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'''
Estimating parameters about vertices: shape para, exp para, pose para(s, R, t)
'''
import numpy as np
from .. import mesh
''' TODO: a clear document.
Given: image_points, 3D Model, Camera Matrix(s, R, t2d)
Estimate: shape parameters, expression parameters
Inference:
projected_vertices = s*P*R(mu + shape + exp) + t2d --> image_points
s*P*R*shape + s*P*R(mu + exp) + t2d --> image_poitns
# Define:
X = vertices
x_hat = projected_vertices
x = image_points
A = s*P*R
b = s*P*R(mu + exp) + t2d
==>
x_hat = A*shape + b (2 x n)
A*shape (2 x n)
shape = reshape(shapePC * sp) (3 x n)
shapePC*sp : (3n x 1)
* flatten:
x_hat_flatten = A*shape + b_flatten (2n x 1)
A*shape (2n x 1)
--> A*shapePC (2n x 199) sp: 199 x 1
# Define:
pc_2d = A* reshape(shapePC)
pc_2d_flatten = flatten(pc_2d) (2n x 199)
=====>
x_hat_flatten = pc_2d_flatten * sp + b_flatten ---> x_flatten (2n x 1)
Goals:
(ignore flatten, pc_2d-->pc)
min E = || x_hat - x || + lambda*sum(sp/sigma)^2
= || pc * sp + b - x || + lambda*sum(sp/sigma)^2
Solve:
d(E)/d(sp) = 0
2 * pc' * (pc * sp + b - x) + 2 * lambda * sp / (sigma' * sigma) = 0
Get:
(pc' * pc + lambda / (sigma'* sigma)) * sp = pc' * (x - b)
'''
def estimate_shape(x, shapeMU, shapePC, shapeEV, expression, s, R, t2d, lamb = 3000):
'''
Args:
x: (2, n). image points (to be fitted)
shapeMU: (3n, 1)
shapePC: (3n, n_sp)
shapeEV: (n_sp, 1)
expression: (3, n)
s: scale
R: (3, 3). rotation matrix
t2d: (2,). 2d translation
lambda: regulation coefficient
Returns:
shape_para: (n_sp, 1) shape parameters(coefficients)
'''
x = x.copy()
assert(shapeMU.shape[0] == shapePC.shape[0])
assert(shapeMU.shape[0] == x.shape[1]*3)
dof = shapePC.shape[1]
n = x.shape[1]
sigma = shapeEV
t2d = np.array(t2d)
P = np.array([[1, 0, 0], [0, 1, 0]], dtype = np.float32)
A = s*P.dot(R)
# --- calc pc
pc_3d = np.resize(shapePC.T, [dof, n, 3]) # 199 x n x 3
pc_3d = np.reshape(pc_3d, [dof*n, 3])
pc_2d = pc_3d.dot(A.T.copy()) # 199 x n x 2
pc = np.reshape(pc_2d, [dof, -1]).T # 2n x 199
# --- calc b
# shapeMU
mu_3d = np.resize(shapeMU, [n, 3]).T # 3 x n
# expression
exp_3d = expression
#
b = A.dot(mu_3d + exp_3d) + np.tile(t2d[:, np.newaxis], [1, n]) # 2 x n
b = np.reshape(b.T, [-1, 1]) # 2n x 1
# --- solve
equation_left = np.dot(pc.T, pc) + lamb * np.diagflat(1/sigma**2)
x = np.reshape(x.T, [-1, 1])
equation_right = np.dot(pc.T, x - b)
shape_para = np.dot(np.linalg.inv(equation_left), equation_right)
return shape_para
def estimate_expression(x, shapeMU, expPC, expEV, shape, s, R, t2d, lamb = 2000):
'''
Args:
x: (2, n). image points (to be fitted)
shapeMU: (3n, 1)
expPC: (3n, n_ep)
expEV: (n_ep, 1)
shape: (3, n)
s: scale
R: (3, 3). rotation matrix
t2d: (2,). 2d translation
lambda: regulation coefficient
Returns:
exp_para: (n_ep, 1) shape parameters(coefficients)
'''
x = x.copy()
assert(shapeMU.shape[0] == expPC.shape[0])
assert(shapeMU.shape[0] == x.shape[1]*3)
dof = expPC.shape[1]
n = x.shape[1]
sigma = expEV
t2d = np.array(t2d)
P = np.array([[1, 0, 0], [0, 1, 0]], dtype = np.float32)
A = s*P.dot(R)
# --- calc pc
pc_3d = np.resize(expPC.T, [dof, n, 3])
pc_3d = np.reshape(pc_3d, [dof*n, 3])
pc_2d = pc_3d.dot(A.T)
pc = np.reshape(pc_2d, [dof, -1]).T # 2n x 29
# --- calc b
# shapeMU
mu_3d = np.resize(shapeMU, [n, 3]).T # 3 x n
# expression
shape_3d = shape
#
b = A.dot(mu_3d + shape_3d) + np.tile(t2d[:, np.newaxis], [1, n]) # 2 x n
b = np.reshape(b.T, [-1, 1]) # 2n x 1
# --- solve
equation_left = np.dot(pc.T, pc) + lamb * np.diagflat(1/sigma**2)
