Mathematical Morphology in Image Processing

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Mathematical MorphologyinImage Processing

• Dr.K.V.Pramod
• Dept. of Computer Applications
• Cochin University of Sc. Technology

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• What is Mathematical Morphology ?

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An (imprecise) Mathematical answer

• A mathematical tool for investigating geometric
  structure in binary and grayscale images.

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A (precise) Mathematical Answer
Algebra Complete Lattices
Topology Hit-or-Miss
Mathematical Morphology
Operators Erosions-Dilations
Geometry Convexity - Connectivity Distance

• A mathematical tool that studies operators on
  complete lattices

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• With in Biology, the term morphology is used for
  the study of the shape and structure of animals
  and plants.
• In image processing Mathematical morphology is
  theoretical framework for representation,
  description and pre-processing

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• Built on Minkowski set theory.
• Part of the theory of finite lattices.
• A mathematical theory or methodology for
  nonlinear image processing.
• A technique for image analysis based on shape.
• Extremely useful, yet not often used

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Morphology - advantages

• Preserves edge information
• Works by using shape-based processing
• Can be designed to be idempotent
• Computationally efficient

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Morphology - applications

• Image enhancement
• Image restoration (eg. Removing scratches from
  digital film)
• Edge detection
• Texture analysis
• Noise reduction
• There are many more applications that
  morphology can be applied to. Morphology has
  been widely researched for use in image and video
  processing.

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Some History

• Mathematical morphology was developed in the
  1970s by George Matheron and Jean Serra
• - George Matheron (1975) Random Sets and Integral
  Geometry, John Wiley.
• Jean Serra (1982) Image Analysis and
  Mathematical Morphology, Academic Press.
• Petros Maragos (1985) A Unified Theory of
  Translations-Invariant Systems with Applications
  to Morphological Analysis and Coding of Images,
  Doctoral Thesis, Georgia Tech.

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Why we use these techniques ?Shape Processing
and Analysis

• Visual perception requires transformation of
  images so as to make explicit particular shape
  information.
• Goal Distinguish meaningful shape information
  from irrelevant one.
• The vast majority of shape processing and
  analysis techniques are based on designing a
  shape operator which satisfies desirable
  properties.

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Example

• Image analysis consists of obtaining measurements
  characteristic to images under consideration.
• Geometric measurements (e.g., object location,
  orientation, area, length of perimeter)

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Mathematical Morphology

• Principles
• Such further processing is performed using one or
  a combination of several morphological
  transformations.
• The transformations work in a certain local
  neighborhood of each pixel (similarly to
  convolution) defined by so called Structuring
  Element . The structuring element can be
  square, cross-like or any other shape.

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Mathematical Morphology Binary
ImagesMorphology uses Set Theory as foundation
for many functions. The simplest functions to
implement are Dilation and Erosion

• Dilation
• Dilation replaces zeros neighboring to ones by
  ones.
• Erosion
• Erosion replaces ones neighboring to zeros by
  zeros.

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• Two other basic operations are -
• Closing Opening
• Closing
• Closing is dilation followed by erosion.
• Closing merges dense of ones together, fills
  small
• holes and smoothes boundaries.
• Closing smoothes objects by adding pixels
• Opening
• Opening is erosion followed by dilation.
• Opening removes single ones, thin lines and
  divides objects connected with a narrow path
  (neck).
• Opening smoothes objects by removing pixels

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Dilation (I)

• Brief Description
• One of the two basic operators
• Basic effect
• Gradually enlarge the boundaries of regions of
  foreground pixels on a binary image.

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Dilation (II)

• How It Works
• A set of Euclidean coordinates corresponding to
  the input binary image
• B set of coordinates for the structuring element
• Bx translation of B so that its origin is at x.
• The dilation of A by B is simply the set of all
  points x such that the intersection of Bx with A
  is non-empty.

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• Dilation in 1D is defined as
• A B x (?)x (intersection) A ? ? Üx?B
  Ax
  (1)
• Where A and B are sets in Z. this definition is
  also known as Minkowski Addition. This
  equation simply means that B is moved over A and
  the intersection of B reflected and translated
  with A is found. Usually A will be the signal or
  image being operated on and B will be the
  Structuring Element.

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. The following figure shows how dilation works
on a 1D binary signal.

