[Ch11] Morphology

11.2 Basic ideas

Morphology (형태) : a branch of image processing which is useful for analysis shapes in (binary) images

 

1) Translation

A : a set of pixels in a binary img

w = (x, y) : a particular coordinate point

A_w = { (a, b) + (x, y) : (a, b) ∈ A}  : a set A 'translated' in direction (x, y)

 

(Matrix coordinate : The origin is at the top left → x goes down, y goes across)

 

2) Reflection

A^ = { (-x, -y) : (x, y) ∈ A} : a set obtained by reflecting A in the origin

 

 

11.3 Basic operations : Dilation and Erosion

All operations are built from a combination of Dilation and Erosion

 

1) Dilation (팽창)

A : img being processed

B : a small set of pixels (= a structuring element of a kernel)

Dilation of A by B : Union of all translations of A by every point x ∈ B

⇔ replacing every point (x, y) in A with a copy of B

Effect : thickening the img or increasing the size of object (if only the original object A lies within its dilation) 

>> imdilate(image,kernel)

>> t = imread('text.jpg');
>> sq = ones(3,3);
>> td = imdilate(t,sq);
>> subplot(1,2,1), imshow(t)
>> subplot(1,2,2), imshow(td)

 

2) Erosion (침식)

Erosion of A by B

: move B over A and find all places it will fit → mark down the corresponding (0, 0) point of B

If B contains the origin, then erosion will be a subset of the original object

Effect : thinning the img 

 

>> imerode(image, kernel)

>> c = imread('text.jpg');
>> sq = ones(3,3);
>> ce = imerode(c, sq);
>> subplot(1,2,1), imshow(c)
>> subplot(1,2,2), imshow(ce)

 

Relationship bw erosion and dilation

They are Inverse of each other

⇔ The complement of erosion = dilation of the complement 

 

Application : Bounary Detection

A : an img

B : a small structuring element consisting of point symmetrically places about the origin

The boundary of A :

>> rice = imread('rice.jpg');
>> rice = rgb2gray(rice);
>> r = rice > 110;
>> sq = ones(3,3);
>> re = imerode(r, sq);
>> r_int = r & ~re;
>> subplot(1,2,1), imshow(r)
>> subplot(1,2,2), imshow(r_int);

>> rd = imdilate(r, sq);
>> r_ext = rd & ~r;
>> r_grad = rd & ~re;
>> subplot(1,2,1), imshow(r_ext)
>> subplot(1,2,2), imshow(r_grad)

 

11.4 Second level operations : Opening and Closing

1) Opening

Opening of A by B

: a erosion followed by a dilation (erosion → dilation)

⇔ union of all translation of B which fit inside A

Properties of opening operation

 

2) Closing

: a dilation followed by a erosion (dilation → erosion)

⇔ all translations B_w which contain x have nin-empty intersections with A

Properties of closing operation

 

Relationship bw opening and closing

Complement of an opening = closing of a complement

Complement of an closing = opening of a complement

 

 

Application : Noise Removal 

A : a binary img corrupted by impulse noise

Morphological filtering : Noise Removal method by opening followed by closing (opening → closing)

>> c = imread('rice.jpg');
>> c = rgb2gray(c);
>> x=rand(size(c));
>> d1=find(x<=0.05);
>> d2=find(x>=0.95);
>> c(d1)=0;
>> c(d2)=1;

>> sq = [1 1 1 ; 1 1 1 ; 1 1 1];
>> cr = [0 1 0 ; 1 1 1 ; 0 1 0];

>> cf1=imclose(imopen(c,sq),sq);
>> cf2=imclose(imopen(c,cr),cr);

>> subplot(1,3,1), imshow(c)
>> subplot(1,3,2), imshow(cf1)
>> subplot(1,3,3), imshow(cf2)

 

 

11.5 The hit-or-miss transform

Ex. Text eroded by a hyphen-shaped structuring element

find the hyphen in 'cross-correlation' in the text img (6 lines)

>> b1=ones(1,6);
>> b2=[1 1 1 1 1 1 1 1;1 0 0 0 0 0 0 1; 1 1 1 1 1 1 1 1];
>> tb1=erode(t,b1);
>> tb2=erode(~t,b2);
>> hit_or_miss=tb1&tb2;
>> [x,y]=find(hit_or_miss==1)
>> figure, plot(tb1)

 

 

11.6 Some morphological algorithms

1) Region filling

 

2) Connected components

 

3) Skeletonization

Skeleton : medial axis transform