# Octree¶

An **octree** is a tree data structure where each internal node has eight children. Octrees are commonly used for spatial partitioning of 3D point clouds. Non-empty leaf nodes of an octree contain one or more points that fall within the same spatial subdivision. Octrees are a useful description of 3D space and can be used to quickly find nearby points. Open3D has the geometry type `Octree`

that can be used to create, search, and traverse octrees with a user-specified maximum tree depth,
`max_depth`

.

## From point cloud¶

An octree can be constructed from a point cloud using the method `convert_from_point_cloud`

. Each point is inserted into the tree by following the path from the root node to the appropriate leaf node at depth `max_depth`

. As the tree depth increases, internal (and eventually leaf) nodes represents a smaller partition of 3D space.

If the point cloud has color, the the corresponding leaf node takes the color of the last inserted point. The `size_expand`

parameter increases the size of the root octree node so it is slightly bigger than the original point cloud bounds to accomodate all points.

```
[2]:
```

```
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()),
center=pcd.get_center())
pcd.colors = o3d.utility.Vector3dVector(np.random.uniform(0, 1, size=(N, 3)))
o3d.visualization.draw_geometries([pcd])
print('octree division')
octree = o3d.geometry.Octree(max_depth=4)
octree.convert_from_point_cloud(pcd, size_expand=0.01)
o3d.visualization.draw_geometries([octree])
```

```
input
downloading armadillo mesh
extract armadillo mesh
```

```
octree division
```

## From voxel grid¶

An octree can also be constructed from an Open3D `VoxelGrid`

geometry using the method `create_from_voxel_grid`

. Each voxel of the input `VoxelGrid`

is treated as a point in 3D space with coordinates corresponding to the origin of the voxel. Each leaf node takes the color of its corresponding voxel.

```
[3]:
```

```
print('voxelization')
voxel_grid = o3d.geometry.VoxelGrid.create_from_point_cloud(pcd,
voxel_size=0.05)
o3d.visualization.draw_geometries([voxel_grid])
print('octree division')
octree = o3d.geometry.Octree(max_depth=4)
octree.create_from_voxel_grid(voxel_grid)
o3d.visualization.draw_geometries([octree])
```

```
voxelization
```

```
octree division
```

Additionally, an `Octree`

can be coverted to a `VoxelGrid`

with `to_voxel_grid`

.

## Traversal¶

An octree can be traversed which can be useful for searching or processing subsections of 3D geometry. By providing the `traverse`

method with a callback, each time a node (internal or leaf) is visited, additional processing can be performed.

