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external_sh_func.py
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import cv2
import numpy as np
from PIL import Image
import scipy.io as io
def P_0_0(theta):
return np.ones(theta.shape, dtype=np.float32)
def P_1_0(theta):
return np.cos(theta)
def P_1_1(theta):
return -np.sin(theta)
def P_2_0(theta):
return 0.5 * (3 * np.power(np.cos(theta), 2) - 1)
def P_2_1(theta):
return -3 * np.cos(theta) * np.sin(theta)
def P_2_2(theta):
return 3 * np.power(np.sin(theta), 2.0)
def P_3_0(theta):
return 0.5 * (5 * np.power(np.cos(theta), 3) - 3 * np.cos(theta) )
def P_3_1(theta):
return -1.5 * (5 * np.power(np.cos(theta), 2) - 1) * np.sin(theta)
def P_3_2(theta):
return 15 * np.cos(theta) * np.power(np.sin(theta), 2)
def P_3_3(theta):
return -15 * np.power(np.sin(theta), 3)
def P_4_0(theta):
return 0.125 * (35 * np.power(np.cos(theta), 4) - 30 * np.power(np.cos(theta), 2) + 3)
def P_4_1(theta):
return -2.5 * (7 * np.power(np.cos(theta), 3) - 3 * np.cos(theta) ) * np.sin(theta)
def P_4_2(theta):
return 7.5 * (7 * np.power(np.cos(theta), 2) - 1) * np.power(np.sin(theta), 2)
def P_4_3(theta):
return -105 * np.cos(theta) * np.power(np.sin(theta), 3)
def P_4_4(theta):
return 105 * np.power(np.sin(theta), 4)
def computeK(l, m):
m = np.absolute(m)
l_s_m = l - m
l_a_m = l + m
for n in range(0, len(l) ):
l_s_m[n] = np.math.factorial(l_s_m[n])
l_a_m[n] = np.math.factorial(l_a_m[n])
l_s_m = l_s_m.astype(np.float32)
l_a_m = l_a_m.astype(np.float32)
K2 = (2*l + 1) * l_s_m /l_a_m / 4 / np.pi
return np.sqrt(K2)
def angleToUV(theta, phi):
u = (phi + np.pi) / 2 / np.pi
v = 1 - theta / np.pi
return u, v
def uvToEnvmap(envmap, u, v):
height, width = envmap.shape[0], envmap.shape[1]
c, r = u * (width-1), (1-v) * (height-1)
cs, rs = int(c), int(r)
ce = min(width-1, cs + 1)
re = min(height-1, rs + 1)
wc, wr = c - cs, r - rs
color1 = (1-wc) * envmap[rs, cs, :] + wc * envmap[rs, ce, :]
color2 = (1-wc) * envmap[re, cs, :] + wc * envmap[re, ce, :]
color = (1 - wr) * color1 + wr * color2
return color
def Integration(phi, colors, K, LP):
sampleNum = colors.shape[0]
W = 4 * np.pi / sampleNum
if len(phi.shape) == 1:
phi = phi[:, np.newaxis]
# level 0
y_0 = K[0] * W * LP['P_0_0'] * colors
y_0 = np.sum(y_0, axis=0)[np.newaxis, :]
# level 1
y_1_n1 = np.sqrt(2) * K[1] * np.sin(1 * phi) * W * LP['P_1_1'] * colors
y_1_0 = K[2] * W * LP['P_1_0'] * colors
y_1_p1 = np.sqrt(2) * K[3] * np.cos(1 * phi) * W * LP['P_1_1'] * colors
y_1_n1 = np.sum(y_1_n1, axis=0)[np.newaxis, :]
y_1_0 = np.sum(y_1_0, axis=0)[np.newaxis, :]
y_1_p1 = np.sum(y_1_p1, axis=0)[np.newaxis, :]