• The output is given by (1) and will be set to one
  unless the input is the inverse of the
  structuring element. For example, in the input
  it is 000 would cause the output to be zero, if
  in the SE it 111.

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How dilation works
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• . Dilation has several interesting properties,
  which make it useful for image processing. These
  properties are
• Translation invariant.
• This means that the result of A dilated with B
  translated is the same as A translated dilated
  with B as given by
• (A B)x Ax B (2)
• Order invariant
• This simply means that if several dilations
  are to be done, then the order in which they are
  done is irrelevant. The result will be same
  irrespective.
• (A B) C A (B C) (3)

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• Increasing operator
• This means that if a set A, is a subject of
  another set B, then the dilation of A by C is
  still a subset of B dilated by C
• (A contained in B) A C contained in B
  C (4)
• Scale invariant
• This means that the input and structuring
  element can be scaled, then dilated and will give
  the same as scaling the dilated output
• rA rB r(A B) (5)
• Where r is a scale factor.
• These properties can be very useful in image
  processing and can result in some operations
  being simplified.

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Dilation (III)

• Guideline for Use

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Dilation (IV)

• Example Binary dilation
• Note that the corners have been rounded off.
• When dilating by a disk shaped structuring
  element, convex boundaries will become rounded,
  and concave boundaries will be preserved as they
  are.

Result of two Dilation passes using a disk shaped
structuring element of 11 pixels radius
Original image
Thresholded image
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Dilation (V)

• Example Binary dilation (edge detection)
• Dilation can also be used for edge detection by
  taking the dilation of an image and then
  subtracting away the original image.

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Dilation (VI)

• Example Binary dilation (Region Filling)
• Dilation is also used as the basis for many other
  mathematical morphology operators, often in
  combination with some logical operators.

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Dilation (VII)

• Conditional dilation
• Combination of the dilation operator and a
  logical operator
• Region filling applies logical NOT, logical AND
  and dilation iteratively.

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Dilation (VIII)

X0 One pixel which lies inside the region
Dilate the left image
AND
Step 1 Result
Negative of the boundary
Original image
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Dilation (IX)

Dilate the left image
X1
AND
Step 2 Result
Negative of the boundary
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Dilation (X)

• Repeating dilation and with the inverted
  boundary until convergence, yields

Step 4 Result
Step 5 Result
Step 6 Result
Step 3 Result
OR
Final Result
Original image
Result of Region Filling
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Dilation (XI)

• Grayscale Dilation
• Generally brighten the image
• Bright regions surrounded by dark regions grow in
  size, and dark regions surrounded by bright
  regions shrink in size.
• The effect of dilation using a disk shaped
  structuring element

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Dilation (XII)

• Example Grayscale dilation (brighten the image)
• The highlights on the bulb surface have increased
  in size.
• The dark body of the cube has shrunk in size
  since it is darker than its surroundings.

Original image
Two passes Dilation by 33 flat square
structuring element
Five passes Dilation by 33 flat square
structuring element
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Dilation (XIII)
Pepper noise image
Dilation by 33 flat square structuring element
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Morphological Erosion
Structuring Element
Pablo Picasso, Pass with the Cape, 1960
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Erosion (I)

• Brief Description
• Erosion is one of the basic operators in the area
  of mathematical morphology.
• Basic effect
• Erode away the boundaries of regions of
  foreground pixels (i.e. white pixels, typically).

Common names Erode, Shrink, Reduce
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Erosion (II)

• How It Works
• A set of Euclidean coordinates corresponding to
  input binary image
• B set of coordinates for the structuring
  element
• Bx translation of B so that its origin is at x
  
• The erosion of A by B is simply the set of all
  points x such that Bx is a subset of A

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The Opposite of Dilation is known as Erosion

• This is defined as
• A ? B x (B)x contained in A
  ?x?B Ax
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  (6)
• This definition is also known as Minkowski
  Substration. The equation simply says, erosion
  of A by B is the set of points x such that B
  translated by x is contained in A. The following
  Figure shows how erosion works on a 1D binary
  signal. This works in exactly the same way as
  dilation. However (6) essentially says that for
  the output to be a one, all of the inputs must be
  the same as the structuring element.

