reach_for_the_spheres
ReachForTheSpheresState
An object to keep state during the iterations of the Reach for the
Spheres method.
Meant to be used in conjunction with reach_for_the_spheres_iteration
.
Parameters:
Name  Type  Description  Default 

V 
(n,dim) numpy double array

Matrix of mesh vertices of the initial mesh 
required 
F 
(m,dim) numpy int array

Matrix of triangle indices into V of the initlal mesh 
required 
U 
(k,dim) numpy double array, optional (default

Matrix of SDF sample positions. The sdf will be sampled at these points 
None

S 
(k,dim) nummpy double array, optional (default

Matrix of SDF samples at the sample positions in U. If this is not provided, it will be computed using sdf. 
None

sdf 
The signed distance function to be reconstructed. This function must take in a (ks,dim) matrix of sample points and return a (ks,) array of SDF values at these points. 
None


V_active 
(na,dim) numpy double array, optional (default

Matrix of mesh vertices active during iteration 
None

F_active 
(ma,dim) numpy int array, optional (default

Matrix of triangle indices into V_active of the mesh active during iteration 
None

V_inactive 
(na,dim) numpy double array, optional (default

Matrix of mesh vertices inactive during iteration 
None

F_inactive 
(ma,dim) numpy int array, optional (default

Matrix of triangle indices into V_inactive of the mesh inactive during iteration 
None

rng 
numpy random generator, optional (default

numpy rng object used to create randomness during iterations 
None

h 
float (default None)

The method's initial target mesh length for the reconstructed mesh. This will change during iteration, set min_h if you want to control the minimum edge length overall. 
None

min_h 
float (default None)

The method's minimal target edge length for the reconstructed mesh. 
None

best_performance 
float (default None)

Remembers the method's best performance 
None

convergence_counter 
int (default None)

The method's convergence counter, used to track for how long progress has not been made. 
None

best_avg_error 
float (default None)

Remembers the method's average error. 
None

resample_counter 
int (default None)

Tracks how often the SDF has been resampled. 
None

V_last_converged 
(nl,dim) numpy double array, optional (default

Matrix of mesh vertices for the last known converged mesh. 
None

F_last_converged 
(ml,dim) numpy int array, optional (default

Matrix of triangle indices into V_last_converged for the last known converged mesh. 
None

U_batch 
(kb,dim) numpy double array, optional (default

Matrix of SDF sample positions as used during the last batching operation. 
None

S_batch 
(kb,dim) nummpy double array, optional (default

Matrix of SDF samples at the sample positions in U as used during the last batching operation. 
None

Notes
If you create this object yourself, you should supply V, F, U, S, sdf. Supply all other parameters only as explicitly needed.
Examples:
import gpytoolbox as gpy
import numpy as npy
# Get a signed distance function
V,F = gpy.read_mesh("my_mesh.obj")
# Create an SDF for the mesh
j = 20
sdf = lambda x: gpy.signed_distance(x, V, F)[0]
gx, gy, gz = np.meshgrid(np.linspace(1.0, 1.0, j+1), np.linspace(1.0, 1.0, j+1), np.linspace(1.0, 1.0, j+1))
U = np.vstack((gx.flatten(), gy.flatten(), gz.flatten())).T
# Create ReachForTheSpheresState to use in iterative method later
V0, F0 = gpy.icosphere(2)
state = ReachForTheSpheresState(V=V0, F=F0, sdf=sdf, U=U)
# Run one iteration of Reach for the Spheres
converged = gpy.reach_for_the_spheres_iteration(state)
# Reconstruct triangle mesh
Vr,Fr = state.V,state.F
#The reconstructed mesh after one iteration is now Vr,Fr
Source code in src/gpytoolbox/reach_for_the_spheres.py
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reach_for_the_spheres(U, sdf, V, F, S=None, return_U=False, verbose=False, max_iter=None, tol=None, h=None, min_h=None, linesearch=None, min_t=None, max_t=None, dt=None, inside_outside_test=None, resample=None, resample_samples=None, feature_detection=None, output_sensitive=None, remesh_iterations=None, batch_size=None, fix_boundary=None, clamp=None, pseudosdf_interior=None)
Creates a mesh from a signed distance function (SDF) using the "Reach for the Spheres" method of S. Sellán, C. Batty, and O. Stein [2023].
This method takes in an sdf, sample points (do not need to be on a grid), and an initial mesh. It then flows this initial mesh into a reconstructed mesh that fulfills the signed distance function.
This method works in dimension 2 (dim==2), where it reconstructs a polyline, and in dimension 3 (dim==3), where it reconstructs a triangle mesh.
Parameters:
Name  Type  Description  Default 