# level 2
y_2_n2 = np.sqrt(2) * K[4] * np.sin(2 * phi) * W * LP['P_2_2'] * colors
y_2_n1 = np.sqrt(2) * K[5] * np.sin(1 * phi) * W * LP['P_2_1'] * colors
y_2_0 = K[6] * W * LP['P_2_0'] * colors
y_2_p1 = np.sqrt(2) * K[7] * np.cos(1 * phi) * W * LP['P_2_1'] * colors
y_2_p2 = np.sqrt(2) * K[8] * np.cos(2 * phi) * W * LP['P_2_2'] * colors
y_2_n2 = np.sum(y_2_n2, axis=0)[np.newaxis, :]
y_2_n1 = np.sum(y_2_n1, axis=0)[np.newaxis, :]
y_2_0 = np.sum(y_2_0, axis=0)[np.newaxis, :]
y_2_p1 = np.sum(y_2_p1, axis=0)[np.newaxis, :]
y_2_p2 = np.sum(y_2_p2, axis=0)[np.newaxis, :]
# level 3
y_3_n3 = np.sqrt(2) * K[9 ] * np.sin(3 * phi) * W * LP['P_3_3'] * colors
y_3_n2 = np.sqrt(2) * K[10] * np.sin(2 * phi) * W * LP['P_3_2'] * colors
y_3_n1 = np.sqrt(2) * K[11] * np.sin(1 * phi) * W * LP['P_3_1'] * colors
y_3_0 = K[12] * W * LP['P_3_0'] * colors
y_3_p1 = np.sqrt(2) * K[13] * np.cos(1 * phi) * W * LP['P_3_1'] * colors
y_3_p2 = np.sqrt(2) * K[14] * np.cos(2 * phi) * W * LP['P_3_2'] * colors
y_3_p3 = np.sqrt(2) * K[15] * np.cos(3 * phi) * W * LP['P_3_3'] * colors
y_3_n3 = np.sum(y_3_n3, axis=0)[np.newaxis, :]
y_3_n2 = np.sum(y_3_n2, axis=0)[np.newaxis, :]
y_3_n1 = np.sum(y_3_n1, axis=0)[np.newaxis, :]
y_3_0 = np.sum(y_3_0, axis=0)[np.newaxis, :]
y_3_p1 = np.sum(y_3_p1, axis=0)[np.newaxis, :]
y_3_p2 = np.sum(y_3_p2, axis=0)[np.newaxis, :]
y_3_p3 = np.sum(y_3_p3, axis=0)[np.newaxis, :]
# level 4
y_4_n4 = np.sqrt(2) * K[16] * np.sin(4 * phi) * W * LP['P_4_4'] * colors
y_4_n3 = np.sqrt(2) * K[17] * np.sin(3 * phi) * W * LP['P_4_3'] * colors
y_4_n2 = np.sqrt(2) * K[18] * np.sin(2 * phi) * W * LP['P_4_2'] * colors
y_4_n1 = np.sqrt(2) * K[19] * np.sin(1 * phi) * W * LP['P_4_1'] * colors
y_4_0 = K[20] * W * LP['P_4_0'] * colors
y_4_p1 = np.sqrt(2) * K[21] * np.cos(1 * phi) * W * LP['P_4_1'] * colors
y_4_p2 = np.sqrt(2) * K[22] * np.cos(2 * phi) * W * LP['P_4_2'] * colors
y_4_p3 = np.sqrt(2) * K[23] * np.cos(3 * phi) * W * LP['P_4_3'] * colors
y_4_p4 = np.sqrt(2) * K[24] * np.cos(4 * phi) * W * LP['P_4_4'] * colors
y_4_n4 = np.sum(y_4_n4, axis=0)[np.newaxis, :]
y_4_n3 = np.sum(y_4_n3, axis=0)[np.newaxis, :]
y_4_n2 = np.sum(y_4_n2, axis=0)[np.newaxis, :]
y_4_n1 = np.sum(y_4_n1, axis=0)[np.newaxis, :]
y_4_0 = np.sum(y_4_0, axis=0)[np.newaxis, :]
y_4_p1 = np.sum(y_4_p1, axis=0)[np.newaxis, :]
y_4_p2 = np.sum(y_4_p2, axis=0)[np.newaxis, :]
y_4_p3 = np.sum(y_4_p3, axis=0)[np.newaxis, :]
y_4_p4 = np.sum(y_4_p4, axis=0)[np.newaxis, :]
return np.concatenate([y_0,
y_1_n1, y_1_0, y_1_p1,
y_2_n2, y_2_n1, y_2_0, y_2_p1, y_2_p2,
y_3_n3, y_3_n2, y_3_n1, y_3_0, y_3_p1, y_3_p2, y_3_p3,