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How Erosion Works
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Erosion, like dilation also contains properties
that are useful for image processing

• Translation invariant.
• This means that the result of A eroded with B
  translated is the same as A translated eroded
  with B as given by
• (A ? B)x Ax ? B (7)
• Order invariant
• This simply means that if several erosions are
  to be done, then the order in which they are done
  is irrelevant. The result will be same
  irrespective.
• (A ? B) ? C A ? (B ? C) (8)

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• Increasing operator
• This means that if a set, A, is a subject of
  another set, B, then the erosion of A by C is
  still a subset of B eroded by C
• (A ? B) ? C A ? C contained in B ? C (9)
• Scale invariant
• This means that the input and structuring
  element can eb scaled, then eroded and will give
  the same as scaling the eroded output
• rA ? rB r(A ? B) (10)
• Where r is a scale factor.

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Erosion (III)

• Guideline for Use

Effect of erosion using a 33 square structuring
element
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Erosion (IV)

• Example Binary erosion
• It shows that the hole in the middle of the image
  increases in size as the border shrinks.
• Erosion using a disk shaped structuring element
  will tend to round concave boundaries, but will
  preserve the shape of convex boundaries.

Original thresholded image
Result of erosion four times with a disk shaped
structuring element of 11 pixels in diameter
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Erosion (V)

• Example Binary erosion (separate touching
  objects)

Original image (a number of dark disks)
Inverted image after thresholding
The result of eroding twice using a disk shaped
structuring element 11 pixels in diameter
Using 99 square structuring element leads to
distortion of the shapes
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Erosion (VI)

• Grayscale erosion
• Generally darken the image
• Bright regions surrounded by dark regions shrink
  in size, and dark regions surrounded by bright
  regions grow in size.
• The effect of erosion using a disk shaped
  structuring element

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Erosion (VII)

• Example Grayscale erosion (darken the image)
• The highlights have disappeared.
• The body of the cube has grown in size since it
  is darker than its surroundings.

Two passes Erosion by 33 flat square structuring
element
Five passes Erosion by 33 flat square
structuring element
Original image
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Erosion (VIII)

• Example Grayscale erosion (Remove salt noise)
• The noise has been removed.
• The rest of the image has been degraded
  significantly.

Erosion by 33 flat square structuring element
Salt noise image
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Opening (I)

• Brief Description
• The Basic Effect
• Somewhat like erosion in that it tends to remove
  some of the foreground(bright) pixels from the
  edges of regions of foreground pixels.
• To preserve foreground regions that have a
  similar shape to structuring element.

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Opening (II)

• How It works
• Opening is defined as an erosion followed by a
  dilation.
• Gray-level opening consists simply of a
  gray-level erosion followed by a gray-level
  dilation.
• Opening is the dual of closing.
• Opening the foreground pixels with a particular
  structuring element is equivalent to closing the
  background pixels with the same element.

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• Both dilation and erosion have interesting and
  useful properties. However, it would be useful
  to have the properties of both in one function.
  This can be done in two ways. The first method,
  Opening is defined as
• A ? B (A ? B) B (11)
• This simply erodes the signal and then dilates
  the result as shown in Figure 3. As can be seen,
  the zeros are opened up. Any ones that are
  shorter than the structuring element are removed,
  but the rest of the signal is left unchanged.
• This is a very useful property as it means that
  if the filter is applied once, no more changes to
  the signal will result from repeated applications
  is known as Idempotent
• (A ? B) ? B A ? B (12)

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Opening (III)

• Guidelines for Use
• Idempotence
• After the opening has been carried out, the new
  boundaries of foreground regions will all be such
  that the structuring element fits inside them.
• So further openings with the same element have no
  effect
• Effect of opening using a 33 square structuring
  element

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Opening (IV)

• Binary Opening Example
• Separate out the circles from the lines
• The lines have been almost completely removed
  while the circles remain almost completely
  unaffected.

Opening with a disk shaped structuring
element with 11 pixels in diameter
A mixture of circle and lines
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Opening (V)

• Binary Opening Example
• Extract the horizontal and vertical lines
  separately
• There are a few glitches in rightmost image where
  the diagonal lines cross vertical lines.
• These could easily be eliminated, however, using
  a slightly longer structuring element.

