Introduction to Digital Image Processing with MATLAB® Asia Edition
McAndrew‧Wang‧Tseng
Chapter 12:
Shapes and
Boundaries
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© 2010 Cengage Learning
Engineering. All Rights Reserved.
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12.1 Introduction
• How do we tell if two objects have the same
shape?
• How can we classify shape?
• How can we describe the shape of an object?
Formal means of describing shapes are called
shape descriptors, which may include size,
symmetry, and length of perimeter
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Engineering. All Rights Reserved.
12.2 Chain Codes and Shape Numbers
• The idea of a chain code is quite straightforward:
 We walk around the boundary of an object, taking
note of the direction we take
 The resulting list of directions is the chain code
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FIGURE 12.2
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12.2 Chain Codes and Shape Numbers
• Suppose we walk along the boundary in a counterclockwise direction starting at the leftmost point in the
top row and list the directions as we go
333233000101112122
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665660001222344
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Engineering. All Rights Reserved.
12.2 Chain Codes and Shape Numbers
• A simple boundary-following algorithm has been
given by Sonka et al. [35]
• The version for 4-connected boundaries:
 Start by finding the pixel in the object that has the
left-most value in the topmost row; call this pixel P0
 Define a variable dir (for direction), and set it equal
to 3
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12.2 Chain Codes and Shape Numbers
 Traverse the 3 × 3 neighborhood of the current pixel
in a counterclockwise direction, beginning the search
at the pixel in direction
 This simply sets the current direction to the first
direction counterclockwise from dir:
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12.2 Chain Codes and Shape Numbers
 The first foreground pixel will be the new boundary
element. Update dir
 Stop when the current boundary element Pn is equal
to the second element P1 and the previous boundary
pixel Pn−1 is equal to the first boundary element P0
im is a binary image
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FIGURE 12.4
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FIGURE 12.5
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12.2 Chain Codes and Shape Numbers
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12.2 Chain Codes and Shape Numbers
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12.2 Chain Codes and Shape Numbers
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12.2 Chain Codes and Shape Numbers
• 12.2.1 Normalization of Chain Codes
• There are two problems with the definition of
the chain code as given in previous sections:
 The chain code is dependent on the starting pixel
 The chain code is dependent on the orientation of
the object.
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FIGURE 12.7
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12.2 Chain Codes and Shape Numbers
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Engineering. All Rights Reserved.
12.2 Chain Codes and Shape Numbers
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Engineering. All Rights Reserved.
12.2 Chain Codes and Shape Numbers
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Engineering. All Rights Reserved.
12.2 Chain Codes and Shape Numbers
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Engineering. All Rights Reserved.
FIGURE 12.8
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Engineering. All Rights Reserved.
12.2 Chain Codes and Shape Numbers
• 12.2.2 Shape Numbers
 We now consider the problem of defining a chain
code that is independent of the orientation of the
object
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12.2 Chain Codes and Shape Numbers
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12.2 Chain Codes and Shape Numbers
 Suppose we rotate the shape so that it has a
different orientation (we use the rot90 function,
which rotates a matrix by 90°)
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12.2 Chain Codes and Shape Numbers
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FIGURE 12.9
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FIGURE 12.10
(normalized)
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12.2 Chain Codes and Shape Numbers
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12.2 Chain Codes and Shape Numbers
This is already normalized
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12.2 Chain Codes and Shape Numbers
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12.3 Fourier Descriptors
• The idea is this: suppose we walk around an object, but
instead of writing down the directions, we write down
the boundary coordinates
• The final list of (x, y) coordinates can be turned into a
list of complex numbers z = x + yi
• The Fourier transform of this list of numbers is a Fourier
descriptor of the object
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12.3 Fourier Descriptors
• We can easily modify our function chaincode4.m to
boundary4.m by replacing the lines
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12.3 Fourier Descriptors
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12.3 Fourier Descriptors
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FIGURE 12.11
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12.3 Fourier Descriptors
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FIGURE 12.12
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12.3 Fourier Descriptors
• The Fourier transform of c contains only three
nonzero terms
• Only two terms of the transform are enough to begin
to get some idea of the shape, size, and symmetry of
the object
• Even though the shape itself has been greatly
hanged—a square has become a circle—many shape
descriptors are still little changed (such as size and
symmetry)
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FIGURE 12.13
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FIGURE 12.14
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FIGURE 12.14
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Engineering. All Rights Reserved.