Morphological Operators
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More on Homogeneous Intensity
Regions
• Morphology
– Refers to “form and structure”
– Defined as set operations
• Pixels that make up a region are the elements of the set
• Pixels not in the region are the elements outside the set
– Operate on binary images where connected
components have been identified
– Basic implementation is similar to that of
convolution with a kernel
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Pixels as Sets
1
2
5
1 Set of pixels (white)
3
4
5 Sets of pixels
(connected components with boundaries)
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Morphology Operations
• Inputs
– Binary image (connected components)
– Structuring element (kernel)
• Output
– Processed regions
Region
Morphological
Operation
Structuring
Element
Region
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Structuring Element
• Analogous to a convolution kernel
• Acts as a “probe” of the binary image
• Designate an origin of the element
– May be the central pixel
– May be some other pixel
• Can be any shape or size
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Structuring Element
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1
1
1
1
1
1
1 1 1 1 1
1 1 1
1 1 1
1 1 1
1
1 1 1
1 1 1 1 1
1 1 1
1
1 1 1 1 1
1
1
1
1
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Structuring Element
• Notice the empty spaces on the elements
– These spaces are treated as “don’t cares”
– Clearly we must represent them as something
when writing code
– You choose what the values should be and make
sure you handle them consistently
– More on this later
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Basic Operations
• Five common Morphological Operations
1.
2.
3.
4.
5.
Dilation
Erosion
Closing
Opening
Hit And Miss
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Dilation
• Used to enlarge a region
B  S   Sb
bB
• Sweep the structuring element over the image (as you did with
convolution)
• Each time the origin lands on a pixel within the region…
– Write a 1 wherever the structuring element is a 1
– That is, replicate the structuring element for every region
pixel
• The effect will be to thicken lines, fill gaps and holes, and
grow object boundaries
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Dilation
With a 3x3 structure element of all 1’s:
INPUT
OUTPUT
DIFFERENCE
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Erosion
• Used to shrink a region
B  S  b | b  s  Bs  s
• Sweep the structuring element over the image
• When all structuring elements (the 1’s) cover region pixels
– Place a 1 at the origin pixel of the output image
• The effect will be to thin lines and shrink object boundaries
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Erosion
With a 3x3 structure element of all 1’s:
INPUT
OUTPUT
DIFFERENCE
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Closing
• Used to close internal holes and eliminate concavities
from regions
B  S  B  S   S
• Concatenation of dilation/erosion operations
• Perform a dilation on the input image to produce an
intermediate image
• Perform an erosion on the intermediate image to
produce the output image
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Closing
With a 3x3 structure element of all 1’s:
INPUT
OUTPUT
DIFFERENCE
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Opening
• Used to remove “tails” from regions
B  S  B  S   S
• Concatenation of erosion/dilation operations
• Perform an erosion on the input image to produce an
intermediate image
• Perform an dilation on the intermediate image to
produce the output image
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Opening
With a 3x3 structure element of all 1’s:
INPUT
OUTPUT
DIFFERENCE
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Another Example
INPUT
DILATE
OPEN
ERODE
CLOSE
With a 3x3 structure element of all 1’s:
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Hit And Miss
• Searches for specific patterns in the image
• Sweep the structuring element over the image
• When all structuring elements match
underlying image pixels exactly
• Consider structure element 1’s (region pixels), 0’s (nonregion pixels), and “don’t cares”
– Logical OR the image pixel corresponding to the
structure element origin into the output image (i.e.
write a value to the output where the origin of the
structuring element is)
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Hit And Miss
• Note that the Structuring Element contains three types
of values
– Foreground (region) indicators
– Background (non-region) indicators
– “don’t care” indicators
• Foreground indicators must land on region pixels
• Background indicators must land on non-region
pixels
• Don’t care indicators can land on anything
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Hit And Miss
INPUT
STRUCTURING
ELEMENT
OUTPUT
1
0 1 1
0 0
There is a white pixel here
(lower left corner of the square)
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Compound Operations
• Advanced operations can be created by
combining the basic operations
• Skeletonization (thinning)
– Thin(i, j) = I(i, j) ∩ HitAndMiss(I(i, j))
– This will take a “wide” object and produce a thin
object
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From Edges to Lines
The Hough Transform
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From Edges To Lines
• Edges are a good start
– They contain some important information and
reduce the amount of raw data
– We don’t have to deal with all the pixel intensities
that may not be all that informative
• But now we must do something with the edges
– As we saw last week, computing texture was one
of the things we can do
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Hough Transform
Hough
Transform
Binarized edgels (edge pixels),
not a line!
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Hough Transform
Hough
Transform
Binarized edgels (edge pixels),
not a line!
