L3 Morphology

7/31/2019 L3 Morphology

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GV12/3072Image Processing.

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Morphological Operations


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Outline

Basic concepts: Erode and dilate

Open and close.

Granulometry

Hit and miss transform

Thinning and thickening

Skeletonization and the medial axis transform

Introduction to gray level morphology.


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What Are Morphological

Operators? Local pixel transformations for processing

region shapes

Most often used on binary images

Logical transformations based on

comparison of pixel neighbourhoods with a

pattern.


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Simple Operations - Examples

Eight-neighbour erode

a.k.a. Minkowsky subtraction

Erase any foreground pixel that has one

eight-connected neighbour that is

background.


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8-neighbour erode

Erode 1 Erode 2 Erode 5

Threshold


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8-neighbour dilate

Eight-neighbour dilate

a.k.a. Minkowsky addition

Paint any background pixel that has one

eight-connected neighbour that is foreground.


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8-neighbour dilate

Dilate 1 Dilate 2 Dilate 5


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Why?

Smooth region boundaries for shape

analysis.

Remove noise and artefacts from an

imperfect segmentation.

Match particular pixel configurations in an

image for simple object recognition.


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Structuring Elements

Morphological operations take two

arguments:

A binary image

A structuring element.

Compare the structuring element to the

neighbourhood of each pixel. This determines the output of the

morphological operation.


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Structuring elements

The structuring element is also a binary

array.

A structuring element has an origin.

1 1 1

1 1 1

1 1 1

0 1 0

1 1 1

0 1 0

0 1 0

1 0 1

0 1 0

1

1

0


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Binary images as sets

We can think of the binary image and the

structuring element as sets containing the

pixels with value 1.

1 1 1

0 1 1

0 0 0

0

0

0

0

0

00 0 0 1 0

0 0 0 0 0

I= {(1,1), (2,1), (3,1),

(2,2), (3,2), (4,4)}


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Some sets notation

Union and

intersection:

Complement

Difference

We use for the

empty set .

2121

2121

and:or:

IxIxxIIIxIxxII

IxxIC :

2121 and:\ IxIxxII


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More sets notation

The symmetrical setofS with respect to

point o is

SxxoS :

1 1 1 1

1 1 1

0 1 1

0 0 0

0 0 0

1 1 0

1 1 1

0 1 1

1 1 1

1 1 1

1

1

1

0

1

1

1 1 1 1 1

0 1 1 1 0

0 1 1

1 1 11 1 1

1

11

0

11

1 1 1 1 1

0 1 1 1 0


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Fitting, Hitting and Missing

S fitsIatx if

S hitsIatx if

S missesIatx if

ISssxyy },:{

ISssxyy },:{

ISssxyy },:{


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Fitting, Hitting and Missing

0 1 0

1 0 1

0 1 1

1

1

1

1

1

0

1

1

0

1

1

0

1

1

0

0 1 0 0 0 0 1 00 1 0 1 0 1 0 0

0 1 1 1 1 0 0 0

0 1 1 1 0 0 0 0

0 1 0 0 0 0 0 0

0 1 0

1 1 10 1 0

Image Structuring

element


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Erosion

The imageE=I S is the erosion of

imageIby structuring element S.

otherwise0

atfitsif1)(

xISxE

SsIsxxE everyfor:


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Implementation (nave)% I is the input image, S is a structuring element

% with origin (ox, oy). E is the output image.

function E = erode(I, S, ox, oy)

[X,Y] = size(I);

[SX, SY] = size(S);

E = ones(X, Y);

for x=1:X; for y=1:Y

for i=1:SX; for j=1:SY

if(S(i,j))

E(x,y) = E(x,y) & ImageEntry(I, x+i-ox, y+j-oy);

end

end; end

end; end


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Example

Structuring

element

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


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Example

Structuring

element

0 1 1

1 1 1

1 1 1

1

1

1

0

1

1

1 1 1 1 1

0 1 1 1 0


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Example

Structuring

element

1 0 0

0 1 0

0 0 1

0

0

0

0

0

0

0 0 0 1 0

0 0 0 0 1


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Dilation

The imageD =I S is the dilation of


otherwise0

athitsif1)(

xISxD

SsIysxxD and,:


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Example

Structuring

element

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


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Example

Structuring

element

0 1 1

1 1 1

1 1 1

1

1

1

0

1

1

1 1 1 1 1

0 1 1 1 0


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Example

Structuring

element

1 0 0

0 1 0

0 0 1

0

0

0

0

0

0

0 0 0 1 0

0 0 0 0 1


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Erosion and dilation

Erosion and dilation are dual operations:

Commutativity and associativity

SISI CC

)(

ISSI

ISSI

)()(

)()(

TSITSI

TSITSI


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Opening and Closing

The opening ofIby S is

The closing ofIby S is

SSISI )(

SSISI )(


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Opening and Closing

Opening and closing are dual

transformations:

Opening and closing are idempotent

operations:

SISI CC )(

SSISI

SSISI

)(

)(


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Example

Structuring

element

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

close

open


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Example

Structuring

element

0 1 1

1 1 1

1 1 1

1

1

1

0

1

1

1 1 1 1 1

0 1 1 1 0

close

open


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Example

Structuring

element

1 0 0

0 1 0

0 0 1

0

0

0

0

0

0

0 0 0 1 0

0 0 0 0 1

close

open


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Morphological filtering

To remove holes in the foreground and islands inthe background, do both opening and closing.

The size and shape of the structuring elementdetermine which features survive.

In the absence of knowledge about the shape offeatures to remove, use a circular structuringelement.


