Voxelization

Point clouds and triangle meshes are very flexible, but irregular geometry types. The voxel grid is another geometry type in 3D that is defined on a regular 3D grid, whereas a voxel can be thought of as the 3D counterpart to the pixel in 2D. Open3D has the geometry type VoxelGrid that can be used to work with voxel grids.

From triangle mesh

Open3D provides the method create_from_triangle_mesh that creates a voxel grid from a triangle mesh. It returns a voxel grid where all voxels that are intersected by a triangle are set to 1, all others are set to 0. The argument voxel_size defines the resolution of the voxel grid.

print('input')
mesh = o3dtut.get_bunny_mesh()
# fit to unit cube
mesh.scale(1 / np.max(mesh.get_max_bound() - mesh.get_min_bound()))
o3d.visualization.draw_geometries([mesh])

print('voxelization')
voxel_grid = o3d.geometry.VoxelGrid.create_from_triangle_mesh(mesh,
                                                              voxel_size=0.05)
o3d.visualization.draw_geometries([voxel_grid])

input

voxelization

From point cloud

The voxel grid can also be created from a point cloud using the method create_from_point_cloud. A voxel is occupied if at least one point of the point cloud is within the voxel. The color of the voxel is the average of all the points within the voxel. The argument voxel_size defines the resolution of the voxel grid.

print('input')
N = 2000
pcd = o3dtut.get_armadillo_mesh().sample_points_poisson_disk(N)
# fit to unit cube
pcd.scale(1 / np.max(pcd.get_max_bound() - pcd.get_min_bound()))
pcd.colors = o3d.utility.Vector3dVector(np.random.uniform(0,1,size=(N,3)))
o3d.visualization.draw_geometries([pcd])

print('voxelization')
voxel_grid = o3d.geometry.VoxelGrid.create_from_point_cloud(pcd,
                                                            voxel_size=0.05)
o3d.visualization.draw_geometries([voxel_grid])

input

voxelization

Inclusion test

The voxel grid can also be used to test if points are within an occupied voxel. The method check_if_included takes a (n,3) array as input and outputs an bool array.

queries = np.asarray(pcd.points)
output = voxel_grid.check_if_included(o3d.utility.Vector3dVector(queries))
print(output[:10])

[True, True, True, True, True, True, True, True, True, True]

Voxel carving

The methods create_from_point_cloud and create_from_triangle_mesh create occupied voxels only on the surface of the geometry. It is however possible to carve a voxel grid from a number of depth maps or silhouettes. Open3D provides the methods carve_depth_map and carve_silhouette for voxel carving.

The code below demostrates the usage by first rendering depthmaps from a geometry and using those depthmaps to carve a dense voxel grid. The result is a filled voxel grid of the given shape.

def xyz_spherical(xyz):
    x = xyz[0]
    y = xyz[1]
    z = xyz[2]
    r = np.sqrt(x * x + y * y + z * z)
    r_x = np.arccos(y / r)
    r_y = np.arctan2(z, x)
    return [r, r_x, r_y]

def get_rotation_matrix(r_x, r_y):
    rot_x = np.asarray([[1, 0, 0], [0, np.cos(r_x), -np.sin(r_x)],
                        [0, np.sin(r_x), np.cos(r_x)]])
    rot_y = np.asarray([[np.cos(r_y), 0, np.sin(r_y)], [0, 1, 0],
                        [-np.sin(r_y), 0, np.cos(r_y)]])
    return rot_y.dot(rot_x)

def get_extrinsic(xyz):
    rvec = xyz_spherical(xyz)
    r = get_rotation_matrix(rvec[1], rvec[2])
    t = np.asarray([0, 0, 2]).transpose()
    trans = np.eye(4)
    trans[:3, :3] = r
    trans[:3, 3] = t
    return trans

def preprocess(model):
    min_bound = model.get_min_bound()
    max_bound = model.get_max_bound()
    center = min_bound + (max_bound - min_bound) / 2.0
    scale = np.linalg.norm(max_bound - min_bound) / 2.0
    vertices = np.asarray(model.vertices)
    vertices -= center
    model.vertices = o3d.utility.Vector3dVector(vertices / scale)
    return model

def voxel_carving(mesh,
                  output_filename,
                  camera_path,
                  cubic_size,
                  voxel_resolution,
                  w=300,
                  h=300,
                  use_depth=True,
                  surface_method='pointcloud'):
    mesh.compute_vertex_normals()
    camera_sphere = o3d.io.read_triangle_mesh(camera_path)

