# modified from https://github.com/Profactor/continuous-remeshing
import torch
import torch.nn.functional as tfunc
import torch_scatter
from typing import Tuple
def prepend_dummies(
vertices:torch.Tensor, #V,D
faces:torch.Tensor, #F,3 long
)->Tuple[torch.Tensor,torch.Tensor]:
"""prepend dummy elements to vertices and faces to enable "masked" scatter operations"""
V,D = vertices.shape
vertices = torch.concat((torch.full((1,D),fill_value=torch.nan,device=vertices.device),vertices),dim=0)
faces = torch.concat((torch.zeros((1,3),dtype=torch.long,device=faces.device),faces+1),dim=0)
return vertices,faces
def remove_dummies(
vertices:torch.Tensor, #V,D - first vertex all nan and unreferenced
faces:torch.Tensor, #F,3 long - first face all zeros
)->Tuple[torch.Tensor,torch.Tensor]:
"""remove dummy elements added with prepend_dummies()"""
return vertices[1:],faces[1:]-1
def calc_edges(
faces: torch.Tensor, # F,3 long - first face may be dummy with all zeros
with_edge_to_face: bool = False
) -> Tuple[torch.Tensor, ...]:
"""
returns Tuple of
- edges E,2 long, 0 for unused, lower vertex index first
- face_to_edge F,3 long
- (optional) edge_to_face shape=E,[left,right],[face,side]
o-<-----e1 e0,e1...edge, e0<e1
| /A L,R....left and right face
| L / | both triangles ordered counter clockwise
| / R | normals pointing out of screen
V/ |
e0---->-o
"""
F = faces.shape[0]
# make full edges, lower vertex index first
face_edges = torch.stack((faces,faces.roll(-1,1)),dim=-1) #F*3,3,2
full_edges = face_edges.reshape(F*3,2)
sorted_edges,_ = full_edges.sort(dim=-1) #F*3,2
# make unique edges
edges,full_to_unique = torch.unique(input=sorted_edges,sorted=True,return_inverse=True,dim=0) #(E,2),(F*3)
E = edges.shape[0]
face_to_edge = full_to_unique.reshape(F,3) #F,3
if not with_edge_to_face:
return edges, face_to_edge
is_right = full_edges[:,0]!=sorted_edges[:,0] #F*3
edge_to_face = torch.zeros((E,2,2),dtype=torch.long,device=faces.device) #E,LR=2,S=2
scatter_src = torch.cartesian_prod(torch.arange(0,F,device=faces.device),torch.arange(0,3,device=faces.device)) #F*3,2
edge_to_face.reshape(2*E,2).scatter_(dim=0,index=(2*full_to_unique+is_right)[:,None].expand(F*3,2),src=scatter_src) #E,LR=2,S=2
edge_to_face[0] = 0
return edges, face_to_edge, edge_to_face
def calc_edge_length(
vertices:torch.Tensor, #V,3 first may be dummy
edges:torch.Tensor, #E,2 long, lower vertex index first, (0,0) for unused
)->torch.Tensor: #E
full_vertices = vertices[edges] #E,2,3
a,b = full_vertices.unbind(dim=1) #E,3
return torch.norm(a-b,p=2,dim=-1)
def calc_face_normals(
vertices:torch.Tensor, #V,3 first vertex may be unreferenced
faces:torch.Tensor, #F,3 long, first face may be all zero
normalize:bool=False,
)->torch.Tensor: #F,3
"""
n
|
c0 corners ordered counterclockwise when
/ \ looking onto surface (in neg normal direction)
c1---c2
"""
full_vertices = vertices[faces] #F,C=3,3
v0,v1,v2 = full_vertices.unbind(dim=1) #F,3
face_normals = torch.cross(v1-v0,v2-v0, dim=1) #F,3
if normalize:
face_normals = tfunc.normalize(face_normals, eps=1e-6, dim=1)
return face_normals #F,3
def calc_vertex_normals(
vertices:torch.Tensor, #V,3 first vertex may be unreferenced
faces:torch.Tensor, #F,3 long, first face may be all zero
face_normals:torch.Tensor=None, #F,3, not normalized
)->torch.Tensor: #F,3
F = faces.shape[0]
if face_normals is None:
face_normals = calc_face_normals(vertices,faces)
