Unit 18: Morphological Operations - TU Wien...Essential part of morphological operations is the StructuringElement used to probe the input image. Two‐dimensional structuringelements

Einführung in Visual Computing Unit 18: Morphological Operations

Content: Introduction Structuring Element Erosion Dilation Opening Closing Hit‐and‐Miss Thinning

http://www.caa.tuwien.ac.at/cvl/teaching/sommersemester/evc

1 Robert Sablatnig, Computer Vision Lab, EVC‐18: Morphological Operations

Introduction

Morphology denotes branch of biology that deals with form and structure of animals and plants.

Same word in the context of Mathematical Morphology: Tool for extracting image components that are useful in representation and description of region shape, such as boundaries, skeletons etc.

Language of Mathematical Morphology is set theory.

Sets in Mathematical Morphology represent objects in image.

Motive: Extract features from shape


Introduction

Morph means “Shape”

We do Morphology for Shape Analysis & Shape Study.

Shape analysis easy in case of binary images, pixel locations describe the shape.

Digital Morphology is a way to describe or analyze the shape of objects in digital images


Introduction

Value of each pixel in the output image is based on a comparisonof the corresponding pixel in the input image with its neighbors.

By choosing size and shape of neighborhood, you can construct a morphological operation that is sensitive to specific shapes in the input image.

Morphologic operations are especially suited to the processing of binary and greyscale images.

Good for, e.g., Noise removal in background Removal of holes in foreground / background


Basic Concepts From Set Theory

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Examples


Dilation and Erosion

Dilation and Erosion are the two fundamentalmorphological operations.

Dilation adds pixels to the boundaries of objects in an image, while erosion removes pixels on object boundaries.

In 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 Dilation or Erosion.


A first Example: Erosion

Erosion is an important morphological operation

Applied Structuring Element:


Example for Erosion

1 0 0 0 1 1 1 0 1 1Input image

Structuring Element

0Output Image

111


Example for Erosion

1 0 0 0 1 1 1 0 1 1

0 0

111

Input image

Structuring Element

Output Image


Example for Erosion

1 0 0 0 1 1 1 0 1 1

0 0 0

111

Input image

Structuring Element

Output Image


Example for Erosion

1 0 0 0 1 1 1 0 1 1

0 0 0 0

111

Input image

Structuring Element

Output Image


Example for Erosion

1 0 0 0 1 1 1 0 1 1

0 0 0 0 1

111

Input image

Structuring Element

Output Image


Example for Erosion

1 0 0 0 1 1 1 0 1 1

0 0 0 0 1 0

111

Input image

Structuring Element

Output Image


Example for Erosion

1 0 0 0 1 1 1 0 1 1

0 0 0 0 1 0 0

111

Input image

Structuring Element

Output Image


Example for Erosion

1 0 0 0 1 1 1 0 1 1

0 0 0 0 1 0 0 0

111

Input image

Structuring Element

Output Image


Dilation & Erosion

Basic operations

Are dual to each other: Erosion shrinks foreground, enlarges Background

Dilation enlarges foreground, shrinks background


Structuring Element

Structuring Element

Essential part of morphological operations is the Structuring Element used to probe the input image.

Two‐dimensional structuring elements consist of a matrix of 0’s and 1’s, much smaller than the image being processed.

Structuring Elements can have varyingsizes

Element values are 0,1 and none (!) Structural Elements have origin (anchor pixel)


Structuring Elements

The center pixel of the structuring element, called the origin, identifies the pixel of interest—the pixel being processed.

The pixels in the structuring element containing 1’s define the neighborhood of the structuring element.


Example of Structuring Elements (Kernels)

Empty spots in the Structuring Elements are don’t care’s!

Box

Disc


Erosion

Erosion

Erosion is the set of all points in the image, where the structuringelement “fits into”.

Consider each foreground pixel in the input image If the structuring element fits in, write a “1” at the origin of the structuring element!

Simple application of pattern matching Input: Binary Image (Gray value) Structuring Element, containing only 1s!


Erosion

Brief Description To delete or to reduce something Pixels matching a given pattern are deleted from the image. Basic effect

Erode away the boundaries of regions of foreground pixels (i.e. white pixels, typically).

Common names: Erode, Shrink, Reduce


Erosion

How It Works A & B are sets in Z², the erosion of A by B, denoted as

In words, this equation indicates that the erosion of A by B is the set of all points z such that B, translated by z, is contained in A.

