Morphological Image Processing

Morphological Image Processing

Introduction Morphology: a branch of biology that deals with the form and structure of animals and plants Morphological image processing is used to extract image components for representation and description of region shape, such as boundaries, skeletons, and the convex hull 2/27/2014 2

Preliminaries (1) Reflection The reflection of a set B, denoted B, is defined as B = { w w= b,for b B} Translation The translation of a set B by point z = ( z, z ), denoted ( B), is defined as ( B) = { c c= b+ z,for b B} Z 1 2 Z 2/27/2014 3

Example: Reflection and Translation 2/27/2014 4

Preliminaries (2) Structure elements (SE) Small sets or sub-images used to probe an image under study for properties of interest 2/27/2014 5

Examples: Structuring Elements (1) origin 2/27/2014 6

Examples: Structuring Elements (2) Accommodate the entire structuring elements when its origin is on the border of the original set A Origin of B visits every element of A At each location of the origin of B, if B is completely contained in A, then the location is a member of the new set, otherwise it is not a member of the new set. 2/27/2014 7

Erosion 2 With A and B as sets in Z, the erosion of A by B, denoted A B, defined { } A B= z ( B) Z A The set of all points z such that B, translated by z, is contained by A. { c ( ) } Z A B= z B A = 2/27/2014 8

Example of Erosion (1) 2/27/2014 9

Example of Erosion (2) 2/27/2014 10

Dilation 2 With and as sets in, the dilation of by, denoted A B Z A B A B, is defined as A B= { z ( B ) A } z The set of all displacements z, the translated B and A overlap by at least one element. { ( ) } A B= z B A A z 2/27/2014 11

Examples of Dilation (1) 2/27/2014 12

Examples of Dilation (2) 2/27/2014 13

Duality Erosion and dilation are duals of each other with respect to set complementation and reflection and ( ) c c = A B A B ( ) c c = A B A B 2/27/2014 14

Duality Erosion and dilation are duals of each other with respect to set complementation and reflection c { } A B = z B A ( ) ( ) { ( ) c z B A } { ( ) c z B A } c = A B Z = = Z = Z c c 2/27/2014 15

Duality Erosion and dilation are duals of each other with respect to set complementation and reflection ( ) c { ( ) } A B = z B A { ( ) } c = z B A = c = A B Z Z c 2/27/2014 16

Opening and Closing Opening generally smoothes the contour of an object, breaks narrow isthmuses, and eliminates thin protrusions Closing tends to smooth sections of contours but it generally fuses narrow breaks and long thin gulfs, eliminates small holes, and fills gaps in the contour 2/27/2014 17

Opening and Closing The opening of set A by structuring element B, denoted A B, is defined as A B= A B B ( ) The closing of set A by structuring element B, denoted AB, is defined as AB = A B B ( ) 2/27/2014 18

Opening The opening of set A by structuring element B, denoted A B, is defined as {( ) ( ) } A B= B B A Z Z 2/27/2014 19

Example: Opening 2/27/2014 20

Example: Closing 2/27/2014 21

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Duality of Opening and Closing Opening and closing are duals of each other with respect to set complementation and reflection ( ) c c = AB ( A B) ( ) c c A B = ( A B) 2/27/2014 23

The Properties of Opening and Closing Properties of Opening (a) A B is a subset (subimage) of A (b) if C is a subset of D, then C B is a subset of D B (c) ( A B) B= A B Properties of Closing (a) Ais subset (subimage) of AB (b) If Cis a subset of D, then CB is a subset of DB (c) ( AB ) B= AB 2/27/2014 24

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The Hit-or-Miss Transformation if B denotes the set composed of D and its background,the match (or set of matches) of B in A, denoted A B, c ( ) ( ) A* B= A D A W D B B B 1 2 = ( B, B ) 1 2 : object : background ( ) A B= A B ( A B ) c 1 2 2/27/2014 26

Some Basic Morphological Algorithms (1) Boundary Extraction The boundary of a set A, can be obtained by first eroding A by B and then performing the set difference between A and its erosion. β ( A) = A A B ( ) 2/27/2014 27

Example 1 2/27/2014 28

Example 2 2/27/2014 29

Some Basic Morphological Algorithms (2) Hole Filling A hole may be defined as a background region surrounded by a connected border of foreground pixels. Let A denote a set whose elements are 8-connected boundaries, each boundary enclosing a background region (i.e., a hole). Given a point in each hole, the objective is to fill all the holes with 1s. 2/27/2014 30

Some Basic Morphological Algorithms (2) Hole Filling 1. Forming an array X 0 of 0s (the same size as the array containing A), except the locations in X 0 corresponding to the given point in each hole, which we set to 1. 2. X k = (X k-1 + B) A c k=1,2,3, Stop the iteration if X k = X k-1 2/27/2014 31

Example 2/27/2014 32

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Some Basic Morphological Algorithms (3) Extraction of Connected Components Central to many automated image analysis applications. Let A be a set containing one or more connected components, and form an array X 0 (of the same size as the array containing A) whose elements are 0s, except at each location known to correspond to a point in each connected component in A, which is set to 1. 2/27/2014 34

Some Basic Morphological Algorithms (3) Extraction of Connected Components Central to many automated image analysis applications. X = ( X + B) A k k 1 B : structuring element until X k = X k-1 2/27/2014 35

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