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from numpy import ndarray
from typing import *
if TYPE_CHECKING:
import scipy.sparse as sp
from scipy.sparse import csr_array
from .transforms import unproject_cv, angle_between
from .utils import lookup, csr_matrix_from_dense_indices
from .segment_ops import segment_roll
__all__ = [
'triangulate_mesh',
'compute_face_corner_angles',
'compute_face_corner_normals',
'compute_face_corner_tangents',
'compute_face_normals',
'compute_face_tangents',
'compute_vertex_normals',
'remove_corrupted_faces',
'merge_duplicate_vertices',
'remove_unused_vertices',
'subdivide_mesh',
'mesh_edges',
'mesh_half_edges',
'mesh_connected_components',
'graph_connected_components',
'mesh_adjacency_graph',
'flatten_mesh_indices',
'create_cube_mesh',
'create_icosahedron_mesh',
'create_square_mesh',
'create_camera_frustum_mesh',
'merge_meshes',
# 'calc_quad_candidates',
# 'calc_quad_distortion',
# 'calc_quad_direction',
# 'calc_quad_smoothness',
# 'solve_quad',
# 'solve_quad_qp',
# 'tri_to_quad'
]
def triangulate_mesh(
faces: ndarray,
vertices: ndarray = None,
method: Literal['fan', 'strip', 'diagonal'] = 'fan',
return_face_indices: bool = False,
) -> ndarray:
"""
Triangulate a polygonal mesh.
## Parameters
faces (ndarray): [L, P] polygonal faces
vertices (ndarray, optional): [N, 3] 3-dimensional vertices.
If given, the triangulation is performed according to the distance
between vertices. Defaults to None.
method (str, optional): triangulation method. Defaults to 'fan'.
- 'fan': connect the first vertex to all other vertex pairs
- 'strip': create a triangle strip
- 'diagonal': for quad faces only, split according to the shorter diagonal
return_face_indices (bool, optional): whether to return the original face indices for each triangle. Defaults to False.
## Returns
(ndarray): [L * (P - 2), 3] triangular faces
"""
if faces.shape[-1] == 3:
return faces
P = faces.shape[-1]
if method == 'fan':
i = np.arange(P - 2, dtype=int)
triangle_loop = np.stack([np.zeros_like(i), i + 1, i + 2], axis=1) # (P - 2, 3)
triangles = faces[:, triangle_loop].reshape((-1, 3))
if return_face_indices:
triangle_face_indices = np.repeat(np.arange(faces.shape[0], dtype=int), len(triangle_loop))
elif method == 'strip':
i = np.arange(P - 2, dtype=int)
j = i // 2
loop_indices = np.where(
(i % 2 == 0)[:, None],
np.stack([(P - j) % P, j + 1, P - j - 1], axis=1),
np.stack([j + 1, j + 2, P - j - 1], axis=1)
)
triangles = faces[:, loop_indices].reshape((-1, 3))
if return_face_indices:
triangle_face_indices = np.repeat(np.arange(faces.shape[0], dtype=int), len(triangle_loop))
elif method == 'diagonal':
assert faces.shape[-1] == 4, "Diagonal-aware method is only supported for quad faces"
assert vertices is not None, "Vertices must be provided for diagonal method"
backslash = np.linalg.norm(vertices[faces[:, 0]] - vertices[faces[:, 2]], axis=-1) < \
np.linalg.norm(vertices[faces[:, 1]] - vertices[faces[:, 3]], axis=-1)
triangles = np.where(
backslash[:, None],
faces[:, [0, 1, 2, 0, 2, 3]],
faces[:, [0, 1, 3, 3, 1, 2]]
).reshape((-1, 3))
if return_face_indices:
triangle_face_indices = np.repeat(np.arange(faces.shape[0], dtype=int), 2)
if return_face_indices:
return triangles, triangle_face_indices
else:
return triangles
def compute_face_corner_angles(
vertices: ndarray,
faces: Optional[ndarray] = None,
) -> ndarray:
"""
Compute face corner angles of a mesh
## Parameters
- `vertices` (ndarray): `(..., N, 3)` vertices if `faces` is provided, or `(..., F, P, 3)` if `faces` is None
- `faces` (ndarray, optional): `(F, P)` face vertex indices, where P is the number of vertices per face
## Returns
- `angles` (ndarray): `(..., F, P)` face corner angles
"""
if faces is not None:
vertices = vertices[..., faces, :] # (..., T, P, 3)
edges = np.roll(vertices, -1, axis=-2) - vertices # (..., T, P, 3)
angles = angle_between(-np.roll(edges, 1, axis=-2), edges)
return angles
def compute_face_corner_normals(
vertices: ndarray,
faces: Optional[ndarray] = None,
normalize: bool = True
) -> ndarray:
"""
Compute the face corner normals of a mesh
## Parameters
- `vertices` (ndarray): `(..., N, 3)` vertices if `faces` is provided, or `(..., F, P, 3)` if `faces` is None
- `faces` (ndarray, optional): `(F, P)` face vertex indices, where P is the number of vertices per face
- `normalize` (bool): whether to normalize the normals to unit vectors. If not, the normals are the raw cross products.
## Returns
- `normals` (ndarray): (..., F, P, 3) face corner normals
"""
if faces is not None:
vertices = vertices[..., faces, :] # (..., T, P, 3)
edges = np.roll(vertices, -1, axis=-2) - vertices # (..., T, P, 3)
normals = np.cross(np.roll(edges, 1, axis=-2), edges)
if normalize:
normals /= np.linalg.norm(normals, axis=-1, keepdims=True) + np.finfo(vertices.dtype).eps
return normals
def compute_face_corner_tangents(
vertices: ndarray,
uv: ndarray,
faces_vertices: Optional[ndarray] = None,
faces_uv: Optional[ndarray] = None,
normalize: bool = True
) -> ndarray:
"""
Compute the face corner tangent (and bitangent) vectors of a mesh
## Parameters
- `vertices` (ndarray): `(..., N, 3)` if `faces` is provided, or `(..., F, P, 3)` if `faces_vertices` is None
- `uv` (ndarray): `(..., N, 2)` if `faces` is provided, or `(..., F, P, 2)` if `faces_uv` is None
- `faces_vertices` (ndarray, optional): `(F, P)` face vertex indices
- `faces_uv` (ndarray, optional): `(F, P)` face UV indices
- `normalize` (bool): whether to normalize the tangents to unit vectors. If not, the tangents (dX/du, dX/dv) matches the UV parameterized manifold.
## Returns
- `tangents` (ndarray): `(..., F, P, 3, 2)` face corner tangents (and bitangents),
where the last dimension represents the tangent and bitangent vectors.
"""
if faces_vertices is not None:
vertices = vertices[..., faces_vertices, :]
if faces_uv is not None:
uv = uv[..., faces_uv, :]
edge_xyz = np.roll(vertices, -1, axis=-2) - vertices # (..., F, P, 3)
edge_uv = np.roll(uv, -1, axis=-2) - uv # (..., F, P, 2)
tangents = np.stack([np.roll(edge_xyz, 1, axis=-2), edge_xyz], axis=-1) \
@ np.linalg.inv(np.stack([np.roll(edge_uv, 1, axis=-2), edge_uv], axis=-1))
if normalize:
tangents /= np.linalg.norm(tangents, axis=-1, keepdims=True) + np.finfo(tangents.dtype).eps
return tangents
def compute_face_normals(
vertices: ndarray,
faces: Optional[ndarray] = None,
) -> ndarray:
"""
Compute face normals of a mesh
## Parameters
- `vertices` (ndarray): `(..., N, 3)` vertices if `faces` is provided, or `(..., F, P, 3)` if `faces` is None
- `faces` (ndarray, optional): `(F, P)` face vertex indices, where P is the number of vertices per face
## Returns
- `normals` (ndarray): `(..., F, 3)` face normals. Always normalized.
