Representation

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Representation Description

• Representation of a segmented region
• In terms of its external characteristics
  (boundary)
• In terms of its internal characteristics
  (pixels comprising the region)

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Representation Description

• Description of the region based on the chosen
  representation
• e.g. boundary length, orientation of straight
  line joining the extreme points, number of
  concavities.
• Descriptors should be insensitive to changes in
  size, translation, rotation,

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Representation Schemes
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Representation Schemes

• Chain Codes
• Polygonal Approximations
• Signatures
• Boundary Segments
• The Skeleton of a Region

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Chain Codes

• Represent boundaries by a connected sequence of
  straight-line segments of specified length and
  direction.
• Based on 4- or 8-connectivity
• Direction of each segment coded by a numbering
  scheme

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Representation Description
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Chain Codes

• Problems
• Long chains of codes
• Easily disturbed by noise, and sidetracked
• Solution
• Resampling using larger grid spacing
• Normalizations

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Representation Description
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Polygonal Approximations

• To capture the essence of the boundary shape with
  the fewest possible polygonal segments.
• Various methods
• Minimum perimeter polygons
• Merging techniques (least square error line fit)

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Representation Description
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Polygonal Approximations

• Merging technique problem corners
• Solution
• Splitting to subdivide a segment successively
  into two parts until a given criterion is
  satisfied.
• Objective seeking prominent inflection points.

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Representation Description
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Signatures

• 1-D functional representation of a boundary
• To generate
• Plot the distance from the centroid to the
  boundary as a function of angle.

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Representation Description
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Signatures

• Invariable to translation, but depends on
  rotation scaling
• normalization of the procedure is necessary
• e.g. select the same starting point

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Signatures

• Changes in size change the amplitude values
• Scaling the functions so that they always span
  the same range of values, e.g. 0,1 might help.
• Disadvantage it only depends on minimum
  maximum values.

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Signatures

• Another way to generate signatures
• Plot the angle between a line tangent to the
  boundary and a reference line as a function of
  position along the boundary
• e.g. horizontal segments would correspond to
  straight lines as the tangent angle would be
  constant there.

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Signatures

• Another way to generate signatures
• Slope-density function a histogram of
  tangent-angle values.

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Boundary Segments

• Decomposition of a boundary into segments reduces
  the boundarys complexity and simplifies the
  description process.
• Convex hull H of a set S is the smallest convex
  containing S
• H-S convex deficiency D of S
• Scheme independent of size and orientation.

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Representation Description
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Boundary Segments

• Prior to partitioning, smooth the boundary
• e.g. by replacing each pixel by the average
  coordinates of m of its neighbors along the
  boundary
• or use a polygonal approximation prior to finding
  the convex deficiency

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The Skeleton of a Region

• To reduce a plane region to a graph
• by e.g. obtaining the skeleton of the region via
  thinning.

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The Skeleton of a Region

• Find medial axis transformation (MAT)
• The MAT of region R with border B is found as
• For each point p in R, we find its closest
  neighbor in B.
• If p has more than one, it belongs to the Medial
  Axis (skeleton) of R.

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Representation Description
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The Skeleton of a Region

• To improve computational efficiency, in essence
  we perform thinning
• Edge points of a region are iteratively deleted
  if
• End points are not deleted
• Connectedness is not broken
• No excessive erosion is caused

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Representation Description
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Boundary Descriptors

• Length
• Number of pixels
• Number of vertical and horizontal components v2
  times the number of diagonal components

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Boundary Descriptors

• Diameter
• D distance measure pi,pj boundary points
• Major axis (connecting the two extreme points)

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Boundary Descriptors

• Curvature
• Rate of change of slope
• i.e. using the difference between the slopes of
  adjacent boundary segments, which have been
  represented as straight lines, as a descriptor of
  curvature at the point of intersection of the
  segments.

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Boundary Descriptors

• Curvature (cont.)
• Convex segment change in slope at p is
  nonnegative
• Concave segment change in slope at p is negative
• Ranges in the change of slope
• Less than 10 ? line
• More than 90 ? corner

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Boundary Descriptors

• Other Boundary Descriptors
• Shape numbers
• Fourier descriptors
• Moments

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Representation Description
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Chapter 11 Representation Description
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Regional Descriptors

• Area of pixels within the boundary
• Perimeter length of boundary
• Can be used with area to measure compactness
  (perimeter2/area)
• Compactness is
• Dimensionless, and thus insensitive to scale
  changes
• Insensitive to orientation

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Regional Descriptors

• Principal axes
• Eigenvectors of the covariance matrix
• Ratio of large to small eigenvalue insensitive
  to scale and rotation

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Regional Descriptors

• Other descriptors
• Mean and median of gray levels
• Min. and max. gray-level values
• pixels with values above and below the mean

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Topological Descriptors

• Topology
• properties of figures that are unaffected by
  deformations
• no tearing or joining though ? rubber sheet
  distortions.

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Topological Descriptors

• Examples
• of holes H
• of connected components C
• A subset of maximal size such that any two points
  can be joined by a connected curve lying entirely
  within the subset.
• Euler number E E C-H
• also a topological property

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Representation Description
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Topological Descriptors

• Regions represented by straight line segments
  (polygonal network)

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Representation Description
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Representation Description
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Morphology

• Morphology deals with form and structure
• Mathematical morphology is a tool for extracting
  image components useful in
• representation and description of region shape
• preprocessing (filtering, thinning, etc.)

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Basic Morphological Algorithms

• Purpose
• to extract image components that are useful in
  the representation and description of shape.
• Boundary Extraction

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Basic Morphological Algorithms

• Extraction of Connected Components

Where X0p ? when XkXk-1 the algorithm has
converged.
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Relational Descriptors

• To organize boundaries regions to exploit any
  structural relationships that may exist between
  them.
• Example

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Relational Descriptors

• In the previous example
• Recursive relationship involving the primitive
  elements a and b.
• Rewriting rules
• S ? aA
• A ? bS, and
• A ? b

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Relational Descriptors

• When dealing with disjoint structures, tree
  descriptors are used
• A tree T is a finite set of one or more nodes for
  which
• There is a unique node designated the root
• The remaining nodes are partitioned into m
  disjointed sets T1, , Tm, each of which in turn
  is a tree called a subtree of T.

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Relational Descriptors

• The tree frontier is the set of nodes at the
  bottom of the tree (leaves), taken in order from
  left to right.

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Relational Descriptors

• Two types of information are important (in a
  tree)
• Information about a node stored as a set of words
  describing the node
• Information relating a node to its neighbors
  stored as a set of pointers to those neighbors

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Representation Description
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Representation Description
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Representation Description
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The End

• (or is it?)