Mathematical Morphology. Mathematical morphology (matematická morfologie) A special image analysis discipline based on morphological transformations of the image (usually a binary image). Developed: In early 1980’s by the group of Jean Serra Centre de Morphologie Mathématique (CMM)
In early 1980’s by the group of Jean Serra
Centre de Morphologie Mathématique (CMM)
Binary image (obtained e.g. using thresholding or edge detection followed by edge linking and object filling) often needs further processing such as overlapping object separation, filling of holes, removal of narrow protrusions and other morphological (i.e. shape) changes.
Original Erosion with a 3x3 mask
Opening with a 3x3 mask Opening with a 5x5 mask
Original Dilation with a 3x3 mask
Closing with a 3x3 mask Closing with a 5x5 mask
Opening + Closing (3x3 mask) Opening + Closing (5x5 mask)
Closing + Opening (3x3 mask) Closing + Opening (5x5 mask)
1978 by Ch. Lantuéjoul (CMM, Fontainebleau, France)
1979 by S. Beucher (CMM, Fontainebleau, France)
Ch. Lantuéjoul, PhD thesis, CMM, 1978
Flood simulation by increasing the water level step by step.The grey-scale image is considered as a topographic surface.Creation of catchment basins and watershed lines.
Example of the determination of markers and watershed
Original grey-scale Thresholding (green) Watershed result
Distance transform Segmentation (red)
Markers (red) Markers (black)
1996 by E.M.M.Manders (Delft, The Netherlands)
Cytometry 23: 15-21 (1996)
Iterative region growing around each local intensity maximum.Works on 2D as well as 3D images.
1) Noise filtering (average or median),the kernel size depends on noise type and magnitude.
2) Determination of local maxima and minimal intensity in the image (GlobalMinI)
3) Calculation of centers of local maxima, i.e. reduction of local maxima to 1 pixel.
4) For each local maximum do
4.1) Current threshold (CurT) = (Max intensity (LocalMaxI) + GlobalMinI) / 2
4.2) Number of iterations (I) = 0
4.3) Perform region growing from the center of the local maximum so that all neighboring pixels are included whose intensity values CurT.
4.4) N = the number of centers inside this region.
4.5) I = I + 1
4.6) If I = BitDepth, finish iteration for the given center and go to step 5.
4.4) If N>1, CurT = (CurT + previous higher threshold) / 2
4.5) If N=1, CurT = (CurT + previous lower threshold) / 2
4.6) Go to step 4.3.
5) If N>1, Final threshold (FinT) = CurT+1If N=1, Final threshold (FinT) = CurT
Original grey-scale Gaussian filter 7x7 Centers of local maxima
LocalMaxI=207 (left top)
LocalMaxI=183 (bot. right)
Example (continued for the bottom right cell)
Input CurT=105 CurT=144 CurT=163 CurT=153
CurT=158 CurT=160 CurT=161 CurT=162 FinT=162