Original image
The result of an Opening with 39 vertically
oriented structuring element
The result of an Opening with 93 horizontally
oriented structuring element
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Opening (VI)

• Example Binary opening

Original image
Inverted image after thresholding
The result of an Opening with 11 pixel circular
structuring element
The result of an Opening with 7 pixel circular
structuring element
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Opening (VII)

• Example Grayscale opening
• The important thing to notice here is the way in
  which bright features smaller than the
  structuring element have been greatly reduced in
  intensity.
• The fine grained hair and whiskers have been much
  reduced in intensity.

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Opening (VIII)

• Compare Opening with Erosion
• Opening
• The noise has been entirely removed with
  relatively little degradation of the underlying
  image.
• Erosion
• The noise has been removed.
• The rest of the image has been degraded
  significantly.

Erosion by 33 flat square structuring element
Salt noise Image
Opening with 33 square structuring element
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Opening (IX)

• Example Grayscale opening (pepper noise)
• No noise has been removed.
• At some places where two nearby noise pixels have
  merged into one larger point, the noise level has
  even been increased.

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Closing (I)

• Brief Description
• Closing is one of the two important operators
  from mathematical morphology.
• Closing is similar in some ways to dilation in
  that it tends to enlarge the boundaries of
  foreground (bright) regions in an image.

Common names Closing
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Closing (II)

• How It works
• Closing is defined as a dilation followed by an
  erosion.
• Closing is the dual of opening.
• Closing the foreground pixels with a particular
  structuring element is equivalent to opening the
  background pixels with the same element.

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Closing

• the opposite of opening is Closing defined by
• A ? B (A B) ? B (13)
• It can be seen that this closes gaps in the
  signal in the same way as opening opened up gaps.
  Closing also has the property of being
  idempotent.

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Closing (III)

• Guidelines for Use
• Idempotence
• After the closing has been carried out, the
  background region will be such that the
  structuring element can be made to cover any
  point in the background without any part of it
  also covering a foreground point.
• So further closings will have no effect.

Effect of closing using a 33 square
structuring element
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Closing (IV)

• Example Binary closing
• If it is desired to remove the small holes while
  retaining the large holes, then we can simply
  perform a closing with a disk-shaped structuring
  element with a diameter larger than the smaller
  holes, but smaller than the larger holes.

Original image
Result of a closing with a 22 pixel diameter disk
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Closing (V)

• Example Binary closing
• Enhance binary images of objects obtained from
  thresholding.
• We can see that skeleton (B) is less complex, and
  it better represents the shape of the object.

Result of closing with a circular structuring
element of size 20
Original image
Thresholded image
(B) The skeleton of the image produced by
the closing operator
(A) The skeleton of the image which was only
thresholded
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Closing (VI)

• Example Grayscale closing
• Gray-level closing can similarly be used to
  select and preserve particular intensity patterns
  while attenuating others.
• Notice how the dark specks in between the bright
  spots in the hair have been largely filled in to
  the same color as the bright spots.
• But, the more uniformly colored nose area is
  largely the same intensity as before.

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Closing (VII)

• Compare Closing with Dilation
• Closing
• The noise has been completely removed with only a
  little degradation to the image.
• Dilation
• Note that although the noise has been effectively
  removed, the image has been degraded
  significantly.

Closing with 33 square structuring element
Dilation by 33 flat square structuring element
Pepper noise image
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Closing (VIII)

• Example Grayscale closing (salt noise)
• No noise has been removed.
• The noise has even been increased at locations
  where two nearby noise pixels have merged
  together into one larger spot.

Salt noise image
Result of Closing
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Open-close Close-open filters

• Both Open and Close filters again have
  interesting properties that would be nice to have
  in one filter. The opening and closing can be
  combined to merge these properties. There are
  two ways of combining these, the first of which
  is known as an Open-Close filter and is defined
  by
• A O? B (A ? B) ? B (14)
• The signal is first opened and the result is then
  closed. The opposite can also be done by closing
  and then opening. This is called a Close-Open
  filter and is defined by
• A ?O B (A ? B) ? B (15)

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Extending to Grey scale and Extending to 2D 3D

• For morphology to be of use in image
  processing, it needs to be extended to non-binary
  signals. There are various ways in which this
  can be done . The chosen method uses very simple
  functions, which allow them to be implemented in
  an efficient way. One method of implementing is
  Gray scale morphology and further extending to 2D
  3D.

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• THANK YOU