"""
if faces is not None:
vertices = vertices[..., faces, :] # (..., F, P, 3)
if vertices.shape[-2] == 3:
normals = np.cross(
vertices[..., 1, :] - vertices[..., 0, :],
vertices[..., 2, :] - vertices[..., 0, :]
)
else:
normals = compute_face_corner_normals(vertices, normalize=False)
normals = np.mean(normals, axis=-2)
normals /= np.linalg.norm(normals, axis=-1, keepdims=True) + np.finfo(vertices.dtype).eps
return normals
def compute_face_tangents(
vertices: ndarray,
uv: ndarray,
faces_vertices: Optional[ndarray] = None,
faces_uv: Optional[ndarray] = None,
normalize: bool = True
) -> ndarray:
"""
Compute the face corner tangent (and bitangent) vectors of a mesh
## Parameters
- `vertices` (ndarray): `(..., N, 3)` if `faces` is provided, or `(..., F, P, 3)` if `faces_vertices` is None
- `uv` (ndarray): `(..., N, 2)` if `faces` is provided, or `(..., F, P, 2)` if `faces_uv` is None
- `faces_vertices` (ndarray, optional): `(F, P)` face vertex indices
- `faces_uv` (ndarray, optional): `(F, P)` face UV indices
## Returns
- `tangents` (ndarray): `(..., F, 3, 2)` face corner tangents (and bitangents),
where the last dimension represents the tangent and bitangent vectors.
"""
if faces_vertices is not None:
vertices = vertices[..., faces_vertices, :] # (..., F, P, 3)
if faces_uv is not None:
uv = uv[..., faces_uv, :] # (..., F, P, 2)
if vertices.shape[-2] == 3:
tangents = np.stack([vertices[..., 1, :] - vertices[..., 0, :], vertices[..., 2, :] - vertices[..., 0, :]], axis=-1) \
@ np.linalg.inv(np.stack([uv[..., 1, :] - uv[..., 0, :], uv[..., 2, :] - uv[..., 0, :]], axis=-1))
else:
tangents = compute_face_corner_tangents(vertices, uv, normalize=False)
tangents = np.mean(tangents, axis=-2)
if normalize:
tangents /= np.linalg.norm(tangents, axis=-1, keepdims=True) + np.finfo(vertices.dtype).eps
return tangents
def compute_vertex_normals(
vertices: ndarray,
faces: ndarray,
weighted: Literal['uniform', 'area', 'angle'] = 'uniform'
) -> ndarray:
"""
Compute vertex normals of a triangular mesh by averaging neighboring face normals
## Parameters
vertices (ndarray): [..., N, 3] 3-dimensional vertices
faces (ndarray): [T, P] face vertex indices, where P is the number of vertices per face
## Returns
normals (ndarray): [..., N, 3] vertex normals (already normalized to unit vectors)
"""
face_corner_normals = compute_face_corner_normals(vertices, faces, normalize=False)
if weighted == 'uniform':
face_corner_normals /= np.linalg.norm(face_corner_normals, axis=-1, keepdims=True) + np.finfo(vertices.dtype).eps
elif weighted == 'area':
pass
elif weighted == 'angle':
face_corner_angle = compute_face_corner_angles(vertices, faces)
face_corner_normals *= face_corner_angle[..., None]
vertex_normals = np.zeros_like(vertices, dtype=vertices.dtype)
np.add.at(
vertex_normals,
(..., faces[..., None], np.arange(3)),
face_corner_normals
)
vertex_normals /= np.linalg.norm(vertex_normals, axis=-1, keepdims=True) + np.finfo(vertices.dtype).eps
return vertex_normals
def remove_corrupted_faces(faces: ndarray) -> ndarray:
"""
Remove corrupted faces (faces with duplicated vertices)
## Parameters
faces (ndarray): [T, 3] triangular face indices
## Returns
ndarray: [T_, 3] triangular face indices
"""
corrupted = (faces[:, 0] == faces[:, 1]) | (faces[:, 1] == faces[:, 2]) | (faces[:, 2] == faces[:, 0])
return faces[~corrupted]
def merge_duplicate_vertices(
vertices: ndarray,
faces: ndarray,
tol: float = 1e-6
) -> Tuple[ndarray, ndarray]:
"""
Merge duplicate vertices of a triangular mesh.
Duplicate vertices are merged by selecte one of them, and the face indices are updated accordingly.
## Parameters
vertices (ndarray): [N, 3] 3-dimensional vertices
faces (ndarray): [T, 3] triangular face indices
tol (float, optional): tolerance for merging. Defaults to 1e-6.
## Returns
vertices (ndarray): [N_, 3] 3-dimensional vertices
faces (ndarray): [T, 3] triangular face indices
"""
vertices_round = np.round(vertices / tol)
_, uni_i, uni_inv = np.unique(vertices_round, return_index=True, return_inverse=True, axis=0)
vertices = vertices[uni_i]
faces = uni_inv[faces]
return vertices, faces
def remove_unused_vertices(
faces: ndarray,
*vertice_attrs,
return_indices: bool = False
) -> Tuple[ndarray, ...]:
"""
Remove unreferenced vertices of a mesh.
Unreferenced vertices are removed, and the face indices are updated accordingly.
## Parameters
faces (ndarray): [T, P] face indices
*vertice_attrs: vertex attributes
## Returns
faces (ndarray): [T, P] face indices
*vertice_attrs: vertex attributes
indices (ndarray, optional): [N] indices of vertices that are kept. Defaults to None.
"""
P = faces.shape[-1]
fewer_indices, inv_map = np.unique(faces, return_inverse=True)
faces = inv_map.astype(np.int32).reshape(-1, P)
ret = [faces]
for attr in vertice_attrs:
ret.append(attr[fewer_indices])
if return_indices:
ret.append(fewer_indices)
return tuple(ret)
def subdivide_mesh(
vertices: ndarray,
faces: ndarray,
level: int = 1
) -> Tuple[ndarray, ndarray]:
"""
Subdivide a triangular mesh by splitting each triangle into 4 smaller triangles.
NOTE: All original vertices are kept, and new vertices are appended to the end of the vertex list.
## Parameters
vertices (ndarray): [N, 3] 3-dimensional vertices
faces (ndarray): [T, 3] triangular face indices
level (int, optional): level of subdivisions. Defaults to 1.
## Returns
vertices (ndarray): [N_, 3] subdivided 3-dimensional vertices
faces (ndarray): [(4 ** level) * T, 3] subdivided triangular face indices
"""
for _ in range(level):
edges = np.stack([faces[:, [0, 1]], faces[:, [1, 2]], faces[:, [2, 0]]], axis=0)
edges = np.sort(edges, axis=2)
uni_edges, uni_inv = np.unique(edges.reshape(-1, 2), return_inverse=True, axis=0)
uni_inv = uni_inv.reshape(3, -1)
midpoints = (vertices[uni_edges[:, 0]] + vertices[uni_edges[:, 1]]) / 2
n_vertices = vertices.shape[0]
vertices = np.concatenate([vertices, midpoints], axis=0)
faces = np.concatenate([
np.stack([faces[:, 0], n_vertices + uni_inv[0], n_vertices + uni_inv[2]], axis=1),
np.stack([faces[:, 1], n_vertices + uni_inv[1], n_vertices + uni_inv[0]], axis=1),
np.stack([faces[:, 2], n_vertices + uni_inv[2], n_vertices + uni_inv[1]], axis=1),
np.stack([n_vertices + uni_inv[0], n_vertices + uni_inv[1], n_vertices + uni_inv[2]], axis=1),
], axis=0)
return vertices, faces
@overload
def flatten_mesh_indices(faces1: ndarray, attr1: ndarray, *more_faces_attrs_pairs: ndarray) -> Tuple[ndarray, ...]:
"""
Rearrange the indices of a mesh to a flattened version. Vertices will be no longer shared.
