1. Digital Image Processing Lecture 20: Representation & Description
Prof. Charlene Tsai
* Materials from Gonzales chapter 11

2. Representation
After segmentation, we get pixels along the boundary or pixels contained in a region.
Representing a region involves two choices:
Using external characteristics, e.g. boundary
Using internal characteristics, e.g. color and texture

3. Common Representation
Common external representation methods are:
Chain code
Polygonal approximation
Signature
Boundary segments
Skeleton (medial axis)
We’ll discuss the first 3.

4. Method 1: Chain Codes
Represent a boundary by a connected sequence of straight-line segments of specified length and direction.
Directions are coded using the numbering scheme
4-connectivity
8 connectivity

5. Generation of Chain Codes
Walking along the boundary in clockwise direction, for every pair of pixels assign the direction.
Problems:
Long chain
Sensitive to noise
Remedies?
Resampling using larger
grid spacing.

6. Resampling for Chain Codes
0766…12
0033…01
(8-directional)
(4-directional)
Spacing of the sampling grid affect the accuracy of the representation

7. Normalization for Chain Codes
With respect to starting point:
Make the chain code a circular sequence
Redefine the starting point which gives an integer of minimum magnitude
E.g normalized to
For rotation:
Using first difference (FD) obtained by counting the number of direction changes in counterclockwise direction
E.g FD of a 4-direction chain code is
For scaling:
Altering the size of the resampling grid.

8. Method 2: Polygonal Approximation
Approximate a digital boundary with arbitrary accuracy by a polygon.
Goal: to capture the “essence” of the boundary shape with the fewest possible polygonal segments.

9. Minimum Perimeter Polygons
Two techniques: merging & splitting

10. Merging Technique Repeat the following steps:
Merging unprocessed points along the boundary until the least-square error line fit exceed the threshold.
Store the parameters of the line
Reset the LS error to 0
Intersections of adjacent line segments form vertices of the polygon.
Problem: vertices of the polygon do not always correspond to inflections (e.g. corners) in the original boundary.

11. Splitting Technique
Subdivide a segment successively into two parts until a specified criterion is met.
For example

12. Method 3: Signature
Reduce the boundary representation to a 1D function
For example, distance vs. angle

13. (con’d) For the signature just described
Invariant to translation (everything w.r.t. the centroid)
Depending on rotation and scaling
How to remove the dependence on rotation?
Selecting point farthest from the centroid
Farthest point on the eigen axis (more robust)
Using the chain code
How to remove the dependence on scaling?
Scale all functions to span the range, say, [0,1] (not very robust)
Normalize by the variance

14. Descriptors
Boundary information can be used directly, or converted into features that describe properties of a region, such as
Length of the boundary
Orientation of straight line joining its extreme points
Number of concavities in the boundary

15. Boundary Descriptors (Properties)
Length: approximated by the # of pixels
Diameter:
Basic rectangle: defined by
Major axis: the line giving the diameter
Minor axis: the line perpendicular to major axis
Eccentricity: ratio of the 2 axes

16. Fourier Descriptor
Given a set of boundary points (x0, y0), (x1, y1), (x2, y2), …,. (xK-1, yK-1).
Represent the sequence of coordinates as s(k)=[xk, yk], for k=0, 1, 2, …, K-1.
Using complex representation for each point:
s(k) = xk + jyk
Advantage: reducing 2D to 1D problem

17. 1D Fourier (Review) DFT of s(k) is
The complex coefficients a(u), for u=0,1, 2, …, K-1, are called the Fourier descriptor of the boundary
The inverse Fourier restores s(k)

18. Fourier Approximation
Using only the first P coefficients (a(u)=0 for u>P)
Please note that k still ranges from 0 to K-1
The smaller P becomes, the more detail that is lost on the boundary
There are ways to make the descriptor insensitive to translation, rotation, and scale changes. Please refer to Gonzalez pg for details.

19. Example

20. Region Descriptors Simple descriptors: Topological descriptors:
Area: # of pixels in the region
Perimeter: length of its boundary
Compactness: (perimeter)2/area
Q: What shape gives the minimal compactness
Mean and median of the gray levels.
Topological descriptors:
# of holes
# of connected components

21. Summary Common external representation methods are: Descriptors
Chain code
Polygonal approximation
Signature
Descriptors
Boundary descriptor
Fourier descriptor
Region descriptor