y_4_n4, y_4_n3, y_4_n2, y_4_n1, y_4_0, y_4_p1, y_4_p2, y_4_p3, y_4_p4], axis=0)
def projection(phi, theta, K, coef):
phi = phi[:, np.newaxis]
p_0_0 = P_0_0(theta)[:, np.newaxis]
p_1_0 = P_1_0(theta)[:, np.newaxis]
p_1_1 = P_1_1(theta)[:, np.newaxis]
p_2_0 = P_2_0(theta)[:, np.newaxis]
p_2_1 = P_2_1(theta)[:, np.newaxis]
p_2_2 = P_2_2(theta)[:, np.newaxis]
p_3_0 = P_3_0(theta)[:, np.newaxis]
p_3_1 = P_3_1(theta)[:, np.newaxis]
p_3_2 = P_3_2(theta)[:, np.newaxis]
p_3_3 = P_3_3(theta)[:, np.newaxis]
p_4_0 = P_4_0(theta)[:, np.newaxis]
p_4_1 = P_4_1(theta)[:, np.newaxis]
p_4_2 = P_4_2(theta)[:, np.newaxis]
p_4_3 = P_4_3(theta)[:, np.newaxis]
p_4_4 = P_4_4(theta)[:, np.newaxis]
# level 0
y_0 = np.matmul(K[0] * p_0_0, coef[0:1, :])
# level 1
y_1_n1 = np.matmul(np.sqrt(2) * K[1] * np.sin(1 * phi) * p_1_1, coef[1:2, :])
y_1_0 = np.matmul(K[2] * p_1_0, coef[2:3, :])
y_1_p1 = np.matmul(np.sqrt(2) * K[3] * np.cos(1 * phi) * p_1_1, coef[3:4, :])
# level 2
y_2_n2 = np.matmul(np.sqrt(2) * K[4] * np.sin(2 * phi) * p_2_2, coef[4:5, :])
y_2_n1 = np.matmul(np.sqrt(2) * K[5] * np.sin(1 * phi) * p_2_1, coef[5:6, :])
y_2_0 = np.matmul(K[6] * p_2_0, coef[6:7, :])
y_2_p1 = np.matmul(np.sqrt(2) * K[7] * np.cos(1 * phi) * p_2_1, coef[7:8, :])
y_2_p2 = np.matmul(np.sqrt(2) * K[8] * np.cos(2 * phi) * p_2_2, coef[8:9, :])
# level 3
y_3_n3 = np.matmul(np.sqrt(2) * K[9 ] * np.sin(3 * phi) * p_3_3, coef[ 9:10, :])
y_3_n2 = np.matmul(np.sqrt(2) * K[10] * np.sin(2 * phi) * p_3_2, coef[10:11, :])
y_3_n1 = np.matmul(np.sqrt(2) * K[11] * np.sin(1 * phi) * p_3_1, coef[11:12, :])
y_3_0 = np.matmul(K[12] * p_3_0, coef[12:13, :])
y_3_p1 = np.matmul(np.sqrt(2) * K[13] * np.cos(1 * phi) * p_3_1, coef[13:14, :])
y_3_p2 = np.matmul(np.sqrt(2) * K[14] * np.cos(2 * phi) * p_3_2, coef[14:15, :])
y_3_p3 = np.matmul(np.sqrt(2) * K[15] * np.cos(3 * phi) * p_3_3, coef[15:16, :])
# level 4
y_4_n4 = np.matmul(np.sqrt(2) * K[16] * np.sin(4 * phi) * p_4_4, coef[16:17, :])
y_4_n3 = np.matmul(np.sqrt(2) * K[17] * np.sin(3 * phi) * p_4_3, coef[17:18, :])
y_4_n2 = np.matmul(np.sqrt(2) * K[18] * np.sin(2 * phi) * p_4_2, coef[18:19, :])
y_4_n1 = np.matmul(np.sqrt(2) * K[19] * np.sin(1 * phi) * p_4_1, coef[19:20, :])
y_4_0 = np.matmul(K[20] * p_4_0, coef[20:21, :])
y_4_p1 = np.matmul(np.sqrt(2) * K[21] * np.cos(1 * phi) * p_4_1, coef[21:22, :])
y_4_p2 = np.matmul(np.sqrt(2) * K[22] * np.cos(2 * phi) * p_4_2, coef[22:23, :])
y_4_p3 = np.matmul(np.sqrt(2) * K[23] * np.cos(3 * phi) * p_4_3, coef[23:24, :])
y_4_p4 = np.matmul(np.sqrt(2) * K[24] * np.cos(4 * phi) * p_4_4, coef[24:25, :])
img = y_0 \
+ y_1_n1 + y_1_0 + y_1_p1 \
