@ -2,15 +2,19 @@ This blog post is the first part of a multi-post series on using quadtrees in Py
This post goes over quadtrees' basics and how you can implement a basic point quadtree in Python.
This post goes over quadtrees' basics and how you can implement a basic point quadtree in Python.
Future posts aim to apply quadtrees in image segmentation and analysis.
Future posts aim to apply quadtrees in image segmentation and analysis.
A quadtree is a data structure where each node has exactly four children. This property makes it particularly suitable for spatial searching.
In a point-quadtree, leaf nodes are a single unit of spatial information. A quadtree is constructed by continuously dividing each node until each leaf node only has a single node inside of it.
However, this partitioning can be modified so that each leaf node only contains at most K elements or that each cell can be at a maximum X large.
A quadtree is a data structure where each node has exactly four children. This property makes it particularly suitable for spatial searching.
Quadtrees are generalized as "k-d/k-dimensional" trees when you have more than 4 divisions at each node.
In a point-quadtree, leaf nodes are a single unit of spatial information.
A quadtree is constructed by continuously dividing each node until each leaf node only has a single node inside of it.
However, this partitioning can be modified so that each leaf node contains no more than K elements or that each cell can be at a maximum X large.
This stopping criterion is similar to that of the stopping criteria when creating a decision tree.
Although usually used in two-dimensions, quadtrees can be expanded to an arbitrary amount of dimensions. The lovely property of quadtrees is that it is a "dimensional reduction" algorithm. Rather than operating in O(n^2) for a traditional linear search in two dimensions, a quadtree can accomplish close to O(log n) time for most operations.
Although usually used in two-dimensions, quadtrees can be expanded to an arbitrary amount of dimensions.
The lovely property of quadtrees is that it is a "dimensional reduction" algorithm. Rather than operating in O(n^2) for a traditional linear search in two dimensions, a quadtree can accomplish close to O(log n) time for most operations.
# Implementing a Point Quadtree
# Implementing a Point Quadtree
To implement a quadtree, we only need a few pieces. First, we need some way to represent our spacial information.
To implement a quadtree, we only need a few pieces. First, we need some way to represent our spatial information.
In this application, we are only using points; however, we may choose to associate data with each point for an application.
In this application, we are only using points; however, we may choose to associate data with each point for an application.
```python
```python
@ -20,7 +24,7 @@ class Point():
self.y = y
self.y = y
```
```
The second thing that we need is a tree representation.
The second thing that we need is a tree data structure.
Like all tree nodes, it has children; however, what is unique about a quadtree is that each node represents a geometric region.
Like all tree nodes, it has children; however, what is unique about a quadtree is that each node represents a geometric region.
This geometric region has a shape represented by a location and a width and height. Additionally, if this is a leaf node, we need to have our node store the region's points.
This geometric region has a shape represented by a location and a width and height. Additionally, if this is a leaf node, we need to have our node store the region's points.
@ -44,7 +48,7 @@ This geometric region has a shape represented by a location and a width and heig
return self.points
return self.points
```
```
To generate the quadtree, we will be taking a top-down approach were we recursively divide the node into four regions until a certain threshold has been satisfied.
To generate the quadtree, we will be taking a top-down approach where we recursively divide the node into four regions until a certain threshold has been satisfied.
In this case, we are stopping division when each node contains less than k nodes.
In this case, we are stopping division when each node contains less than k nodes.
```python
```python
@ -94,7 +98,7 @@ def find_children(node):
The QTree class is used to tie together all the data associated with creating a quadtree.
The QTree class is used to tie together all the data associated with creating a quadtree.
This class is also used to generate dummy data and graph it using matplotlib.
This class is also used to generate dummy data and graph it using matplotlib.
```python
```Python
import random
import random
import matplotlib.pyplot as plt # plotting libraries
import matplotlib.pyplot as plt # plotting libraries
import matplotlib.patches as patches
import matplotlib.patches as patches
@ -136,11 +140,11 @@ Creating a quadtree where each cell can only contain at the most section will pr
If we change the hyperparameter to split until there is at most two objects per cell, we get larger cells.
If we change the hyperparameter to split until there are at most two objects per cell, we get larger cells.
# Future Work
# Future Work
In the near future, I plan on making a post on how you can use quadtrees to do image compression.
In the future, I plan on making a post on how you can use quadtrees to do image compression.
