MATHEMATICS OF BINARY MORPHOLOGY

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MATHEMATICS OF BINARY MORPHOLOGY and
APPLICATIONS IN Vision
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In general what is Morphology?

• The science of form and structure
• the science of form, that of the outer form,
  inner structure, and development of living
  organisms and their parts
• about changing/counting regions/shapes
• Among other applications it is used to pre- or
  post-process images
• via filtering, thinning and pruning

• Smooth region edges
• create line drawing of face
• Force shapes onto region edges
• curve into a square

• Count regions (granules)
• number of black regions
• Estimate size of regions
• area calculations

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What is Morphology in computer vision ?

1. Morphology generally concerned with shape and
   properties of objects.
2. Used for segmentation and feature extraction.
3. Segmentation used for cleaning binary objects.
4. Two basic operations
5. erosion (opening)
6. dilation (closing)

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Morphological operations and algebras

1. Different definitions in the textbooks
2. Different implementations in the image processing
   programs.
3. The original definition, based on set theory, is
   made by J. Serra in 1982.
4. Defined for binary images - binary operations
   (boolean, set-theoretical)
5. Can be used on grayscale images - multiple-valued
   logic operations

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Morphological operations on a PC

• Various but slightly different implementations in
• Scion
• Paint Shop Pro
• Adope Photoshop
• Corel Photopaint
• mm

Try them, it is a lot of fun
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Mathematical Morphology - Set-theoretic
representation for binary shapes
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Binary Morphology

• Morphological operators are used to prepare
  binary (thresholded) images for object
  segmentation/recognition
• Binary images often suffer from noise
  (specifically salt-and-pepper noise)
• Binary regions also suffer from noise (isolated
  black pixels in a white region). Can also have
  cracks, picket fence occlusions, etc.
• Dilation and erosion are two binary morphological
  operations that can assist with these problems.

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Goals of morphological operations

• 1. Simplifies image data
• 2. Preserves essential shape characteristics
• 3. Eliminates noise
• 4. Permits the underlying shape to be identified
  and optimally reconstructed from their distorted,
  noisy forms

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What is the mathematical morphology ?

• 1. An approach for processing digital image based
  on its shape
• 2. A mathematical tool for investigating
  geometric structure in image
• The language of morphology is set theory.
• Mathematical morphology is extension to set
  theory.

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Importance of Shape in Processing and Analysis

• Shape is a prime carrier of information in
  machine vision
• For instance, the following directly correlate
  with shape
• identification of objects
• object features
• assembly defects

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Binary Morphology
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Shape Operators

• Shapes are usually combined by means of

• Set Difference based on Set intersection
  (occluded objects)

Set difference
Set intersection
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Morphological Operations based on combining base
operations

• The primary morphological operations are dilation
  and erosion
• More complicated morphological operators can be
  designed by means of combining erosions and
  dilations

We will use combinations of union, complement,
intersection, erosion, dilation, translation...
Let us illustrate them and explain how to combine
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Libraries of Structuring Elements

• Application specific structuring elements created
  by the user

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Notation
x
-2 -1 0 1 2
-2 -1 0 1 2
B
y
A special set the structuring element
Origin at center in this case, but not
necessarily centered nor symmetric
X
No necessarily compact nor filled
33 structuring element, see next slide how it
works
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Dilation
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Explanation of Dilation
Dilation x (x1,x2) such that if we center B
on them, then the so translated B
intersects X.
X
difference
dilation
B
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Notation for Dilation
Dilation x (x1,x2) such that if we center B
on them, then the so translated B
intersects X. How to formulate this definition ?
1) Literal translation
Mathematical definition of dilation
2) Better from Minkowskis sum of sets
Another Mathematical definition of dilation uses
the concept of Minkowskis sum
B is ingeneral not the same as B
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The Concept of Minkowski Sum
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Minkowskis Sum
Definition of Minkowskis sum of sets S and B
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Minkowskis Sum
l
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Another View at Dilation
Dilation
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Dilation
Dilation
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Comparison of Dilation and Minkowski sum
Dilation
Bx
x and b are points
Minkowski sum
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It is like dilation but we are not going around
, we go only to top and to right
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Dilation and Minkowski Set
Dilation and Minkowski Set are denoted by or by
? No unified notation
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Dilation is not the Minkowskis sum
Minkowskis Sum
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Dilation is not the Minkowskis sum
Dilation
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Dilation
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B is not the same as B
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Dilation with other structuring elements
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Dilation with other structuring elements
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Dilation vs SE

