18 | python hks.py --input homer.off --t 5 --output hks5.off
19 |
20 |
23 | python hks.py --input homer.off --t 20 --output hks20.off
24 |
25 |
28 | python hks.py --input homer.off --t 200 --output hks200.off
29 |
30 | 
36 | 
39 | 
42 |
17 |
18 | python hks.py --input homer.off --t 5 --output hks5.off
19 |
20 | |
21 |
22 |
23 | python hks.py --input homer.off --t 20 --output hks20.off
24 |
25 | |
26 |
27 |
28 | python hks.py --input homer.off --t 200 --output hks200.off
29 |
30 | |
31 |
|
35 |
36 | |
37 |
38 |
39 | |
40 |
41 |
42 | |
43 |
44 |
45 |
--------------------------------------------------------------------------------
/trimesh.py:
--------------------------------------------------------------------------------
1 | import numpy as np
2 | from scipy import sparse
3 |
4 |
5 | def load_off(filename):
6 | """
7 | Load in an OFF file, assuming it's a triangle mesh
8 | Parameters
9 | ----------
10 | filename: string
11 | Path to file
12 | Returns
13 | -------
14 | VPos : ndarray (N, 3)
15 | Array of points in 3D
16 | VColors : ndarray(N, 3)
17 | Array of RGB colors
18 | ITris : ndarray (M, 3)
19 | Array of triangles connecting points, pointing to vertex indices
20 | """
21 | fin = open(filename, 'r')
22 | nVertices = 0
23 | nFaces = 0
24 | lineCount = 0
25 | face = 0
26 | vertex = 0
27 | divideColor = False
28 | VPos = np.zeros((0, 3))
29 | VColors = np.zeros((0, 3))
30 | ITris = np.zeros((0, 3))
31 | for line in fin:
32 | lineCount = lineCount+1
33 | fields = line.split() #Splits whitespace by default
34 | if len(fields) == 0: #Blank line
35 | continue
36 | if fields[0][0] in ['#', '\0', ' '] or len(fields[0]) == 0:
37 | continue
38 | #Check section
39 | if nVertices == 0:
40 | if fields[0] == "OFF" or fields[0] == "COFF":
41 | if len(fields) > 2:
42 | fields[1:4] = [int(field) for field in fields]
43 | [nVertices, nFaces, nEdges] = fields[1:4]
44 | #Pre-allocate vertex arrays
45 | VPos = np.zeros((nVertices, 3))
46 | VColors = np.zeros((nVertices, 3))
47 | ITris = np.zeros((nFaces, 3))
48 | if fields[0] == "COFF":
49 | divideColor = True
50 | else:
51 | fields[0:3] = [int(field) for field in fields]
52 | [nVertices, nFaces, nEdges] = fields[0:3]
53 | VPos = np.zeros((nVertices, 3))
54 | VColors = np.zeros((nVertices, 3))
55 | ITris = np.zeros((nFaces, 3))
56 | elif vertex < nVertices:
57 | fields = [float(i) for i in fields]
58 | P = [fields[0],fields[1], fields[2]]
59 | color = np.array([0.5, 0.5, 0.5]) #Gray by default
60 | if len(fields) >= 6:
61 | #There is color information
62 | if divideColor:
63 | color = [float(c)/255.0 for c in fields[3:6]]
64 | else:
65 | color = [float(c) for c in fields[3:6]]
66 | VPos[vertex, :] = P
67 | VColors[vertex, :] = color
68 | vertex = vertex+1
69 | elif face < nFaces:
70 | #Assume the vertices are specified in CCW order
71 | fields = [int(i) for i in fields]
72 | ITris[face, :] = fields[1:fields[0]+1]
73 | face = face+1
74 | fin.close()
75 | VPos = np.array(VPos, np.float64)
76 | VColors = np.array(VColors, np.float64)
77 | ITris = np.array(ITris, np.int32)
78 | return (VPos, VColors, ITris)
79 |
80 | def save_off(filename, VPos, VColors, ITris):
81 | """
82 | Save a .off file