U 
(k,dim) numpy double array

Matrix of SDF sample positions. The sdf will be sampled at these points 
required 
sdf 
The signed distance function to be reconstructed. This function must take in a (ks,dim) matrix of sample points and return a (ks,) array of SDF values at these points. 
required  
V 
(n,dim) numpy double array

Matrix of mesh vertices of the initial mesh 
required 
F 
(m,dim) numpy int array

Matrix of triangle indices into V of the initlal mesh 
required 
S 
(k,dim) nummpy double array, optional (default

Matrix of SDF samples at the sample positions in U. If this is not provided, it will be computed using sdf. 
None

return_U 
bool, optional (default

Whether to return the matrix of SDF sample positions along with the reconstructed mesh. 
False

verbose 
bool (default false)

Whether to print method statistics during operation. 
False

max_iter 
int (default None)

The maximum number of iterations to perform for the method. If not supplied, a sensible default is used. 
None

tol 
float (default None)

The method's tolerance for the sphere tangency test. If not supplied, a sensible default is used. 
None

h 
float (default None)

The method's initial target mesh length for the reconstructed mesh. This will change during iteration, set min_h if you want to control the minimum edge length overall. If not supplied, a sensible default is used. 
None

min_h 
float (default None)

The method's minimal target edge length for the reconstructed mesh. If not supplied, a sensible default is used. 
None

linesearch 
bool (default None)

Whether to use a linesearchlike heuristic for the method's timestep. If not supplied, linesearch is used. 
None

min_t 
float (default None)

The method's minimum timestep. If not supplied, a sensible default is used. 
None

max_t 
float (default None)

The method's minimum timestep. If not supplied, a sensible default is used. 
None

dt 
float (default None)

The method's default timestep. If not supplied, a sensible default is used. 
None

inside_outside_test 
bool (default None)

Whether to use insideoutside testing when projecting points to be tangent to the sphere. Turn this off if your distance function is unsigned. If not supplied, insideoutside test is used 
None

resample 
int (default None)

How often to resample the SDF after convergence to extract more information. If not supplied, resampling is not performed. 
None

resample_samples 
int (default None)

How many samples to use when resampling. If not supplied, a sensible default is used. 
None

feature_detection 
string (default None)

Which feature detection mode to use. If not supplied, aggressive feature detection is used. 
None

output_sensitive 
bool (default None)

Whether to use outputsensitive remeshing. If not supplied, remeshing is outputsensitive. 
None

remesh_iterations 
int (default None)

How many iterations of the remesher to run each step. If not supplied, a sensible default is used. 
None

batch_size 
int (default None)

For large amounts of sample points, the method is sped up using sample point batching. This parameter specifies how many samples to take for each batch. Set it to 0 to disable batching. If not supplied, a sensible default is used. 
None

fix_boundary 
bool (default None)

Whether to fix the boundary of the mesh during iteration. If not supplied, the boundary is not fixed. 
None

clamp 
float (default None)

If sdf is a clamped SDF, the clamp value to use. np.inf for no clamping. If not supplied, there is no clamping. 
None

pseudosdf_interior 
bool (default None)

If enabled, treats every negative SDF value as a bound on the signed distance, as opposed to an exact signed distance, for use in SDFs resulting from CSG unions as described by Marschner et al. "Constructive Solid Geometry on Neural Signed Distance Fields" [2023]. If not supplied, this feature is disabled. 
None

Returns 
required  
Vr 
(nr,dim) numpy double array

Matrix of mesh vertices of the reconstructed triangle mesh 
required 
Fr 
(mr,dim) numpy int array

Matrix of triangle indices into Vr of the reconstructed mesh 
required 
Ur 
(kr,dim) numpy double array, if requested

Matrix of SDF sample positions. This can be different from the supplied Ur if the method is set to resample. 
required 
See Also
marching_squares, marching_cubes
Notes
This method has a number of limitations that are described in the paper. E.g., the method will only work for SDFs that describe surfaces with the same topology as the initial surface.
Examples:
import gpytoolbox as gpy
import numpy as npy
# Get a signed distance function
V,F = gpy.read_mesh("my_mesh.obj")
# Create an SDF for the mesh
j = 20
sdf = lambda x: gpy.signed_distance(x, V, F)[0]
gx, gy, gz = np.meshgrid(np.linspace(1.0, 1.0, j+1), np.linspace(1.0, 1.0, j+1), np.linspace(1.0, 1.0, j+1))
U = np.vstack((gx.flatten(), gy.flatten(), gz.flatten())).T
# Choose an initial surface for reach_for_the_spheres
V0, F0 = gpy.icosphere(2)
# Reconstruct triangle mesh
Vr,Fr = gpy.reach_for_the_spheres(U, sdf, V0, F0)
#The reconstructed triangle mesh is now Vr,Fr.
Source code in src/gpytoolbox/reach_for_the_spheres.py
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reach_for_the_spheres_iteration(state, max_iter=None, tol=None, linesearch=None, min_t=None, max_t=None, dt=None, inside_outside_test=None, resample=None, resample_samples=None, feature_detection=None, output_sensitive=None, remesh_iterations=None, batch_size=None, fix_boundary=None, clamp=None, pseudosdf_interior=None, verbose=False)
Performs one iteration of the "Reach for the Spheres" method of S. Sellán, C. Batty, and O. Stein [2023]. This method is used to create a mesh from a signed distance function (SDF).
This method takes in the current state of the method in the form of a ReachForTheSpheresState object, and stores its results as well as any temporary information needed in that state object.
This method works in dimension 2 (dim==2), where it reconstructs a polyline, and in dimension 3 (dim==3), where it reconstructs a triangle mesh.
Parameters:
Name  Type  Description  Default 