• Erosion shrinks
• Dilation expands binary regions
• Can be used to fill in gaps or cracks in binary
  images

structuring Element ( SE )

• If the point at the origin of the structuring
  element is set in the underlying image, then all
  the points that are set in the SE are also set in
  the image
• Basically, its like ORing the SE into the image

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Dilation fills holes

• Fills in holes.
• Smoothes object boundaries.
• Adds an extra outer ring of pixels onto object
  boundary, ie, object becomes slightly larger.

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Main Applications of Dilation
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Dilation example
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Possible problems with Morphological Operators

• Erosion and dilation clean image but leave
  objects either smaller or larger than their
  original size.
• Opening and closing perform same functions as
  erosion and dilation but object size remains the
  same.

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More Erode and Dilate Examples
Input Image Dilated Eroded
Made in Paint Shop Pro
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Dilation explained pixed by pixel
Denotes origin of B i.e. its (0,0)
Denotes origin of A i.e. its (0,0)
B
A
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Dilation explained by shape of A
Shape of A repeated without shift
B
Shape of A repeated with shift
A
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Properties of Dilation
objects are light (white in binary)
Dilation does the following

• 1. fills in valleys between spiky regions
• 2. increases geometrical area of object
• 3. sets background pixels adjacent to object's
  contour to object's value
• 4. smoothes small negative grey level regions

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Structuring Element for Dilation
Length 6
Length 5
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Structuring Element for Dilation
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Structuring Element for Dilation
Single point in Image replaced with this in
the Result
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Structuring Element for Dilation
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Definition of Dilation Mathematically

• Dilation is the operation that combines two sets
  using vector addition of set elements.
• Let A and B are subsets in 2-D space. A image
  undergoing analysis, B Structuring element,
  denotes dilation

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Dilation versus translation

• Let A be a Subset of and .
• The translation of A by x is defined as
• The dilation of A by B can be computed as the
  union of translation of A by the elements of B

x is a vector
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Dilation versus translation, illustrated
Shift vector (0,1)
Element (0,0)
Shift vector (0,0)
B
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Dilation using Union Formula
Center of the circle
This circle will create one point
This circle will create no point
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Example of Dilation with various sizes of
structuring elements
Pablo Picasso, Pass with the Cape, 1960
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Mathematical Properties of Dilation

• Commutative
• Associative
• Extensivity
• Dilation is increasing

Illustrated in next slide
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Illustration of Extensitivity of Dilation
A
B
Replaced with
Here 0 does not belong to B and A is not included
in A? B
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More Properties of Dilation

• Translation Invariance
• Linearity
• Containment
• Decomposition of structuring element

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Dilation

1. The dilation operator takes two pieces of data as
   input
2. A binary image, which is to be dilated
3. A structuring element (or kernel), which
   determines the behavior of the morphological
   operation
4. Suppose that X is the set of Euclidean
   coordinates of the input image, and K is the set
   of coordinates of the structuring element
5. Let Kx denote the translation of K so that its
   origin is at x.
6. The DILATION of X by K is simply the set of all
   points x such that the intersection of Kx with X
   is non-empty

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Dilation
Example Suppose that the structuring element is
a 3x3 square with the origin at its center
(-1,-1), (0,-1), (1,-1), (-1,0), (0,0),
(1,0), ( 1,1), (0,1), (1,1)
K
X
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Dilation
Example Suppose that the structuring element is
a 3x3 square with the origin at its center
Note Foreground pixels are represented by a
color and background pixels are empty
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Dilation
Structuring element
Input
output
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Dilation
output
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Dilation
output
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Dilation
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Dilation
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Erosion
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Example of Erosion
Erosion x (x1,x2) such that if we center B on
them, then the so translated B is
contained in X.
difference
erosion
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Notation for Erosion
Erosion x (x1,x2) such that if we center B on
them, then the so translated B is
contained in X. How to formulate this definition
? 1) Literal translation
Erosion
2) Better from Minkowskis substraction of sets
Minkowskis substraction
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Notation for Erosion
BINARY MORPHOLOGY
Minkowskis substraction of sets
Erosion
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Erosion with other structuring elements
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Erosion with other structuring elements
Did not belong to X
?
?
When the new SE is included in old SE then a
larger area is created
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Common structuring elements shapes
origin
x
y
circle
disk
segments 1 pixel wide
Note that here
points
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Problem in BINARY MORPHOLOGY using Minkowski Sum
First we calculate the operation in parentheses
to obtain a diamond
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PROBLEM BINARY MORPHOLOGY
next we calculate the external operation to
obtain a hexagon
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ANOTHER EXAMPLE OF EROSION
Where d is a diameter
Problem
ltd/2
d/2
d
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Implementation of dilation very low
computational cost
0
1 (or gt0)
Logical or
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Implementation of erosion very low
computational cost
0
1
Logical and
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More on Erosion
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Erosion