83 | Parameters
84 | ----------
85 | filename: string
86 | Path to which to write .off file
87 | VPos : ndarray (N, 3)
88 | Array of points in 3D
89 | VColors : ndarray(N, 3)
90 | Array of RGB colors
91 | ITris : ndarray (M, 3)
92 | Array of triangles connecting points, pointing to vertex indices
93 | """
94 | nV = VPos.shape[0]
95 | nF = ITris.shape[0]
96 | fout = open(filename, "w")
97 | if VColors.size == 0:
98 | fout.write("OFF\n%i %i %i\n"%(nV, nF, 0))
99 | else:
100 | fout.write("COFF\n%i %i %i\n"%(nV, nF, 0))
101 | for i in range(nV):
102 | fout.write("%g %g %g"%tuple(VPos[i, :]))
103 | if VColors.size > 0:
104 | fout.write(" %g %g %g"%tuple(VColors[i, :]))
105 | fout.write("\n")
106 | for i in range(nF):
107 | fout.write("3 %i %i %i\n"%tuple(ITris[i, :]))
108 | fout.close()
109 |
110 |
111 | def sample_by_area(VPos, ITris, npoints, colPoints = False):
112 | """
113 | Randomly sample points by area on a triangle mesh. This function is
114 | extremely fast by using broadcasting/numpy operations in lieu of loops
115 |
116 | Parameters
117 | ----------
118 | VPos : ndarray (N, 3)
119 | Array of points in 3D
120 | ITris : ndarray (M, 3)
121 | Array of triangles connecting points, pointing to vertex indices
122 | npoints : int
123 | Number of points to sample
124 | colPoints : boolean (default True)
125 | Whether the points are along the columns or the rows
126 |
127 | Returns
128 | -------
129 | (Ps : NDArray (npoints, 3) array of sampled points,
130 | Ns : Ndarray (npoints, 3) of normals at those points )
131 | """
132 | ###Step 1: Compute cross product of all face triangles and use to compute
133 | #areas and normals (very similar to code used to compute vertex normals)
134 |
135 | #Vectors spanning two triangle edges
136 | P0 = VPos[ITris[:, 0], :]
137 | P1 = VPos[ITris[:, 1], :]
138 | P2 = VPos[ITris[:, 2], :]
139 | V1 = P1 - P0
140 | V2 = P2 - P0
141 | FNormals = np.cross(V1, V2)
142 | FAreas = np.sqrt(np.sum(FNormals**2, 1)).flatten()
143 |
144 | #Get rid of zero area faces and update points
145 | ITris = ITris[FAreas > 0, :]
146 | FNormals = FNormals[FAreas > 0, :]
147 | FAreas = FAreas[FAreas > 0]
148 | P0 = VPos[ITris[:, 0], :]
149 | P1 = VPos[ITris[:, 1], :]
150 | P2 = VPos[ITris[:, 2], :]
151 |
152 | #Compute normals
153 | NTris = ITris.shape[0]
154 | FNormals = FNormals/FAreas[:, None]
155 | FAreas = 0.5*FAreas
156 | FNormals = FNormals
157 | VNormals = np.zeros_like(VPos)
158 | VAreas = np.zeros(VPos.shape[0])
159 | for k in range(3):
160 | VNormals[ITris[:, k], :] += FAreas[:, None]*FNormals
161 | VAreas[ITris[:, k]] += FAreas
162 | #Normalize normals
163 | VAreas[VAreas == 0] = 1
164 | VNormals = VNormals / VAreas[:, None]
165 |
166 | ###Step 2: Randomly sample points based on areas
167 | FAreas = FAreas/np.sum(FAreas)
168 | AreasC = np.cumsum(FAreas)
169 | samples = np.sort(np.random.rand(npoints))
170 | #Figure out how many samples there are for each face
171 | FSamples = np.zeros(NTris, dtype=np.int32)
172 | fidx = 0
173 | for s in samples:
174 | while s > AreasC[fidx]:
175 | fidx += 1
176 | FSamples[fidx] += 1
177 | #Now initialize an array that stores the triangle sample indices