    # setup dense voxel grid
    voxel_carving = o3d.geometry.VoxelGrid.create_dense(
        width=cubic_size,
        height=cubic_size,
        depth=cubic_size,
        voxel_size=cubic_size / voxel_resolution,
        origin=[-cubic_size / 2.0, -cubic_size / 2.0, -cubic_size / 2.0])

    # rescale geometry
    camera_sphere = preprocess(camera_sphere)
    mesh = preprocess(mesh)

    # setup visualizer to render depthmaps
    vis = o3d.visualization.Visualizer()
    vis.create_window(width=w, height=h, visible=False)
    vis.add_geometry(mesh)
    vis.get_render_option().mesh_show_back_face = True
    ctr = vis.get_view_control()
    param = ctr.convert_to_pinhole_camera_parameters()

    # carve voxel grid
    pcd_agg = o3d.geometry.PointCloud()
    centers_pts = np.zeros((len(camera_sphere.vertices), 3))
    for cid, xyz in enumerate(camera_sphere.vertices):
        # get new camera pose
        trans = get_extrinsic(xyz)
        param.extrinsic = trans
        c = np.linalg.inv(trans).dot(np.asarray([0, 0, 0, 1]).transpose())
        centers_pts[cid, :] = c[:3]
        ctr.convert_from_pinhole_camera_parameters(param)

        # capture depth image and make a point cloud
        vis.poll_events()
        vis.update_renderer()
        depth = vis.capture_depth_float_buffer(False)
        pcd_agg += o3d.geometry.PointCloud.create_from_depth_image(
            o3d.geometry.Image(depth),
            param.intrinsic,
            param.extrinsic,
            depth_scale=1)

        # depth map carving method
        if use_depth:
            voxel_carving.carve_depth_map(o3d.geometry.Image(depth), param)
        else:
            voxel_carving.carve_silhouette(o3d.geometry.Image(depth), param)
        print("Carve view %03d/%03d" % (cid + 1, len(camera_sphere.vertices)))
    vis.destroy_window()

    # add voxel grid survace
    print('Surface voxel grid from %s' % surface_method)
    if surface_method == 'pointcloud':
        voxel_surface = o3d.geometry.VoxelGrid.create_from_point_cloud_within_bounds(
            pcd_agg,
            voxel_size=cubic_size / voxel_resolution,
            min_bound=(-cubic_size / 2, -cubic_size / 2, -cubic_size / 2),
            max_bound=(cubic_size / 2, cubic_size / 2, cubic_size / 2))
    elif surface_method == 'mesh':
        voxel_surface = o3d.geometry.VoxelGrid.create_from_triangle_mesh_within_bounds(
            mesh,
            voxel_size=cubic_size / voxel_resolution,
            min_bound=(-cubic_size / 2, -cubic_size / 2, -cubic_size / 2),
            max_bound=(cubic_size / 2, cubic_size / 2, cubic_size / 2))
    else:
        raise Exception('invalid surface method')
    voxel_carving_surface = voxel_surface + voxel_carving

    return voxel_carving_surface, voxel_carving, voxel_surface

mesh = o3dtut.get_armadillo_mesh()

output_filename = os.path.abspath("../../TestData/voxelized.ply")
camera_path = os.path.abspath("../../TestData/sphere.ply")
visualization = True
cubic_size = 2.0
voxel_resolution = 128.0

voxel_grid, voxel_carving, voxel_surface = voxel_carving(
    mesh, output_filename, camera_path,
    cubic_size, voxel_resolution)

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Surface voxel grid from pointcloud

print("surface voxels")
print(voxel_surface)
o3d.visualization.draw_geometries([voxel_surface])

print("carved voxels")
print(voxel_carving)
o3d.visualization.draw_geometries([voxel_carving])

print("combined voxels (carved + surface)")
print(voxel_grid)
o3d.visualization.draw_geometries([voxel_grid])

surface voxels
geometry::VoxelGrid with 17215 voxels.

carved voxels
geometry::VoxelGrid with 48370 voxels.

combined voxels (carved + surface)
geometry::VoxelGrid with 50786 voxels.