vertex_normals = torch.zeros((vertices.shape[0],3,3),dtype=vertices.dtype,device=vertices.device) #V,C=3,3
vertex_normals.scatter_add_(dim=0,index=faces[:,:,None].expand(F,3,3),src=face_normals[:,None,:].expand(F,3,3))
vertex_normals = vertex_normals.sum(dim=1) #V,3
return tfunc.normalize(vertex_normals, eps=1e-6, dim=1)
def calc_face_ref_normals(
faces:torch.Tensor, #F,3 long, 0 for unused
vertex_normals:torch.Tensor, #V,3 first unused
normalize:bool=False,
)->torch.Tensor: #F,3
"""calculate reference normals for face flip detection"""
full_normals = vertex_normals[faces] #F,C=3,3
ref_normals = full_normals.sum(dim=1) #F,3
if normalize:
ref_normals = tfunc.normalize(ref_normals, eps=1e-6, dim=1)
return ref_normals
def pack(
vertices:torch.Tensor, #V,3 first unused and nan
faces:torch.Tensor, #F,3 long, 0 for unused
)->Tuple[torch.Tensor,torch.Tensor]: #(vertices,faces), keeps first vertex unused
"""removes unused elements in vertices and faces"""
V = vertices.shape[0]
# remove unused faces
used_faces = faces[:,0]!=0
used_faces[0] = True
faces = faces[used_faces] #sync
# remove unused vertices
used_vertices = torch.zeros(V,3,dtype=torch.bool,device=vertices.device)
used_vertices.scatter_(dim=0,index=faces,value=True,reduce='add')
used_vertices = used_vertices.any(dim=1)
used_vertices[0] = True
vertices = vertices[used_vertices] #sync
# update used faces
ind = torch.zeros(V,dtype=torch.long,device=vertices.device)
V1 = used_vertices.sum()
ind[used_vertices] = torch.arange(0,V1,device=vertices.device) #sync
faces = ind[faces]
return vertices,faces
def split_edges(
vertices:torch.Tensor, #V,3 first unused
faces:torch.Tensor, #F,3 long, 0 for unused
edges:torch.Tensor, #E,2 long 0 for unused, lower vertex index first
face_to_edge:torch.Tensor, #F,3 long 0 for unused
splits, #E bool
pack_faces:bool=True,
)->Tuple[torch.Tensor,torch.Tensor]: #(vertices,faces)
# c2 c2 c...corners = faces
# . . . . s...side_vert, 0 means no split
# . . .N2 . S...shrunk_face
# . . . . Ni...new_faces
# s2 s1 s2|c2...s1|c1
# . . . . .
# . . . S . .
# . . . . N1 .
# c0...(s0=0)....c1 s0|c0...........c1
#
# pseudo-code:
# S = [s0|c0,s1|c1,s2|c2] example:[c0,s1,s2]
# split = side_vert!=0 example:[False,True,True]
# N0 = split[0]*[c0,s0,s2|c2] example:[0,0,0]
# N1 = split[1]*[c1,s1,s0|c0] example:[c1,s1,c0]
# N2 = split[2]*[c2,s2,s1|c1] example:[c2,s2,s1]
V = vertices.shape[0]
F = faces.shape[0]
S = splits.sum().item() #sync
if S==0:
return vertices,faces
edge_vert = torch.zeros_like(splits, dtype=torch.long) #E
edge_vert[splits] = torch.arange(V,V+S,dtype=torch.long,device=vertices.device) #E 0 for no split, sync
side_vert = edge_vert[face_to_edge] #F,3 long, 0 for no split
split_edges = edges[splits] #S sync
#vertices
split_vertices = vertices[split_edges].mean(dim=1) #S,3
vertices = torch.concat((vertices,split_vertices),dim=0)
#faces
side_split = side_vert!=0 #F,3
shrunk_faces = torch.where(side_split,side_vert,faces) #F,3 long, 0 for no split
new_faces = side_split[:,:,None] * torch.stack((faces,side_vert,shrunk_faces.roll(1,dims=-1)),dim=-1) #F,N=3,C=3
faces = torch.concat((shrunk_faces,new_faces.reshape(F*3,3))) #4F,3
if pack_faces:
mask = faces[:,0]!=0
mask[0] = True
faces = faces[mask] #F',3 sync
return vertices,faces
def collapse_edges(
vertices:torch.Tensor, #V,3 first unused
faces:torch.Tensor, #F,3 long 0 for unused
edges:torch.Tensor, #E,2 long 0 for unused, lower vertex index first
