# Presentaci髇 de PowerPoint

Document Sample

```					         MATHEMATICAL MORPHOLOGY

I.    INTRODUCTION

II.   BINARY MORPHOLOGY

III. GREY-LEVEL MORPHOLOGY
INTRODUCTION

Mathematical morphology

• Self-sufficient framework for image processing and
analysis, created at the École des Mines
(Fontainebleau) in 70’s by Jean Serra, Georges
Mathéron, from studies in science materials

• Conceptually simple operations combined to define
others more and more complex and powerful

• Simple because operations often have geometrical
meaning

• Powerful for image analysis
INTRODUCTION

• Binary and grey-level images seen as sets

X

Xc
X = { (x, y, z) , z  f (x,y) }

f (x,y)
INTRODUCTION

• Operations defined as interaction of images with a
special set, the structuring element
MATHEMATICAL MORPHOLOGY

I.    INTRODUCTION

II.   BINARY MORPHOLOGY

III. GREY-LEVEL MORPHOLOGY
BINARY MORPHOLOGY

1. Erosion and dilation
2. Common structuring elements
3. Opening, closing
4. Properties
5. Hit-or-miss
6. Thinning, thickenning
7. Other useful transforms :
i.   Contour
ii. Convex-hull
iii. Skeleton
iv. Geodesic influence zones
BINARY MORPHOLOGY

Notation
x
-2 -1 0 1    2

-2
-1
0
1
2
B
y
A special set :
the structuring        Origin at center in this
X                                     case, but not necessarily
element
No necessarily compact                          centered nor symmetric
nor filled
BINARY MORPHOLOGY

Dilation : x = (x1,x2) such that if we center B on them,
then the so translated B intersects X.

X                                        difference

B
BINARY MORPHOLOGY

Dilation : x = (x1,x2) such that if we center B on them,
then the so translated B intersects X.

How to formulate this definition ?

1) Literal translation

2) Better : from Minkowski’s sum of sets
BINARY MORPHOLOGY

Minkowski’s sum of sets :

l

l
BINARY MORPHOLOGY

Dilation :

l

Dilation
BINARY MORPHOLOGY

Dilation is not the Minkowski’s sum

l
BINARY MORPHOLOGY

l

l

b b bb           l
BINARY MORPHOLOGY

Dilation with other structuring elements
BINARY MORPHOLOGY

Dilation with other structuring elements
BINARY MORPHOLOGY

Erosion : x = (x1,x2) such that if we center B on them,
then the so translated B is contained in X.

difference
BINARY MORPHOLOGY

Erosion : x = (x1,x2) such that if we center B on them,
then the so translated B is contained in X.

How to formulate this definition ?

1) Literal translation

2) Better : from Minkowski’s substraction of sets
BINARY MORPHOLOGY
BINARY MORPHOLOGY

Erosion with other structuring elements
BINARY MORPHOLOGY

Erosion with other structuring elements
Did not belong to X
BINARY MORPHOLOGY

Common structuring elements shapes
= origin

x

y

disk           circle

segments 1 pixel wide

Note :
points
BINARY MORPHOLOGY

Problem :
BINARY MORPHOLOGY
BINARY MORPHOLOGY

Problem :
<d/2

d/2
d
BINARY MORPHOLOGY

Implementation : very low computational cost

0
1 (or >0)

Logical or
BINARY MORPHOLOGY

Implementation : very low computational cost

0
1

Logical and
BINARY MORPHOLOGY

Opening :
also

Supresses :                    difference
• small islands
• ithsmus (narrow unions)
• narrow caps
BINARY MORPHOLOGY

Opening with other structuring elements
BINARY MORPHOLOGY

Closing :
also

Supresses :
• small lakes (holes)
• channels (narrow separations)
• narrow bays
BINARY MORPHOLOGY

Closing with other structuring elements
BINARY MORPHOLOGY

Application : shape smoothing and noise filtering
BINARY MORPHOLOGY

Application : segmentation of microstructures (Matlab Help)

original
negated
threshold
closing
opening
and with
threshold
BINARY MORPHOLOGY

Properties

• all of them are increasing :

• opening and closing are idempotent :
BINARY MORPHOLOGY

• dilation and closing are extensive
erosion and opening are anti-extensive :
BINARY MORPHOLOGY
• duality of erosion-dilation, opening-closing,...
BINARY MORPHOLOGY
• structuring elements decomposition

operations with big structuring elements can be done
by a succession of operations with small s.e’s
BINARY MORPHOLOGY

Hit-or-miss :

Bi-phase structuring element

“Hit” part   “Miss” part
(white)       (black)
BINARY MORPHOLOGY

Looks for pixel configurations :

background
foreground
doesn’t matter
BINARY MORPHOLOGY

isolated points at
4 connectivity
BINARY MORPHOLOGY

Thinning :

Thickenning :

Depending on the structuring elements (actually, series
of them), very different results can be achieved :

• Prunning
• Skeletons
• Zone of influence
• Convex hull
• ...
BINARY MORPHOLOGY

Prunning at 4 connectivity : remove end points by a
sequence of thinnings

1 iteration =
BINARY MORPHOLOGY

1st iteration

2nd iteration     3rd iteration: idempotence
BINARY MORPHOLOGY

What does the following sequence ?

background            doesn’t
matter
foreground
BINARY MORPHOLOGY

1. Erosion and dilation
2. Common structuring elements
3. Opening, closing
4. Properties
5. Hit-or-miss
6. Thinning, thickenning
7. Other useful transforms :
i.   Contour
ii. Convex-hull
iii. Skeleton
iv. Geodesic influence zones
BINARY MORPHOLOGY

i. Contours of binary regions :
BINARY MORPHOLOGY

8-connectivity         4-connectivity
contour                contour

4-connectivity       8-connectivity

Important for perimeter computation.
BINARY MORPHOLOGY

ii. Convex hull : union of thickenings, each up to idempotence
BINARY MORPHOLOGY
BINARY MORPHOLOGY

iii. Skeleton :

Maximal disk : disk centered at x, Dx, such that
Dx  X and no other Dy contains it .

Skeleton : union of centers of maximal disks.
BINARY MORPHOLOGY

Problems :

• Instability : infinitessimal variations in the border of X
cause large deviations of the skeleton

• not necessarily connex even though X connex

• good approximations provided by thinning with
special series of structuring elements
BINARY MORPHOLOGY

1st
iteration
BINARY MORPHOLOGY

result of
1st iteration

2nd iteration reaches
idempotence
BINARY MORPHOLOGY

20 iterations
thinning

40 iterations
thickening
BINARY MORPHOLOGY

Application : skeletonization for OCR by graph matching
BINARY MORPHOLOGY

Application : skeletonization for OCR by graph matching

Hit-or-Miss

and 3 rotations
BINARY MORPHOLOGY

iv. Geodesic zones of influence :

X set of n connex components {Xi}, i=1..n .
The zone of influence of Xi , Z(Xi) , is the set of
points closer to some point of Xi than to a point of
any other component. Also, Voronoi partition.
Dual to skeleton.
BINARY MORPHOLOGY
thr     erosion
7x7

GZI               and     opening
5x5
BINARY MORPHOLOGY
BINARY MORPHOLOGY
BINARY MORPHOLOGY
BINARY MORPHOLOGY
BINARY MORPHOLOGY
BINARY MORPHOLOGY
thr             erosion
7x7

dist

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 11 posted: 11/14/2010 language: pages: 63