BA

ABzBA z )(


Erosion

Robert Sablatnig, Computer Vision Lab, EVC‐18: Morphological Operations26

A 3×3 square structuring element

Set of coordinate points ={ (‐1, ‐1), (0, ‐1), (1, ‐1),(‐1, 0), (0, 0), (1, 0),(‐1, 1), (0, 1), (1, 1) }

Effect of erosion using a 3×3 square structuring element

Erosion

Erosion By a Rectangular Structuring Element

A

B

BA


Another Example of Erosion

White = 0, black = 1, dual property, image as a result of erosion gets darker


Matlab Programming: Erosion

%Read imageI = imread(‘ford.tiff');

figure('Name', 'original');imshow(I);

%create structuring elements% 11‐by‐11 squarese1 = strel('square',11);

%Apply erosion operationfigure('Name', 'Erode');Ierode1 = imerode(I,se1);

%Show the result imagesubplot(1,1,1), imshow(Ierode1), title('11x11 square');


Iterative Operation of Erosion

Original Image after 1 erosion

after 5 erosions after inf erosions


Counting Coins

Counting coins is difficult because they touch each other! Solution: Binarization and Erosion separates them!


Dilation

Dilation

Dilation is the set of all points in the image, where the structuring element “touches” the foreground.

Consider each pixel in the input image If the structuring element touches the foreground image, write a “1” at the origin of the structuring element!

Input: Binary Image Structuring Element, containing only 1s!!


Dilation

Brief Description

To expand or to increase something.

Basic effect: Gradually enlarge the boundaries of regions of foreground pixels on a binary image.

Common names: Dilate, Grow, Expand


Dilation

How It Works: A & B are sets in Z², the dilation of A by B, denoted as

This equation is based on obtaining the reflection of B about its origin and shifting this reflection by z.

The dilation of A by B is the set of all displacements z, such that Band A overlap by at least one element. Based on this interpretation the equation can be written as:

BA

AABzBA z )(


Dilation


Effect of dilation using a 3×3 square structuring element

3×3 square structuring element

Set of coordinate points ={ (‐1, ‐1), (0, ‐1), (1, ‐1),(‐1, 0), (0, 0), (1, 0),(‐1, 1), (0, 1), (1, 1) }

Example for Dilation

1 0 0 0 1 1 1 0 1 1

1

111

Input image

Structuring Element

Output Image



1 0 0 0 1 1 1 0 1 1

1 0

111

Input image

Structuring Element

Output Image



1 0 0 0 1 1 1 0 1 1

1 0 1

111

Input image

Structuring Element

Output Image



1 0 0 0 1 1 1 0 1 1

1 0 1 1

111

Input image

Structuring Element

Output Image



1 0 0 0 1 1 1 0 1 1

1 0 1 1 1

111

Input image

Structuring Element

Output Image



1 0 0 0 1 1 1 0 1 1

1 0 1 1 1 1

111

Input image

Structuring Element

Output Image



1 0 0 0 1 1 1 0 1 1

1 0 1 1 1 1 1

111

Input image

Structuring Element

Output Image



1 0 0 0 1 1 1 0 1 1

1 0 1 1 1 1 1 1

111

Input image

Structuring Element

Output Image


Dilation

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

Region filling is a morphological algorithm

Original image Region filling


Another Dilation Example

Image get lighter, more uniform intensity


Matlab Programming: Dilation




%Apply dilation operationfigure('Name', 'Dilate');Idilate1 = imdilate(I,se1);

%Show the result imagesubplot(1,1,1), imshow(Idilate1), title('11x11 square');


Iterative Operation of Dilation

Original Image after 1 dilation

after 5 dilations after inf dilations


Dilation

Dilation By A Rectangular Structuring Element


A

A BA

B

Edge Detection

Edge Detection1. Dilate input image2. Subtract input image from dilated image3. Edges remain!


Opening

Opening & Closing

Important operations Derived from the fundamental operations Dilation Erosion

Usually applied to binary images, but gray value images are also possible

Opening and closing are dual operations


Combining Dilation and Erosion

Morphological Opening Opening is defined as an erosion, followed by a dilation.

Morphological Closing Closing is defined as a dilation, followed by an erosion.


Opening

Similar to Erosion Spot and noise removal Less destructive

Erosion next Dilation Same structuring element for both operations.

Input: Binary Image Structuring Element, containing only 1s!


Opening

Take the structuring element and slide it around inside each foreground region. All pixels which can be covered by the SE with the SE being entirely within the foreground region will be preserved.

All foreground pixels which can not be reached by the structuring element without lapping over the edge of the foreground object will be eroded away!

Opening is idempotent: Repeated application has no further effects!


Opening

Brief Description Opening tends to “open” small gaps or spaces between touching objects in an image.

Opening is also used to remove noise (Pepper noise). The erosion step in an opening will remove isolated pixels as well as boundaries of object and the dilation step will restore most of the boundary pixels without restoring the noise.

Common names: Opening


Opening

How It works Opening is defined as an erosion followed by a 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.

