mesh_reconstruction/remesh.py

# 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))