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discrete_differential_operators.py
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discrete_differential_operators.py
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import os
import numpy as np
from numpy import linalg as LA
import pylab as plt
import mesh_calc
import io_off_model
import visualization
def calc_operators(mesh, verbose=False):
if 'opers' in mesh.keys():
return
if verbose:
print('calc_operators')
mesh_calc.calc_vertices_area(mesh, verbose=False)
mesh_calc.calc_triangles_area(mesh, verbose=False)
mesh_calc.calc_face_normals(mesh, verbose=False)
face_norms = mesh['face_normals']
At = mesh['faces_area']
edge0 = mesh['vertices'][mesh['faces'][:, 2], :] - mesh['vertices'][mesh['faces'][:, 1], :]
edge1 = mesh['vertices'][mesh['faces'][:, 0], :] - mesh['vertices'][mesh['faces'][:, 2], :]
edge2 = mesh['vertices'][mesh['faces'][:, 1], :] - mesh['vertices'][mesh['faces'][:, 0], :]
edge0_90 = np.cross(face_norms, edge0)
edge1_90 = np.cross(face_norms, edge1)
edge2_90 = np.cross(face_norms, edge2)
edge_90_x = np.hstack((edge0_90[:, 0:1], edge1_90[:, 0:1], edge2_90[:, 0:1]))
edge_90_y = np.hstack((edge0_90[:, 1:2], edge1_90[:, 1:2], edge2_90[:, 1:2]))
edge_90_z = np.hstack((edge0_90[:, 2:3], edge1_90[:, 2:3], edge2_90[:, 2:3]))
Ex = np.zeros((mesh['n_vertices'], mesh['n_faces']))
Ey = np.zeros((mesh['n_vertices'], mesh['n_faces']))
Ez = np.zeros((mesh['n_vertices'], mesh['n_faces']))
for ee, edge in zip([Ex, Ey, Ez], [edge_90_x, edge_90_y, edge_90_z]):
for i in range(3):
e_idxs = mesh['faces'][:, i]
f_idxs = np.arange(mesh['n_faces'], dtype=np.int)
ee[e_idxs, f_idxs] = edge[:, i]
Ex = Ex.T
Ey = Ey.T
Ez = Ez.T
E = np.vstack((Ex, Ey, Ez))
Gf = np.diag(np.hstack((mesh['faces_area'], mesh['faces_area'], mesh['faces_area'])))
GfInv = np.diag(1 / np.hstack((mesh['faces_area'], mesh['faces_area'], mesh['faces_area'])))
Gv = np.diag(mesh['vertices_area'])
GvInv = np.diag( 1 / mesh['vertices_area'])
W_ = 0.25 * np.dot(E.T, GfInv)
W = np.dot(W_, E)
mesh['opers'] = {'E': E, # |F| x 3 , |V|
'Gf': Gf, # |F| x 3 , |F| x 3
'Gv': Gv, # |V| , |V|
'GfInv': GfInv, # |F| x 3 , |F| x 3
'GvInv': GvInv, # |V| , |V|
'W': W, # |V| , |V|
}
def grad(mesh, f_vertices):
calc_operators(mesh)
grad_op = 0.5 * np.dot(mesh['opers']['GfInv'], mesh['opers']['E'])
grad = np.dot(grad_op, f_vertices[:, None])
grad = np.hstack((grad[:mesh['n_faces']],
grad[ mesh['n_faces']:mesh['n_faces']*2],
grad[ mesh['n_faces']*2:]))
return grad
def divergence(mesh, vector_field):
# vector_field on faces to scalars on vertices
grad_op = 0.5 * np.dot(mesh['opers']['GfInv'], mesh['opers']['E'])
div_op_1 = -np.dot(mesh['opers']['GvInv'], grad_op.T)
div_op = np.dot(div_op_1, mesh['opers']['Gf'])
vector_field_ = np.vstack((vector_field[:, 0:1], vector_field[:, 1:2], vector_field[:, 2:3]))
diverg = np.dot(div_op, vector_field_)
return diverg
def laplacian(mesh, f_vertices):
# Scalars on vertices to scalars on vertices
lap_oper = np.dot(mesh['opers']['GvInv'], mesh['opers']['W'])
lap = np.dot(lap_oper, f_vertices)
return lap
def remove_some_faces(mesh, n_faces_to_keep=20, start_face=0):
faces_order = mesh_calc.bfsdfs(mesh['faces_graph'], start_face, bfs_flag=True)
idxs = faces_order[:n_faces_to_keep]