## Parameters:
- `faces1`: [T, P] face indices of the first attribute
- `attr1`: [N1, ...] attributes of the first mesh
Optionally, more pairs of faces and attributes can be provided:
- `faces2`: ...
- `attr2`: ...
- ...
## Returns
- `faces`: [T, P] flattened face indices, contigous from 0 to T * P - 1
- `attr1`: [T * P, ...] attributes of the first mesh, where every P values correspond to a face
- `attr2`: ...
- ...
"""
def flatten_mesh_indices(*args: ndarray) -> Tuple[ndarray, ...]:
assert len(args) % 2 == 0, "The number of arguments must be even."
T, P = args[0].shape
assert all(arg.shape[0] == T and arg.shape[1] == P for arg in args[::2]), "The faces must have the same shape."
attr_flat = []
for faces_, attr_ in zip(args[::2], args[1::2]):
attr_flat_ = attr_[faces_].reshape(-1, *attr_.shape[1:])
attr_flat.append(attr_flat_)
faces_flat = np.arange(T * P, dtype=np.int32).reshape(T, P)
return faces_flat, *attr_flat
def create_square_mesh(tri: bool = False) -> Tuple[ndarray, ndarray]:
"""
Create a square mesh of area 1 centered at origin in the xy-plane.
## Returns
vertices (ndarray): shape (4, 3)
faces (ndarray): shape (1, 4)
"""
vertices = np.array([
[0.5, 0.5, 0], [-0.5, 0.5, 0], [-0.5, -0.5, 0], [0.5, -0.5, 0] # v0-v1-v2-v3
], dtype=np.float32)
if tri:
faces = np.array([[0, 1, 2], [0, 2, 3]], dtype=np.int32)
else:
faces = np.array([[0, 1, 2, 3]], dtype=np.int32)
return vertices, faces
def create_cube_mesh(tri: bool = False) -> Tuple[ndarray, ndarray]:
"""
Create a cube mesh of size 1 centered at origin.
### Parameters
tri (bool, optional): return triangulated mesh. Defaults to False, which returns quad mesh.
### Returns
vertices (ndarray): shape (8, 3)
faces (ndarray): shape (12, 3)
"""
vertices = np.array([
[0.5, 0.5, 0.5], [-0.5, 0.5, 0.5], [-0.5, -0.5, 0.5], [0.5, -0.5, 0.5], # v0-v1-v2-v3
[0.5, 0.5, -0.5], [-0.5, 0.5, -0.5], [-0.5, -0.5, -0.5], [0.5, -0.5, -0.5] # v4-v5-v6-v7
], dtype=np.float32).reshape((-1, 3))
faces = np.array([
[0, 1, 2, 3], # (front)
[5, 4, 7, 6], # (back)
[4, 5, 1, 0], # (top)
[2, 6, 7, 3], # (bottom)
[1, 5, 6, 2], # (left)
[4, 0, 3, 7], # (right)
], dtype=np.int32)
if tri:
faces = triangulate_mesh(faces, vertices=vertices)
return vertices, faces
def create_camera_frustum_mesh(extrinsics: ndarray, intrinsics: ndarray, depth: float = 1.0) -> Tuple[ndarray, ndarray, ndarray]:
"""
Create a triangle mesh of camera frustum.
"""
assert extrinsics.shape == (4, 4) and intrinsics.shape == (3, 3)
vertices = unproject_cv(
np.array([[0, 0], [0, 0], [0, 1], [1, 1], [1, 0]], dtype=np.float32),
np.array([0] + [depth] * 4, dtype=np.float32),
intrinsics,
extrinsics
).astype(np.float32)
edges = np.array([
[0, 1], [0, 2], [0, 3], [0, 4],
[1, 2], [2, 3], [3, 4], [4, 1]
], dtype=np.int32)
faces = np.array([
[0, 1, 2],
[0, 2, 3],
[0, 3, 4],
[0, 4, 1],
[1, 2, 3],
[1, 3, 4]
], dtype=np.int32)
return vertices, edges, faces
def create_icosahedron_mesh():
"""
Create an icosahedron mesh of centered at origin.
"""
A = (1 + 5 ** 0.5) / 2
vertices = np.array([
[0, 1, A], [0, -1, A], [0, 1, -A], [0, -1, -A],
[1, A, 0], [-1, A, 0], [1, -A, 0], [-1, -A, 0],
[A, 0, 1], [A, 0, -1], [-A, 0, 1], [-A, 0, -1]
], dtype=np.float32)
faces = np.array([
[0, 1, 8], [0, 8, 4], [0, 4, 5], [0, 5, 10], [0, 10, 1],
[3, 2, 9], [3, 9, 6], [3, 6, 7], [3, 7, 11], [3, 11, 2],
[1, 6, 8], [8, 9, 4], [4, 2, 5], [5, 11, 10], [10, 7, 1],
[2, 4, 9], [9, 8, 6], [6, 1, 7], [7, 10, 11], [11, 5, 2]
], dtype=np.int32)
return vertices, faces
def merge_meshes(meshes: List[Tuple[ndarray, ...]]) -> Tuple[ndarray, ...]:
"""
Merge multiple meshes into one mesh. Vertices will be no longer shared.
## Parameters
- `meshes`: a list of tuple (faces, vertices_attr1, vertices_attr2, ....)
## Returns
- `faces`: [sum(T_i), P] merged face indices, contigous from 0 to sum(T_i) * P - 1
- `*vertice_attrs`: [sum(T_i) * P, ...] merged vertex attributes, where every P values correspond to a face
"""
faces_merged = []
attrs_merged = [[] for _ in meshes[0][1:]]
vertex_offset = 0
for f, *attrs in meshes:
faces_merged.append(f + vertex_offset)
vertex_offset += len(attrs[0])
for attr_merged, attr in zip(attrs_merged, attrs):
attr_merged.append(attr)
faces_merged = np.concatenate(faces_merged, axis=0)
attrs_merged = [np.concatenate(attr_list, axis=0) for attr_list in attrs_merged]
return (faces_merged, *attrs_merged)
def mesh_edges(
faces: Union[ndarray, Tuple[ndarray, ndarray], 'csr_array'],
return_face2edge: bool = False,
return_edge2face: bool = False,
return_counts: bool = False
) -> Tuple[ndarray, Union[ndarray, 'csr_array'], 'csr_array', 'ndarray']:
"""Get undirected edges of a mesh. Optionally return additional mappings.
## Parameters
- `faces` (ndarray): polygon faces, which can be in 3 formats:
- Regular mesh in regular array: `(F, P)`, where each face has `P` vertices.
- Irregular mesh in segmented array: tuple of `(vertex_indices, offsets)`, where vertex_indices[offsets[i]:offsets[i+1]] are the vertex indices of face i.
- Irregular mesh in CSR array: `(F, V)` binary CSR array of indices, each row corresponds to the vertices of a face.
(Note that segmented array is almost equivalent to csr array: `vertex_indices` ~ `faces.indices`, `offsets` ~ `faces.indptr`.)
- `return_face2edge` (bool): whether to return the face to edge mapping
- `return_edge2face` (bool): whether to return the edge to face mapping
- `return_counts` (bool): whether to return the counts of edges
## Returns
- `edges` (ndarray): `(E, 2)` unique edges' vertex indices
If `return_face2edge`, `return_edge2face`, `return_opposite_edge`, or `return_counts` is True, the corresponding outputs will be appended in order:
- `face2edge` (ndarray | csr_array): mapping from faces to the indices of edges
- `(F, P)` if input `faces` is a dense array
- `(F, E)` if input `faces` is segmented array
- `edge2face` (csr_array): `(E, F)` binary sparse CSR matrix of edge to face.
