Lecture 3

Question 1

Translation

Answer 1

object is moved down

Question 2

Reflection

Answer 2

(-1,1) (0,1) to (1,-1)(0,-1)

Question 3

Complement

Answer 3

All 1s become 0s and all 0s become 1s

Question 4

Union

Answer 4

Two objects merged together

Question 5

Intersection

Answer 5

the overlap of two objects

Question 6

Difference

Answer 6

the two images subtracted

Question 7

Cardinality

Answer 7

The number of elements in a set

Question 8

Set operations

Answer 8

Given two sets, how they can be transformed

Question 9

Binary morphological operations

Answer 9

Applies to binary images: pixels either black or white

Question 10

Dilation

Answer 10

Binary operation
Make the image wider
I + S = {x|Sx^r intersection I=o}
The set of points for which the intersection of the image I with the refected version of structuring element S translated x is not empty
All pixels that dont have all neighbors will get one

Question 11

Erosion

Answer 11

Binary operation
Removal of edges: All neighboring pixels on the borders are deleted. I-S={x|SxI}
That is, the set of points for which the structuring element S translated over x is completely contained in the image I

Question 12

Opening

Answer 12

Binary operation
I*S=(I-S)+S
An erosion followed by a dilation using the same structuring element.
Eliminates details smaller than the structering element outside the main object

Question 13

Closing

Answer 13

Binary operation
I*S=(I+S)-S
Dilation followed by erosion using same structuring element.
Eliminates details smaller than the structuring element inside the main object

Question 14

Structuring elements

Answer 14

Can have any shape but the symmetrical 3x2 shape is used most. They can be decomposed

Question 15

Decomposition

Answer 15

Iterative decomposition such that you can calculate it easier

Question 16

Gray-scale morpholocial operations

Answer 16

Apply to gray-scale images

Question 17

Umbra

Answer 17

Gray scale dilation: I + S = U-1[U(I)+U(S)] or (I+S)(x)=max{I(x-p)+S(p)}
Binary dilation of Umbra U(I) of gray-scale image I with umbra U(S) of grayscale structuring element S turned back into gray-scale.
The umbra represents each column the colour brightness of the corresponding element I. The brightest element has highest U(I)

Question 18

Edge detection

Answer 18

Want one pixel thick outline of all objects in image: I+S-I

Question 19

Reconstruction

Answer 19

• Image containing selected objects only: Create marker image R0 containing seed pixels from each selected object and iteratively compute Ri=(Ri-1 + S) intersection I until Ri=Ri-1
• Remove objects that are only partly in image: Take boundary pixels B of input I as seeds. Compute reconstruction R from those seeds and subtract the results from the input.
• Fill all holes in all objects in the image: Take the complement Ic of image I, take the boundary pixels of Ic as seeds, compute reconstruction R of Ic from those seeds. Take the complement Rc.

Question 20

Distance transformation

Answer 20

Find distance of object pixels to background. Denote input image I as I0 and iteratively compute Ii=Ii-1 - S while setting all pixels eroded in iteration i to value i in output image D

Question 21

Ultimate erosion and dilation

Answer 21

• Find representative center points for all the objects: compute distance transform of the image and find all local maxima. This is the same as keeping only the last object pixels before final erosion

Question 22

Inner and outer gradient

Answer 22

Difference between dilated and the eroded image
Outer gradient (D-I) and inner gradient (I-E)
Sum outer and inner gradient gives more information L=D+E-2I

Question 23

morphological smoothing

Answer 23

Suppressing image structures of specific size:

• high values image structures are removed by grayscale opening
• low valued image structures are removed by gray-scale closing

Question 24

Morphological Laplacean

Answer 24

L =D +E-2I

difference between outer and inner gradient

Question 25

Object separation

Answer 25

- Seperate touching objects: Perform ultimate erosion and perform a construction of the results with the additional constraints that objects may not merge. Elongated objects may result in multiple local maxima.

Question 26

Skeletonization

Answer 26

Iteratively apply conditional erosion that does not break the connectivity of the result and does not remove single pixels or end pixels. Results in one pixel thick configurations

Question 27

Granulometry

Answer 27

Obtain the size distribution of binary objects: Define a family of structuring elements Sk of increasing size for increasing k having a shape that matches with the expected object shape.

Question 28

Background removal using top-hat filtering

Answer 28

Enhance bright objects of interest in a dark background.

Question 29

H-dome filtering

Answer 29

Finding the regional maxima using morphological reconstruction: create seed image R0 by substracting from image I a constant value h and iteratively compute Ri=min((Ri-1+S),I) until Ri=Ri-1 Substract the reconstructed image from the input image. The result is teh h-dome image Dh which contains all the object domes corresponding to the specified peak height h.

Question 30

Mathematical morphology

Answer 30

Math of processing and analyzing object shapes. Seperating touching objects and clean up noise.

Question 31

Gray scale erosion

Answer 31

I - S = U^-1[U(I)-U(S)] | Binary erosion of umbra U(I) of gray scale image I with umbra U(S) turned back into grayscale

Question 32

Opening grayscale

Answer 32

I*S=(I-S)+S | Gray scale erosion then dilation

Question 33

Closing grayscale

Answer 33

I*S=(I+S)-S