priorities:torch.Tensor, #E float
stable:bool=False, #only for unit testing
)->Tuple[torch.Tensor,torch.Tensor]: #(vertices,faces)
V = vertices.shape[0]
# check spacing
_,order = priorities.sort(stable=stable) #E
rank = torch.zeros_like(order)
rank[order] = torch.arange(0,len(rank),device=rank.device)
vert_rank = torch.zeros(V,dtype=torch.long,device=vertices.device) #V
edge_rank = rank #E
for i in range(3):
torch_scatter.scatter_max(src=edge_rank[:,None].expand(-1,2).reshape(-1),index=edges.reshape(-1),dim=0,out=vert_rank)
edge_rank,_ = vert_rank[edges].max(dim=-1) #E
candidates = edges[(edge_rank==rank).logical_and_(priorities>0)] #E',2
# check connectivity
vert_connections = torch.zeros(V,dtype=torch.long,device=vertices.device) #V
vert_connections[candidates[:,0]] = 1 #start
edge_connections = vert_connections[edges].sum(dim=-1) #E, edge connected to start
vert_connections.scatter_add_(dim=0,index=edges.reshape(-1),src=edge_connections[:,None].expand(-1,2).reshape(-1))# one edge from start
vert_connections[candidates] = 0 #clear start and end
edge_connections = vert_connections[edges].sum(dim=-1) #E, one or two edges from start
vert_connections.scatter_add_(dim=0,index=edges.reshape(-1),src=edge_connections[:,None].expand(-1,2).reshape(-1)) #one or two edges from start
collapses = candidates[vert_connections[candidates[:,1]] <= 2] # E" not more than two connections between start and end
# mean vertices
vertices[collapses[:,0]] = vertices[collapses].mean(dim=1)
# update faces
dest = torch.arange(0,V,dtype=torch.long,device=vertices.device) #V
dest[collapses[:,1]] = dest[collapses[:,0]]
faces = dest[faces] #F,3
c0,c1,c2 = faces.unbind(dim=-1)
collapsed = (c0==c1).logical_or_(c1==c2).logical_or_(c0==c2)
faces[collapsed] = 0
return vertices,faces
def calc_face_collapses(
vertices:torch.Tensor, #V,3 first unused
faces:torch.Tensor, #F,3 long, 0 for unused
edges:torch.Tensor, #E,2 long 0 for unused, lower vertex index first
face_to_edge:torch.Tensor, #F,3 long 0 for unused
edge_length:torch.Tensor, #E
face_normals:torch.Tensor, #F,3
vertex_normals:torch.Tensor, #V,3 first unused
min_edge_length:torch.Tensor=None, #V
area_ratio = 0.5, #collapse if area < min_edge_length**2 * area_ratio
shortest_probability = 0.8
)->torch.Tensor: #E edges to collapse
E = edges.shape[0]
F = faces.shape[0]
# face flips
ref_normals = calc_face_ref_normals(faces,vertex_normals,normalize=False) #F,3
face_collapses = (face_normals*ref_normals).sum(dim=-1)<0 #F
# small faces
if min_edge_length is not None:
min_face_length = min_edge_length[faces].mean(dim=-1) #F
min_area = min_face_length**2 * area_ratio #F
face_collapses.logical_or_(face_normals.norm(dim=-1) < min_area*2) #F
face_collapses[0] = False
# faces to edges
face_length = edge_length[face_to_edge] #F,3
if shortest_probability<1:
#select shortest edge with shortest_probability chance
randlim = round(2/(1-shortest_probability))
rand_ind = torch.randint(0,randlim,size=(F,),device=faces.device).clamp_max_(2) #selected edge local index in face
sort_ind = torch.argsort(face_length,dim=-1,descending=True) #F,3
local_ind = sort_ind.gather(dim=-1,index=rand_ind[:,None])
else:
local_ind = torch.argmin(face_length,dim=-1)[:,None] #F,1 0...2 shortest edge local index in face
edge_ind = face_to_edge.gather(dim=1,index=local_ind)[:,0] #F 0...E selected edge global index
edge_collapses = torch.zeros(E,dtype=torch.long,device=vertices.device)