BBABA


Opening

Structuring element: 3x3 square1 1 1

1 1 1

1 1 1


Opening

Binary Opening Example Separate out the circles from the lines

The lines have been almost completely removed while the circles remain almost completely unaffected.

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


Opening Example

Opening with a 9 pixel diameter disc


1

1 1 1

1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1

1 1 1

1

Opening Example

3x9 and 9x3 Structuring Element

3*9

9*3

1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1


Opening

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 3×9 vertically oriented structuring element

The result of an Opening with 9×3 horizontally oriented

structuring element


Matlab Programming: Opening




%Apply the open oprationfigure('Name', 'Open');Iopen1 = imopen(I,se1);

%Show the result imagesubplot(1,1,1), imshow(Iopen1), title('11x11 square');


Iterative Operation of Opening

Original Image after 1 opening

after 5 openings after inf openings


Use Opening for Separating Blobs

Use large structuring element that fits into the big blobs Structuring Element: 11 pixel disc


Closing

Closing

Similar to Dilation Removal of holes Tends to enlarge regions, shrink background

Closing is defined as a Dilatation, followed by an Erosion using the same structuring element for both operations.

Dilation next Erosion!

Input: Binary Image Structuring Element, containing only 1s!


Closing

Take the structuring element and slide it around outside each foreground region. All background pixels which can be covered by the SE with the SE being entirely within the background region will be preserved.

All background pixels which can not be reached by the structuring element without lapping over the edge of the foreground object will be turned into a foreground.

Opening is idempotent: Repeated application has no further effects!


Closing

Structuring element: 3x3 square1 1 1

1 1 1

1 1 1


Closing Example

Closing operation with a 22 pixel disc Closes small holes in the foreground


Threshold Closing with disc of size 20

Closing Example

Thresholded closed


Matlab Programming: Closing




%Apply close operationfigure('Name', ‘Close');Iclose1 = imclose(I,se1);

%Show the result imagesubplot(1,1,1), imshow(Iclose1), title('11x11 square');


Iterative Operation of Closing

Original Image after 1 opening

after 5 openings after inf openings


Opening & Closing

Opening is the dual of closing i.e. opening the foreground pixels with a particular structuring element

is equivalent to closing the background pixels with the same element.


Hit and Miss

Hit‐and‐Miss Transform

Used to look for particular patterns of foreground and background pixels

Very simple object recognition All other morphological operations can be derived from it!!

Input: Binary Image Structuring Element, containing 0s and 1s!!



How It Works The structuring element used in the hit‐and‐miss can contain bothforeground and background pixels.

OperationsIf the foreground and background pixels in the structuring elementexactly match foreground and background pixels in the image, thenthe pixel underneath the origin of the structuring element is set tothe foreground color.If it doesn't match, then that pixel is set to the background color.



Effect of the hit‐and‐miss based right angle convex corner detector


Four structuring elements used for corner finding in binary images


Guidelines for Use The hit‐and‐miss transform is used to look for occurrences of particular binary patterns.

It can be used to look for several patterns: Simply by running successive transforms using different structuring elements, and then ORing the results together.

The operations of erosion, dilation, opening, closing, thinning and thickening can all be derived from the hit‐and‐miss transform in conjunction with simple set operations.




The triple points (points where three lines meet) of the skeleton

Hit‐and‐miss transform outputs single foreground pixels at each triple point

Image was dilated once using a cross‐shaped structuring element in order to mark these isolated points clearly, and this was then ORedwith the original skeleton.

Basic Thinning

Thinning

1. Used to remove selected foreground pixels from binary images

2. After edge detection, lines are often thicker than one pixel.

3. Thinning can be used to thin those line to one pixel width.


Thinning

Brief Description

Remove selected foreground pixels from binary images, somewhat like erosion or opening.

Thinning is normally applied only to binary images.

Common names: Thinning


Thinning

Operation Translate origin of the structuring element to each possible pixel position in the image and compare it with the underlying image pixels.

If foreground and background pixels in the structuring element exactly match foreground and background pixels in the image, then the image pixel underneath the origin of the structuring element is set to background (zero).

Otherwise, it is left unchanged.


Thinning

Example: Thinning (character recognition) 1) shows the structuring element used in combination with thinning to obtain the skeleton.

2) was used in combination with thinning to prune the skeleton and with the hit‐and‐miss operator to find the end points of the skeleton.

Each structuring element was used in each of its 45°rotations.


Thinning

A simple way to obtain the skeleton of the character is to thin the image until convergence.The line is broken at some locations, which might cause problems during the recognition process.

Original image Thresholded image Thinning result


Example Thinning

We use two Hit‐and‐miss Transforms


Unit 18: Morphological Operations - TU Wien...Essential part of morphological operations is the StructuringElement used to probe the input image. Two‐dimensional structuringelements

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