178 | tidx = np.zeros(npoints, dtype=np.int64)
179 | idx = 0
180 | for i in range(len(FSamples)):
181 | tidx[idx:idx+FSamples[i]] = i
182 | idx += FSamples[i]
183 | N = np.zeros((npoints, 3)) #Allocate space for normals
184 | idx = 0
185 |
186 | #Vector used to determine if points need to be flipped across parallelogram
187 | V3 = P2 - P1
188 | V3 = V3/np.sqrt(np.sum(V3**2, 1))[:, None] #Normalize
189 |
190 | #Randomly sample points on each face
191 | #Generate random points uniformly in parallelogram
192 | u = np.random.rand(npoints, 1)
193 | v = np.random.rand(npoints, 1)
194 | Ps = u*V1[tidx, :] + P0[tidx, :]
195 | Ps += v*V2[tidx, :]
196 | #Flip over points which are on the other side of the triangle
197 | dP = Ps - P1[tidx, :]
198 | proj = np.sum(dP*V3[tidx, :], 1)
199 | dPPar = V3[tidx, :]*proj[:, None] #Parallel project onto edge
200 | dPPerp = dP - dPPar
201 | Qs = Ps - dPPerp
202 | dP0QSqr = np.sum((Qs - P0[tidx, :])**2, 1)
203 | dP0PSqr = np.sum((Ps - P0[tidx, :])**2, 1)
204 | idxreg = np.arange(npoints, dtype=np.int64)
205 | idxflip = idxreg[dP0QSqr < dP0PSqr]
206 | u[idxflip, :] = 1 - u[idxflip, :]
207 | v[idxflip, :] = 1 - v[idxflip, :]
208 | Ps[idxflip, :] = P0[tidx[idxflip], :] + u[idxflip, :]*V1[tidx[idxflip], :] + v[idxflip, :]*V2[tidx[idxflip], :]
209 |
210 | #Step 3: Compute normals of sampled points by barycentric interpolation
211 | Ns = u*VNormals[ITris[tidx, 1], :]
212 | Ns += v*VNormals[ITris[tidx, 2], :]
213 | Ns += (1-u-v)*VNormals[ITris[tidx, 0], :]
214 |
215 | if colPoints:
216 | return (Ps.T, Ns.T)
217 | return (Ps, Ns)
218 |
219 |
220 |
221 | def get_edges(VPos, ITris):
222 | """
223 | Given a list of triangles, return an array representing the edges
224 | Parameters
225 | ----------
226 | VPos : ndarray (N, 3)
227 | Array of points in 3D
228 | ITris : ndarray (M, 3)
229 | Array of triangles connecting points, pointing to vertex indices
230 | Returns: I, J
231 | Two parallel 1D arrays with indices of edges
232 | """
233 | N = VPos.shape[0]
234 | M = ITris.shape[0]
235 | I = np.zeros(M*6)
236 | J = np.zeros(M*6)
237 | V = np.ones(M*6)
238 | for shift in range(3):
239 | #For all 3 shifts of the roles of triangle vertices
240 | #to compute different cotangent weights
241 | [i, j, k] = [shift, (shift+1)%3, (shift+2)%3]
242 | I[shift*M*2:shift*M*2+M] = ITris[:, i]
243 | J[shift*M*2:shift*M*2+M] = ITris[:, j]
244 | I[shift*M*2+M:shift*M*2+2*M] = ITris[:, j]
245 | J[shift*M*2+M:shift*M*2+2*M] = ITris[:, i]
246 | L = sparse.coo_matrix((V, (I, J)), shape=(N, N)).tocsr()
247 | return L.nonzero()
248 |
--------------------------------------------------------------------------------
/hks.py:
--------------------------------------------------------------------------------
1 | import numpy as np
2 | from scipy import sparse
3 | from scipy.sparse.linalg import lsqr, cg, eigsh
4 | import matplotlib.pyplot as plt
5 | import argparse
6 | from trimesh import load_off, save_off
7 |
8 | def get_cotan_laplacian(VPos, ITris, anchorsIdx = [], anchorWeights = 1):
9 | """
10 | Quickly compute sparse Laplacian matrix with cotangent weights and Voronoi areas
11 | by doing many operations in parallel using NumPy
