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Digital Image Processing ee.sharif.edu/~dip

Representation and Description

• Representation and Description
– Representing regions in 2 ways:
• Based on their external characteristics (its boundary):
– Shape characteristics
• Based on their internal characteristics (its region):
– Regional properties: color, texture, and …
• Both

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E. Fatemizadeh, Sharif University of Technology, 2011



• Representation and Description
– Describes the region based on a selected
representation:
• Representation  boundary or textural features
• Description  length, orientation, the number of
concavities in the boundary, statistical measures of
region.

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• Invariant Description:
– Size (Scaling)
– Translation
– Rotation

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• Boundary (Border) Following:
– We need the boundary as a ordered sequence of points.

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• Moore Boundary Tracking Algorithm:
1. Let the starting point, b0 , be the uppermost, leftmost point in the
image that is labeled 1. Denote by c0 the west neighbor of b0 . Clearly
c0 always will be a background point. Then, examine the 8 neighbors
of b0 , starting at c0 and proceeding in clockwise direction. Let b1
denote the first pixel encountered whose value is 1, and let c1 be the
points immediately preceding b1 in the sequence. Store the location
of b0 and b1.
2. Let b=b1 and c=c1.
3. Let the 8 neighbors of b , starting at c and proceeding in a clockwise
direction be denoted by n1,n2,... , nk . Find the first nk whose value is
1.
4. Let b=nk and c=nk-1.
5. Repeat steps 3 and 4 until b=b0 and the next boundary point found
is b1. 5
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• Example:

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• Freeman Chain Code:
– Code the 4 or 8 connectivity

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• Chain Code Problems:
– Long Code
– Low noise Robustness
• Solution: Resampling
– Starting Point:
• Solution: Rotary/Circular shift until forms a minimum integer
– 10103322 01033221

– Angle normalization:
• First difference (Counterclockwise)
– 10103322  3133030 or 33133030 (transition between last and first)
• Useful for integer multiple of used chain code (45° or 90°)

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• Example:
– Resampling
– 4 and 8 chain codes

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• Example:

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• Polygon Approximation:
– Minimum‐Perimeter Polygons
• Read Pages 801‐807.

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• Example (Cont.):

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• Example:

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– Merging:
• Start from a seed point
• Continue on a line based on local average error (e.g. linear
regression)
• Stop if error exceeds a threshold
• Continue from the last point
• ....
– No guarantee for corner detection

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– Splitting:
• Segments to two parts based on a criteria (e.g. maximum internal
distance)
• Check each segment for splitting based on another criteria (e.g.
linearity error)
• ….

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• Signatures:
– A 1‐D functional representation of a boundary
• Distance vs. Angle (in the polar representation):
– Invariant to translation
– Non‐Invariant to rotation (may be achieved by start point selection)
» Farthest point from centroid
» The point on eigen axis
» Use chain code solution for the start point

• Line tangent angle
• Histogram of tangent angle

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• Example:

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• Example:

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• Boundary Segments:
– Convex Hull: Smallest convex set containing the S
– Convex Deficiency: The set difference D=H‐S
– Problem of noise:
• Smoothing (K‐neighbour average)
• Polygon approximation
– Convex Hull of Polygon

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• Boundary Segments:
• Skeletonization / Thinning:
– Medial Axis Transform (MAT)
• Point p is belong to medial (Region R and Border B):
– Has more than one closest neighbor in B
– Chapter 9 and Page 813

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• Example of Thinning:

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• Boundary Descriptor:
– Simple one:
• Diameter:
– Lead to Major axis and minor axis (Perpendicular to major)
– Basic Rectangle
• Curvature

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• Shape Number
– Smallest integers of first difference circular chain code.

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• Example:

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• Fourier Descriptors:

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• Example:

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• Problem of invariance:
– Rotation

– Normalization will solve!

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– Translation

– Translation to centroid will solve!

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– Scaling

– Normalization will solve

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– Starting Point

– Normalization will solve

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• Properties in a single table:

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• Statistical Moments:
– Boundary Representation

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• Regional Descriptor:
– The simple one:
• Area (Number of pixels)
• Perimeter (Length of boundary)
• Compactness (Perimeter2/Area)
• Circularity: Ratio of the area to the area of a circle with same
perimeter
• Mean, median, max, min, ratio pixels above/below … from
intensity data.

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• Example:
– Normalized area:
• Ratio of light pixels to total light pixels

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• Topology Descriptor:
– Number of holes (H):
• Invariants to several operators.

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– Number of connected components (C) :

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– Euler number (E=C‐H) :

0 ‐1

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– Polygonal net:
• V: # of vertices (7)
• Q: # of edges (11)
• F: # of faces (2)
• E=C‐H=V‐Q+F
– C=1 , H=3
– E=‐2

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Original Threshold
• Example:

Largest Connected Region Skeleton 39
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• Texture:
– No formal definition
Smooth Coarse Regular

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• Statistical Approaches
– 1st order grey level statistics
• From normalised histogram
– One pixel gray level repeat n times
– 2nd order grey level statistics
• From GLCM (Grey Level Co‐ocurrence Matrix)
– Repeatation of two pixels in a pre‐defined neighbourhood
• Needs:
– A Positioning Operator, P.
– GLCM(i,j): # of times that points with gray level Zi occure relative to
points with gray level Zj

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• Texture feature from 1st order statistics:

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• Example:
Smooth Coarse Regular

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• Problem with 1st order histogram
– Lack of spatial information

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• Gray Level Co‐Occurence Matrix (GLCM):

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• Texture feature from GLCM

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• Example:
– One pixel immediately to he right

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• Example:

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• Effect of Horizontal offset:
– Correlation index
– Horizontal distance between neighbors

Noisy Sin Circuit board

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• Spectral approaches:
– Peaks and Frequencies related to periodicity:
– S(r,θ), Sr(θ),Sθ(r):

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• Spectral approaches:

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• Spectral approach:

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• Moments of Image as a 2D pdf:

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• Digital Data:

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• Moments:

• A set of invariant moments (7 by Hu)

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• Hu Moments:

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• Example:

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• Results of invariant moments:

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• Complex Zernike Moments:

• Where Vmn are Zernike polynomials:

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• Some Zernike Polynomials:

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• How to Compute Zernike Polynomials:
– Scale and Translation invariance transform:
• Shift to center of gravity (m10 = m01= 0)
• Fix m00 at a predetermined value (= ):

– Map Image to the unit disc using polar coordinates.
– The center of the image is the origin of the unit disc
– Those pixels falling outside the unit disc are not used in
the computation.

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• How to Compute Zernike Polynomials:

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• Legendre Moments:

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• Fourier‐Mellin Moments:
– Fourier‐Mellin Transform:

– Fourier‐Mellin Moments:

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• Principal Component Analysis:
– Consider a multi (value/Channel/Spectral) images:

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• Karhunen–Loève or Hotelling Transform:
– Analysis:

– Complete Synthesis:

– Optimal Synthesis:

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• Example:

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• Data Preparation:

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• Eigenvalues Analysis:

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• PCA Analysis:
– Eigneimages

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• Example:
– Using 2 components

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• Example:
– Difference

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• PCA in Boundary description:

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• PCA in Boundary description:

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