Presentacin de PowerPoint

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MATHEMATICAL MORPHOLOGY

• INTRODUCTION
• BINARY MORPHOLOGY
• GREY-LEVEL MORPHOLOGY

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INTRODUCTION

• Mathematical morphology
• Self-sufficient framework for image processing
  and analysis, created at the École des Mines
  (Fontainebleau) in 70s 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

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INTRODUCTION

• Binary and grey-level images seen as sets

X
Xc
X (x, y, z) , z ? f (x,y)
f (x,y)
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INTRODUCTION

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

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MATHEMATICAL MORPHOLOGY

• INTRODUCTION
• BINARY MORPHOLOGY
• GREY-LEVEL MORPHOLOGY

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BINARY MORPHOLOGY

• Erosion and dilation
• Common structuring elements
• Opening, closing
• Properties
• Hit-or-miss
• Thinning, thickenning
• Other useful transforms
• Contour
• Convex-hull
• Skeleton
• Geodesic influence zones

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BINARY MORPHOLOGY
Notation
x
-2 -1 0 1 2
-2 -1 0 1 2
B
y
A special set the structuring element
Origin at center in this case, but not
necessarily centered nor symmetric
X
No necessarily compact nor filled
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BINARY MORPHOLOGY
Dilation x (x1,x2) such that if we center B
on them, then the so translated B
intersects X.
X
difference
B
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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 Minkowskis sum of sets
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Minkowskis sum of sets
l
l
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Dilation
l
Dilation
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Dilation is not the Minkowskis sum
l
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l
l
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Dilation with other structuring elements
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Dilation with other structuring elements
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Erosion x (x1,x2) such that if we center B on
them, then the so translated B is
contained in X.
difference
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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 Minkowskis substraction of sets
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BINARY MORPHOLOGY
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BINARY MORPHOLOGY
Erosion with other structuring elements
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Erosion with other structuring elements
Did not belong to X
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Common structuring elements shapes
origin
x
y
circle
disk
segments 1 pixel wide
Note
points
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Problem
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Problem
ltd/2
d/2
d
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Implementation very low computational cost
0
1 (or gt0)
Logical or
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Implementation very low computational cost
0
1
Logical and
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Opening
also
difference

• Supresses
• small islands
• ithsmus (narrow unions)
• narrow caps

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Opening with other structuring elements
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Closing
also

• Supresses
• small lakes (holes)
• channels (narrow separations)
• narrow bays

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Closing with other structuring elements
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Application shape smoothing and noise filtering
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Application segmentation of microstructures
(Matlab Help)
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• Properties
• all of them are increasing
• opening and closing are idempotent

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• dilation and closing are extensive erosion and
  opening are anti-extensive

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• duality of erosion-dilation, opening-closing,...

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• structuring elements decomposition

operations with big structuring elements can be
done by a succession of operations with small
s.es
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Hit-or-miss
Bi-phase structuring element
Hit part (white)
Miss part (black)
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Looks for pixel configurations
background
foreground
doesnt matter
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isolated points at4 connectivity
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Thinning
Thickenning

• Depending on the structuring elements (actually,
  series
• of them), very different results can be achieved
  
• Prunning
• Skeletons
• Zone of influence
• Convex hull
• ...

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Prunning at 4 connectivity remove end points by
a sequence of thinnings
1 iteration
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1st iteration
2nd iteration
3rd iteration idempotence
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What does the following sequence ?
doesnt matter
background
foreground
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• Erosion and dilation
• Common structuring elements
• Opening, closing
• Properties
• Hit-or-miss
• Thinning, thickenning
• Other useful transforms
• Contour
• Convex-hull
• Skeleton
• Geodesic influence zones

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i. Contours of binary regions
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Important for perimeter computation.
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ii. Convex hull union of thickenings, each up
to idempotence
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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.
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• 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

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1st iteration
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result of 1st iteration
2nd iteration reaches idempotence
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20 iterations thinning
40 iterations thickening
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Application skeletonization for OCR by graph
matching
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Application skeletonization for OCR by graph
matching
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iv. Geodesic zones of influence
X set of n connex components Xi, i1..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.
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dist