- `counts` (ndarray): `(E,)` counts of each edge
"""
from scipy.sparse import csr_array
if isinstance(faces, (tuple, csr_array)):
if isinstance(faces, csr_array):
vertex_indices, offsets = faces.indices, faces.indptr
else:
vertex_indices, offsets = faces
edges = np.stack([vertex_indices, segment_roll(vertex_indices, offsets, -1)], axis=-1) # (nzz, 2)
else:
edges = np.stack([faces, np.roll(faces, -1, axis=-1)], axis=-1).reshape(-1, 2) # (F * P, 2)
a, b = edges[:, 0], edges[:, 1]
edges = np.stack([np.minimum(a, b), np.maximum(a, b)], axis=-1)
unique = np.unique(edges, return_inverse=return_face2edge or return_edge2face, return_counts=return_counts or return_edge2face, axis=0)
edges: ndarray = unique[0] if isinstance(unique, tuple) else unique
if return_face2edge or return_edge2face:
inv_map: ndarray = unique[1]
if return_counts or return_edge2face:
counts: ndarray = unique[-1]
ret = (edges,)
if return_face2edge:
if isinstance(faces, tuple):
face2edge = csr_array((np.ones_like(inv_map, dtype=bool), inv_map, offsets), shape=(len(offsets) - 1, edges.shape[0]))
else:
face2edge = inv_map.reshape(faces.shape)
ret += (face2edge,)
if return_edge2face:
if isinstance(faces, (tuple, csr_array)):
lengths = np.diff(offsets)
edge2face = csr_array((
np.ones_like(inv_map, dtype=bool),
np.repeat(np.arange(len(lengths)), lengths)[np.argsort(inv_map)],
np.concatenate([np.array([0], dtype=counts.dtype), np.cumsum(counts)]),
), shape=(edges.shape[0], len(lengths)))
ret += (edge2face,)
else:
edge2face = csr_array((
np.ones_like(inv_map, dtype=bool),
np.argsort(inv_map) // faces.shape[1],
np.concatenate([np.array([0], dtype=counts.dtype), np.cumsum(counts)]),
), shape=(edges.shape[0], faces.shape[0]))
ret += (edge2face,)
if return_counts:
ret += (counts,)
return ret[0] if len(ret) == 1 else ret
def mesh_half_edges(
faces: Union[ndarray, Tuple[ndarray, ndarray], 'csr_array'],
return_face2edge: bool = False,
return_edge2face: bool = False,
return_twin: bool = False,
return_next: bool = False,
return_prev: bool = False,
return_counts: bool = False
) -> Tuple[ndarray, Union[ndarray, 'csr_array'], 'csr_array', ndarray, ndarray, ndarray, ndarray]:
"""Get half edges of a mesh. Optionally return additional mappings.
## Parameters
- `faces` (ndarray): polygon faces, which can be in 3 formats:
- `faces` (ndarray): polygon faces, which can be in 3 formats:
- Regular mesh in regular array: `(F, P)`, where each face has `P` vertices.
- Irregular mesh in segmented array: tuple of `(vertex_indices, offsets)`, where vertex_indices[offsets[i]:offsets[i+1]] are the vertex indices of face i.
- Irregular mesh in CSR array: `(F, V)` binary CSR array of indices, each row corresponds to the vertices of a face.
(Note that segmented array is almost equivalent to csr array: `vertex_indices` ~ `faces.indices`, `offsets` ~ `faces.indptr`.)
- `return_face2edge` (bool): whether to return the face to edge mapping
- `return_edge2face` (bool): whether to return the edge to face mapping
- `return_twin` (bool): whether to return the mapping from one edge to its opposite/twin edge
- `return_next` (bool): whether to return the mapping from one edge to its next edge in the face loop
- `return_prev` (bool): whether to return the mapping from one edge to its previous edge in the face loop
- `return_counts` (bool): whether to return the counts of edges
## Returns
- `edges` (ndarray): `(E, 2)` unique edges' vertex indices
If `return_face2edge`, `return_edge2face`, `return_opposite_edge`, or `return_counts` is True, the corresponding outputs will be appended in order:
- `face2edge` (ndarray | csr_array): mapping from faces to the indices of edges
- `(F, P)` if input `faces` is a dense array
- `(F, E)` if input `faces` is a sparse csr array
- `edge2face` (csr_array): `(E, F)` binary sparse CSR matrix of edge to face.
- `twin` (ndarray): `(E,)` mapping from edges to indices of opposite edges. -1 if not found.
- `next` (ndarray): `(E,)` mapping from edges to indices of next edges in the face loop.
- `prev` (ndarray): `(E,)` mapping from edges to indices of previous edges in the face loop.
- `counts` (ndarray): `(E,)` counts of each half edge
NOTE: If the mesh is not manifold, `twin`, `next`, and `prev` can point to arbitrary one of the candidates.
"""
from scipy.sparse import csr_array
if isinstance(faces, (tuple, csr_array)):
if isinstance(faces, csr_array):
vertex_indices, offsets = faces.indices, faces.indptr
else:
vertex_indices, offsets = faces
edges = np.stack([vertex_indices, segment_roll(vertex_indices, offsets, -1)], axis=-1) # (nzz, 2)
else:
edges = np.stack([faces, np.roll(faces, -1, axis=-1)], axis=-1).reshape(-1, 2) # (F * P, 2)
requires_inv_map = return_face2edge or return_edge2face or return_next or return_prev
requires_counts = return_counts or return_edge2face
unique = np.unique(edges, return_inverse=requires_inv_map, return_counts=requires_counts, axis=0)
edges: ndarray = unique[0] if isinstance(unique, tuple) else unique
if requires_inv_map:
inv_map: ndarray = unique[1]
if requires_counts:
counts: ndarray = unique[-1]
ret = (edges,)
if return_face2edge or return_next or return_prev:
if isinstance(faces, (tuple, csr_array)):
face2edge = csr_array(
(np.ones_like(inv_map, dtype=bool), inv_map, offsets),
shape=(len(offsets) - 1, edges.shape[0])
)
else:
face2edge = inv_map.reshape(faces.shape)
if return_face2edge:
ret += (face2edge,)
if return_edge2face:
if isinstance(faces, (tuple, csr_array)):
lengths = np.diff(offsets)
edge2face = csr_array((
np.ones_like(inv_map, dtype=bool),
np.repeat(np.arange(len(lengths)), lengths)[np.argsort(inv_map)],
np.concatenate([np.array([0], dtype=counts.dtype), np.cumsum(counts, axis=0)]),
), shape=(edges.shape[0], len(lengths)))
ret += (edge2face,)
else:
edge2face = csr_array((
np.ones_like(inv_map, dtype=bool),
np.argsort(inv_map) // faces.shape[1],
np.concatenate([np.array([0], dtype=counts.dtype), np.cumsum(counts, axis=0)]),
), shape=(edges.shape[0], faces.shape[0]))
ret += (edge2face,)
if return_twin:
twin_edge = lookup(edges, np.flip(edges, -1))
ret += (twin_edge,)
if return_next or return_prev:
if isinstance(face2edge, csr_array):
face2edge_indices = face2edge.indices
face2edge_indices_next = segment_roll(face2edge.indices, face2edge.indptr, -1)
else:
face2edge_indices = face2edge.reshape(-1)
face2edge_indices_next = np.roll(face2edge, -1, axis=-1).reshape(-1)
if return_next:
next_edge = np.full(edges.shape[0], -1, dtype=np.int32)
np.put(next_edge, face2edge_indices, face2edge_indices_next)
ret += (next_edge,)
if return_prev:
prev_edge = np.full(edges.shape[0], -1, dtype=np.int32)
np.put(prev_edge, face2edge_indices_next, face2edge_indices)
ret += (prev_edge,)
if return_counts:
ret += (counts,)
return ret[0] if len(ret) == 1 else ret
def mesh_connected_components(
faces: Optional[Union[ndarray, Tuple[ndarray, ndarray], 'csr_array']] = None,
num_vertices: Optional[int] = None
) -> Union[ndarray, Tuple[ndarray, ndarray]]:
"""
Compute connected faces of a mesh.
