

%matplotlib inline

import sys import numpy as np import matplotlib.pyplot as plt %matplotlib inline

In [2]:

sys.path.append('../project_viz')
from umath import LTC, spherical_plot

sys.path.append('../project_viz') from umath import LTC, spherical_plot

Original Spherical Distributions¶

• Distributions:

$$ D_o(\omega_o(x_o, y_o, z_o)) $$

• These distributions blow are all normalized:

$$ \int_{\Omega} D_o(\omega_o) \mathrm{d} \omega_o = 1 $$

Uniform Spherical Distribution¶

$$ D_o = \frac{1}{4 \pi} $$

In [3]:

def do_uniform_sphere(w):
    return 1/(4*np.pi)

def do_uniform_sphere(w): return 1/(4*np.pi)

In [4]:

fig = plt.figure(figsize=[10, 10])
ax = fig.add_subplot(1, 1, 1, projection='3d')
    
spherical_plot.heatmap(do_uniform_sphere, ax)

plt.title('Uniform Spherical Distribution')
plt.show()

fig = plt.figure(figsize=[10, 10]) ax = fig.add_subplot(1, 1, 1, projection='3d') spherical_plot.heatmap(do_uniform_sphere, ax) plt.title('Uniform Spherical Distribution') plt.show()

Uniform Hemispherical Distribution¶

$$ D_{o}(\omega_{o})= \begin{cases} \frac{1}{2 \pi} & \text { if } \quad z_o \geq 0 \\ 0 & \text { if } \quad z_o<0 \end{cases} $$

In [5]:

def do_uniform_hemisphere(w):
    if w[2]<0:
        return 0
    else:
        return 1 / (2*np.pi)

def do_uniform_hemisphere(w): if w[2]<0: return 0 else: return 1 / (2*np.pi)

In [6]:

fig = plt.figure(figsize=[10, 10])
ax = fig.add_subplot(1, 1, 1, projection='3d')

spherical_plot.heatmap(do_uniform_hemisphere, ax)

plt.title('Uniform Spherical Distribution')
plt.show()

fig = plt.figure(figsize=[10, 10]) ax = fig.add_subplot(1, 1, 1, projection='3d') spherical_plot.heatmap(do_uniform_hemisphere, ax) plt.title('Uniform Spherical Distribution') plt.show()

Clamped Cosine Distribution¶

$$ D_o(\omega_o)= \begin{cases} \frac{z_o}{\pi} & \text{ if } \quad z_o \geq 0 \\ 0 & \text{ if } \quad z_o<0 \end{cases} $$

• 注：打眼一看，取 Z 值，然后归一化，和 Cosine 有啥关系？
  • 回想一下球面坐标的转换就明了了

$$ \begin{align} x =& r \sin(\phi) \cos(\theta) \\ y =& r \sin(\phi) \sin(\theta) \\ z =& r \cos(\phi) \end{align} $$

In [7]:

def do_clamped_cosine(w):
    return np.maximum(0, w[2]/np.pi)

def do_clamped_cosine(w): return np.maximum(0, w[2]/np.pi)

In [8]:

fig = plt.figure(figsize=[10, 10])
ax = fig.add_subplot(1, 1, 1, projection='3d')

spherical_plot.heatmap(do_clamped_cosine, ax)

plt.title('Clamped Cosine Distribution')
plt.show()

fig = plt.figure(figsize=[10, 10]) ax = fig.add_subplot(1, 1, 1, projection='3d') spherical_plot.heatmap(do_clamped_cosine, ax) plt.title('Clamped Cosine Distribution') plt.show()

Squared Clamped Cosine Distribution¶

$$ D_o(\omega_o)= \begin{cases} \frac{3 z_o^2}{2 \pi} & \text{ if } \quad z_o \geq 0 \\ 0 & \text{ if } \quad z_o<0 \end{cases} $$

In [9]:

def do_squared_clamped_cosine(w):
    return np.maximum(0, w[2]**2*3 / (2*np.pi))

def do_squared_clamped_cosine(w): return np.maximum(0, w[2]**2*3 / (2*np.pi))

In [10]:

fig = plt.figure(figsize=[10, 10])
ax = fig.add_subplot(1, 1, 1, projection='3d')

spherical_plot.heatmap(do_squared_clamped_cosine,ax)

plt.title('Clamped Cosine$^2$ Distribution')
plt.show()

fig = plt.figure(figsize=[10, 10]) ax = fig.add_subplot(1, 1, 1, projection='3d') spherical_plot.heatmap(do_squared_clamped_cosine,ax) plt.title('Clamped Cosine$^2$ Distribution') plt.show()

Linearly Transformed Distributions¶

$$ D(\omega) = D_{o}\left(\frac{M^{-1} \omega}{\left\|M^{-1} \omega\right\|}\right) \frac{\left|M^{-1}\right|}{\left\|M^{-1} \omega\right\|^{3}} $$

In [11]:

amplitude = 1
forward = np.array([0, 1, 0], dtype='float')
up = np.array([0, 0, 1], dtype='float')
scale = np.array([0.75, 1, 1], dtype='float')
skew = 0.5

ltc = LTC.LTC(amplitude, forward, up, scale, skew)

fig = plt.figure(figsize=[10, 10])
ax = fig.add_subplot(1, 1, 1, projection='3d')

spherical_plot.heatmap(lambda w: ltc.evaluate(w), ax)

plt.title('Linearly Transformed Cosines')
plt.show()

amplitude = 1 forward = np.array([0, 1, 0], dtype='float') up = np.array([0, 0, 1], dtype='float') scale = np.array([0.75, 1, 1], dtype='float') skew = 0.5 ltc = LTC.LTC(amplitude, forward, up, scale, skew) fig = plt.figure(figsize=[10, 10]) ax = fig.add_subplot(1, 1, 1, projection='3d') spherical_plot.heatmap(lambda w: ltc.evaluate(w), ax) plt.title('Linearly Transformed Cosines') plt.show()

BRDF Fitting¶

• TODO

Reference¶

• Real-Time Polygonal-Light Shading with Linearly Transformed Cosines