Morphological Operators

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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
3
2
5
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1 Set of pixels (white)
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
Region
Morphological Operation
Structuring Element
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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 dont 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
• Dilation
• Erosion
• Closing
• Opening
• Hit And Miss

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Dilation

• Used to enlarge a region
• 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 1s
INPUT
OUTPUT
DIFFERENCE
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Erosion

• Used to shrink a region
• Sweep the structuring element over the image
• When all structuring elements (the 1s) 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 1s
INPUT
OUTPUT
DIFFERENCE
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Closing

• Used to close internal holes and eliminate
  concavities from regions
• 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 1s
INPUT
OUTPUT
DIFFERENCE
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Opening

• Used to remove tails from regions
• 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 1s
INPUT
OUTPUT
DIFFERENCE
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Another Example
With a 3x3 structure element of all 1s
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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 1s (region pixels),
  0s (non-region pixels), and dont 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
• dont care indicators
• Foreground indicators must land on region pixels
• Background indicators must land on non-region
  pixels
• Dont care indicators can land on anything

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Hit And Miss
STRUCTURING ELEMENT
OUTPUT
INPUT
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) n 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 dont 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
• Its 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 (?, ?) in Hough space represents line
  in image space
• ? is the perpendicular distance from the line to
  the origin
• ? 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
• for all values of ?. For example

Xp, Yp
Xp, Yp
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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 ?
• Calculate ?
• Increment the (?, ?) bin

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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 wont 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 wont always be perfectly
  collinear and therefore the Hough Space
  sinusoidal waves wont 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)
• Well look at various color representations
• I especially like this topic ?