+ y_2_n2 + y_2_n1 + y_2_0 + y_2_p1 + y_2_p2 \
+ y_3_n3 + y_3_n2 + y_3_n1 + y_3_0 + y_3_p1 + y_3_p2 + y_3_p3 \
+ y_4_n4 + y_4_n3 + y_4_n2 + y_4_n1 + y_4_0 + y_4_p1 + y_4_p2 + y_4_p3 + y_4_p4
return img
def reconstImageFromSH(coef, nrows, ncols, K = None, isClip = True):
if K is None:
larr = np.array([0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, \
4, 4, 4, 4, 4, 4, 4, 4, 4], dtype=np.int32)
marr = np.array([0, -1, 0, 1, -2, -1, 0, 1, 2, -3, -2, -1, 0, 1, 2, 3, \
-4, -3, -2, -1, 0, 1, 2, 3, 4], dtype=np.int32)
K = computeK(larr, marr)
x, y = np.meshgrid(np.linspace(-1, 1, ncols+1), np.linspace(0, 1, nrows+1) )
phi, theta = np.pi * x, np.pi * y
phi, theta = phi[0:nrows, 0:ncols], theta[0:nrows, 0:ncols]
img = projection(phi.reshape(-1), theta.reshape(-1), K, coef)
img = img.reshape([nrows, ncols, 3] )
if isClip:
img = np.clip(img, 0, 1)
return img
def reconstImageFromSHAfterRotate(coef, cameraLoc, cameraUp, nrows=512, ncols=1024, K = None, isClip = True, isInv = False):
if K is None:
larr = np.array([0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, \
4, 4, 4, 4, 4, 4, 4, 4, 4], dtype=np.int32)
marr = np.array([0, -1, 0, 1, -2, -1, 0, 1, 2, -3, -2, -1, 0, 1, 2, 3, \
-4, -3, -2, -1, 0, 1, 2, 3, 4], dtype=np.int32)
K = computeK(larr, marr)
x, y = np.meshgrid(np.linspace(-1, 1, ncols+1), np.linspace(0, 1, nrows+1) )
phi, theta = np.pi * x, np.pi * y
phi, theta = phi[0:nrows, 0:ncols], theta[0:nrows, 0:ncols]
envmap = projection(phi.reshape(-1), theta.reshape(-1), K, coef)
envmap = envmap.reshape([nrows, ncols, 3] )
if isClip:
envmap = np.clip(envmap, 0, 1)
cameraLoc = np.asarray(cameraLoc, dtype=np.float32)
cameraUp = np.asarray(cameraUp, dtype=np.float32)
cameraLoc = cameraLoc / np.sqrt(np.sum(cameraLoc * cameraLoc), dtype=np.float32)
cameraUp = cameraUp / np.sqrt(np.sum(cameraUp * cameraUp), dtype=np.float32)
rz, ry = cameraLoc, cameraUp
rx = np.cross(ry, rz)
rx = rx / np.sqrt(np.sum(rx * rx) )
ry = np.cross(rz, rx)
ry = ry / np.sqrt(np.sum(ry * ry) )
if isInv == True:
rx, ry, rz = rx[:, np.newaxis], ry[:, np.newaxis], rz[:, np.newaxis]
rotMatrix = np.concatenate([rx, ry, rz], axis=1)
rotMatrix = rotMatrix.transpose([1, 0] )
rx, ry, rz = rotMatrix[:, 0], rotMatrix[:, 1], rotMatrix[:, 2]
envmapRot = np.zeros(envmap.shape, dtype=np.float32)
height, width = envmapRot.shape[0], envmapRot.shape[1]
for r in range(0, height):
for c in range(0, width):
theta = r / float(height-1) * np.pi
phi = c / float(width) * np.pi * 2 - np.pi
x = np.sin(theta) * np.cos(phi)
y = np.sin(theta) * np.sin(phi)
z = np.cos(theta)
coord = x * rx + y * ry + z * rz
nx, ny, nz = coord[0], coord[1], coord[2]