state 
Stores all needed information about the current state of the method. 
required  
max_iter 
int (default None)

The maximum number of iterations to perform for the method. If not supplied, a sensible default is used. 
None

tol 
float (default None)

The method's tolerance for the sphere tangency test. If not supplied, a sensible default is used. 
None

linesearch 
bool (default None)

Whether to use a linesearchlike heuristic for the method's timestep. If not supplied, linesearch is used. 
None

min_t 
float (default None)

The method's minimum timestep. If not supplied, a sensible default is used. 
None

max_t 
float (default None)

The method's minimum timestep. If not supplied, a sensible default is used. 
None

dt 
float (default None)

The method's default timestep. If not supplied, a sensible default is used. 
None

inside_outside_test 
bool (default None)

Whether to use insideoutside testing when projecting points to be tangent to the sphere. Turn this off if your distance function is unsigned. If not supplied, insideoutside test is used 
None

resample 
int (default None)

How often to resample the SDF after convergence to extract more information. If not supplied, resampling is not performed. 
None

resample_samples 
int (default None)

How many samples to use when resampling. If not supplied, a sensible default is used. 
None

feature_detection 
string (default None)

Which feature detection mode to use. If not supplied, aggressive feature detection is used. 
None

output_sensitive 
bool (default None)

Whether to use outputsensitive remeshing. If not supplied, remeshing is outputsensitive. 
None

remesh_iterations 
int (default None)

How many iterations of the remesher to run each step. If not supplied, a sensible default is used. 
None

batch_size 
int (default None)

For large amounts of sample points, the method is sped up using sample point batching. This parameter specifies how many samples to take for each batch. Set it to 0 to disable batching. If not supplied, a sensible default is used. 
None

fix_boundary 
bool (default None)

Whether to fix the boundary of the mesh during iteration. If not supplied, the boundary is not fixed. 
None

clamp 
float (default None)

If sdf is a clamped SDF, the clamp value to use. np.inf for no clamping. If not supplied, there is no clamping. 
None

pseudosdf_interior 
bool (default None)

If enabled, treats every negative SDF value as a bound on the signed distance, as opposed to an exact signed distance, for use in SDFs resulting from CSG unions as described by Marschner et al. "Constructive Solid Geometry on Neural Signed Distance Fields" [2023]. If not supplied, this feature is disabled. 
None

verbose 
bool (default false)

Whether to print method statistics during operation. 
False

Returns 
required  
converged 
bool

Whether the method has converged after this iteration or not. 
required 
See Also
reach_for_the_spheres, marching_squares, marching_cubes
Notes
This method has a number of limitations that are described in the paper. E.g., the method will only work for SDFs that describe surfaces with the same topology as the initial surface.
Examples:
import gpytoolbox as gpy
import numpy as npy
# Get a signed distance function
V,F = gpy.read_mesh("my_mesh.obj")
# Create an SDF for the mesh
j = 20
sdf = lambda x: gpy.signed_distance(x, V, F)[0]
gx, gy, gz = np.meshgrid(np.linspace(1.0, 1.0, j+1), np.linspace(1.0, 1.0, j+1), np.linspace(1.0, 1.0, j+1))
U = np.vstack((gx.flatten(), gy.flatten(), gz.flatten())).T
# Create ReachForTheSpheresState to use in iterative method later
V0, F0 = gpy.icosphere(2)
state = ReachForTheSpheresState(V=V0, F=F0, sdf=sdf, U=U)
# Run one iteration of Reach for the Spheres
converged = gpy.reach_for_the_spheres_iteration(state)
# Reconstruct triangle mesh
Vr,Fr = state.V,state.F
#The reconstructed mesh after one iteration is now Vr,Fr
Source code in src/gpytoolbox/reach_for_the_spheres.py
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