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Hough Transform
Hough
Transform
Binarized edgels (edge pixels),
not a line!
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Hough Transform
Hough
Transform
Binarized edgels (edge pixels),
not a line!
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Hough Transform
Hough
Transform
Binarized edgels (edge pixels),
not a line!
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Hough Transform
Hough
Transform
Binarized edgels (edge pixels),
not a line!
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Hough Transform
Hough
Transform
Binarized edgels (edge pixels),
not a line!
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Hough Transform
Hough
Transform
Binarized edgels (edge pixels),
not a line!
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Hough Transform
• What exactly is going on?
• The Hough Transform is a technique for
finding straight lines in a mass of disconnected
points
• It’s a “model-based” technique
– It utilizes a mathematical equation (a model) of a
line and then tries to fit points to the equation
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Hough Transform
• Input image is [typically] a binarized edge magnitude
map
– You could define it to use edge magnitudes in a manner
similar to the Canny edge detector
• The process maps image space edge points to Hough
Space coordinates
• The output image is the location of lines in image
space
– What I showed was a rendering of Hough space which is
not really the desired output
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Hough Space
• A line in image space maps to a point in Hough Space
• A point in image space maps to multiple points in
Hough Space
• Each point (ρ, θ) in Hough space represents line in
image space
– ρ is the perpendicular distance from the line to the origin
– θ is the orientation of the line relative to a specified axis
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Hough Transform
• A single edge point in image space can be part
of an infinite number of lines in image space
– Since “infinite” is a bit difficult to deal with in the
digital computer world we quantize the orientation
of the lines into convenient bins
• To compute the Hough Transform of a single
point in image space we measure the distance
from the origin to every possible line through
the point
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Hough Transform
• Specifically, we compute
  x cos( )  y sin( )
for all values of θ. For example:
Xp, Yp
Xp, Yp
ρ
ρ
θ
θ
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Hough Space
• The result is a sinusoidal waveform
in Hough space
There is a single
dot in here
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Implementation
• Basic data structure is the accumulator matrix
• Similar to intensity
histogram creation
• Initialize each location to 0
• Set θ
• Calculate ρ
• Increment the (ρ, θ) bin
ρ
θ
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Mechanization
• When multiple dots contribute to the same line
(are colinear) in image space the sin waves
will intersect in a single point in Hough Space
• The coordinates of that point represent the
parameters of the line
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One Dot
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Two Collinear Dots
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Three Collinear Dots
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Four Collinear Dots
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Implementation Considerations
• Must settle on an origin
–
–
–
–
–
Upper left image corner
Image center
Lower left image corner
Whatever
Just choose one and document it
• Must select a quantization of the orientation
– Too fine of a quantization and the sinusoidal waves won’t
intersect in a single point
– Too coarse of a quantization and near parallel lines are
accumulated into a single bin
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Implementation Considerations
• The image points won’t always be perfectly
collinear and therefore the Hough Space
sinusoidal waves won’t always intersect in a
single point
– Angle quantization
– Non-maximal peak suppression on the Hough
accumulator array may be employed to reduce
clustering effects (similar to the process used by
Canny)
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Implementation Considerations
The answer is in here somewhere
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Implementation Considerations
• After non-maximal peak suppression you may
want to binarize (threshold) the accumulator
array leaving only the strongest peaks
– Those represent locations where the most sinusoids
intersected
– This is where the most number of points
contributed to a given line (high degree of
colinearity)
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Implementation Considerations
• The basic Hough Transform does not keep
track of which points contribute to a line
– Make each accumulator bin a linked-list (or array)
of image space point locations if you want to do
this
• Edge orientation information is completely
ignored
– May want to store this or use it to weight a
particular edges contribution to the accumulator
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“Gotchas”
• Make sure your accumulator array is initialized
to 0 each time you compute a Hough
Transform
• Make sure your accumulator array is big
enough to hold all possible distances
– Distances can be as far as the diagonal length of
the image
– Distances can be positive or negative dependent on
how you choose your origin
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Generalized Hough Transform
• The only thing specific to a straight line is the
parameterization – the process is general
• Basically, any shape that can be quantitatively
described can be detected with the Hough
Transform
– Many approaches to detecting circles have been
published – especially fixed radius circles
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Remainder of Class Period
• Open for questions regarding current or past
programming assignments
• Write code to perform Morphology operations:
–
–
–
–
–
Erosion
Dilation
Open
Close
Hit and Miss
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Things To Do
• Write code to perform
– The five basic morphological operations
– The Hough Transform
– Due 2 weeks from now
• Reading for Next week
– Chapter 6 – Color
• Why now? – because it is used to do segmentation, region
description, and other mid-level vision stuff (as well as low-level
operations)
• We’ll look at various color representations
• I especially like this topic 
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