12 |
13 | Parameters
14 | ----------
15 | VPos : ndarray (N, 3)
16 | Array of vertex positions
17 | ITris : ndarray (M, 3)
18 | Array of triangle indices
19 | anchorsIdx : list
20 | A list of vertex indices corresponding to the anchor vertices
21 | (for use in Laplacian mesh editing; by default none)
22 | anchorWeights : float
23 |
24 |
25 | Returns
26 | -------
27 | L : scipy.sparse (NVertices+anchors, NVertices+anchors)
28 | A sparse Laplacian matrix with cotangent weights
29 | """
30 | N = VPos.shape[0]
31 | M = ITris.shape[0]
32 | #Allocate space for the sparse array storage, with 2 entries for every
33 | #edge for eves ry triangle (6 entries per triangle); one entry for directed
34 | #edge ij and ji. Note that this means that edges with two incident triangles
35 | #will have two entries per directed edge, but sparse array will sum them
36 | I = np.zeros(M*6)
37 | J = np.zeros(M*6)
38 | V = np.zeros(M*6)
39 |
40 | #Keep track of areas of incident triangles and the number of incident triangles
41 | IA = np.zeros(M*3)
42 | VA = np.zeros(M*3) #Incident areas
43 | VC = 1.0*np.ones(M*3) #Number of incident triangles
44 |
45 | #Step 1: Compute cotangent weights
46 | for shift in range(3):
47 | #For all 3 shifts of the roles of triangle vertices
48 | #to compute different cotangent weights
49 | [i, j, k] = [shift, (shift+1)%3, (shift+2)%3]
50 | dV1 = VPos[ITris[:, i], :] - VPos[ITris[:, k], :]
51 | dV2 = VPos[ITris[:, j], :] - VPos[ITris[:, k], :]
52 | Normal = np.cross(dV1, dV2)
53 | #Cotangent is dot product / mag cross product
54 | NMag = np.sqrt(np.sum(Normal**2, 1))
55 | cotAlpha = np.sum(dV1*dV2, 1)/NMag
56 | I[shift*M*2:shift*M*2+M] = ITris[:, i]
57 | J[shift*M*2:shift*M*2+M] = ITris[:, j]
58 | V[shift*M*2:shift*M*2+M] = cotAlpha
59 | I[shift*M*2+M:shift*M*2+2*M] = ITris[:, j]
60 | J[shift*M*2+M:shift*M*2+2*M] = ITris[:, i]
61 | V[shift*M*2+M:shift*M*2+2*M] = cotAlpha
62 | if shift == 0:
63 | #Compute contribution of this triangle to each of the vertices
64 | for k in range(3):
65 | IA[k*M:(k+1)*M] = ITris[:, k]
66 | VA[k*M:(k+1)*M] = 0.5*NMag
67 |
68 | #Step 2: Create laplacian matrix
69 | L = sparse.coo_matrix((V, (I, J)), shape=(N, N)).tocsr()
70 | #Create the diagonal by summing the rows and subtracting off the nondiagonal entries
71 | L = sparse.dia_matrix((L.sum(1).flatten(), 0), L.shape) - L
72 | #Scale each row by the incident areas TODO: Fix this
73 | """
74 | Areas = sparse.coo_matrix((VA, (IA, IA)), shape=(N, N)).tocsr()
75 | Areas = Areas.todia().data.flatten()
76 | Areas[Areas == 0] = 1
77 | Counts = sparse.coo_matrix((VC, (IA, IA)), shape=(N, N)).tocsr()
78 | Counts = Counts.todia().data.flatten()
79 | RowScale = sparse.dia_matrix((3*Counts/Areas, 0), L.shape)
80 | L = L.T.dot(RowScale).T
81 | """
82 |
83 | #Step 3: Add anchors
84 | L = L.tocoo()
85 | I = L.row.tolist()
86 | J = L.col.tolist()
87 | V = L.data.tolist()
88 | I = I + list(range(N, N+len(anchorsIdx)))
89 | J = J + anchorsIdx
90 | V = V + [anchorWeights]*len(anchorsIdx)
91 | L = sparse.coo_matrix((V, (I, J)), shape=(N+len(anchorsIdx), N)).tocsr()
92 | return L
93 |
94 | def get_umbrella_laplacian(VPos, ITris, anchorsIdx = [], anchorWeights = 1):
