By Dr Rajeev Srivastava What is Morphology Definition The filters can be described using set theoretic notation A set is a collection of pixels in the context of an image Morphological Operations ID: 491779
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Slide1
Morphological Image Processing
By
Dr. Rajeev SrivastavaSlide2
What is Morphology?Slide3
DefinitionSlide4Slide5
The filters can be described using set theoretic notation . A set is a collection of pixels in the context of an image.Slide6
Morphological OperationsSlide7
OperationsSlide8
Let A and B be sets. If a is the index of a pixel in A, then we write
If a is not in A we write
If every element that is in A is also in B then A is a subset of B, written
This is equivalent to the
statement
.
Slide9
In the morphological dilation and erosion operations, the state of any given pixel in the output image is determined by applying a rule to the corresponding pixel and its neighbors in the input image.
The rule used to process the pixels defines the operation as a dilation or an erosion. Slide10
Operation
Rule
Dilation
The value of the output pixel is the maximum value of all the pixels in the input pixels neighborhood. In
a binary image, if any of the pixels is set to the value 1, the output pixel is set to 1
Erosion
The value of the output pixel is the minimum value of all the pixels in the input pixels neighborhood. In
a binary image, if any of the pixels is set to the value 0, the output pixel is set to 0
The table lists the rules for both dilation and erosion.Slide11
Dilation
It can be applied to binary as well as grey-scale images
Effect of this operator on a binary image is , it gradually increases the boundaries of the region, while the small holes in image becomes smaller
Assume that A and B are two set of pixels, then the dilation of A by B is denoted by
Slide12
It means that A is translated by every point of the set B.
Dilation can be considered as a union operation of all the translations of the image A caused by the elements specified in the structuring element B
Slide13
Erosion
The objective of erosion is to make an object smaller by removing its outer layer of pixels.
This operator takes the image and structuring element as inputs and thins the object
Slide14
Algorithms for dilation and erosion
Let the number of pixels in structuring element be k
Let the number of pixels of value 1 in the input image be z
Let the pixel coordinates beneath the origin of the structuring element be (
)
Slide15
For dilation, the output is given by
For erosion, the output is given by
These algorithms can also be extended to modified dilation and erosion, which involves the use of a threshold m.
The threshold is user-controlled, based on the requirement
Slide16
Modified dilation and erosion
The algorithm for dilation can be modified using the threshold value m
The mapping functions are given by
For modified erosion, the mapping function is given by
Slide17
An another approach to morphological operations is to consider them as binary correlation operations involving logical elements
The structuring element is placed on a binary imageSlide18
Properties of dilation and erosion
The dilation and erosion shows the following properties
Communicative property
Associative property
Distributive property
Duality property
Translation property
Decomposition propertySlide19
Combining Dilation and Erosion
Opening and ClosingSlide20
Opening and Closing Operations
The opening is defined as erosion followed by a dilation operator
The opening operation satisfies the following properties
1)
This is called idempotent property
Opening is useful for smoothing the edges, breaking the narrow joints and thinning the protrusions that are present in the image
Slide21
Closing is a dilation operation followed by an erosion operation
Slide22
Properties of Opening and Closing
They show the following properties
Dual transformation
Ordering relationship
Increasing transformation
Transform invariance
Idempotence
Slide23
Hit-or-Miss transform
It is a general binary morphological operation that can be used to look for particular patterns of fore-G and background pixels of an image
The algorithm is as shown below
Translate the
centre
of the structuring element to all the points of the input image
Compare the structuring element with the image pixelsSlide24
If there is a complete match, then
the pixel underneath the structuring element is set to foreground, its called a hit.
Else
the pixel underneath the structuring element is set to background color, its called a missSlide25
Uses of morphological operations
They are immensely useful in a variety of imaging applications which are discussed as…
Boundary extraction
Noise removal
Thinning
Thickening
Convex hull
Skeletonization
Medical axis transform and distance transform
Region filling
Extraction of connected component
PruningSlide26
Grey-Scale Morphology
Similar to binary morphological operations, the mask moves across the image
The pixel-by-pixel process is done and the resultant is produced in the output image.
The structuring element can be a square matrix of size 3*3, 5*5, or larger depending upon the applicationSlide27
The erosion mask is shown as
Slide28
Comparison of grey-scale
erosion and dilation
Erosion
Dilation
Reduces the size of the objects
wrt
. background
Increases the six=
ze
of the objects
Eliminates noise spikes and ragged edges
It also eliminates noise spikes and ragged edges
Darkens the bright objects
Brighten the objects
Increases the size of holes and sharpen
cornersConnects objects, bridge gaps,smoothenes edges, fill holes and creates outline in an imageSlide29
The
morphological gradient highlights sharp transitions in the input image. It depends less on edge directionality than the
Sobel
operator and is useful for locating faint but large scale structures. The morphological gradient is defined by
,
Slide30
Gradient…
It is thus the difference between a dilated image and an eroded image. Dilation removes small scale dark features and erosion removes small scale bright features. Dilation brightens the image and erosion darkens it.