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Example

Structuring

element

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

Close then open

Open then close


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Example

Structuring

element

0 1 1

1 1 1

1 1 1

1

1

1

0

1

1

1 1 1 1 1

0 1 1 1 0

Close then open

Open then close


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Count the Red Blood Cells


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Granulometry

Provides a size distribution of distinct

regions or granules in the image.

We open the image with increasingstructuring element size and count the

number of regions after each operation.

Creates a morphological sieve.


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Granulometryfunction gSpec = granulo(I, T, maxRad)

% Segment the image I

B = (I>T);

% Open the image at each structuring element size up to a

% maximum and count the remaining regions.

for x=1:maxRad

O = imopen(B,strel(disk,x));

numRegions(x) = max(max(connectedComponents(O)));

end

gSpec = diff(numRegions);


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Count the Red Blood Cells


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Threshold and Label


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Disc(11)


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Disc(19)


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Disc(59)


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Number of Regions

Struct. Elt. radius

Numberofregions


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Granulometric Pattern

Spectrum

Numberofregions

Struct. Elt. radius


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Hit-and-miss transform

Searches for an exact match of the

structuring element.

H = I S is the hit-and-miss transform of


Simple form of template matching.


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Hit-and-miss transform1 0 1

1 1 0

1 0 0

1

1

1

1

0

1

1 1 0 1 1

1 0 1 0 1

1 0 1 =

0 1 0

0 0 1

0 0 0

0

0

0

0

0

0

0 0 1 0 0

0 1 0 1 0

1 0 1

1 1 0

1 0 0

1

1

1

1

0

1

1 1 0 1 1

1 0 1 0 1

* 1 =

0 1 0

0 0 0

0 1 0

0

0

0

0

1

0

0 0 1 0 0

0 0 0 0 0

1 0


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Upper-Right Corner Detector

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Thinning and Thickening

Defined in terms of the hit-and-miss

transform:

The thinning ofIby S is

The thickening ofIby S is

Dual operations:

)(\ SIISI

)( SIISI

SISI CC )(


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Sequential

Thinning/Thickening These operations are often performed in

sequence with a selection of structuring

elements S1, S2, , Sn. Sequential thinning:

Sequential thickening:

))...)(((,...,1: 21 ni SSSIniSI

))...)(((,...,1: 21 ni SSSIniSI


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Sequential

Thinning/Thickening Several sequences of structuring elements

are useful in practice

These are usually the set of rotations of asingle structuring element.

Sometimes called the Golay alphabet.


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Golay element L

0 0 *

0 1 1

* 1 *

L8

0 * 1

0 1 1

0 * 1

L7

* 1 *

0 1 1

0 0 *

L6

1 1 1

* 1 *

0 0 0

L5

* 1 *

1 1 0

* 0 0

L4

1 * 0

1 1 0

1 * 0

L3

* 0 0

1 1 0

* 1 *

L2

0 0 0

* 1 *

1 1 1

L1


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Sequential Thinning

See bwmorph in matlab.

0 iterations 1 iteration 2 iterations 5 iterations Inf iterations


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Sequential Thickening

2 iterations

1 iteration

Inf

iterations

5 iterations


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Skeletonization and the Medial

Axis Transform The skeleton and medial axis transform

(MAT) are stick-figure representations of a

regionX 2

.

Start a grassfire at the boundary of theregion.

The skeleton is the set of points at whichtwo fire fronts meet.


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Skeletons


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Medial axis transform

Alternative skeleton definition: The skeleton is the union of centres of maximal

discs withinX.

A maximal disc is a circular subset ofXthat touches

the boundary in at least two places.

The MAT is the skeleton with the maximal

disc radius retained at each point.


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Medial axis transform


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Skeletonization using

morphology Use structuring element

The n-th skeleton subset is

The skeleton is the union of

all the skeleton subsets:

1

)()(n

n XSXS

0 1 0

1 1 1

0 1 0

B =

BBXBXXS nnn )(\)()( n denotes n

successive

erosions.


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Reconstruction

We can reconstruct regionXfrom its

skeleton subsets:

We can reconstructXfrom the MAT.

We cannot reconstructXfrom S(X).

0

)(n

nn BXSX


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DiFi: Fast 3D Distance Field Computation Using Graphics Hardware

Sud, Otaduy, Manocha, Eurographics 2004


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MAT in 3D

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from Transcendata Europe Medial Object

Price, Stops, Butlin Transcendata Europe Ltd


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Applications and Problems

The skeleton/MAT provides a stick figure

representing the region shape

Used in object recognition, in particular, characterrecognition.

Problems: Definition of a maximal disc is poorly defined on a digital grid.

Sensitive to noise on the boundary.

Sequential thinning output sometimes preferred to

skeleton/MAT.


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Example

Skeletons:

Thinned:


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Gray-level Morphology

Erosion, dilation

Opening and closing

The image and the structuring element aregray level arrays.

Used to remove speckle noise. Also for smoothing, edge detection and

segmentation.


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Umbra

f U(f)

)(:),()( xfyyxfU


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Top surface

T(R)R

RzxzyyxRT ),(allfor:),()(


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Gray level erode and dilate

Erosion:

Dilation:

Open and close defined as before.

))()(( kUfUTkf

))()(( kUfUTkf


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f U(f)

k

))()(( kUfUT )()( kUfU

)(kU


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Erosion

S=ones(3,3) S=ones(5,5) S=ones(7,7)


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Dilation

S=ones(3,3) S=ones(5,5) S=ones(7,7)


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Speckle removal

Salt and pepper noise

Dilate

Erode

Close

Open


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Favorites

E = medfilt2()

[D,L] = bwdist()

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Summary

Simple morphological operations Erode and dilate

Open and close Applications:

Granulometry

Thinning and thickening

Skeletons and the medial axis transform

Gray level morphology


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L3 Morphology