95 | """
96 | Quickly compute sparse Laplacian matrix with "umbrella weights" (unweighted)
97 | by doing many operations in parallel using NumPy
98 |
99 | Parameters
100 | ----------
101 | VPos : ndarray (N, 3)
102 | Array of vertex positions
103 | ITris : ndarray (M, 3)
104 | Array of triangle indices
105 | anchorsIdx : list
106 | A list of vertex indices corresponding to the anchor vertices
107 | (for use in Laplacian mesh editing; by default none)
108 | anchorWeights : float
109 |
110 |
111 | Returns
112 | -------
113 | L : scipy.sparse (NVertices+anchors, NVertices+anchors)
114 | A sparse Laplacian matrix with umbrella weights
115 | """
116 | N = VPos.shape[0]
117 | M = ITris.shape[0]
118 | I = np.zeros(M*6)
119 | J = np.zeros(M*6)
120 | V = np.ones(M*6)
121 |
122 | #Step 1: Set up umbrella entries
123 | for shift in range(3):
124 | #For all 3 shifts of the roles of triangle vertices
125 | #to compute different cotangent weights
126 | [i, j, k] = [shift, (shift+1)%3, (shift+2)%3]
127 | I[shift*M*2:shift*M*2+M] = ITris[:, i]
128 | J[shift*M*2:shift*M*2+M] = ITris[:, j]
129 | I[shift*M*2+M:shift*M*2+2*M] = ITris[:, j]
130 | J[shift*M*2+M:shift*M*2+2*M] = ITris[:, i]
131 |
132 | #Step 2: Create laplacian matrix
133 | L = sparse.coo_matrix((V, (I, J)), shape=(N, N)).tocsr()
134 | L[L > 0] = 1
135 | #Create the diagonal by summing the rows and subtracting off the nondiagonal entries
136 | L = sparse.dia_matrix((L.sum(1).flatten(), 0), L.shape) - L
137 |
138 | #Step 3: Add anchors
139 | L = L.tocoo()
140 | I = L.row.tolist()
141 | J = L.col.tolist()
142 | V = L.data.tolist()
143 | I = I + list(range(N, N+len(anchorsIdx)))
144 | J = J + anchorsIdx
145 | V = V + [anchorWeights]*len(anchorsIdx)
146 | L = sparse.coo_matrix((V, (I, J)), shape=(N+len(anchorsIdx), N)).tocsr()
147 | return L
148 |
149 |
150 |
151 |
152 | def get_laplacian_spectrum(VPos, ITris, K):
153 | """
154 | Given a mesh, to compute first K eigenvectors of its Laplacian
155 | and the corresponding eigenvalues
156 | Parameters
157 | ----------
158 | VPos : ndarray (N, 3)
159 | Array of points in 3D
160 | ITris : ndarray (M, 3)
161 | Array of triangles connecting points, pointing to vertex indices
162 | K : int
163 | Number of eigenvectors to compute
164 | Returns
165 | -------
166 | (eigvalues, eigvectors): a tuple of the eigenvalues and eigenvectors
167 | """
168 | L = get_cotan_laplacian(VPos, ITris)
169 | (eigvalues, eigvectors) = eigsh(L, K, which='LM', sigma = 0)
170 | return (eigvalues, eigvectors)
171 |
172 |
173 | def get_heat(eigvalues, eigvectors, t, initialVertices, heatValue = 100.0):
174 | """
175 | Simulate heat flow by projecting initial conditions
176 | onto the eigenvectors of the Laplacian matrix, and then sum up the heat
177 | flow of each eigenvector after it's decayed after an amount of time t
178 | Parameters
179 | ----------
180 | eigvalues : ndarray (K)
181 | Eigenvalues of the laplacian
182 | eigvectors : ndarray (N, K)
183 | An NxK matrix of corresponding laplacian eigenvectors
184 | Number of eigenvectors to compute
185 | t : float
186 | The time to simulate heat flow
187 | initialVertices : ndarray (L)
