1. Gray-level images

2. Image thresholding
During the thresholding process, individual pixels in an image are marked as “object” pixels if their value is greater than some threshold value (assuming an object to be brighter than the background) and as “background” pixels otherwise. This convention is known as threshold above.
Variants include threshold below, which is opposite of threshold above;
threshold inside, where a pixel is labeled "object" if its value is between two thresholds; and threshold outside, which is the opposite of threshold inside (Shapiro, et al. 2001:83).
Typically, an object pixel is given a value of “1” while a background pixel is given a value of “0.”
Finally, a binary image is created by coloring each pixel white or black, depending on a pixel's label.

3. Image thresholding
Sezgin and Sankur (2004) categorize thresholding methods into the following six groups based on the information the algorithm manipulates (Sezgin et al., 2004):
histogram shape-based methods, where, for example, the peaks, valleys and curvatures of the
smoothed histogram are analyzed
clustering-based methods, where the gray-level samples are clustered in two parts as background
and foreground (object), or alternately are modeled as a mixture of two Gaussians
entropy-based methods result in algorithms that use the entropy of the foreground and background
regions, the cross-entropy between the original and binarized image, etc.
object attribute-based methods search a measure of similarity between the gray-level and the
binarized images, such as fuzzy shape similarity, edge coincidence, etc.
spatial methods [that] use higher-order probability distribution and/or correlation between pixels
local methods adapt the threshold value on each pixel to the local image characteristics."

4. Mathematical morphology
Mathematical Morphology was born in 1964 from the collaborative work of
Georges Matheron and Jean Serra, at the École des Mines de Paris, France.
Matheron supervised the PhD thesis of Serra, devoted to the quantification
of mineral characteristics from thin cross sections, and this work resulted
in a novel practical approach, as well as theoretical advancements in
integral geometry and topology.

5. In 1968, the Centre de Morphologie Mathématique was founded by the École des
Mines de Paris in Fontainebleau, France, lead by Matheron and Serra.
During the rest of the 1960's and most of the 1970's, MM dealt essentially with
binary images, treated as sets, and generated a large number of binary operators
and techniques: Hit-or-miss transform, dilation, erosion, opening, closing,
granulometry, thinning, skeletonization, ultimate erosion, conditional bisector, and
others.
A random approach was also developed, based on novel image models. Most of the
work in that period was developed in Fontainebleau.
The name ‘Mathematical Morphology’ for the new discipline was
coined in a pub.

6. From mid-1970's to mid-1980's, MM was generalized to grayscale functions
and images as well.
Besides extending the main concepts (such as dilation, erosion, etc...) to
functions, this generalization yielded new operators, such as morphological
gradients, top-hat transform and the Watershed (MM's main segmentation
approach).
In the 1980's and 1990's, MM gained a wider recognition, as research
centers in several countries began to adopt and investigate the method.
MM started to be applied to a large number of imaging problems and
applications.

7. In 1986, Jean Serra further generalized MM, this time to a theoretical
framework based on complete lattices.
This generalization brought flexibility to the theory, enabling its application
to a much larger number of structures, including color images, video,
graphs, meshes, etc...
At the same time, Matheron and Serra also formulated a theory for
morphological filtering, based on the new lattice framework.
The 1990's and 2000's also saw further theoretical advancements, including
the concepts of connections and levelings.

29. When the structuring element B has a center (e. g
When the structuring element B has a center (e.g., B is a disk or a square), and this center is located on the origin of E, then the erosion of A by B can be understood as the locus of points reached by the center of B when B moves inside A.

31. Erosion and dilation in Matlab

32. The erosion of the dark-blue square by a disk, resulting in the light-blue square.

41. Dilation
Alternative way to "see it"

46. Opening

47. Opening
Alternative way to "see it"

49. Closing

65. Grayscale Dilation
Grayscale Dilation: A grayscale image F dilated by a grayscale SE K is defined as:
It generally brighten the source image.
Bright regions surrounded by dark regions grow in size, and dark regions surrounded by bright regions shrink in size. Small dark spots in images will disappear as they get `filled in' to the surrounding intensity value. Small bright spots will become larger spots. The effect is most marked at places in the image where the intensity changes rapidly and regions of fairly uniform intensity will be largely unchanged except at their edges.

66. Grayscale Dilation
Bright regions surrounded by dark regions grow in size
Dark regions surrounded by bright regions shrink in size.
Small dark spots in images will disappear as they get `filled in' to the
surrounding intensity value.
Small bright spots will become larger spots.
The effect is most marked at places in the image where the intensity
changes rapidly and regions of fairly uniform intensity will be largely
unchanged except at their edges.