1. (Minkowski subtraction)
2. The contraction of a binary region (aka, region
   shrinking)
3. Use a structuring element on binary image data to
   produce a new binary image
4. Structuring elements (SE) are simply patterns
   that are matched in the image
5. It is useful to explain operation of erosion and
   dilation in different ways.

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Typical Uses of Erosion

1. Removes isolated noisy pixels.
2. Smoothes object boundary.
3. Removes the outer layer of object pixels, ie,
   object becomes slightly smaller.

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Properties of Erosion

• Erosion removes spiky edges
• objects are light (white in binary)
• decreases geometrical area of object
• sets contour pixels of object to background value
• smoothes small positive grey level regions

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Erosion Example
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Erosion explained pixel by pixel
A
B
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Structuring Element in Erosion Example
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How It Works?

• During erosion, a pixel is turned on at the image
  pixel under the structuring element origin only
  when the pixels of the structuring element match
  the pixels in the image
• Both ON and OFF pixels should match.
• This example erodes regions horizontally from the
  right.

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Structuring Element in Erosion Example
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Structuring Element in Erosion Example
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Structuring Element in Erosion Example
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Structuring Element in Erosion Example
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Structuring Element in Erosion Example
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Structuring Element in Erosion Example
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Mathematical Definition of Erosion

1. Erosion is the morphological dual to dilation.
2. It combines two sets using the vector subtraction
   of set elements.
3. Let denotes the erosion of A by B

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Erosion in terms of other operations

• Erosion can also be defined in terms of
  translation
• In terms of intersection

Observe that vector here is negative
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Reminder - this was A
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Erosion illustrated in terms of intersection and
negative translation
Observe negative translation
Because of negative shift the origin is here
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Erosion formula and intuitive example
Center of B is here and adds a point
Center here will not add a point to the Result
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Example of Erosions with various sizes of
structuring elements
Structuring Element
Pablo Picasso, Pass with the Cape, 1960
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Properties of Erosion

• Erosion is not commutative!
• Extensivity
• Erosion is dereasing
• Chain rule

The chain rule is as sharp operator in Cube
Calculus used in logic synthesis. There are more
similarities of these algebras
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Properties of Erosion

• Translation Invariance
• Linearity
• Containment
• Decomposition of structuring element

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Duality Relationship between erosion and dilation

• Dilation and Erosion transformation bear a marked
  similarity, in that what one does to image
  foreground and the other does for the image
  background.
• , the reflection of B, is
  defined as
• Erosion and Dilation Duality Theorem

Observe negative value which is mirror image
reflection of B
Similar but not identical to De Morgan rule in
Boolean Algebra
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Erosion as Dual of Dilation

• Erosion is the dual of dilation
• i.e. eroding foreground pixels is equivalent to
  dilating the background pixels.

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Duality Relationship between erosion and dilation

• Easily visualized on binary image
• Template created with known origin
• Template stepped over entire image
• similar to correlation
• Dilation
• if origin 1 -gt template unioned
• resultant image is large than original
• Erosion
• only if whole template matches image
• origin 1, result is smaller than original

Another look at duality
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One more view at Erosion with examples

1. To compute the erosion of a binary input image by
   the structuring element
2. For each foreground pixel superimpose the
   structuring element
3. If for every pixel in the structuring element,
   the corresponding pixel in the image underneath
   is a foreground pixel, then the input pixel is
   left as it is
4. Otherwise, if any of the corresponding pixels in
   the image are background, however, the input
   pixel is set to background value