188 | indices of the verticies that have an initial amount of heat
189 | heatValue : float
190 | The value to put at each of the initial vertices at the beginning of time
191 |
192 | Returns
193 | -------
194 | heat : ndarray (N) holding heat values at each vertex on the mesh
195 | """
196 | N = eigvectors.shape[0]
197 | I = np.zeros(N)
198 | I[initialVertices] = heatValue
199 | coeffs = I[None, :].dot(eigvectors)
200 | coeffs = coeffs.flatten()
201 | coeffs = coeffs*np.exp(-eigvalues*t)
202 | heat = eigvectors.dot(coeffs[:, None])
203 | return heat
204 |
205 | def get_hks(VPos, ITris, K, ts):
206 | """
207 | Given a triangle mesh, approximate its curvature at some measurement scale
208 | by recording the amount of heat that remains at each vertex after a unit impulse
209 | of heat is applied. This is called the "Heat Kernel Signature" (HKS)
210 |
211 | Parameters
212 | ----------
213 | VPos : ndarray (N, 3)
214 | Array of points in 3D
215 | ITris : ndarray (M, 3)
216 | Array of triangles connecting points, pointing to vertex indices
217 | K : int
218 | Number of eigenvalues/eigenvectors to use
219 | ts : ndarray (T)
220 | The time scales at which to compute the HKS
221 |
222 | Returns
223 | -------
224 | hks : ndarray (N, T)
225 | A array of the heat kernel signatures at each of N points
226 | at T time intervals
227 | """
228 | L = get_cotan_laplacian(VPos, ITris)
229 | (eigvalues, eigvectors) = eigsh(L, K, which='LM', sigma = 0)
230 | res = (eigvectors[:, :, None]**2)*np.exp(-eigvalues[None, :, None]*ts.flatten()[None, None, :])
231 | return np.sum(res, 1)
232 |
233 |
234 | def saveHKSColors(filename, VPos, hks, ITris, cmap = 'gray'):
235 | """
236 | Save the mesh as a .coff file using a divergent colormap, where
237 | negative curvature is one one side and positive curvature is on the other
238 | """
239 | c = plt.get_cmap(cmap)
240 | x = (hks - np.min(hks))
241 | x /= np.max(x)
242 | np.array(np.round(x*255.0), dtype=np.int32)
243 | C = c(x)
244 | C = C[:, 0:3]
245 | save_off(filename, VPos, C, ITris)
246 |
247 | if __name__ == '__main__':
248 | parser = argparse.ArgumentParser()
249 | parser.add_argument("--input", type=str, required=True, help="Path to OFF file for triangle mesh on which to compute the HKS")
250 | parser.add_argument("--output", type=str, required=True, help="Path to OFF file which holds a colored mesh showing the HKS")
251 | parser.add_argument("--t", type=float, required=True, help="Time parameter for the HKS")
252 | parser.add_argument("--neigvecs", type=int, required=False, default = 200, help="Number of eigenvectors to use")
253 |
254 | opt = parser.parse_args()
255 | (VPos, VColors, ITris) = load_off(opt.input)
256 | neigvecs = min(VPos.shape[0], opt.neigvecs)
257 | hks = get_hks(VPos, ITris, neigvecs, np.array([opt.t]))
258 | saveHKSColors(opt.output, VPos, hks[:, 0], ITris)
259 |
--------------------------------------------------------------------------------
/LICENSE:
--------------------------------------------------------------------------------
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2 | Version 2.0, January 2004
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