医学图像处理 第二版教学课件7

• Morphology can provide boundaries of objects, their skeletons, and their convex hulls. It is also useful for many pre- and post-processing techniques, especially in edge thinning and pruning.
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References
[1] Luc Vincent, Pierre Soille. Watersheds in Digital Spaces: An Efficient Algorithm Based on Immersion Simulations[J].IEEE Trans. PAMI, 1991 ,13(6), 583-598.
Mathematic Morphology Operation
• Generally speaking most morphological operations are based on simple expanding and shrinking operations.
• The primary application of morphology occurs in binary images, though it is also used on grey level images.
• The simplest way to construct dams separated sets of binary points is to use morphological dilation.
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Mathematic Morphology Operation
• Mathematical Morphology is a tool for extracting image components that are useful for representation and description.
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Watershed: Catchment Basin
f(x,y)
Minima Watershed line Watershed
W1
M1 W2 L1 M2 W3 L2
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What is morphological watershed?
• The principal objectives of morphological watershed is to find the watershed lines.
Medical Image Processing: Watershed
Mingyue Ding
School of Life Science and Technology Huazhong University of Science and
Technology, Wuhan, Hubei 430074, China 2022/2/25
Set Operations
• The two basic morphological set transformations are erosion and dilation
• These transformations involve the interaction between an image A (the object of interest) and a structuring set B, called the structuring element.
[2] P.R. Hill, N. Canagarajah, D. R. Bull. Image segmentation using a texture gradient based on watershed transform [J].IEEE Trans. Image Processing,2003,12(12), 1618-1633.
• Consider the example where A is a rectangle and B is a disc centred on the origin. (Note that if B is not centred on the origin we will get a translation of the object as well.) Since B is symmetric, Bˆ is B.
from distinct catchment basins are the watershed lines. – The flooding will continue until the highest value of
image is reached.
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Schematic views of Morphological Watershed
• An explanation based on immersion simulation:
– Suppose that a hole is punched in each regional minimum.
– Let water rise through the holes at a uniform level rate. – The dams built to prevent the merging of the water
• It can also be useful on range images. (A range image is one where grey levels represent the distance from the sensor to the objects in the scene rather than the intensity of light reflected from them).
• The technique was originally developed by Matheron and Serra at the Ecole des Mines in Paris.
• It is a set-theoretic method of image analysis providing a quantitative description of geometrical structures. (At the Ecole des Mines they were interested in analysing geological data and the structure of materials).
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What is morphological watershed?
• Morphological watershed is a segmentation approach.
• The concept of watersheds is based on visualizing an image in three dimensions through Digital Elevation Models (DEM’s):
– Row and column correspond to two geographical coordinates.
– Gray level=height.
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What is watershed?
• Rainfall:
(1) Points belonging to a regional minimum. (2) Points at which a drop of water, if placed at the location of
• For a particular regional minimum, the set of points (2) is called watershed of that minimum.The set of points (3) is called divide lines or watershed lines. Actually, the boundaries of regions.
Gray scale image
Flooding in the lowest region
DEM
Flooding in the 2nd lowest region
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Schematic views of Morphological Watershed
The form of first dam
More dams are formed
• Let Cn-1(M1) and Cn-1(M2) be the set of the points in the catchment basin associated with these two minima at stage n-1.
• Let C[n-1] be the union of Cn-1(M1) and Cn-1(M2) .
Example of dilation
Set operations: Erosion
• Erosion of the object A by a structuring element B is given by
Dam Construction
• Let M1 and M2 denote the regional minima of the two catchment basins.
• Let q be the merged basin at stage n.
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Dam Construction Example
• Typically the structuring element B is a circular disc in the plane, but it can be any shape. The image and structuring element sets need not be restricted to sets in the 2D plane, but could be defined in 1, 2, 3 (or higher) dimensions.
Set Operations
• Let A and B be subsets of Z2. The translation of A by x is denoted by Ax and is defined as
• The reflection of B is defined as
• The complement of A is denoted Ac, and the difference of two sets A and B is denoted A - B.
Setlation of the object A by the structuring element B is given by
• The result is a new set made up of all points generated by obtaining the reflection of B about its origin and then shifting this reflection by x.
any those points, would fall with certainty to a single minimum. (3) Points at which water would be equally likely to fall to more than one such minimum.
The form of 2nd dam
Final boundaries
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Demo Movie of Watershed
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Dam Construction
• Dam construction is based on binary images, which is members of 2-D integer space Z2 .