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from gaussian import Gaussian
from aabb import process_aabb, aabb_merge, center, solve_scale, aabb_size
from utils import quat2rot, rot2quat, sq2cov
from scipy.optimize import minimize
import numpy as np
import random
import time
def merge_4(gaussian: Gaussian, id1: int, id2: int):
scale_1 = gaussian.scales[id1]
scale_2 = gaussian.scales[id2]
weight_1 = gaussian.opacity[id1] * np.prod(scale_1)
weight_2 = gaussian.opacity[id2] * np.prod(scale_2)
weight = weight_1 + weight_2
position = (weight_1 * gaussian.positions[id1] + weight_2 * gaussian.positions[id2]) / weight
cov_1 = sq2cov(scale_1, gaussian.rotations[id1])
cov_2 = sq2cov(scale_2, gaussian.rotations[id2])
cov = (weight_1 * cov_1 + weight_2 * cov_2) / weight
eigenvalues, eigenvectors = np.linalg.eig(cov)
scale = np.sqrt(eigenvalues)
rotation = rot2quat(eigenvectors.T)
sh = (gaussian.sh[id1] + gaussian.sh[id2]) / 2.
features_dc = sh[0, :]
features_rest = sh[1:, :]
opacity = (weight_1 * gaussian.opacity[id1] + weight_2 * gaussian.opacity[id2]) / weight
return position, scale, rotation, features_dc, features_rest, opacity
def merge_3(gaussian: Gaussian, id1: int, id2: int) -> float:
def make_gaussian(mu, s, r, o):
def calc_inner(pos: np.ndarray) -> float:
x = pos - mu
S = np.diag(s)
R = quat2rot(r)
M = S @ R
Sigma = np.linalg.inv(M.T @ M)
return o * np.exp(-0.5 * np.dot(x, Sigma @ x))
return calc_inner
opacity_f = gaussian.opacity[id1]
opacity_g = gaussian.opacity[id2]
mu_f = gaussian.positions[id1]
mu_g = gaussian.positions[id2]
cov_f = sq2cov(gaussian.scales[id1], gaussian.rotations[id1])
cov_g = sq2cov(gaussian.scales[id2], gaussian.rotations[id2])
sqrt_det_cov_f = np.sqrt(np.linalg.det(cov_f))
sqrt_det_cov_g = np.sqrt(np.linalg.det(cov_g))
def c_without_2pi(mu1, mu2, cov1, cov2):
mu = mu1 - mu2
cov = cov1 + cov2
return np.exp(-0.5 * np.dot(mu, cov @ mu)) / np.sqrt(np.linalg.det(cov))
f2integral = opacity_f * opacity_f * sqrt_det_cov_f
g2integral = opacity_g * opacity_g * sqrt_det_cov_g
fgintegral = (2 ** 1.5) * opacity_f * opacity_g * sqrt_det_cov_f * sqrt_det_cov_g * c_without_2pi(mu_f, mu_g, cov_f, cov_g)
gaussian_f = make_gaussian(gaussian.positions[id1], gaussian.scales[id1], gaussian.rotations[id1], opacity_f)
gaussian_g = make_gaussian(gaussian.positions[id2], gaussian.scales[id2], gaussian.rotations[id2], opacity_g)
mu_0, s_0, r_0, o_0, point_min, basis = merge_geo(gaussian, id1, id2)
x_0 = np.concatenate((mu_0, s_0, r_0, o_0), axis=0)
basis_x = basis[:, 0]
basis_y = basis[:, 1]
basis_z = basis[:, 2]
def target_inner_analytical(features) -> float:
mu = features[:3]
s = features[3:6]
r = features[6:10]
o = features[10]
cov = sq2cov(s, r)
sqrt_det_cov = np.sqrt(np.linalg.det(cov))
h2integral = o * o * sqrt_det_cov
fhintegral = (2 ** 1.5) * opacity_f * o * sqrt_det_cov_f * sqrt_det_cov * c_without_2pi(mu_f, mu, cov_f, cov)
ghintegral = (2 ** 1.5) * opacity_g * o * sqrt_det_cov_g * sqrt_det_cov * c_without_2pi(mu_g, mu, cov_g, cov)
return (f2integral + g2integral + h2integral + 2 * fgintegral - 2 * fhintegral - 2 * ghintegral) * (np.pi ** 1.5)
def target_inner(features) -> float:
mu = features[:3]
s = features[3:6]
r = features[6:10]
o = features[10]
gaussian_h = make_gaussian(mu, s, r, o)
N = 64
sum = 0.