edge_collapses.scatter_add_(dim=0,index=edge_ind,src=face_collapses.long())
return edge_collapses.bool()
def flip_edges(
vertices:torch.Tensor, #V,3 first unused
faces:torch.Tensor, #F,3 long, first must be 0, 0 for unused
edges:torch.Tensor, #E,2 long, first must be 0, 0 for unused, lower vertex index first
edge_to_face:torch.Tensor, #E,[left,right],[face,side]
with_border:bool=True, #handle border edges (D=4 instead of D=6)
with_normal_check:bool=True, #check face normal flips
stable:bool=False, #only for unit testing
):
V = vertices.shape[0]
E = edges.shape[0]
device=vertices.device
vertex_degree = torch.zeros(V,dtype=torch.long,device=device) #V long
vertex_degree.scatter_(dim=0,index=edges.reshape(E*2),value=1,reduce='add')
neighbor_corner = (edge_to_face[:,:,1] + 2) % 3 #go from side to corner
neighbors = faces[edge_to_face[:,:,0],neighbor_corner] #E,LR=2
edge_is_inside = neighbors.all(dim=-1) #E
if with_border:
# inside vertices should have D=6, border edges D=4, so we subtract 2 for all inside vertices
# need to use float for masks in order to use scatter(reduce='multiply')
vertex_is_inside = torch.ones(V,2,dtype=torch.float32,device=vertices.device) #V,2 float
src = edge_is_inside.type(torch.float32)[:,None].expand(E,2) #E,2 float
vertex_is_inside.scatter_(dim=0,index=edges,src=src,reduce='multiply')
vertex_is_inside = vertex_is_inside.prod(dim=-1,dtype=torch.long) #V long
vertex_degree -= 2 * vertex_is_inside #V long
neighbor_degrees = vertex_degree[neighbors] #E,LR=2
edge_degrees = vertex_degree[edges] #E,2
#
# loss = Sum_over_affected_vertices((new_degree-6)**2)
# loss_change = Sum_over_neighbor_vertices((degree+1-6)**2-(degree-6)**2)
# + Sum_over_edge_vertices((degree-1-6)**2-(degree-6)**2)
# = 2 * (2 + Sum_over_neighbor_vertices(degree) - Sum_over_edge_vertices(degree))
#
loss_change = 2 + neighbor_degrees.sum(dim=-1) - edge_degrees.sum(dim=-1) #E
candidates = torch.logical_and(loss_change<0, edge_is_inside) #E
loss_change = loss_change[candidates] #E'
if loss_change.shape[0]==0:
return
edges_neighbors = torch.concat((edges[candidates],neighbors[candidates]),dim=-1) #E',4
_,order = loss_change.sort(descending=True, stable=stable) #E'
rank = torch.zeros_like(order)
rank[order] = torch.arange(0,len(rank),device=rank.device)
vertex_rank = torch.zeros((V,4),dtype=torch.long,device=device) #V,4
torch_scatter.scatter_max(src=rank[:,None].expand(-1,4),index=edges_neighbors,dim=0,out=vertex_rank)
vertex_rank,_ = vertex_rank.max(dim=-1) #V
neighborhood_rank,_ = vertex_rank[edges_neighbors].max(dim=-1) #E'
flip = rank==neighborhood_rank #E'
if with_normal_check:
# cl-<-----e1 e0,e1...edge, e0<e1
# | /A L,R....left and right face
# | L / | both triangles ordered counter clockwise
# | / R | normals pointing out of screen
# V/ |
# e0---->-cr
v = vertices[edges_neighbors] #E",4,3
v = v - v[:,0:1] #make relative to e0
e1 = v[:,1]
cl = v[:,2]
cr = v[:,3]
n = torch.cross(e1,cl) + torch.cross(cr,e1) #sum of old normal vectors
flip.logical_and_(torch.sum(n*torch.cross(cr,cl),dim=-1)>0) #first new face
flip.logical_and_(torch.sum(n*torch.cross(cl-e1,cr-e1),dim=-1)>0) #second new face
flip_edges_neighbors = edges_neighbors[flip] #E",4
flip_edge_to_face = edge_to_face[candidates,:,0][flip] #E",2
flip_faces = flip_edges_neighbors[:,[[0,3,2],[1,2,3]]] #E",2,3
faces.scatter_(dim=0,index=flip_edge_to_face.reshape(-1,1).expand(-1,3),src=flip_faces.reshape(-1,3))