## Parameters
- `faces` (ndarray): polygon faces, which can be in 3 formats:
- Regular mesh in regular array: `(F, P)`, where each face has `P` vertices.
- Irregular mesh in segmented array: tuple of `(vertex_indices, offsets)`, where vertex_indices[offsets[i]:offsets[i+1]] are the vertex indices of face i.
- Irregular mesh in CSR array: `(F, V)` binary CSR array of indices, each row corresponds to the vertices of a face.
(Note that segmented array is almost equivalent to csr array: `vertex_indices` ~ `faces.indices`, `offsets` ~ `faces.indptr`.)
- `num_vertices` (int, optional): total number of vertices. If not given, only presented vertices in `faces` are considered.
## Returns
If `num_vertices` is given, return:
- `labels` (ndarray): (N,) component labels of each vertex
If `num_vertices` is None, return:
- `vertices_ids` (ndarray): (N,) vertex indices that are in the edges
- `labels` (ndarray): (N,) int32 component labels corresponding to `vertices_ids`
"""
edges = mesh_edges(faces, directed=False)
return graph_connected_components(edges, num_vertices)
def graph_connected_components(
edges: ndarray,
num_vertices: Optional[int] = None
) -> Union[ndarray, Tuple[ndarray, ndarray]]:
"""
Compute connected components of an undirected graph.
Using scipy.sparse.csgraph.connected_components as backend.
## Parameters
- `edges` (ndarray): (E, 2) edge indices
## Returns
If `num_vertices` is given, return:
- `labels` (ndarray): (N,) component labels of each vertex
If `num_vertices` is None, return:
- `vertices_ids` (ndarray): (N,) vertex indices that are in the edges
- `labels` (ndarray): (N,) int32 component labels corresponding to `vertices_ids`
"""
from scipy.sparse.csgraph import connected_components
if num_vertices is None:
# Re-index edges
vertices_ids, edges = np.unique(edges.reshape(-1), return_inverse=True)
edges = edges.reshape(-1, 2)
num_connected_components, labels = connected_components(
sp.coo_array(
(np.ones(edges.shape[0]), edges.T),
shape=(num_vertices or len(vertices_ids), num_vertices or len(vertices_ids))
),
directed=False
)
if num_vertices is None:
return vertices_ids, labels
else:
return labels
def mesh_adjacency_graph(
adjacency: Literal[
'vertex2edge',
'vertex2face',
'edge2vertex',
'edge2face',
'face2edge',
'face2vertex',
'vertex2edge2vertex',
'vertex2face2vertex',
'edge2vertex2edge',
'edge2face2edge',
'face2edge2face',
'face2vertex2face',
],
faces: Optional[Union[ndarray, Tuple[ndarray, ndarray], 'csr_array']] = None,
edges: Optional[ndarray] = None,
num_vertices: Optional[int] = None,
self_loop: bool = False,
) -> 'csr_array':
"""
Get adjacency graph of a mesh.
## Parameters
- `adjacency` (str): type of adjacency graph. Options:
- `'vertex2edge'`: vertex to adjacent edges. Returns (V, E) csr
- `'vertex2face'`: vertex to adjacent faces. Returns (V, F) csr
- `'edge2vertex'`: edge to adjacent vertices. Returns (E, V) csr
- `'edge2face'`: edge to its adjacent faces. Returns (E, F) csr
- `'face2edge'`: face to its adjacent edges. Returns (F, E) csr
- `'face2vertex'`: face to its adjacent vertices. Returns (F, V) csr
- `'vertex2edge2vertex'`: vertex to adjacent vertices if they share an edge. Returns (V, V) csr
- `'vertex2face2vertex'`: vertex to adjacent vertices if they share a face. Returns (V, V) csr
- `'edge2vertex2edge'`: edge to adjacent edges if they share a vertex. Returns (E, E) csr
- `'edge2face2edge'`: edge to adjacent edges if they share a face. Returns (E, E) csr
- `'face2edge2face'`: face to adjacent faces if they share an edge. Returns (F, F) csr
- `'face2vertex2face'`: face to adjacent faces if they share a vertex. Returns (F, F) csr
- `faces` (ndarray): polygon faces
- `(F, P)` dense array of indices, where each face has `P` vertices.
- `(F, V)` binary sparse csr array of indices, each row corresponds to the vertices of a face.
- `edges` (ndarray, optional): (E, 2) edge indices. NOTE: assumed to be undirected edges.
- `num_vertices` (int, optional): total number of vertices.
- `self_loop` (bool): whether to include self-loops in the adjacency graph. Defaults to False.
## Returns
- `graph` (csr_array): adjacency graph in csr format
"""
from scipy.sparse import csr_array
if isinstance(faces, csr_array):
if num_vertices is None:
num_vertices = faces.shape[1]
else:
assert num_vertices == faces.shape[1], f'num_vertices ({num_vertices}) does not match csr array faces.shape[1] ({faces.shape[1]})'
if adjacency == 'vertex2edge':
assert edges is not None and num_vertices is not None
return mesh_adjacency_graph('edge2vertex', edges=edges, num_vertices=num_vertices).transpose().tocsr()
elif adjacency == 'vertex2face':
return mesh_adjacency_graph('face2vertex', faces=faces, num_vertices=num_vertices).transpose().tocsr()
elif adjacency == 'edge2vertex':
assert edges is not None and num_vertices is not None
return csr_matrix_from_dense_indices(edges, num_vertices)
elif adjacency == 'edge2face':
assert edges is not None and faces is not None
return mesh_adjacency_graph('face2edge', faces=faces, edges=edges).transpose().tocsr()
elif adjacency == 'face2edge':
assert edges is not None and faces is not None
if isinstance(faces, (tuple, csr_array)):
vertex_indices, offsets = faces if isinstance(faces, tuple) else (faces.indices, faces.indptr)
face_edges = np.stack([vertex_indices, segment_roll(vertex_indices, offsets, -1)], axis=-1) # (nzz, 2)
else:
face_edges = np.stack([faces, np.roll(faces, -1, axis=-1)], axis=-1).reshape(-1, 2) # (F * P, 2)
a, b = face_edges[:, 0], face_edges[:, 1]
face_edges = np.stack([np.minimum(a, b), np.maximum(a, b)], axis=-1)
indices = lookup(edges, face_edges)
return csr_array((
np.ones_like(indices, dtype=bool),
indices,
offsets if isinstance(faces, (tuple, csr_array)) else np.arange(0, faces.size + 1, faces.shape[1])
), shape=(faces.shape[0], edges.shape[0])).tocsr()
elif adjacency == 'face2vertex':
assert faces is not None
if isinstance(faces, csr_array):
return faces
elif isinstance(faces, tuple):
assert num_vertices is not None
return csr_array(
(np.ones_like(faces[0], dtype=bool), vertex_indices, offsets),
shape=(len(offsets) - 1, num_vertices)
)
else:
assert num_vertices is not None
return csr_matrix_from_dense_indices(faces, num_vertices)
elif adjacency == 'vertex2edge2vertex':
e2v = mesh_adjacency_graph('edge2vertex', edges=edges, num_vertices=num_vertices)
v2e2v = (e2v.transpose() @ e2v).tocsr()
if not self_loop:
v2e2v.setdiag(0)
v2e2v.eliminate_zeros()
