lecture #18engr.case.edu/merat_francis/eecs490f07/lectures/lecture18.pdf · eecs490: digital image...

EECS490: Digital Image Processing

Lecture #18

• Connectivity: convex hull, thinning, thickening,skeletons, end point location

• Geodesic dilation and erosion

• Morphological reconstruction

• Automated hole filling, edge object removal

• Summary of binary morphology

• Morphological operations in MATLAB

• Gray scale morphology: image functions and SE’s

• Basic gray scale operations: erosion, dilation,opening, closing

• Advanced operations: smoothing, gradient, top-ha


Connectivity

© 2002 R. C. Gonzalez & R. E. Woods

The convex hull built objects up.We use a thinning operator basedupon 8-connectivity to reduceobjects to their basic structure.

Structuring elements. X’s indicate “don’t cares”.

1

1

1

11

Basically, if thestructuring elementmatches we remove thatpixel. In this case of 61’s B1 matches so weremove the center pixel.

A⊕B=A-(A B) =A (A B)C

1

Converted to m-connectivity


Connectivity


Thickening is the dual of thinning. The result of thinning thecomplement of A is the boundary of the thickened object.Thickening can result in some disconnected points which need tobe removed.

A 1. Complement A to get AC

2. Thin AC 3. Complement thinned AC

4. Remove unconnected points toget thickened object


Skeletons


A skeleton is the set of maximumdisks which fit inside A andtouch the boundary of A in twoor more places.

This is theunion of allskeletonsegments

S A( ) = Sk A( )k=0

K

Sk A( ) = A kB( ) A kB( ) B

Mathematically, a skeleton canbe written as a series ofopenings and closing where kindicates k successive erosionsof A by B

Note that this maximum disk(which is smaller) also touchesthe boundary at two points.


Skeletons

© 2002 R. C. Gonzalez & R. E. Woods 1. S

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K=0 row: no erosions ordilations since k=0

k=1 row: one erosion ordilation

k=2 row: two successiveerosions or dilations

Stops at k=2 since anothererosion would simply givethe empty set.

Originalobject

Reconstructedobject

Strick skeletoningdoes not guaranteeconnectivity


Connectivity


Want to remove this

{B} specifies usingB1 to B8 in sequence

2. Identify end pointsby matching

X2 = X1«B

k( )k=1

8X1 = A B{ }

3. Dilation from endpoints conditioned onA

X3 = X2 H( ) A

H is a 3x3 of 1’s

4. Union of thinnedand conditionallydilated imageseliminates parasiticbranches

X4 = X1 X3

1. Thin using Bi



Geodesic Dilation

1. Dilatemarker image Fby structuringelement B

2. Intersect with mask Gto form the geodesicdilation DG

(1)(F)

The marker imagecontains thestarting point(s)

The maskconstrains thetransformation

The structuringelement definesthe connectivity

This operation can beperformed iterativelywith the iterationdetermined by thesuperscript.



Geodesic ErosionErode marker image F bystructuring element B

Union with mask G toform geodesic erosionEG

(1)(F)

The geodesicerosion is similar tothe geodesicdilation in that it isa erosion of amarker which isconstrained by amask

This operation can beperformed iterativelywith the iterationdetermined by thesuperscript.



Morphological Reconstruction

Dilate DG(1)(F) by B;AND

with mask G to get DG(2)(F)

Dilate DG(2)(F) by B; AND


Dilate DG(3)(F) by B; AND


When DG(j+1)(F)= DG

(j)(F) we call this RGD(F) the

reconstruction of the mask by dilation

Geodesic dilation can be iterated until the image does not change. Theresulting image is called the reconstruction of the mask by dilation.



Locating Tall Characters

Original image A;average characterheight is 50 pixels

1. Erosion of A with astructuring element ofsize 51x1 to locate tallletters. This is themask F

Opening of A with astructuring element ofsize 51x1 simply givesthe vertical stroke ofeach tall character

Opening byreconstruction of Fwith a structuringelement of size 51x1

Goal is to find characters with tall, vertical strokes so wechoose a structuring element of 51x1

OR

n( ) F( ) = RFD FsnB[ ]

Do n erosions of F byB followed bymorphologicalreconstruction bydilation. This exampleis for n=1

Morphological reconstruction is a very powerfultechnique for finding complete objects in an image.



