Quadtrees, Octrees and their Applications in Digital Image Processing

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Quadtrees, Octrees and their Applications in
Digital Image Processing
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Hierarchical Data Structures for Computer Vision
and Image Processing
Definition of pyramids Explanation of Quadtrees
and Octrees Techniques used for
generation Applications
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What is a pyramid?
A(0)
A(1)
2 1 2 1
A(2)
2 2 2 2
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Example of a four layer pyramid
2 0 2 0
2 1 2 1
2 2 2 2
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Example of a four layer pyramid
Layer 3
2 3 2 3
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How partitioning is done?
1 partitioned to 2 2 4
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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?

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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
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The partitioning Algorithm
Cell (j,k) at level i-1
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The partitioning Algorithm
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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
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Types of pyramids quadtree and octree
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Recursive Tree Decomposition
Think how to write this software in Lisp
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Construction of the quadtree
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Advantages of the quadtree
Trees can be well manipulated in software, for
instance in Lisp
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Disadvantages of the quadtree
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Structure of an Octree
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Structure of an Octree
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Advantages of the Octrees
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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

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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

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Application to pattern recognition
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Tree Decomposition in Pattern Classification
Tree decomposition can be used not only in image
space but in transform space or feature space
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Tree Decomposition in Pattern Classification
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Tree Decomposition in Pattern Classification
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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.

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

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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

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Extracting compact objects
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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

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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.

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Using Quadtrees to Smooth Images
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Using Quadtrees to Smooth Images
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Using Quadtrees to Smooth Images
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Using Quadtrees to Smooth Images
Using Quadtrees to Smooth Images
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Using Quadtrees to Smooth Images
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
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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
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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
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Quadtree Compression
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G goto ground
Wwhite Bblack
w
w
Second level
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Hierarchical Coding of Binary Images
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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

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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.

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Problems to solve

• Use octree to represent the space for robot
  manipulator
• Use this space description to plan precise
  assembling operations.

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References
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References
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References
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References