return v2e2v
elif adjacency == 'vertex2face2vertex':
f2v = mesh_adjacency_graph('face2vertex', faces=faces, num_vertices=num_vertices)
v2f2v = (f2v.transpose() @ f2v).tocsr()
if not self_loop:
v2f2v.setdiag(0)
v2f2v.eliminate_zeros()
return v2f2v
elif adjacency == 'edge2vertex2edge':
# num_vertices is optional here
if num_vertices is None:
vertices_id, edges = np.unique(edges.reshape(-1), return_inverse=True)
edges = edges.reshape(-1, 2)
num_vertices = len(vertices_id)
e2v = mesh_adjacency_graph('edge2vertex', edges=edges, num_vertices=num_vertices)
e2v2e = (e2v @ e2v.transpose()).tocsr()
if not self_loop:
e2v2e.setdiag(0)
e2v2e.eliminate_zeros()
return e2v2e
elif adjacency == 'edge2face2edge':
e2f = mesh_adjacency_graph('edge2face', faces=faces, edges=edges)
e2f2e = (e2f @ e2f.transpose()).tocsr()
if not self_loop:
e2f2e.setdiag(0)
e2f2e.eliminate_zeros()
return e2f2e
elif adjacency == 'face2edge2face':
if edges is None:
_, f2e = mesh_edges(faces, directed=False, return_face2edge=True)
else:
f2e = mesh_adjacency_graph('face2edge', faces=faces, edges=edges)
f2e2f = (f2e @ f2e.transpose()).tocsr()
if not self_loop:
f2e2f.setdiag(0)
f2e2f.eliminate_zeros()
return f2e2f
elif adjacency == 'face2vertex2face':
if isinstance(faces, csr_array):
f2v = faces
elif isinstance(faces, tuple):
vertex_indices, offsets = faces
unique_vertices, vertex_indices = np.unique(vertex_indices, return_inverse=True)
f2v = csr_array(
(np.ones_like(vertex_indices, dtype=bool), vertex_indices, offsets),
shape=(len(offsets) - 1, len(unique_vertices))
)
else:
unique_vertices, inv = np.unique(faces.reshape(-1), return_inverse=True)
faces = inv.reshape(-1, faces.shape[1])
f2v = csr_matrix_from_dense_indices(faces, len(unique_vertices))
f2v2f = (f2v @ f2v.transpose()).tocsr()
if not self_loop:
f2v2f.setdiag(0)
f2v2f.eliminate_zeros()
return f2v2f
else:
raise ValueError(f'Unknown adjacency type: {adjacency}')
# def calc_quad_candidates(
# edges: ndarray,
# face2edge: ndarray,
# edge2face: ndarray,
# ):
# """
# Calculate the candidate quad faces.
# ## Parameters
# edges (ndarray): [E, 2] edge indices
# face2edge (ndarray): [T, 3] face to edge relation
# edge2face (ndarray): [E, 2] edge to face relation
# ## Returns
# quads (ndarray): [Q, 4] quad candidate indices
# quad2edge (ndarray): [Q, 4] edge to quad candidate relation
# quad2adj (ndarray): [Q, 8] adjacent quad candidates of each quad candidate
# quads_valid (ndarray): [E] whether the quad corresponding to the edge is valid
# """
# E = edges.shape[0]
# T = face2edge.shape[0]
# quads_valid = edge2face[:, 1] != -1
# Q = quads_valid.sum()
# quad2face = edge2face[quads_valid] # [Q, 2]
# quad2edge = face2edge[quad2face] # [Q, 2, 3]
# flag = quad2edge == np.arange(E)[quads_valid][:, None, None] # [Q, 2, 3]
# flag = flag.argmax(axis=-1) # [Q, 2]
# quad2edge = np.stack([
# quad2edge[np.arange(Q)[:, None], np.arange(2)[None, :], (flag + 1) % 3],
# quad2edge[np.arange(Q)[:, None], np.arange(2)[None, :], (flag + 2) % 3],
# ], axis=-1).reshape(Q, 4) # [Q, 4]
# quads = np.concatenate([
# np.where(
# (edges[quad2edge[:, 0:1], 1:] == edges[quad2edge[:, 1:2], :]).any(axis=-1),
# edges[quad2edge[:, 0:1], [[0, 1]]],
# edges[quad2edge[:, 0:1], [[1, 0]]],
# ),
# np.where(
# (edges[quad2edge[:, 2:3], 1:] == edges[quad2edge[:, 3:4], :]).any(axis=-1),
# edges[quad2edge[:, 2:3], [[0, 1]]],
# edges[quad2edge[:, 2:3], [[1, 0]]],
# ),
# ], axis=1) # [Q, 4]
# quad2adj = edge2face[quad2edge] # [Q, 4, 2]
# quad2adj = quad2adj[quad2adj != quad2face[:, [0,0,1,1], None]].reshape(Q, 4) # [Q, 4]
# quad2adj_valid = quad2adj != -1
# quad2adj = face2edge[quad2adj] # [Q, 4, 3]
# quad2adj[~quad2adj_valid, 0] = quad2edge[~quad2adj_valid]
# quad2adj[~quad2adj_valid, 1:] = -1
# quad2adj = quad2adj[quad2adj != quad2edge[..., None]].reshape(Q, 8) # [Q, 8]
# edge_valid = -np.ones(E, dtype=np.int32)
# edge_valid[quads_valid] = np.arange(Q)
# quad2adj_valid = quad2adj != -1
# quad2adj[quad2adj_valid] = edge_valid[quad2adj[quad2adj_valid]] # [Q, 8]
# return quads, quad2edge, quad2adj, quads_valid
# def calc_quad_distortion(
# vertices: ndarray,
# quads: ndarray,
# ):
# """
# Calculate the distortion of each candidate quad face.
# ## Parameters
# vertices (ndarray): [N, 3] 3-dimensional vertices
# quads (ndarray): [Q, 4] quad face indices
# ## Returns
# distortion (ndarray): [Q] distortion of each quad face
# """
# edge0 = vertices[quads[:, 1]] - vertices[quads[:, 0]] # [Q, 3]
# edge1 = vertices[quads[:, 2]] - vertices[quads[:, 1]] # [Q, 3]
# edge2 = vertices[quads[:, 3]] - vertices[quads[:, 2]] # [Q, 3]
# edge3 = vertices[quads[:, 0]] - vertices[quads[:, 3]] # [Q, 3]
# cross = vertices[quads[:, 0]] - vertices[quads[:, 2]] # [Q, 3]
# len0 = np.maximum(np.linalg.norm(edge0, axis=-1), 1e-10) # [Q]
# len1 = np.maximum(np.linalg.norm(edge1, axis=-1), 1e-10) # [Q]
# len2 = np.maximum(np.linalg.norm(edge2, axis=-1), 1e-10) # [Q]
# len3 = np.maximum(np.linalg.norm(edge3, axis=-1), 1e-10) # [Q]
# len_cross = np.maximum(np.linalg.norm(cross, axis=-1), 1e-10) # [Q]
# angle0 = np.arccos(np.clip(np.sum(-edge0 * edge1, axis=-1) / (len0 * len1), -1, 1)) # [Q]
# angle1 = np.arccos(np.clip(np.sum(-edge1 * cross, axis=-1) / (len1 * len_cross), -1, 1)) \
# + np.arccos(np.clip(np.sum(cross * edge2, axis=-1) / (len_cross * len2), -1, 1)) # [Q]
# angle2 = np.arccos(np.clip(np.sum(-edge2 * edge3, axis=-1) / (len2 * len3), -1, 1)) # [Q]
# angle3 = np.arccos(np.clip(np.sum(-edge3 * -cross, axis=-1) / (len3 * len_cross), -1, 1)) \
# + np.arccos(np.clip(np.sum(-cross * edge0, axis=-1) / (len_cross * len0), -1, 1)) # [Q]
# normal0 = np.cross(edge0, edge1) # [Q, 3]
# normal1 = np.cross(edge2, edge3) # [Q, 3]
# normal0 = normal0 / np.maximum(np.linalg.norm(normal0, axis=-1, keepdims=True), 1e-10) # [Q, 3]
# normal1 = normal1 / np.maximum(np.linalg.norm(normal1, axis=-1, keepdims=True), 1e-10) # [Q, 3]
# angle_normal = np.arccos(np.clip(np.sum(normal0 * normal1, axis=-1), -1, 1)) # [Q]
# D90 = np.pi / 2
# D180 = np.pi
# D360 = np.pi * 2
# ang_eng = (np.abs(angle0 - D90)**2 + np.abs(angle1 - D90)**2 + np.abs(angle2 - D90)**2 + np.abs(angle3 - D90)**2) / 4 # [Q]
# dist_eng = np.abs(angle0 - angle2)**2 / np.minimum(np.maximum(np.minimum(angle0, angle2), 1e-10), np.maximum(D180 - np.maximum(angle0, angle2), 1e-10)) \
# + np.abs(angle1 - angle3)**2 / np.minimum(np.maximum(np.minimum(angle1, angle3), 1e-10), np.maximum(D180 - np.maximum(angle1, angle3), 1e-10)) # [Q]
