Lecture 3 Flashcards
Translation
object is moved down
Reflection
(-1,1) (0,1) to (1,-1)(0,-1)
Complement
All 1s become 0s and all 0s become 1s
Union
Two objects merged together
Intersection
the overlap of two objects
Difference
the two images subtracted
Cardinality
The number of elements in a set
Set operations
Given two sets, how they can be transformed
Binary morphological operations
Applies to binary images: pixels either black or white
Dilation
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
Erosion
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
Opening
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
Closing
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
Structuring elements
Can have any shape but the symmetrical 3x2 shape is used most. They can be decomposed
Decomposition
Iterative decomposition such that you can calculate it easier
Gray-scale morpholocial operations
Apply to gray-scale images
Umbra
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)
Edge detection
Want one pixel thick outline of all objects in image: I+S-I
Reconstruction
- 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.
Distance transformation
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
Ultimate erosion and dilation
- 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
Inner and outer gradient
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
morphological smoothing
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
Morphological Laplacean
L =D +E-2I
difference between outer and inner gradient
Object separation
- 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.
Skeletonization
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
Granulometry
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.
Background removal using top-hat filtering
Enhance bright objects of interest in a dark background.
H-dome filtering
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.
Mathematical morphology
Math of processing and analyzing object shapes. Seperating touching objects and clean up noise.
Gray scale erosion
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
Opening grayscale
I*S=(I-S)+S
Gray scale erosion then dilation
Closing grayscale
I*S=(I+S)-S