thetaNew = np.arccos(nz)
nx = nx / (np.sqrt(1-nz*nz) + 1e-12)
ny = ny / (np.sqrt(1-nz*nz) + 1e-12)
nx = np.clip(nx, -1, 1)
ny = np.clip(ny, -1, 1)
nz = np.clip(nz, -1, 1)
phiNew = np.arccos(nx)
if ny < 0:
phiNew = - phiNew
u, v = angleToUV(thetaNew, phiNew)
color = uvToEnvmap(envmap, u, v)
envmapRot[r, c, :] = color
return envmapRot
def computeSHFromImage(im):
nSampleX, nSampleY = im.shape[1], im.shape[0]
angles = np.zeros([nSampleX * nSampleY, 2] )
for r in range(nSampleY):
for c in range(nSampleX):
ind = r * nSampleX + c
y = (r + np.random.random() ) / float(nSampleY)
x = (c + np.random.random() ) / float(nSampleX)
phi = 2 * np.pi * x - np.pi
theta = 2 * np.arccos( np.sqrt(1 - y) )
angles[ind, 0] = theta
angles[ind, 1] = phi
# Compute the coef for spherical Harmonics
larr = np.array([0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, \
4, 4, 4, 4, 4, 4, 4, 4, 4], dtype=np.int32)
marr = np.array([0, -1, 0, 1, -2, -1, 0, 1, 2, -3, -2, -1, 0, 1, 2, 3, \
-4, -3, -2, -1, 0, 1, 2, 3, 4], dtype=np.int32)
karr = computeK(larr, marr)
# Compute Associated Legendre Polynomial
LP = {}
LP['P_0_0'] = P_0_0(angles[:, 0])[:, np.newaxis]
LP['P_1_0'] = P_1_0(angles[:, 0])[:, np.newaxis]
LP['P_1_1'] = P_1_1(angles[:, 0])[:, np.newaxis]
LP['P_2_0'] = P_2_0(angles[:, 0])[:, np.newaxis]
LP['P_2_1'] = P_2_1(angles[:, 0])[:, np.newaxis]
LP['P_2_2'] = P_2_2(angles[:, 0])[:, np.newaxis]
LP['P_3_0'] = P_3_0(angles[:, 0])[:, np.newaxis]
LP['P_3_1'] = P_3_1(angles[:, 0])[:, np.newaxis]
LP['P_3_2'] = P_3_2(angles[:, 0])[:, np.newaxis]
LP['P_3_3'] = P_3_3(angles[:, 0])[:, np.newaxis]
LP['P_4_0'] = P_4_0(angles[:, 0])[:, np.newaxis]
LP['P_4_1'] = P_4_1(angles[:, 0])[:, np.newaxis]
LP['P_4_2'] = P_4_2(angles[:, 0])[:, np.newaxis]
LP['P_4_3'] = P_4_3(angles[:, 0])[:, np.newaxis]
LP['P_4_4'] = P_4_4(angles[:, 0])[:, np.newaxis]
colors = []
# Bilinear interpolate the color in Enivronment map
for n in range(angles.shape[0] ):
theta, phi = angles[n, 0], angles[n, 1]
u, v = angleToUV(theta, phi)
color = uvToEnvmap(im, u, v).squeeze()[np.newaxis, :]
colors.append(color)
colors = np.concatenate(colors, axis=0)
coef = Integration(angles[:, 1], colors, karr, LP)
return coef
def computeSHFromImageAfterRotate(envmap, cameraLoc, cameraUp, isInv=False):
cameraLoc = np.asarray(cameraLoc, dtype=np.float32)
cameraUp = np.asarray(cameraUp, dtype=np.float32)
cameraLoc = cameraLoc / np.sqrt(np.sum(cameraLoc * cameraLoc), dtype=np.float32)
cameraUp = cameraUp / np.sqrt(np.sum(cameraUp * cameraUp), dtype=np.float32)
rz, ry = cameraLoc, cameraUp
rx = np.cross(ry, rz)
rx = rx / np.sqrt(np.sum(rx * rx) )
