图像处理技术讲座（11） Digital Image Processing (11) 灰度的数学形态学(3) Mathematical morphology in gray scale (3) 顾力栩上海交通大学计算机系 2006.6

图像处理技术讲座（ 11 ）Digital Image Processing (11)

灰度的数学形态学 (3) Mathematical morphology in gray scale (3)

顾力栩顾力栩上海交通大学计算机系2006.6

Lixu Gu @ 2005 copyright reserved

Conditional Dilation

• Conditional Dilation : a special recursive dilation operation. It is also known as Geodesic Dilation or Morphological Reconstruction used for restoring destroyed objective regions. Let M and V (M V) be two binary images defined as

“marker” and “mask”, respectively. Conditional dilation Ri(M,V) is defined as:

Marker M is only allowed to grow in the region restricted by mask V.

1

( , ) ( )( , ) ( ,

,)

i

i

i i

R M V M K VR M V R M Vuntil


How it works


Morphological Reconstruction Algorithm for binary reconstruction:

Original (V) Opened (M) Reconstructed (T)

1. M = V o K , where K is any SE.2. T = M,3. M= M Ki , where i=4 or i=8,

4. M = M ∩ V, [Take only those pixels from M that are also in V .]5. if M T then go to 2,

6. else stop;


Conditional Dilation


Grayscale Reconstruction

• Grayscale Reconstruction: Step1: perform grayscale dilation on marker g.Step2: check every gray value in dilated result is not exceed

the restriction of mask f.Step3: repeat step1 and 2, until g getting stable.





• Grayscale Opening by Reconstruction (OBR):1. Smooth image or detect seeds by grayscale opening operation2. Recover the objective regions by grayscale reconstruction

f f °g r30X1Kline OBR

http://www.mmorph.com/html/fig/img_mm_openrec1.jpg





• Grayscale Closing by Reconstruction(CBR):1. Smooth image by grayscale closing operation2. Recover the objective regions by grayscale reconstruction

http://www.mmorph.com/html/fig/img_mm_closerec1.jpg

http://www.mmorph.com/html/fig/img_mm_closerec2.jpg



Source

OBR

CBR

r10Kdisk r20Kdisk



OBR CBR

r5Kdisk

r10Kdisk



Source

OBR by r10Kdisk CBR by r10Kdisk




Geodesic Distance

• Geodesic Distance: Let Fa be a connected region. The geodesic distance dFa(x,y) between two pixels x and y in Fa is the infimum of the length of the paths P which join x and y and are totally included in Fa:

where, “” stands for the infimum and l(P) is the length of the path P.

( , ) { ( )}aFd x y l P


Geodesic Distance vs. Euclidean Distance

Euclidean distance

Geodesic distance


Geodesic Influence Zone Geodesic Influence Zone (IZ): Suppose a region

A contains a set B made of several connected components B1,B2,…, Bk. IZ is denoted by izA(Bi) and defined as:

( ) { , [1, ] /{ }, ( , ) ( , )}A i A i A jiz B p A j k i d p B d p B

izA(Bi) is the locus of the points of A whose geodesic distance to Bi is smaller than the distances to any other components of B.

SKIZ


SKIZ

Skeleton by Influence Zone (SKIZ): the points of A which do not belong to any IZ, constitute the SKIZ of B inside A, denoted as SKIZA(B) :

[1. ]

( ) / ( )

( ) ( )A A

A A ii k

SKIZ B A IZ B

with IZ B iz B

“ A/B” means all members in A except those in B (BA)


Watershed

The Great Divide


Watershed – Catchment Basin


Watershed - Dam


Watershed• How it works:

1. Sort all the pixels in an image by their intensities2. Find the minimum intensity pixel as the start point

(it’s gray level as the initial threshold level)3. Increase threshold level by 1:

1) If find another local minimum point, add it to minimum list, and calculate the SKIZ with other existing minimum points

2) Otherwise, calculate SKIZ within existing minimum list.4. Repeat 3., until all the pixels are sorted to basins or

threshold exceed maximum intensity.


Watershed Demo


Marked Watershed - Demo


Watershed - Example

Source Image Watershed

Over-segmented


Watershed - Example

Merged by increase flood level

Final Result


Application

• Extract blood cells and separate them:

Source Smoothed Binary Smoothed

Distance Transform Maximum points SKIZ Watershed

http://www.mmorph.com/html/fig/img_mmdcells1.jpg

http://www.mmorph.com/html/fig/img_mmdcells2.jpg


Application

The cell binary image

overlay with watershed lines

Cells separated by watershed

lines

The Cells that touch the frames

are removed

Final


Exercise


Grayscale Operations

* 0 2 2 2 1* 1 2 6 2 1* 0 6 7 2 1* 1 1 6 1 ** 1 0 2 2 1* * * * * *

f

0 33 4

g



0,1 0,0 1,1 1,0max{ 0, 3, 3, 4}f g f f f f

* 0 3 5 5 5 4* 3 5 6 9 6 5* 4 6 9 10 6 5

( , ) * 3 9 10 11 6 5* 4 5 9 10 5 4* 4 5 5 6 6 5* * * * * * *

gD f g



1,0 1, 1 0,0 0, 1min{ 4, 3, 3, 0}f g f f f f $

* 2 1 2 3* 3 2 2 3

( , ) * 3 2 3 ** 4 3 2 ** * * * *

gE f g



( )f g f g g $

* 2 1 2 1 0* 1 2 5 2 1* 0 5 6 2 1* 0 1 2 1 ** 1 0 1 2 ** * * * * *

f g


Projects


Project1• Write your own code to realize dilation,

erosion,opening and closing operations in grayscale.

• Requirement:– Design your own UI and display I/O images– Try to apply fast operations in case


Project2• Write code to realize the next functions:

– Morphological edge detection– Morphological Reconstruction

– Conditional dilation in binary image– Gray scale Reconstruction

– Morphological gradient• Requirement:

– Design your own UI and display I/O images– Try to apply fast operations in case


Discussion

Documents

图像处理技术讲座（11） Digital Image Processing (11) 灰度的数学形态学(3) Mathematical morphology in gray scale (3) 顾 力栩 上海交通大学 计算机系 2006.6

图像处理技术讲座（11） Digital Image Processing (11) 灰度的数学形态学(3) Mathematical morphology in gray scale (3) 顾力栩上海交通大学计算机系 2006.6