mmregmin - Regional Minimum (with generalized dynamics).

Synopsis: y = mmregmin( f, Bc, option )

Input: f: Gray-scale (uint8 or uint16) image.; Bc: Structuring element. (connectivity). Default: Elementary cross (mmsecross).; option: String. Choose one of: BINARY: output a binary image; VALUE: output a grayscale image with points at the regional minimum with the pixel values of the input image; DYNAMICS: output a grayscale image with points at the regional minimum with its dynamics; AREA-DYN: int32 image with the area-dynamics; VOLUME-DYN: int32 image with the volume-dynamics. Default: "binary".

Output: y: Gray-scale (uint8 or uint16) or binary image (logical uint8).

Description: mmregmin creates a binary image f by computing the regional minima of f, according to the connectivity defined by the structuring element Bc. There are three output options: binary image; valued image; and generalized dynamics. The dynamics of a regional minima is the minimum height a pixel has to climb in a walk to reach another regional minima with a lower value in the input topographic surface image. The area-dyn is the minimum area a catchment basin has to raise to reach another regional minima. The volume-dyn is the minimum volume a catchment basin has to raise to reach another regional minima.
The dynamics concept was first introduced in [GR92] and it is the basic notion for the hierarchical or multiscale watershed transform.

Examples

Numerical example:

a = uint8([...
    10   10   10   10   10   10   10;...
    10    9    6   18    6    5   10;...
    10    9    6   18    6    5   10;...
    10    9    9   15    4    9   10;...
    10    9    9   15   12   10   10;...
    10   10   10   10   10   10   10]);
b = mmregmin(a)

b =
     0     0     0     0     0     0     0
     0     0     1     0     0     1     0
     0     0     1     0     0     1     0
     0     0     0     0     1     0     0
     0     0     0     0     0     0     0
     0     0     0     0     0     0     0
b = mmregmin(a,mmsecross,'value')

b =
     0     0     0     0     0     0     0
     0     0     6     0     0     5     0
     0     0     6     0     0     5     0
     0     0     0     0     4     0     0
     0     0     0     0     0     0     0
     0     0     0     0     0     0     0
b = mmregmin(a,mmsecross,'dynamics')

b =
     0     0     0     0     0     0     0
     0     0     4     0     0     1     0
     0     0     4     0     0     1     0
     0     0     0     0    14     0     0
     0     0     0     0     0     0     0
     0     0     0     0     0     0     0

Image example, filtering the regional minima:

Typically, the regional minima is very numerous in a real image. To filter the regional minima, one can use the mmhbasin, mmclose, mmareaclose, or other functions that simplifies basins.

f1=mmreadgray('bloodcells.tif');
m1=mmregmin(f1,mmsebox);
mmshow(f1,m1);
f2=mmhbasin(f1,70);
mmshow(f2);
m2=mmregmin(f2,mmsebox);
mmshow(f2,m2);

(f1,m1)

(f2)

(f2,m2)

To build a pyramid of nested segmentations, use the marker for the watershed only the minima above a given dynamic. Two levels of this pyramid are build below, one with dynamic above 20 and the other, above 40.

f=mmreadgray('cameraman.tif');
g=mmgradm(f);
mh=mmregmin(g,mmsecross,'dynamics');
ws1=mmcwatershed(g, mmthreshad(mh, uint8(20)));
ws2=mmcwatershed(g, mmthreshad(mh, uint8(40)));
mmshow(ws1);
mmshow(ws2);

(ws1)

(ws2)

Equation

Algorithm

function y=mmregmin_equ( f, bc)
 fminus = mmsubm(f,1);
 g = mmsubm(mmsuprec(f,fminus,bc),fminus);
 y = mmunion(mmthreshad(g,1),mmthreshad(f,0,0));

See Also: mmareaclose - Area closing; mmfreedom - Control automatic data type conversion.; mmhbasin - Remove basins with contrast smaller than h.; mmregmax - Regional Maximum.; mmsebox - Create a box structuring element.; mmsecross - Cross structuring element.; mmvbasin - Remove basins with volume smaller than v.

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Copyright (c) 1998-2002 by SDC Information Systems

mmregmin - Regional Minimum (with generalized dynamics).

Numerical example:

Image example, filtering the regional minima:

Multiscale watershed