# plane_eng = np.where(angle_normal < D90/2, np.abs(angle_normal)**2, 1e10) # [Q]
# eng = ang_eng + 2 * dist_eng + 2 * plane_eng # [Q]
# return eng
# def calc_quad_direction(vertices: ndarray, quads: ndarray):
# """
# Calculate the direction of each candidate quad face.
# ## Parameters
# vertices (ndarray): [N, 3] 3-dimensional vertices
# quads (ndarray): [Q, 4] quad face indices
# ## Returns
# direction (ndarray): [Q, 4] direction of each quad face.
# Represented by the angle between the crossing and each edge.
# """
# mid0 = (vertices[quads[:, 0]] + vertices[quads[:, 1]]) / 2 # [Q, 3]
# mid1 = (vertices[quads[:, 1]] + vertices[quads[:, 2]]) / 2 # [Q, 3]
# mid2 = (vertices[quads[:, 2]] + vertices[quads[:, 3]]) / 2 # [Q, 3]
# mid3 = (vertices[quads[:, 3]] + vertices[quads[:, 0]]) / 2 # [Q, 3]
# cross0 = mid2 - mid0 # [Q, 3]
# cross1 = mid3 - mid1 # [Q, 3]
# cross0 = cross0 / np.maximum(np.linalg.norm(cross0, axis=-1, keepdims=True), 1e-10) # [Q, 3]
# cross1 = cross1 / np.maximum(np.linalg.norm(cross1, axis=-1, keepdims=True), 1e-10) # [Q, 3]
# edge0 = vertices[quads[:, 1]] - vertices[quads[:, 0]] # [Q, 3]
# edge1 = vertices[quads[:, 2]] - vertices[quads[:, 1]] # [Q, 3]
# edge2 = vertices[quads[:, 3]] - vertices[quads[:, 2]] # [Q, 3]
# edge3 = vertices[quads[:, 0]] - vertices[quads[:, 3]] # [Q, 3]
# edge0 = edge0 / np.maximum(np.linalg.norm(edge0, axis=-1, keepdims=True), 1e-10) # [Q, 3]
# edge1 = edge1 / np.maximum(np.linalg.norm(edge1, axis=-1, keepdims=True), 1e-10) # [Q, 3]
# edge2 = edge2 / np.maximum(np.linalg.norm(edge2, axis=-1, keepdims=True), 1e-10) # [Q, 3]
# edge3 = edge3 / np.maximum(np.linalg.norm(edge3, axis=-1, keepdims=True), 1e-10) # [Q, 3]
# direction = np.stack([
# np.arccos(np.clip(np.sum(cross0 * edge0, axis=-1), -1, 1)),
# np.arccos(np.clip(np.sum(cross1 * edge1, axis=-1), -1, 1)),
# np.arccos(np.clip(np.sum(-cross0 * edge2, axis=-1), -1, 1)),
# np.arccos(np.clip(np.sum(-cross1 * edge3, axis=-1), -1, 1)),
# ], axis=-1) # [Q, 4]
# return direction
# def calc_quad_smoothness(
# quad2edge: ndarray,
# quad2adj: ndarray,
# quads_direction: ndarray,
# ):
# """
# Calculate the smoothness of each candidate quad face connection.
# ## Parameters
# quad2adj (ndarray): [Q, 8] adjacent quad faces of each quad face
# quads_direction (ndarray): [Q, 4] direction of each quad face
# ## Returns
# smoothness (ndarray): [Q, 8] smoothness of each quad face connection
# """
# Q = quad2adj.shape[0]
# quad2adj_valid = quad2adj != -1
# connections = np.stack([
# np.arange(Q)[:, None].repeat(8, axis=1),
# quad2adj,
# ], axis=-1)[quad2adj_valid] # [C, 2]
# shared_edge_idx_0 = np.array([[0, 0, 1, 1, 2, 2, 3, 3]]).repeat(Q, axis=0)[quad2adj_valid] # [C]
# shared_edge_idx_1 = np.argmax(quad2edge[quad2adj][quad2adj_valid] == quad2edge[connections[:, 0], shared_edge_idx_0][:, None], axis=-1) # [C]
# valid_smoothness = np.abs(quads_direction[connections[:, 0], shared_edge_idx_0] - quads_direction[connections[:, 1], shared_edge_idx_1])**2 # [C]