ry = np.cross(rz, rx)
ry = ry / np.sqrt(np.sum(ry * ry) )
if isInv == True:
rx, ry, rz = rx[:, np.newaxis], ry[:, np.newaxis], rz[:, np.newaxis]
rotMatrix = np.concatenate([rx, ry, rz], axis=1)
rotMatrix = rotMatrix.transpose([1, 0] )
rx, ry, rz = rotMatrix[:, 0], rotMatrix[:, 1], rotMatrix[:, 2]
envmapRot = np.zeros(envmap.shape, dtype=np.float32)
height, width = envmapRot.shape[0], envmapRot.shape[1]
for r in range(0, height):
for c in range(0, width):
theta = r / float(height-1) * np.pi
phi = c / float(width) * np.pi * 2 - np.pi
x = np.sin(theta) * np.cos(phi)
y = np.sin(theta) * np.sin(phi)
z = np.cos(theta)
coord = x * rx + y * ry + z * rz
nx, ny, nz = coord[0], coord[1], coord[2]
thetaNew = np.arccos(nz)
nx = nx / (np.sqrt(1-nz*nz) + 1e-12)
ny = ny / (np.sqrt(1-nz*nz) + 1e-12)
nx = np.clip(nx, -1, 1)
ny = np.clip(ny, -1, 1)
nz = np.clip(nz, -1, 1)
phiNew = np.arccos(nx)
if ny < 0:
phiNew = - phiNew
u, v = angleToUV(thetaNew, phiNew)
color = uvToEnvmap(envmap, u, v)
envmapRot[r, c, :] = color
coef = computeSHFromImage(envmapRot)
return coef
def computeImageAfterRotate(envmap, mat):
rx, ry, rz = mat[:, 0], mat[:, 1], mat[:, 2]
# envmapRot = np.zeros((512, 1024, 3), dtype=np.float32)
envmapRot = np.zeros((300, 600, 3), dtype=np.float32)
height, width = envmapRot.shape[0], envmapRot.shape[1]
for r in range(0, height):
for c in range(0, width):
theta = r / float(height-1) * np.pi
phi = c / float(width) * np.pi * 2 - np.pi
x = np.sin(theta) * np.cos(phi)
y = np.sin(theta) * np.sin(phi)
z = np.cos(theta)
coord = x * rx + y * ry + z * rz
nx, ny, nz = coord[0], coord[1], coord[2]
thetaNew = np.arccos(nz)
nx = nx / (np.sqrt(1-nz*nz) + 1e-12)
ny = ny / (np.sqrt(1-nz*nz) + 1e-12)
nx = np.clip(nx, -1, 1)
ny = np.clip(ny, -1, 1)
nz = np.clip(nz, -1, 1)
phiNew = np.arccos(nx)
if ny < 0:
phiNew = - phiNew
u, v = angleToUV(thetaNew, phiNew)
color = uvToEnvmap(envmap, u, v)
envmapRot[r, c, :] = color
return envmapRot
## Integral Image with Spherical Harmonics
if __name__ == "__main__":
imName = '50Lre.hdr'
im = cv2.imread(imName, cv2.IMREAD_ANYCOLOR | cv2.IMREAD_ANYDEPTH)[:, :, ::-1]
coef = computeSHFromImage(im)
coef[9:25, :] = 0
np.save(imName[0:-4] + 'SH.npy', coef)
reIm = reconstImageFromSH(coef,im.shape[0], im.shape[1], isClip = False)
io.savemat('50Lre.mat', {'im': reIm})
reIm = np.clip(reIm, 0, 1)
imLDR = (255 *(np.clip(im, 0, 1) ** (1/2.2) ) ).astype(np.uint8)
reImLDR = (255 *(np.clip(reIm, 0, 1) ** (1/2.2) ) ).astype(np.uint8)
imLDR = Image.fromarray(imLDR)
reImLDR = Image.fromarray(reImLDR)
imLDR.save(imName[0:-4] + '.png')
reImLDR.save(imName[0:-4] + 're.png')