1. Quadtrees, Octrees and their Applications in Digital Image Processing

2. Hierarchical Data Structures for Computer Vision and Image Processing
Definition of pyramids
Explanation of Quadtrees and Octrees
Techniques used for generation
Applications

3. What is a pyramid?
A(0)
A(1)
2 1 * 2 1
A(2)
2 2 * 2 2

4. Example of a four layer pyramid
2 0 * 2 0
2 1 * 2 1
2 2 * 2 2

5. Example of a four layer pyramid
2 3 * 2 3

6. How partitioning is done?
1 partitioned to 2 2 = 4

7. How pyramid is build? From top to bottom From bottom to top
Always recursively
Good exercise in recursion and arrays
Treat image as a Boolean or discrete function, what is the counterpart of these type of recursions?

8. Another Way of Partitioning
For simplicity, dimension = 2
Partitioning at any level i from i-1 can be done by defining a two-dimensional array A(i) for the i-th level

9. The partitioning Algorithm
Cell (j,k) at level i-1

10. The partitioning Algorithm

11. Pyramids versus trees
Pyramids are interlinked (for instance by indices) sequences of arrays with hierarchy.
Similarly we can create trees to define this hierarchy
Trees can be more convenient for processing

12. Types of pyramids: quadtree and octree

13. Recursive Tree Decomposition
Think how to write this software in Lisp

14. Construction of the quadtree

15. Advantages of the quadtree
Trees can be well manipulated in software, for instance in Lisp

16. Disadvantages of the quadtree

17. Structure of an Octree

18. Structure of an Octree

19. Advantages of the Octrees

20. Applications of these data structures
The quadtree, octree and binary tree decomposition methods are widely used in two and three dimension image processing and computer graphics
Some of the application areas involve:
the image data structure,
region representation,
picture segmentation,
component labeling,
image smoothing,
image enhancement,
data compression

21. Applications of these data structures
Pattern recognition
Shape analysis
Image segmentation
Region matching
Images can be represented with pyramids and thus, both local and global feature extraction is possible

22. Application to pattern recognition

27. Tree Decomposition in Pattern Classification
Tree decomposition can be used not only in image space but in transform space or feature space

28. Tree Decomposition in Pattern Classification

29. Tree Decomposition in Pattern Classification

30. Application to Edge Detection
The edge detection task can be accomplished by applying a point-neighborhood operator or the edge detector to every point of a large matrix
The algorithm for this works as follows:
An edge detector is applied at each point in the starting level
At each point, if the value exceeds a threshold, the operation is applied to the descendants of the point in the next finer level.

31. Application to Feature Detection
Pyramides are used for feature detection
Pyramides limit scope of the search
Pyramides are used for feature extraction

32. Application to Feature Detection
The disadvantage of this method is that the reduction of resolution will affect the visual appearance of edges and small objects
In particular, at a coarser level of resolution:
edges tend to get smeared and
region separation may disappear

33. Extracting compact objects
Many image analysis tasks require the extraction of compact objects from a background, where
the shapes of the desired objects are not known,
except for the fact that they are compact
Image segmentation using pyramids can be applied to extract such objects.
“Spot detectors” are applied to image at each level of the pyramid:
this is equivalent to applying spot detectors of many sizes to full-resolution image

34. Extracting compact objects

35. Extracting compact objects
Three sets of information are represented in the pyramid structure
1. Gray level
2. Edge magnitude and direction
3. Surroundedness
The interaction between the different types of information at each level of the pyramid leads to the final segmentation

36. Using Quadtrees to Smooth Images
Digital images usually contain noise of various kinds.
Most image processing tasks are simplified if noise removed
A general approach to noise removal is to smooth the image
Smoothing done by replacing each pixel value by a new value which is a function of the values in some neighborhood of the pixel.

37. Using Quadtrees to Smooth Images

38. Using Quadtrees to Smooth Images

39. Using Quadtrees to Smooth Images

40. Using Quadtrees to Smooth Images

41. Using Quadtrees to Smooth Images
Method 2
1. Constructs a quadtree from an image
2. Replaces each pixel by the gray level of the leaf to which each corresponds

42. Hierarchical Coding of Binary Images
Hierarchical Coding = to segment a picture into the largest possible uniform areas and to transmit a hierarchical representation of these areas.
Quadtrees can be used for coding
Pictures with large uniform areas can be highly compressed

43. Hierarchical Coding of Binary Images
The transmission result can be recreated by the receiver as soon as sufficient information about transmitted picture has been gathered

44. Quadtree Compression

45. G = goto ground
W=white
B=black w w
Second level

46. Hierarchical Coding of Binary Images

47. Hierarchical Coding of Binary Images
A bit assignment can be selected for the symbols
The coding can also be extended to three dimensions with the use of octrees

48. Problems to solve: Use pyramide for edge detection
Treat a large (12 variables) Karnaugh Map as an image. What is the counterpart of Shannon Decomposition in terms of binary trees?
Generalize to 4-valued logic and show link to quadtrees
Generalize to 8-valued logic and show link to octrees
Disscuss general links between discrete functions, images and compression methods.

49. Problems to solve:
Use octree to represent the space for robot manipulator
Use this space description to plan precise assembling operations.

50. References

51. References

52. References

53. References