# smoothness = np.zeros([Q, 8], dtype=np.float32)
# smoothness[quad2adj_valid] = valid_smoothness
# return smoothness
# def solve_quad(
# face2edge: ndarray,
# edge2face: ndarray,
# quad2adj: ndarray,
# quads_distortion: ndarray,
# quads_smoothness: ndarray,
# quads_valid: ndarray,
# ):
# """
# Solve the quad mesh from the candidate quad faces.
# ## Parameters
# face2edge (ndarray): [T, 3] face to edge relation
# edge2face (ndarray): [E, 2] edge to face relation
# quad2adj (ndarray): [Q, 8] adjacent quad faces of each quad face
# quads_distortion (ndarray): [Q] distortion of each quad face
# quads_smoothness (ndarray): [Q, 8] smoothness of each quad face connection
# quads_valid (ndarray): [E] whether the quad corresponding to the edge is valid
# ## Returns
# weights (ndarray): [Q] weight of each valid quad face
# """
# import scipy.optimize as opt
# T = face2edge.shape[0]
# E = edge2face.shape[0]
# Q = quads_distortion.shape[0]
# edge_valid = -np.ones(E, dtype=np.int32)
# edge_valid[quads_valid] = np.arange(Q)
# quads_connection = np.stack([
# np.arange(Q)[:, None].repeat(8, axis=1),
# quad2adj,
# ], axis=-1)[quad2adj != -1] # [C, 2]
# quads_connection = np.sort(quads_connection, axis=-1) # [C, 2]
# quads_connection, quads_connection_idx = np.unique(quads_connection, axis=0, return_index=True) # [C, 2], [C]
# quads_smoothness = quads_smoothness[quad2adj != -1] # [C]
# quads_smoothness = quads_smoothness[quads_connection_idx] # [C]
# C = quads_connection.shape[0]
# # Construct the linear programming problem
# # Variables:
# # quads_weight: [Q] weight of each quad face
# # tri_min_weight: [T] minimum weight of each triangle face
# # conn_min_weight: [C] minimum weight of each quad face connection
# # conn_max_weight: [C] maximum weight of each quad face connection
# # Objective:
# # mimi
# c = np.concatenate([
# quads_distortion - 3,
# quads_smoothness*4 - 2,
# quads_smoothness*4,
# ], axis=0) # [Q+C]
# A_ub_triplet = np.concatenate([
# np.stack([np.arange(T), edge_valid[face2edge[:, 0]], np.ones(T)], axis=1), # [T, 3]
# np.stack([np.arange(T), edge_valid[face2edge[:, 1]], np.ones(T)], axis=1), # [T, 3]
# np.stack([np.arange(T), edge_valid[face2edge[:, 2]], np.ones(T)], axis=1), # [T, 3]
# np.stack([np.arange(T, T+C), np.arange(Q, Q+C), np.ones(C)], axis=1), # [C, 3]
# np.stack([np.arange(T, T+C), quads_connection[:, 0], -np.ones(C)], axis=1), # [C, 3]
# np.stack([np.arange(T, T+C), quads_connection[:, 1], -np.ones(C)], axis=1), # [C, 3]
# np.stack([np.arange(T+C, T+2*C), np.arange(Q+C, Q+2*C), -np.ones(C)], axis=1), # [C, 3]
# np.stack([np.arange(T+C, T+2*C), quads_connection[:, 0], np.ones(C)], axis=1), # [C, 3]
# np.stack([np.arange(T+C, T+2*C), quads_connection[:, 1], np.ones(C)], axis=1), # [C, 3]
# ], axis=0) # [3T+6C, 3]
# A_ub_triplet = A_ub_triplet[A_ub_triplet[:, 1] != -1] # [3T', 3]
# A_ub = sp.coo_matrix((A_ub_triplet[:, 2], (A_ub_triplet[:, 0], A_ub_triplet[:, 1])), shape=[T+2*C, Q+2*C]) # [T,
# b_ub = np.concatenate([np.ones(T), -np.ones(C), np.ones(C)], axis=0) # [T+2C]
# bound = np.stack([
# np.concatenate([np.zeros(Q), -np.ones(C), np.zeros(C)], axis=0),
# np.concatenate([np.ones(Q), np.ones(C), np.ones(C)], axis=0),
# ], axis=1) # [Q+2C, 2]
# A_eq = None
# b_eq = None
# print('Solver statistics:')
# print(f' #T = {T}')
# print(f' #Q = {Q}')
# print(f' #C = {C}')
# # Solve the linear programming problem
# last_num_valid = 0
# for i in range(100):
# res_ = opt.linprog(c, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq, bounds=bound)
# if not res_.success:
# print(f' Iter {i} | Failed with {res_.message}')
# break
# res = res_
# weights = res.x[:Q]
# valid = (weights > 0.5)
# num_valid = valid.sum()
# print(f' Iter {i} | #Q_valid = {num_valid}')
# if num_valid == last_num_valid:
# break
# last_num_valid = num_valid
# A_eq_triplet = np.stack([
# np.arange(num_valid),
# np.arange(Q)[valid],
# np.ones(num_valid),
# ], axis=1) # [num_valid, 3]
# A_eq = sp.coo_matrix((A_eq_triplet[:, 2], (A_eq_triplet[:, 0], A_eq_triplet[:, 1])), shape=[num_valid, Q+2*C]) # [num_valid, Q+C]
# b_eq = np.where(weights[valid] > 0.5, 1, 0) # [num_valid]
# # Return the result
# quads_weight = res.x[:Q]
# conn_min_weight = res.x[Q:Q+C]
# conn_max_weight = res.x[Q+C:Q+2*C]
# return quads_weight, conn_min_weight, conn_max_weight
# def solve_quad_qp(
# face2edge: ndarray,
# edge2face: ndarray,
# quad2adj: ndarray,
# quads_distortion: ndarray,
# quads_smoothness: ndarray,
# quads_valid: ndarray,
# ):
# """
# Solve the quad mesh from the candidate quad faces.
# ## Parameters
# face2edge (ndarray): [T, 3] face to edge relation
# edge2face (ndarray): [E, 2] edge to face relation
# quad2adj (ndarray): [Q, 8] adjacent quad faces of each quad face
# quads_distortion (ndarray): [Q] distortion of each quad face
# quads_smoothness (ndarray): [Q, 8] smoothness of each quad face connection
# quads_valid (ndarray): [E] whether the quad corresponding to the edge is valid
# ## Returns
# weights (ndarray): [Q] weight of each valid quad face
# """
# import piqp
# T = face2edge.shape[0]
# E = edge2face.shape[0]
# Q = quads_distortion.shape[0]
# edge_valid = -np.ones(E, dtype=np.int32)
# edge_valid[quads_valid] = np.arange(Q)
# # Construct the quadratic programming problem
# C_smoothness_triplet = np.stack([
# np.arange(Q)[:, None].repeat(8, axis=1)[quad2adj != -1],
# quad2adj[quad2adj != -1],
# 5 * quads_smoothness[quad2adj != -1],
# ], axis=-1) # [C, 3]
# # C_smoothness_triplet = np.concatenate([
# # C_smoothness_triplet,
# # np.stack([np.arange(Q), np.arange(Q), 20*np.ones(Q)], axis=1),
# # ], axis=0) # [C+Q, 3]
# C_smoothness = sp.coo_matrix((C_smoothness_triplet[:, 2], (C_smoothness_triplet[:, 0], C_smoothness_triplet[:, 1])), shape=[Q, Q]) # [Q, Q]
# C_smoothness = C_smoothness.tocsc()
# C_dist = quads_distortion - 20 # [Q]
# A_eq = sp.coo_matrix((np.zeros(Q), (np.zeros(Q), np.arange(Q))), shape=[1, Q]) # [1, Q]\
# A_eq = A_eq.tocsc()
# b_eq = np.array([0])
# A_ub_triplet = np.concatenate([
# np.stack([np.arange(T), edge_valid[face2edge[:, 0]], np.ones(T)], axis=1), # [T, 3]
# np.stack([np.arange(T), edge_valid[face2edge[:, 1]], np.ones(T)], axis=1), # [T, 3]
# np.stack([np.arange(T), edge_valid[face2edge[:, 2]], np.ones(T)], axis=1), # [T, 3]
# ], axis=0) # [3T, 3]
# A_ub_triplet = A_ub_triplet[A_ub_triplet[:, 1] != -1] # [3T', 3]
# A_ub = sp.coo_matrix((A_ub_triplet[:, 2], (A_ub_triplet[:, 0], A_ub_triplet[:, 1])), shape=[T, Q]) # [T, Q]
# A_ub = A_ub.tocsc()
# b_ub = np.ones(T)
# lb = np.zeros(Q)
# ub = np.ones(Q)
# solver = piqp.SparseSolver()
# solver.settings.verbose = True
# solver.settings.compute_timings = True
# solver.setup(C_smoothness, C_dist, A_eq, b_eq, A_ub, b_ub, lb, ub)
# status = solver.solve()
# # x = cp.Variable(Q)
# # prob = cp.Problem(
# # cp.Minimize(cp.quad_form(x, C_smoothness) + C_dist.T @ x),
# # [
# # A_ub @ x <= b_ub,
# # x >= 0, x <= 1,
# # ]
# # )
# # # Solve the quadratic programming problem
# # prob.solve(solver=cp.PIQP, verbose=True)
# # Return the result
# weights = solver.result.x
# return weights
# def tri_to_quad(
# vertices: ndarray,
# faces: ndarray,
# ) -> Tuple[ndarray, ndarray]:
# """
# Convert a triangle mesh to a quad mesh.
# NOTE: The input mesh must be a manifold mesh.
# ## Parameters
# vertices (ndarray): [N, 3] 3-dimensional vertices
# faces (ndarray): [T, 3] triangular face indices
# ## Returns
# vertices (ndarray): [N_, 3] 3-dimensional vertices
# faces (ndarray): [Q, 4] quad face indices
# """
# raise NotImplementedError |