File size: 7,502 Bytes
d56c551 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 |
'''
Functions about lighting mesh(changing colors/texture of mesh).
1. add light to colors/texture (shade each vertex)
2. fit light according to colors/texture & image.
Preparation knowledge:
lighting: https://cs184.eecs.berkeley.edu/lecture/pipeline
spherical harmonics in human face: '3D Face Reconstruction from a Single Image Using a Single Reference Face Shape'
'''
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
import numpy as np
def get_normal(vertices, triangles):
''' calculate normal direction in each vertex
Args:
vertices: [nver, 3]
triangles: [ntri, 3]
Returns:
normal: [nver, 3]
'''
pt0 = vertices[triangles[:, 0], :] # [ntri, 3]
pt1 = vertices[triangles[:, 1], :] # [ntri, 3]
pt2 = vertices[triangles[:, 2], :] # [ntri, 3]
tri_normal = np.cross(pt0 - pt1, pt0 - pt2) # [ntri, 3]. normal of each triangle
normal = np.zeros_like(vertices) # [nver, 3]
for i in range(triangles.shape[0]):
normal[triangles[i, 0], :] = normal[triangles[i, 0], :] + tri_normal[i, :]
normal[triangles[i, 1], :] = normal[triangles[i, 1], :] + tri_normal[i, :]
normal[triangles[i, 2], :] = normal[triangles[i, 2], :] + tri_normal[i, :]
# normalize to unit length
mag = np.sum(normal**2, 1) # [nver]
zero_ind = (mag == 0)
mag[zero_ind] = 1;
normal[zero_ind, 0] = np.ones((np.sum(zero_ind)))
normal = normal/np.sqrt(mag[:,np.newaxis])
return normal
# TODO: test
def add_light_sh(vertices, triangles, colors, sh_coeff):
'''
In 3d face, usually assume:
1. The surface of face is Lambertian(reflect only the low frequencies of lighting)
2. Lighting can be an arbitrary combination of point sources
--> can be expressed in terms of spherical harmonics(omit the lighting coefficients)
I = albedo * (sh(n) x sh_coeff)
albedo: n x 1
sh_coeff: 9 x 1
Y(n) = (1, n_x, n_y, n_z, n_xn_y, n_xn_z, n_yn_z, n_x^2 - n_y^2, 3n_z^2 - 1)': n x 9
# Y(n) = (1, n_x, n_y, n_z)': n x 4
Args:
vertices: [nver, 3]
triangles: [ntri, 3]
colors: [nver, 3] albedo
sh_coeff: [9, 1] spherical harmonics coefficients
Returns:
lit_colors: [nver, 3]
'''
assert vertices.shape[0] == colors.shape[0]
nver = vertices.shape[0]
normal = get_normal(vertices, triangles) # [nver, 3]
sh = np.array((np.ones(nver), n[:,0], n[:,1], n[:,2], n[:,0]*n[:,1], n[:,0]*n[:,2], n[:,1]*n[:,2], n[:,0]**2 - n[:,1]**2, 3*(n[:,2]**2) - 1)) # [nver, 9]
ref = sh.dot(sh_coeff) #[nver, 1]
lit_colors = colors*ref
return lit_colors
def add_light(vertices, triangles, colors, light_positions = 0, light_intensities = 0):
''' Gouraud shading. add point lights.
In 3d face, usually assume:
1. The surface of face is Lambertian(reflect only the low frequencies of lighting)
2. Lighting can be an arbitrary combination of point sources
3. No specular (unless skin is oil, 23333)
Ref: https://cs184.eecs.berkeley.edu/lecture/pipeline
Args:
vertices: [nver, 3]
triangles: [ntri, 3]
light_positions: [nlight, 3]
light_intensities: [nlight, 3]
Returns:
lit_colors: [nver, 3]
'''
nver = vertices.shape[0]
normals = get_normal(vertices, triangles) # [nver, 3]
# ambient
# La = ka*Ia
# diffuse
# Ld = kd*(I/r^2)max(0, nxl)
direction_to_lights = vertices[np.newaxis, :, :] - light_positions[:, np.newaxis, :] # [nlight, nver, 3]
direction_to_lights_n = np.sqrt(np.sum(direction_to_lights**2, axis = 2)) # [nlight, nver]
direction_to_lights = direction_to_lights/direction_to_lights_n[:, :, np.newaxis]
normals_dot_lights = normals[np.newaxis, :, :]*direction_to_lights # [nlight, nver, 3]
normals_dot_lights = np.sum(normals_dot_lights, axis = 2) # [nlight, nver]
diffuse_output = colors[np.newaxis, :, :]*normals_dot_lights[:, :, np.newaxis]*light_intensities[:, np.newaxis, :]
diffuse_output = np.sum(diffuse_output, axis = 0) # [nver, 3]
# specular
# h = (v + l)/(|v + l|) bisector
# Ls = ks*(I/r^2)max(0, nxh)^p
# increasing p narrows the reflectionlob
lit_colors = diffuse_output # only diffuse part here.