67. Grayscale Dilation Matlab code
function [out] = graydil(im,se)
if (~isa(se,'strel'))
se=strel(se);
end
seq=getsequence(se);
lunghezza=size(seq,1);
out=gdil(im,getnhood(seq(1)),2);
for ii=2:lunghezza
out=gdil(out,getnhood(seq(ii)),2);

68. Grayscale Dilation
dilation
dilation over original

69. Grayscale Dilation
Source image
Dilated image

70. Grayscale Erosion
Grayscale Erosion: A grayscale image F eroded by a grayscale SE K is defined as:
It generally darken the image.
Bright regions surrounded by dark regions shrink in size, and dark regions surrounded by bright regions grow in size. Small bright spots in images will disappear as they get eroded away down to the surrounding intensity value, and small dark spots will become larger spots. The effect is most marked at places in the image where the intensity changes rapidly, and regions of fairly uniform intensity will be left more or less unchanged except at their edges.

71. Grayscale Erosion
Bright regions surrounded by dark regions shrink in size
Dark regions surrounded by bright regions grow in size.
Small bright spots in images will disappear as they get eroded away down to
the surrounding intensity value
Small dark spots will become larger spots.
The effect is most marked at places in the image where the intensity changes
rapidly, and regions of fairly uniform intensity will be left more or less
unchanged except at their edges.

72. Grayscale Erosion Matlab code
function [out] = grayero(im,se)
if (~isa(se,'strel'))
se=strel(se);
end
seq=getsequence(se);
lunghezza=size(seq,1);
out=gerod(im,getnhood(seq(1)),2);
for ii=2:lunghezza
out=gerod(out,getnhood(seq(ii)),2);

73. Grayscale Erosion
erosion
erosion under original

74. Grayscale Erosion
Source image
Eroded image

75. Grayscale Opening
Grayscale Opening: A grayscale image F opened by a grayscale SE K is defined as:
It can be used to select and preserve particular intensity patterns while attenuating others
Width of Opening kernel
intensity
intensity
The important thing to notice here is the way in which bright features smaller than the structuring element should be greatly reduced in intensity, while larger features have remained more or less unchanged in intensity.
scan line
scan line

76. Grayscale Opening
opening
opened & original

77. Grayscale Opening
Source image
Opened image

78. Grayscale Closing
Grayscale Closing: A grayscale image F closed by a grayscale SE K is defined as:
It is another way to select and preserve particular intensity patterns while attenuating others.
Width of closing kernel
intensity
intensity
scan line
scan line

79. Grayscale Closing
closing
closing & original

80. Grayscale Closing
Source image
Closed image

81. Grayscale Dilation and Erosion
Grayscale dilation and erosion with disk‐shaped structuring elements of radius r = 2.5, 5.0, 10.0

82. Grayscale Dilation and Erosion
Grayscale dilation and erosion with various free‐form structuring elements

83. Grayscale Opening and Closing
Grayscale opening and closing with disk‐shaped structuring elements of radius r = 2.5, 5.0, 10.0

84. Morphological Edge Detection
Morphological Edge Detection is based on Binary Dilation, Binary Erosion and Image Subtraction.
Morphological Edge Detection Algorithms:
Standard:
External:
Internal:

85. Morphological Edge Detection
F  K F
F  K K

86. Morphological Edge DetectionF

87. Morphological Gradient
Morphological Gradient is calculated by grayscale dilation and grayscale Erosion.
It is quite similar to the standard edge detection
We also have external and internal gradient

88. Morphological Gradient
Standard
External
Internal

89. Morphological Smoothing
Morphological Smoothing is based on the observation that a grayscale opening smoothes a grayscale image from above the brightness surface and the grayscale closing smoothes from below. So we could get both smooth like:

90. Morphological Smoothing
Frequency spectrum, in linear system theory, provides a measure of similarity between a function and the collection of all the possible sinusoids, whereas pattern spectrum, in mathematical morphology, provides a measure of a set and the collection of all the possible structuring elements.

91. Operators In Software VTK: vtkImageContinuousDilate3D()
vtkImageContinuousErode3D()
ITK:
itkGrayscaleDilateImageFilter() (Flat SE)
itkGrayscaleErodeImageFilter() (Flat SE)
itkGrayscaleFunctionDilateImageFilter() (Grayscale SE)
itkGrayscaleFunctionErodeImageFilter() (Grayscale SE)
MATLAB:
mmdil(), mmero(), mmopen(),mmclose(), mmgradm()

92. Top-hat Transform
Top-hat Transform (TT): An efficient segmentation tool for extracting bright (respectively dark) objects from uneven background.
White Top-hat Transform (WTT):
(ri is the scalar of SE)
Black Top-hat Transform (BTT):

93. Top-hat Transform tophat + opened = original
tophat: original - opening

94. Top-hat Transform
BTT
WTT

95. Top-hat Transform
BTT
WTT

96. Application 1 Detecting runways in satellite airport imagery
Source
WTT
Threshold
Detect long feature
Final result
Reconstruction

97. The End