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Erosion
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Erosion example with dilation and negation
We want to calculate this
We dilate with negation
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Erosion
.. And we negate the result
We obtain the same thing as from definition
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Morphological Operations in terms of more general
neighborhoods
This exists in Matlab
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Erode and Dilate in terms of more general
neighborhoods
Yet another loook at Duality Relationship between
erosion and dilation
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Edge detection or Binary Contour
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Boundary Extraction
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Erode and Binary Contour in Matlab
Erosion can be used to find contour Dilation can
be also used for it - think how?
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Edge detection
This subtraction is set theoretical
Dilate - original
Now you need to invert the image There are more
methods for edge detection
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Opening Closing

1. Opening and Closing are two important operators
   from mathematical morphology
2. They are both derived from the fundamental
   operations of erosion and dilation
3. They are normally applied to binary images

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Open and Close
Close Dilate next Erode Open Erode next Dilate
Original image
eroded
dilated
dilated
eroded
Open
Close
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OPENING
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OPENING
Opening
also
OPENING
difference

• Supresses
• small islands
• ithsmus (narrow unions)
• narrow caps

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Opening with other structuring elements
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Open

• An erosion followed by a dilation
• It serves to eliminate noise
• Does not significantly change an objects size

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Comparison of Opening and Erosion

1. Opening is defined as an erosion followed by a
   dilation using the same structuring element
2. The basic effect of an opening is similar to
   erosion
3. Tends to remove some of the foreground pixels
   from the edges of regions of foreground pixels
4. Less destructive than erosion
5. The exact operation is determined by a
   structuring element.

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Opening Example

• What combination of erosion and dilation gives
• cleaned binary image
• object is the same size as in original

Original
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Opening Example Cont

• Erode original image.
• Dilate eroded image.
• Smooths object boundaries, eliminates noise
  (isolated pixels) and maintains object size.

Dilate
Original
Erode
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One more example of Opening

1. Erosion can be used to eliminate small clumps of
   undesirable foreground pixels, e.g. salt noise
2. However, it affects all regions of foreground
   pixels indiscriminately
3. Opening gets around this by performing both an
   erosion and a dilation on the image

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CLOSING
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EXAMPLE OF CLOSING
Closing
also

• Supresses
• small lakes (holes)
• channels (narrow separations)
• narrow bays

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BINARY MORPHOLOGY
Closing previous image with other structuring
elements
With bigger rectangle like this
With smaller cross like this
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Application shape smoothing and noise filtering
Papilary lines recognition
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Application segmentation of microstructures
(Matlab Help)
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PROPERTIES IN BINARY MORPHOLOGY

• Properties
• all of them are increasing
• opening and closing are idempotent

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EXTENSIVE VERSUS ANTI-EXTENSIVE OPERATIONS

• dilation and closing are extensive operations
• erosion and opening are anti-extensive
  operations

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DUALITIES OF MORPHOLOGICAL OPERATORS

• duality of erosion-dilation, opening-closing,...

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Decomposition of structuring elements
operations with big structuring elements can be
done by a succession of operations with small
s.es
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HIT-OR-MISS
Hit-or-miss
Bi-phase structuring element
Hit part (white)
Miss part (black)
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HIT or MISS FOR ISOLATED POINTS
Looks for pixel configurations
background
foreground
doesnt matter
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ISOLATED POINTS
isolated points at4 connectivity
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More examples on Closing
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Close

• Dilation followed by erosion
• Serves to close up cracks in objects and holes
  due to pepper noise
• Does not significantly change object size

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More examples of Closing

• What combination of erosion and dilation gives
• cleaned binary image
• object is the same size as in original

Original
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More examples of Closing cont

• Dilate original image.
• Erode dilated image.
• Smooths object boundaries, eliminates noise
  (holes) and maintains object size.

Erode
Dilate
Original
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Closing as dual to Opening

1. Closing, like its dual operator opening, is
   derived from the fundamental operations of
   erosion and dilation.
2. Normally applied to binary images
3. Tends to enlarge the boundaries of foreground
   regions
4. Less destructive of the original boundary shape
5. The exact operation is determined by a
   structuring element.

