Digital Image ProcessingLecture 15: Morphological Algorithms

Prof. Charlene Tsai

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Before Lecture …

Solution to previous quiz: all erosions

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Overview

Extracting image components that are useful in the representation and description of shape.

We’ll consider: Boundary extraction Region filling Extraction of connected components Skeletonization

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Boundary Extraction Denote the boundary of set A by Step1: eroding A by the structuring

element B Step2: taking the difference between A

and its erosion.

A

BAAA

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Another Illustration

Using B as the structuring element, so boundary is 1 pixel thick.

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Region Filling

A is a set containing a subset whose elements are 8-connected boundary points of a region.

Goal: fill the entire region with 1’s.

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Region Filling (con’d)

ckk ABXX 1

Terminate when Xk=Xk-1

Final

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Some Remarks The dilation process would fill the entire area if left

unchecked. limits the result to inside the region of interest. Conditional dilation Applicable to any finite number of such subsets,

assuming that a point inside each boundary is given.

cA

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Connected Components Extraction of connected components in a binary

image is central to many automated image analysis applications.

Let Y be a connected component in set A and p a point of Y.

ABXX kk 1

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Application of Connected Component

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Skeletonization

We have seen some algorithms for skeletonization when discussing topology.

Review: skeleton of a binary object is a collection of lines and curves that describe the size and shape of the object.

Different algorithms and many possible different skeletons of the same object.

Here we use a combination erosion and opening operations

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(con’d)

Formulation:

with

where

The big K is the last iterative step before A erodes to an empty set.

ASAS k

K

k 0

BBBAkBA ......

BkBAkBAASk

k time

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Skeletonization: Demo

Final skeleton

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Grayscale Morphology – Dilation (Advance)

General formulation:

where f is the grayscale image and b is the structuring element.

In other words, the value of dilation at (x,y) is the maximum of the sum of f and b in the interval spanned by b.

bDtstsbtysxfyxbf , |,,max,

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Example

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1 2 3 4 5 (x)

1

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(y)

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-1 0 1

-1

0

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fb

What is ? 3,4bf

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Effect of Dilation

Two effects: If all values of the structuring element are positive,

the output image tends to be brighter. Dark details either are reduced or eliminated,

depending on how their values and shapes relate to the structuring element.

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Grayscale Morphology - Erosion bDtstsbtysxfyxbf , |,,min,

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30 50 60 60 70

1 2 3 4 5 (x)

1

2

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(y)

1 2 3

4 5 6

7 8 9

-1 0 1

-1

0

1

fb

What is ? 3,4bf

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Demo

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Opening & Closing

bbfbf

bbfbf

Opening:

Closing:

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Demo

original

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Application: Granulometry

Definition: determining the size distribution of particles in an image Useful when objects are overlapping and

clustered. Detection of individual particles are hard.

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(con’d)

Opening operations with structuring elements of increasing size are performed on the original image.

Motivation: opening operations of a particular size have the most effect on regions containing particles of similar size.

Method: For every structuring element, compute the difference btw

the Im-1 and Im, where m is the index of the structuring element.

At the end, normalize the differences => a histogram of particle-size distribution

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Demo