In the following example, an early stopping criterion is used to only process internal/leaf nodes with more than a certain number of points. This early stopping ability can be used to efficiently process spatial regions meeting certain conditions.

```
[4]:
```

```
def f_traverse(node, node_info):
early_stop = False
if isinstance(node, o3d.geometry.OctreeInternalNode):
if isinstance(node, o3d.geometry.OctreeInternalPointNode):
n = 0
for child in node.children:
if child is not None:
n += 1
print(
"{}{}: Internal node at depth {} has {} children and {} points ({})"
.format(' ' * node_info.depth,
node_info.child_index, node_info.depth, n,
len(node.indices), node_info.origin))
# we only want to process nodes / spatial regions with enough points
early_stop = len(node.indices) < 250
elif isinstance(node, o3d.geometry.OctreeLeafNode):
if isinstance(node, o3d.geometry.OctreePointColorLeafNode):
print("{}{}: Leaf node at depth {} has {} points with origin {}".
format(' ' * node_info.depth, node_info.child_index,
node_info.depth, len(node.indices), node_info.origin))
else:
raise NotImplementedError('Node type not recognized!')
# early stopping: if True, traversal of children of the current node will be skipped
return early_stop
```

```
[5]:
```

```
octree = o3d.geometry.Octree(max_depth=4)
octree.convert_from_point_cloud(pcd, size_expand=0.01)
octree.traverse(f_traverse)
```

```
0: Internal node at depth 0 has 8 children and 2000 points ([-2.5296415 30.57793351 2.22678118])
0: Internal node at depth 1 has 4 children and 65 points ([-2.5296415 30.57793351 2.22678118])
1: Internal node at depth 1 has 2 children and 45 points ([-2.0246415 30.57793351 2.22678118])
2: Internal node at depth 1 has 8 children and 402 points ([-2.5296415 31.08293351 2.22678118])
0: Internal node at depth 2 has 2 children and 5 points ([-2.5296415 31.08293351 2.22678118])
1: Internal node at depth 2 has 1 children and 5 points ([-2.2771415 31.08293351 2.22678118])
2: Internal node at depth 2 has 4 children and 43 points ([-2.5296415 31.33543351 2.22678118])
3: Internal node at depth 2 has 1 children and 5 points ([-2.2771415 31.33543351 2.22678118])
4: Internal node at depth 2 has 4 children and 55 points ([-2.5296415 31.08293351 2.47928118])
5: Internal node at depth 2 has 5 children and 96 points ([-2.2771415 31.08293351 2.47928118])
6: Internal node at depth 2 has 4 children and 72 points ([-2.5296415 31.33543351 2.47928118])
7: Internal node at depth 2 has 6 children and 121 points ([-2.2771415 31.33543351 2.47928118])
3: Internal node at depth 1 has 7 children and 363 points ([-2.0246415 31.08293351 2.22678118])
0: Internal node at depth 2 has 1 children and 5 points ([-2.0246415 31.08293351 2.22678118])
2: Internal node at depth 2 has 1 children and 6 points ([-2.0246415 31.33543351 2.22678118])
3: Internal node at depth 2 has 4 children and 11 points ([-1.7721415 31.33543351 2.22678118])
4: Internal node at depth 2 has 4 children and 82 points ([-2.0246415 31.08293351 2.47928118])
5: Internal node at depth 2 has 3 children and 46 points ([-1.7721415 31.08293351 2.47928118])
6: Internal node at depth 2 has 6 children and 105 points ([-2.0246415 31.33543351 2.47928118])
7: Internal node at depth 2 has 6 children and 108 points ([-1.7721415 31.33543351 2.47928118])
4: Internal node at depth 1 has 5 children and 347 points ([-2.5296415 30.57793351 2.73178118])
0: Internal node at depth 2 has 4 children and 60 points ([-2.5296415 30.57793351 2.73178118])
1: Internal node at depth 2 has 4 children and 103 points ([-2.2771415 30.57793351 2.73178118])
2: Internal node at depth 2 has 2 children and 22 points ([-2.5296415 30.83043351 2.73178118])
3: Internal node at depth 2 has 8 children and 141 points ([-2.2771415 30.83043351 2.73178118])
7: Internal node at depth 2 has 1 children and 21 points ([-2.2771415 30.83043351 2.98428118])
5: Internal node at depth 1 has 4 children and 325 points ([-2.0246415 30.57793351 2.73178118])
0: Internal node at depth 2 has 8 children and 178 points ([-2.0246415 30.57793351 2.73178118])
1: Internal node at depth 2 has 1 children and 1 points ([-1.7721415 30.57793351 2.73178118])
2: Internal node at depth 2 has 8 children and 143 points ([-2.0246415 30.83043351 2.73178118])
6: Internal node at depth 2 has 1 children and 3 points ([-2.0246415 30.83043351 2.98428118])
6: Internal node at depth 1 has 6 children and 234 points ([-2.5296415 31.08293351 2.73178118])
7: Internal node at depth 1 has 5 children and 219 points ([-2.0246415 31.08293351 2.73178118])
```

## Find leaf node containing point¶

Using the above traversal mechanism, an octree can be quickly searched for the leaf node that contains a given point. This functionality is provided via the `locate_leaf_node`

method.

```
[6]:
```

```
octree.locate_leaf_node(pcd.points[0])
```

```
[6]:
```

```
(OctreePointColorLeafNode with color [0.611293, 0.0542197, 0.503748] containing 4 points.,
OctreeNodeInfo with origin [-1.96152, 31.1461, 2.79491], size 0.063125, depth 4, child_index 7)
```