lit_colors = np.minimum(np.maximum(lit_colors, 0), 1)
return lit_colors
## TODO. estimate light(sh coeff)
## -------------------------------- estimate. can not use now.
def fit_light(image, vertices, colors, triangles, vis_ind, lamb = 10, max_iter = 3):
[h, w, c] = image.shape
# surface normal
norm = get_normal(vertices, triangles)
nver = vertices.shape[1]
# vertices --> corresponding image pixel
pt2d = vertices[:2, :]
pt2d[0,:] = np.minimum(np.maximum(pt2d[0,:], 0), w - 1)
pt2d[1,:] = np.minimum(np.maximum(pt2d[1,:], 0), h - 1)
pt2d = np.round(pt2d).astype(np.int32) # 2 x nver
image_pixel = image[pt2d[1,:], pt2d[0,:], :] # nver x 3
image_pixel = image_pixel.T # 3 x nver
# vertices --> corresponding mean texture pixel with illumination
# Spherical Harmonic Basis
harmonic_dim = 9
nx = norm[0,:];
ny = norm[1,:];
nz = norm[2,:];
harmonic = np.zeros((nver, harmonic_dim))
pi = np.pi
harmonic[:,0] = np.sqrt(1/(4*pi)) * np.ones((nver,));
harmonic[:,1] = np.sqrt(3/(4*pi)) * nx;
harmonic[:,2] = np.sqrt(3/(4*pi)) * ny;
harmonic[:,3] = np.sqrt(3/(4*pi)) * nz;
harmonic[:,4] = 1/2. * np.sqrt(3/(4*pi)) * (2*nz**2 - nx**2 - ny**2);
harmonic[:,5] = 3 * np.sqrt(5/(12*pi)) * (ny*nz);
harmonic[:,6] = 3 * np.sqrt(5/(12*pi)) * (nx*nz);
harmonic[:,7] = 3 * np.sqrt(5/(12*pi)) * (nx*ny);
harmonic[:,8] = 3/2. * np.sqrt(5/(12*pi)) * (nx*nx - ny*ny);
'''
I' = sum(albedo * lj * hj) j = 0:9 (albedo = tex)
set A = albedo*h (n x 9)
alpha = lj (9 x 1)
Y = I (n x 1)
Y' = A.dot(alpha)
opt function:
||Y - A*alpha|| + lambda*(alpha'*alpha)
result:
A'*(Y - A*alpha) + lambda*alpha = 0
==>
(A'*A*alpha - lambda)*alpha = A'*Y
left: 9 x 9
right: 9 x 1
'''
n_vis_ind = len(vis_ind)
n = n_vis_ind*c
Y = np.zeros((n, 1))
A = np.zeros((n, 9))
light = np.zeros((3, 1))
for k in range(c):
Y[k*n_vis_ind:(k+1)*n_vis_ind, :] = image_pixel[k, vis_ind][:, np.newaxis]
A[k*n_vis_ind:(k+1)*n_vis_ind, :] = texture[k, vis_ind][:, np.newaxis] * harmonic[vis_ind, :]
Ac = texture[k, vis_ind][:, np.newaxis]
Yc = image_pixel[k, vis_ind][:, np.newaxis]
light[k] = (Ac.T.dot(Yc))/(Ac.T.dot(Ac))
for i in range(max_iter):
Yc = Y.copy()
for k in range(c):
Yc[k*n_vis_ind:(k+1)*n_vis_ind, :] /= light[k]
# update alpha
equation_left = np.dot(A.T, A) + lamb*np.eye(harmonic_dim); # why + ?
equation_right = np.dot(A.T, Yc)
alpha = np.dot(np.linalg.inv(equation_left), equation_right)
# update light
for k in range(c):
Ac = A[k*n_vis_ind:(k+1)*n_vis_ind, :].dot(alpha)
Yc = Y[k*n_vis_ind:(k+1)*n_vis_ind, :]
light[k] = (Ac.T.dot(Yc))/(Ac.T.dot(Ac))
appearance = np.zeros_like(texture)
for k in range(c):
tmp = np.dot(harmonic*texture[k, :][:, np.newaxis], alpha*light[k])
appearance[k,:] = tmp.T
appearance = np.minimum(np.maximum(appearance, 0), 1)
return appearance
|