x = np.reshape(x.T, [-1, 1])
equation_right = np.dot(pc.T, x - b)
exp_para = np.dot(np.linalg.inv(equation_left), equation_right)
return exp_para
# ---------------- fit
def fit_points(x, X_ind, model, n_sp, n_ep, max_iter = 4):
'''
Args:
x: (n, 2) image points
X_ind: (n,) corresponding Model vertex indices
model: 3DMM
max_iter: iteration
Returns:
sp: (n_sp, 1). shape parameters
ep: (n_ep, 1). exp parameters
s, R, t
'''
x = x.copy().T
#-- init
sp = np.zeros((n_sp, 1), dtype = np.float32)
ep = np.zeros((n_ep, 1), dtype = np.float32)
#-------------------- estimate
X_ind_all = np.tile(X_ind[np.newaxis, :], [3, 1])*3
X_ind_all[1, :] += 1
X_ind_all[2, :] += 2
valid_ind = X_ind_all.flatten('F')
shapeMU = model['shapeMU'][valid_ind, :]
shapePC = model['shapePC'][valid_ind, :n_sp]
expPC = model['expPC'][valid_ind, :n_ep]
for i in range(max_iter):
X = shapeMU + shapePC.dot(sp) + expPC.dot(ep)
X = np.reshape(X, [int(len(X)/3), 3]).T
#----- estimate pose
P = mesh.transform.estimate_affine_matrix_3d22d(X.T, x.T)
s, R, t = mesh.transform.P2sRt(P)
rx, ry, rz = mesh.transform.matrix2angle(R)
#print('Iter:{}; estimated pose: s {}, rx {}, ry {}, rz {}, t1 {}, t2 {}'.format(i, s, rx, ry, rz, t[0], t[1]))
#----- estimate shape
# expression
shape = shapePC.dot(sp)
shape = np.reshape(shape, [int(len(shape)/3), 3]).T
ep = estimate_expression(x, shapeMU, expPC, model['expEV'][:n_ep,:], shape, s, R, t[:2], lamb = 20)
# shape
expression = expPC.dot(ep)
expression = np.reshape(expression, [int(len(expression)/3), 3]).T
if i == 0 :
sp = estimate_shape(x, shapeMU, shapePC, model['shapeEV'][:n_sp,:], expression, s, R, t[:2], lamb = 40)
return sp, ep, s, R, t
# ---------------- fitting process
def fit_points_for_show(x, X_ind, model, n_sp, n_ep, max_iter = 4):
'''
Args:
x: (n, 2) image points
X_ind: (n,) corresponding Model vertex indices
model: 3DMM
max_iter: iteration
Returns:
sp: (n_sp, 1). shape parameters
ep: (n_ep, 1). exp parameters
s, R, t
'''
x = x.copy().T
#-- init
sp = np.zeros((n_sp, 1), dtype = np.float32)
ep = np.zeros((n_ep, 1), dtype = np.float32)
#-------------------- estimate
X_ind_all = np.tile(X_ind[np.newaxis, :], [3, 1])*3
X_ind_all[1, :] += 1
X_ind_all[2, :] += 2
valid_ind = X_ind_all.flatten('F')
shapeMU = model['shapeMU'][valid_ind, :]
shapePC = model['shapePC'][valid_ind, :n_sp]
expPC = model['expPC'][valid_ind, :n_ep]
s = 4e-04
R = mesh.transform.angle2matrix([0, 0, 0])
t = [0, 0, 0]
lsp = []; lep = []; ls = []; lR = []; lt = []
for i in range(max_iter):
X = shapeMU + shapePC.dot(sp) + expPC.dot(ep)
X = np.reshape(X, [int(len(X)/3), 3]).T
lsp.append(sp); lep.append(ep); ls.append(s), lR.append(R), lt.append(t)
#----- estimate pose
P = mesh.transform.estimate_affine_matrix_3d22d(X.T, x.T)
s, R, t = mesh.transform.P2sRt(P)
lsp.append(sp); lep.append(ep); ls.append(s), lR.append(R), lt.append(t)
#----- estimate shape
# expression
shape = shapePC.dot(sp)
shape = np.reshape(shape, [int(len(shape)/3), 3]).T
ep = estimate_expression(x, shapeMU, expPC, model['expEV'][:n_ep,:], shape, s, R, t[:2], lamb = 20)
lsp.append(sp); lep.append(ep); ls.append(s), lR.append(R), lt.append(t)
# shape
expression = expPC.dot(ep)
expression = np.reshape(expression, [int(len(expression)/3), 3]).T
sp = estimate_shape(x, shapeMU, shapePC, model['shapeEV'][:n_sp,:], expression, s, R, t[:2], lamb = 40)
# print('ls', ls)
# print('lR', lR)
return np.array(lsp), np.array(lep), np.array(ls), np.array(lR), np.array(lt)
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