tot_weight = 0.
random.seed(int(time.time()))
for _ in range(N):
while True:
x = random.random()
y = random.random()
z = random.random()
pos = point_min + x * basis_x + y * basis_y + z * basis_z
f_value = gaussian_f(pos)
g_value = gaussian_g(pos)
if N < 32: # 用 f 的 pdf
weight = f_value / opacity_f
if random.random() < weight:
break
else: # 用 g 的 pdf
weight = g_value / opacity_g
if random.random() < weight:
break
sum += (f_value + g_value - gaussian_h(pos)) ** 2 * weight
tot_weight += weight
return sum / tot_weight
res = minimize(target_inner_analytical, x_0, options={'gtol': 1e-4, 'disp': False})
mu_h = res.x[:3]
s_h = res.x[3:6]
r_h = res.x[6:10]
o_h = res.x[10]
sh = (gaussian.sh[id1] + gaussian.sh[id2]) / 2.
features_dc = sh[0, :]
features_rest = sh[1:, :]
# print(mu_h, s_h, r_h, features_dc, features_rest, o_h)
return mu_h, s_h, r_h, features_dc, features_rest, o_h
# def merge_neo(gaussian: Gaussian, id1: int, id2: int):
# N_f = np.prod(gaussian.scales[id1]) * gaussian.opacity[id1]
# N_g = np.prod(gaussian.scales[id2]) * gaussian.opacity[id2]
# # N_f = gaussian.opacity[id1]
# # N_g = gaussian.opacity[id2]
# mu_f = gaussian.positions[id1]
# mu_g = gaussian.positions[id2]
# Lambda_f = np.diag(gaussian.scales[id1] ** 2)
# Lambda_g = np.diag(gaussian.scales[id2] ** 2)
# U_f = quat2rot(gaussian.rotations[id1])
# U_g = quat2rot(gaussian.rotations[id2])
# N_h = N_f + N_g
# position = (N_f * mu_f + N_g * mu_g) / N_h
# g = U_f.T @ (mu_f - mu_g)
# G = U_f.T @ U_g
# mat = N_f / N_h * Lambda_f + \
# N_g / N_h * G @ Lambda_g @ G.T + \
# N_f * N_g / (N_h * N_h) * g @ g.T
# eigenvalues, eigenvectors = np.linalg.eig(mat)
# U_h = U_f @ eigenvectors
# rotation = rot2quat(U_h)
# scale = np.sqrt(eigenvalues)
# opacity = N_h / np.prod(scale)
# sh = (gaussian.sh[id1] + gaussian.sh[id2]) / 2.
# features_dc = sh[0, :]
# features_rest = sh[1:, :]
# return position, scale, rotation, features_dc, features_rest, opacity
def merge_geo(gaussian: Gaussian, id1: int, id2: int):
rotation = gaussian.rotations[id1] + gaussian.rotations[id2]
rotation = rotation / np.linalg.norm(rotation)
rotation_mat = quat2rot(rotation)
inv_rotation_mat = np.linalg.inv(rotation_mat)
rotation1_mat = inv_rotation_mat @ quat2rot(gaussian.rotations[id1])
rotation2_mat = inv_rotation_mat @ quat2rot(gaussian.rotations[id2])
position1 = inv_rotation_mat @ gaussian.positions[id1]
position2 = inv_rotation_mat @ gaussian.positions[id2]
aabb1 = process_aabb(position1, gaussian.scales[id1], rotation1_mat)
aabb2 = process_aabb(position2, gaussian.scales[id2], rotation2_mat)
aabb = aabb_merge(aabb1, aabb2)