mesh['faces'] = mesh['faces'][idxs]
mesh['face_centers'] = mesh['face_centers'][idxs]
mesh['face_normals'] = mesh['face_normals'][idxs]
return idxs
def objectives():
mesh = get_mesh(1)
mesh_calc.calc_face_centers(mesh)
mesh_calc.calc_interpolation_matrices(mesh)
mesh_calc.add_edges_to_mesh(mesh)
# Gradient on X^2
faces_function = mesh['face_centers'][:, 0] ** 2
vertices_function = np.dot(mesh['interp_matrix_f2v'], faces_function)
vector_field = grad(mesh, vertices_function)
# Divergence on "ones"
vf_for_dvrg = np.ones((mesh['n_faces'], 3))
dvrgn = divergence(mesh, vf_for_dvrg)
dvrgn_on_faces = np.dot(mesh['interp_matrix_v2f'], dvrgn)
# Laplacian on X^2
lap = laplacian(mesh, vertices_function)
lap_on_faces = np.dot(mesh['interp_matrix_v2f'], lap)
if 0:
kept_faces = remove_some_faces(mesh, n_faces_to_keep=80, start_face=83)
faces_function = faces_function[kept_faces]
vector_field = vector_field[kept_faces]
vf_for_dvrg = vf_for_dvrg[kept_faces]
dvrgn_on_faces = dvrgn_on_faces[kept_faces]
visualization.visualize_mesh(mesh, show_tringles=True, faces_function=faces_function, alpha=0.75,
faces_vector_field=vector_field, title='Gradient on X^2', pause=False)
visualization.visualize_mesh(mesh, show_tringles=True, faces_function=dvrgn_on_faces[:, 0], alpha=0.75,
faces_vector_field=vf_for_dvrg, title='Divergence from vector field [1,1,1] as v.f.', pause=False)
visualization.visualize_mesh(mesh, show_tringles=True, faces_function=lap_on_faces, alpha=0.75,
title='Laplacian on X^2', pause=False)
plt.show()
def analysis_1():
mesh = get_mesh(6)
mesh_calc.calc_face_centers(mesh)
mesh_calc.calc_interpolation_matrices(mesh)
mesh_calc.add_edges_to_mesh(mesh)
calc_operators(mesh)
w, v = LA.eig(mesh['opers']['W'])
epsilon = 1e-5
w_is_pos_semi_def = np.all(np.real(w) >= -epsilon)
print('NULL: Sum over rows(all results should be close to zero): ', mesh['opers']['W'].sum(axis=1))
print(' All are close to 0: ', np.all(np.abs(mesh['opers']['W'].sum(axis=1)) < 0.0001))
print('SYM: If W is sym, this will be close to zero: ', np.sum((mesh['opers']['W'] - mesh['opers']['W'].T) ** 2))
print('LOC: Zero / non-zero elements: ', (mesh['opers']['W'] == 0).sum(), (mesh['opers']['W'] != 0).sum())
print('POS: Is all values are zeros? : ', np.all(mesh['opers']['W'] > 0))
print('PSD: Is W positive sem-definite? ', w_is_pos_semi_def)
def analysis_2():
mesh = get_mesh(6)
mesh_calc.calc_interpolation_matrices(mesh, verbose=True)
calc_operators(mesh, verbose=True)
w, v = LA.eig(mesh['opers']['W'])
sorted_idxs = np.argsort(w)
if 0:
N = 9
fig = plt.figure()
for i in range(N):
ax = fig.add_subplot(3, 3, i + 1, projection='3d')
print(w[sorted_idxs[i]])
vertices_function = v[:, sorted_idxs[i]]
faces_function = np.dot(mesh['interp_matrix_v2f'], vertices_function)
visualization.visualize_mesh(mesh, show_tringles=True, faces_function=faces_function, alpha=0.85,
title=str(i + 1) + 'eig vec', pause=False, ax=ax, vmax=0.1, vmin=-0.1) #
vertices_function = np.zeros((mesh['n_vertices'], 1))
vertices_function[26] = 10
faces_function = np.dot(mesh['interp_matrix_v2f'], vertices_function)[:,0]
plot = True
if plot:
fig = plt.figure()
ax = fig.add_subplot(2, 2, 1, projection='3d')
#visualization.visualize_mesh(mesh, show_tringles=True, vertices_function=vertices_function[:, 0],