Hole Filling

The goal of this example is to fill all holes in a simple image.

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Original image Complement image

Marker image Hole filling using geodesicdilation

f x, y( ) =1 I x, y( ) if x, y( ) is on image border

0 elsewhere

This is an example of applying the hole filling algorithm to amore complex image.




Marker image

f x, y( ) =I x, y( ) if x, y( ) is on image border

0 elsewhere

Compute morphologicalreconstruction RI

D(F) usingmarker image to identify objectson edge.

Remove edge objects by simplesubtraction X=I- RI

D(F)

This is an example of removing objects on the edge of images.



Basic Binary Structuring Elements



Basic Morphological Operations


MATLAB/Image Processing Toolbox

MATLAB morphological operations

>> f=imread(‘Fig0911(a)(fingerprint_cleaned).tif’);% load in noisy fingerprint>> se=strel(‘square’,3); % 3x3 structuring element>> fo=imopen(f,se); % open image>> imshow(f), figure, imshow(fo) % show both images

>> foc=imclose(fo,se); % now close the image>> figure, imshow(foc) % show both images

% strel can be used to make structuring elements of common shapes as% well as completely arbitrary shapes

% imopen and imclose implement opening and closing% imerode and imdilate implement erosion and dilation

SEE GWE, Section 9.2 Erosion and DilationGWE, Section 9.3 Combining Erosion and Dilation


MATLAB/Image Processing Toolbox

MATLAB morphological operations

>> h=imread(‘fingerprint_clean.tif’);% load in fingerprint>> h1=bwmorph(f,’thin’,1); % apply thinning once>> h2=bwmorph(f,’thin’,2); % apply thinning twice>> imshow(h1), figure, imshow(h2) % show both images% you can actually apply thinning ‘inf’ times

% the bwmorph function can be used to implement erode, dilate,% open, close, shrink, skeleton, top-hat and many other% morphological operations

SEE GWE, Section 9.3.4 Function bwmorph


gray-scale imagefunction f

Gray Scale Morphology


Dilate by slidingthe structuringelement over thegray-scalefunction. Takingthe max pushesthe function up.

Drawing Error:top should beflat for adistance b

frb( ) s,t( ) = max f s x,t y( ) + b x, y( ) | s x( ) t y( ) Df ; x, y( ) Db{ }

Structuring element b




fsb( ) s,t( ) = min f s + x,t + y( ) b x, y( ) | s + x( ) t + y( ) Df ; x, y( ) Db{ }

Erode by slidingthe structuringelement b overthe gray-scalefunction f.Taking the minpushes thefunction down.



Gray-Scale Morphology

Gray-scalestructuringelements canhave bothgeometry(connectivity)and intensity.




original

dilated

eroded

Erosion1. Tends to darkenpositive images2. Bright details(smaller than SE) arereduced or eliminated

Dilation1. Tends to brighten images2. Dark details (smaller thanSE) are reduced or eliminated




Erosion using a 2-pixel disk SE

Dilation using a 2-pixel disk SE




Opening“sharp” detailsreduced inamplitude andsharpness

original

ClosingRemoves“dark” detailswhile leavingbright alone

Upper profile of all balls

Lower profile of all balls




Opening andclosing withflat-topelement




original opening closing




Opening using a 3-pixel disk SE

Closing using a 5-pixel disk SE




original smoothed

SmoothingOpening followed bya closing removes“bright” and “dark”details




Opening followedby closing(“smoothing”)using diskstructuringelements ofdifferent radii




original “gradient”

g = frb( ) fsb( )

An erosion subtractedfrom a dilation will give amorphological gradient




Dilation

Erosion

g = frb( ) fsb( )morphological gradient




original “top-hat”

h = f f b( )

A top-hat transformationsubtracts an opening fromthe image.

Enhances detail inthe presence ofshading

lecture #18engr.case.edu/merat_francis/eecs490f07/lectures/lecture18.pdf · eecs490: digital image...

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