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Closing is opening in revers

• Closing is opening performed in reverse.
• It is defined simply as a dilation followed by an
  erosion using the same

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One more example of Closing
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Mathematical Definitions of Opening and Closing

• Opening and closing are iteratively applied
  dilation and erosion
• Opening
• Closing

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Relation of Opening and Closing
Difference is only in corners
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Opening and Closing are idempotent

• Their reapplication has not further effects to
  the previously transformed result

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Properties of Opening and Closing

• Translation invariance
• Antiextensivity of opening
• Extensivity of closing
• Duality

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Example of Openings with various sizes of
structuring elements
Pablo Picasso, Pass with the Cape, 1960
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Example of Closings with various sizes of
structuring elements
Example of Closing
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Thinning and Thickening
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Thinning and Thickening
Thinning
Thickenning

• Depending on the structuring elements (actually,
  series
• of them), very different results can be achieved
  
• Prunning
• Skeletons
• Zone of influence
• Convex hull
• ...

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Prunning at 4 connectivity
Prunning at 4 connectivity remove end points by
a sequence of thinnings
This point is removed with dark green neighbors
1 iteration
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IDEMPOTENCE shown as a result of thinning
1st iteration
2nd iteration
3rd iteration idempotence
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Other thinning operations
doesnt matter
background
foreground
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USING EROSION TO FIND CONTOURS
Contours of binary regions
erosion
difference
Contour found with larger mask
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CONTOURS with different connectivity patterns
Important for perimeter computation.
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Use of thickening Convex hull
ii. Convex hull union of thickenings, each up
to idempotence
Original shaper
Thickening with first mask
Union of four thickenings
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Example of using convex hull
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iii. Skeleton
Maximal disk disk centered at x, Dx, such that
Dx ? X and no other Dy contains it . Skeleton
union of centers of maximal disks.
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PROBLEMS with skeletons

• Problems
• Instability infinitessimal variations in the
  border of X
• cause large deviations of the skeleton
• not necessarily connex even though X connex

• good approximations provided by thinning with
• special series of structuring elements

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Example of iterative thinning with 8 masks
1st iteration
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Example of iterative thinning with 8 masks
result of 1st iteration
2nd iteration reaches idempotence
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Thinning with thickening
20 iterations of thinning color white
40 iterations Thickening color white
Some sort of region clustering
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Skeletonization for OCR
BINARY MORPHOLOGY
Application skeletonization for OCR by graph
matching
skeletonization
vectorization
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skeletonization
Application skeletonization for OCR by graph
matching
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Calculation of Geodesic zones of influence (GZI)

1. X set of n connex components Xi, i1..n .
2. The zone of influence of Xi , Z(Xi) , is the set
   of points closer to some point of Xi than to a
   point of any other component.
3. Also, Voronoi partition.
4. Dual to skeleton.

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Calculating and using Geodesic Zones of Influence
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Calculating and using Geodesic Zones of Influence
(cont)
dist
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Skeleton by Maximal Balls
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Example Morphological Processing of Handwritten
Digits
thresholding
thinning
smoothing
opening
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PROGRAMMING OF MORPHOLOGICAL OPERATIONS
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USING LISP
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USING LISP
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USING LISP
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USING LISP
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USING LISP
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Morphological Filtering
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Morphological Filtering

• Main idea of Morphological Filtering
• Examine the geometrical structure of an image by
  matching it with small patterns called
  structuring elements at various locations
• By varying the size and shape of the matching
  patterns, we can extract useful information about
  the shape of the different parts of the image and
  their interrelations.
• Combine set-theoretical and morphological
  operations

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Example 1 Morphological filtering

• Noisy image will break down OCR systems

Noisy image
Clean original image
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Morphological filtering (MF)
Restored image
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Rank Filter Median
Input 1 operation 2 operations
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Postprocessing

• Opening followed by closing.
• Removes noise and smoothes boundaries.

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Postprocessing

• Opening followed by closing.
• Removes noise and smoothes boundaries.

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• Grey Level Morphology

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erosion
dilation
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Change of histogram as a result of dilation
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Removal of Border Objects

• Marker is the border itself

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Summary on Morphological Approaches

• Mathematical morphology is an approach for
  processing digital image based on its shape
• The language of morphology is set theory
• The basic morphological operations are erosion
  and dilation
• Morphological filtering can be developed to
  extract useful shape information
• Methods can be extended to more values and more
  dimensions
• Nice mathematics can be formulated - non-linear

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Conclusion

• Segmentation separates an image into regions.
• Use of histograms for brightness based
  segmentation.
• Peak corresponds to object.
• Height of peak corresponds to size of object.
• If global image histogram is multimodal, local
  image region histogram may be bimodal.
• Local thresholds can give better segmentation.

