Quadtrees, Octrees and their Applications in Digital Image Processing

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

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

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

Example of a four layer pyramid 2 3 * 2 3

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

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?

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

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

The partitioning Algorithm

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

Types of pyramids: quadtree and octree

Recursive Tree Decomposition Think how to write this software in Lisp

Construction of the quadtree

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

Disadvantages of the quadtree

Structure of an Octree

Structure of an Octree

Advantages of the Octrees

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

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

Application to pattern recognition

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

Tree Decomposition in Pattern Classification

Tree Decomposition in Pattern Classification

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.

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

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

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

Extracting compact objects

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

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.

Using Quadtrees to Smooth Images

Using Quadtrees to Smooth Images

Using Quadtrees to Smooth Images

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

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

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

Quadtree Compression

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

Hierarchical Coding of Binary Images

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

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.

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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