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mmregmin
Regional Minimum (with generalized dynamics).

Synopsis

function y = mmregmin ( f , Bc , option )

Input

f Image Gray-scale (uint8 or uint16) image
Bc Structuring Element

Default: 3x3 elementary cross

(connectivity).
option String

Default: 'binary'

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.

Output

y Image Gray-scale (uint8 or uint16) or binary image

Description

mmregmin creates a binary image f by computing the regional minima of f, according to the connectivity defined by the structuring element Bc. A regional minimum is a flat zone not surrounded by flat zones of lower gray values. A flat zone is a maximal connected component of a gray-scale image with same pixel values. 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 higher dynamics. The area-dyn is the minimum area a catchment basin has to raise to reach another regional minima with higher area-dynamics. The volume-dyn is the minimum volume a catchment basin has to raise to reach another regional minima with a higher volume dynamics.

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 mmhmin , mmclose , mmareaclose , or other functions that simplifies basins. 

f1=mmreadgray('bloodcells.tif');

m1=mmregmin(f1,mmsebox);

mmshow(f1,m1);

f2=mmhmin(f1,70);

mmshow(f2);

m2=mmregmin(f2,mmsebox);

mmshow(f2,m2);
f1,m1 f2
f2,m2
Multiscale watershed

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)
 fplus = mmaddm(f,1);
 g = mmsubm(mmsuprec(fplus,f,bc),f);
 y = mmunion(mmthreshad(g,1),mmthreshad(f,0,0));

See also

mmfreedom Control automatic data type conversion.
mmregmax Regional Maximum.
mmsebox Create a box structuring element.
mmsecross Diamond structuring element and elementary 3x3 cross.
mmhmin Remove basins with contrast less than h.
mmareaclose Area closing
mmvmin Remove basins with volume less than v.
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