# title='hat function', pause=False, ax=ax, vmax=1.1, vmin=-0.1)
visualization.visualize_mesh(mesh, show_tringles=True, faces_function=faces_function, alpha=0.85,
title='hat function', pause=False, ax=ax, vmax=1.1, vmin=-0.1)
next_subplot = 2
all_err = []
if plot:
all_k = [1, 20, 100]
else:
all_k = np.arange(1, mesh['n_vertices'], 5)
for k in all_k:
Bi = v[:, sorted_idxs[:k]]
vertices_function_ = np.dot(Bi, np.dot(Bi.T, vertices_function))
err = LA.norm(vertices_function - vertices_function_)
faces_function_ = np.dot(mesh['interp_matrix_v2f'], vertices_function_)[:, 0]
if plot:
ax = fig.add_subplot(2, 2, next_subplot, projection='3d')
visualization.visualize_mesh(mesh, show_tringles=True, faces_function=faces_function_, alpha=0.85,
title='hat function, estimated by k=' + str(k), pause=False, ax=ax, vmax=1.1, vmin=-0.1)
all_err.append(err)
next_subplot += 1
plt.figure()
plt.plot(all_k, all_err, '*-')
plt.ylabel('Reconstruction Error')
plt.xlabel('Number of eigen vector used')
plt.title('Representation using a reduced basis, for ' + mesh['name'])
plt.show()
def get_mesh(idx=0):
if 0:
mesh = io_off_model.get_simple_mesh('one_triangle')
else:
mesh_fns = [r"C:\Users\alon\Downloads\ModelNet40\ModelNet40\car\train\car_0016.off",
r"C:\Users\alon\Downloads\ModelNet40\ModelNet40\bottle\train\bottle_0320.off",
r"C:\Users\alon\Downloads\ModelNet40\ModelNet40\airplane\train\airplane_0169.off",
r"C:\Users\alon\Downloads\ModelNet40\ModelNet40\cone\train\cone_0088.off",
r"C:\Users\alon\Downloads\ModelNet40\ModelNet40\person\train\person_0034.off",
r"C:\Users\alon\Downloads\ModelNet40\ModelNet40\cup\train\cup_0019.off",
'hw2_data/sphere_s0.off',
'hw2_data/phands.off'
]
mesh = io_off_model.read_off(mesh_fns[idx], verbose=True)
mesh['name'] = os.path.split(mesh_fns[idx])[-1]
return mesh
if __name__ == '__main__':
analysis_1()
if __name__ == '-__main__':
mesh = get_mesh()
mesh_calc.add_edges_to_mesh(mesh)
faces_order = mesh_calc.bfsdfs(mesh['faces_graph'], 0, bfs_flag=True)
if 1:
faces_function = np.zeros((mesh['n_faces'],))
faces_function[faces_order[:1]] = 1
if 0:
mesh_calc.calc_interpolation_matrices(mesh)
vertices_function = np.dot(mesh['interp_matrix_f2v'], faces_function)
elif 1:
vertices_function = np.zeros((mesh['n_vertices'],))
vertices_function[mesh['faces'][faces_order[:1]].flatten()] = 1
else:
vertices_function = np.zeros((mesh['n_vertices'],))
vertices_function[2] = 1
vertices_function[1] = 1
vector_field = grad(mesh, vertices_function)
else:
vector_field = grad(mesh, mesh['vertices_area'])
if 0:
faces_function = np.zeros((mesh['n_faces'],))
faces_function[0:N] = 1
else:
faces_function = mesh['faces_area']
divr = laplacian(mesh, vertices_function)
if 0:
idxs = faces_order[:20]
mesh['faces'] = mesh['faces'][idxs]
faces_function = faces_function[idxs]
#faces_function[1] = -10
vector_field = vector_field[idxs]
#visualization.visualize_mesh(mesh, show_tringles=True, faces_vector_field=vector_field,
# faces_function=faces_function, alpha=0.25, normalize_quiver=True)
#visualization.visualize_mesh(mesh, show_tringles=True, vertices_function=vertices_function,
# to_show='text', alpha=0.5, faces_vector_field=vector_field)
visualization.visualize_mesh(mesh, show_tringles=True,
vertices_function=divr, alpha=0.5)