# point_min = aabb[:3]
# vectors = aabb[3:] - point_min
# vector_x = rotation_mat @ np.array([vectors.x, 0, 0])
# vector_y = rotation_mat @ np.array([0, vectors.y, 0])
# vector_z = rotation_mat @ np.array([0, 0, vectors.z])
point_min = np.array([aabb[0], aabb[2], aabb[4]])
point_max = np.array([aabb[1], aabb[3], aabb[5]])
vectors = rotation_mat @ np.diag(point_max - point_min)
# new_aabb = np.concatenate((rotation_mat @ point_min, rotation_mat @ point_max), axis=0)
position = rotation_mat @ center(aabb)
scale = np.array([aabb[1] - aabb[0], aabb[3] - aabb[2], aabb[5] - aabb[4]]) / 2.
opacity = np.array((gaussian.opacity[id1] + gaussian.opacity[id2]) / 2.)
return position, scale, rotation, opacity, rotation_mat @ point_min, vectors
def merge_2(gaussian: Gaussian, id1: int, id2: int):
rotation = gaussian.rotations[id1] + gaussian.rotations[id2]
rotation = rotation / np.linalg.norm(rotation)
rotation_mat = quat2rot(rotation)
inv_rotation_mat = np.linalg.inv(rotation_mat)
rotation1_mat = inv_rotation_mat @ quat2rot(gaussian.rotations[id1])
rotation2_mat = inv_rotation_mat @ quat2rot(gaussian.rotations[id2])
position1 = inv_rotation_mat @ gaussian.positions[id1]
position2 = inv_rotation_mat @ gaussian.positions[id2]
aabb1 = process_aabb(position1, gaussian.scales[id1], rotation1_mat)
aabb2 = process_aabb(position2, gaussian.scales[id2], rotation2_mat)
aabb = aabb_merge(aabb1, aabb2)
position = rotation_mat @ center(aabb)
scale = np.array([aabb[1] - aabb[0], aabb[3] - aabb[2], aabb[5] - aabb[4]]) / 2.
sh = (gaussian.sh[id1] + gaussian.sh[id2]) / 2.
features_dc = sh[0, :]
features_rest = sh[1:, :]
opacity = (gaussian.opacity[id1] + gaussian.opacity[id2]) / 2.
return position, scale, rotation, features_dc, features_rest, opacity
def merge(gaussian: Gaussian, id1: int, id2: int):
aabb1 = process_aabb(
gaussian.positions[id1],
gaussian.scales[id1],
gaussian.rotations[id1])
aabb2 = process_aabb(
gaussian.positions[id2],
gaussian.scales[id2],
gaussian.rotations[id2])
aabb = aabb_merge(aabb1, aabb2)
print(f"merging:\n {aabb1}\n {aabb2}\n\n {aabb}")
position = center(aabb)
rotation = gaussian.rotations[id1] + gaussian.rotations[id2]
rotation = rotation / np.linalg.norm(rotation)
scale = solve_scale(rotation, aabb)
if np.isnan(scale).any():
# 为什么会出现 nan??????
return None, None, None, None, None, None
sh = (gaussian.sh[id1] + gaussian.sh[id2]) / 2.
features_dc = sh[0, :]
features_rest = sh[1:, :]
opacity = (gaussian.opacity[id1] + gaussian.opacity[id2]) / 2.
# opacity = min(gaussian.opacity[id1], gaussian.opacity[id2])
# aabb_test = process_aabb(
# position,
# scale,
# rotation,
# scale_factor=1
# )
# assert(np.allclose(aabb, aabb_test))
return position, scale, rotation, features_dc, features_rest, opacity
# gaussian.replace(id1, id2, position, scale, rotation, sh, opacity)
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