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Conclusion

• Postprocessing uses morphological operators.
• Same as convolution only use Boolean operators
  instead of multiply and add.
• Erosion clears noise, makes smaller.
• Dilation fills in holes, makes larger.
• Postprocessing
• Opening and closing to clean binary images.
• Repeated erosion with special rule produces
  skeleton.

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Problems 1 - 6

• 1. Write LISP or C program for dilation of
  binary images
• 2. Modify it to do erosions (few types)
• 3. Modify it to perform shift and exor operation
  and shift and min operation
• 4. Generalize to multi-valued algebra
• 5. Create a comprehensive theory of multi-valued
  morphological algebra and its algorithms
  (publishable).
• 6. Write a program for inspection of Printed
  Circuit Boards using morphological algebra.

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Problem 7.

• Electric Outlet Extraction has been done using a
  combination of Canny Edge Detection and Hough
  Transforms
• Write a LISP program that will use only basic
  morphological methods for this application.

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Image Processing for electric outlet, how?

• Currently there are many, many ways to approach
  this problem
• Segmentation
• Edge Detection
• DPC compression
• FFT
• IFFT
• DFT
• Thinning
• Growing
• Haar Transform
• Hex Rotate

Alpha filtering DPC compression Perimeter Fractal
Gaussian Filter Band Pass Filter Homomorphic
Filtering Contrast Sharper Least Square
Restoration Warping Dilation
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Image Processing, how?

• Create morphological equivalents of other image
  processing methods.
• New, publishable, use outlet problem as example
  to illustrate

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Problem 8. Openings and Closings as examples.

• The solution here is to follow up one operation
  with the other.
• An opening is defined as an erosion operation
  followed by dilation using the same structuring
  element.
• Similarly, a closing is dilation followed by
  erosion.
• Define and implement other combined operations.

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Problems 9 - 12.

• 9. Generalize binary morphological algebra from 2
  dimensional to 3 dimensional images. What are the
  applications.
• 10. Write software for 9.
• 11. Generalize your generalized multi-valued
  morphological algebra to 3 or more dimensions,
  theoretically, find properties and theorems like
  those from this lecture.
• 12. Write software for 11.

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Problem 13

• Mathematical morphology uses the concept of
  structuring elements to analyze image features.
• A structuring element is a set of pixels in some
  arrangement that can extract shape information
  from an image.
• Typical structuring elements include rectangles,
  lines, and circles.
• Think about other structuring elements and their
  applications.

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Morphological Operations Matlab
BWMORPH Perform morphological operations on
binary image. BW2 BWMORPH(BW1,OPERATION)
applies a specific morphological operation to the
binary image BW1. BW2 BWMORPH(BW1,OPERATION,N)
applies the operation N times. N can be Inf, in
which case the operation is repeated until the
image no longer changes. OPERATION is a string
that can have one of these values 'close'
Perform binary closure (dilation followed by
erosion) 'dilate' Perform dilation using the
structuring elementones(3) 'erode' Perform
erosion using the structuring elementones(3)
'fill' Fill isolated interior pixels (0's
surrounded by1's) 'open' Perform binary
opening (erosion followed bydilation) 'skel'
With N Inf, remove pixels on the boundariesof
objects without allowing objects to break apart

demos/demo9morph/
208
Sources
D.A. Forsyth, University of New Mexico, Qigong
Zheng, Language and Media Processing Lab Center
for Automation Research University of Maryland
College Park October 31, 2000 John Miller Matt
Roach J. W. V. Miller and K. D. Whitehead The
University of Michigan-Dearborn Spencer
Lustor Light Works Inc. C. Rössl, L. Kobbelt,
H.-P. Seidel, Max-Planck Institute for, Computer
Science, Saarbrücken, Germany LBA-PC4 Howard
Schultz Shreekanth Mandayam ECE Department Rowan
University D.A. Forsyth, University of New Mexico
209
More recent Sources

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