used to extract image components that are useful in the
TRANSCRIPT
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Digital Image Processing
Morphological Image processing
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• Used to extract image components that are useful in the representation and description of region shape, such as: – boundaries extraction
– skeletons
– convex hull
– morphological filtering
– thinning
– pruning
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Digital Image Processing
Morphological Image processing
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• Sets in mathematic morphology represent objects in an image:
– binary image (0 = white, 1 = black): • The element of the set is the coordinates (x,y) of pixel belong to
the object Z2
– gray-scaled image: • The element of the set is the coordinates (x,y) of pixel belong to the object and the
gray levels Z3
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Digital Image Processing
Morphological Image processing
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• Preliminary:
– A: A set in Z2 with elements of a=(a1,a2)
– Reflection:
– Translation:
– Used in Structuring Elements (SEs)
ˆ , for B w w b b B
, for z
B c c b z b B
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Digital Image Processing
Morphological Image processing
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• Reflection and Translation by examples:
– Need for a reference point.
Reference point
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Digital Image Processing
Morphological Image processing
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• Structuring Elements:
– Used to structure the objects: • 1 (or balck): Action
• 0 (or white): No Action
• × : Don’t Care
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Digital Image Processing
Morphological Image processing
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• Example:
– An image
– A structural element
– A processing
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Digital Image Processing
Morphological Image processing
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• Two Fundamental Morphological Operators:
– Erosion
– Dilation
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Digital Image Processing
Morphological Image processing
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• Erosion: – Set B: A structural elements.
– Other Names: Shrink, Reduce
z
c
z
A B z B A
A B z B A
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Digital Image Processing
Morphological Image processing
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• Erosion Example:
z
A B z B A
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Digital Image Processing
Morphological Image processing
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• Erosion by example (1):
– B: a 3×3 mask (Full).
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Digital Image Processing
Morphological Image processing
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• Erosion by example (2):
Disk: 11 pixels (Diameter)
Square: 9*9
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Digital Image Processing
Morphological Image processing
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• Erosin by Example (3):
– Applied on white area
Original 11×11
45×45 15×15
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Digital Image Processing
Morphological Image processing
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• Dilation:
– Set B: A structural elements.
– Other Names: Grow, Expanding
– Relation to Convolution mask: • Flipping
• Overlapping
ˆ
ˆ
z
z
A B z B A
A B z B A A
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Digital Image Processing
Morphological Image processing
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• Dilation by example: ˆz
A B z B A
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Digital Image Processing
Morphological Image processing
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• Dilation by example (1):
– B: a 3×3 mask (full).
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Digital Image Processing
Morphological Image processing
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• Application:
– Gap filling
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Morphological Image processing
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• Dilation-Erosion Duality:
ˆ
ˆRemember:
ccc c
z z
c c
z
z
A B z B A z B A
z B A A B
A B z B A
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Digital Image Processing
Morphological Image processing
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• Erosion Application:
– Remove details
1,3,5,7,9, and 15 Erode with 13 Dilate with 13
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Digital Image Processing
Morphological Image processing
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• Opening and Closing: – Dilation expands and Erosion shrinks.
– Opening: • Smooth contour
• Break narrow isthmuses ( تنگه)
• Eliminates thin protrusion ( بیرون زدگی)
– Closing: • Smooth contour
• Fuse narrow breaks, and long thin gulfs.
• Eliminates small holes, and fill gaps.
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Digital Image Processing
Morphological Image processing
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• Opening and Closing:
– Dilation expands and Erosion shrinks.
– Opening: • A erosion followed by a dilation using the same structuring
element for both operations.
– Closing: • A Dilation followed by a erosion using the same structuring
element for both operations.
z zA B A B B B B A
A B A B B
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Digital Image Processing
Morphological Image processing
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• Opening Illustration:
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Morphological Image processing
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• Opening Example:
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Morphological Image processing
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• Closing Illustration:
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Morphological Image processing
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• Closing Example:
Disk
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Morphological Image processing
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• Closing Example:
Original Thresholded Closing
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Morphological Image processing
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• Opening and Closing
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Morphological Image processing
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• Medical Application:
Original Segmentation
Opening 5×5
Closing 5×5
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Morphological Image processing
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Opening and Closing Duality:
• Opening Properties: – A○B is a subset (subimage) of A
– If C is a subset of D, then C○B is a subset of D○B
– (A○B)○B = A○B ↔ Multiple apply has no effect.
• Closing Properties: – A is a subset (subimage) of AB
– If C is a subset of D, then CB is a subset of DB
– (AB)B = AB ↔ Multiple apply has no effect.
ˆc cA B A B
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Digital Image Processing
Morphological Image processing
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Opening
Dilation of Opening Closing of Opening
• Noise
Reduction
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Morphological Image processing
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• Hit-or-Miss
– Shape Detection
– X-Y-X shape
– X enclosed by W
– B1: Object related
– B2: Background related
1 2
1 2ˆ
c
c
A B A D A W D
A B A B A B
A B A B A B
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Morphological Image processing
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• Hit-or-Miss:
– Another application: • Corners
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Morphological Image processing
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• Morphological Operator Applications:
– Boundary Extraction:
A A A B
Eroded Difference
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Morphological Image processing
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• Boundary Extraction:
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Morphological Image processing
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• Hole/Region Filling:
– Start form p inside boundary.
0
1
1
, 1, 2,3,
Until:
c
k k
k k
X p
X X B A k
X X
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Digital Image Processing
Morphological Image processing
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• Region Filling Example:
– Semi-automated to cancel reflection effect
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Digital Image Processing
Morphological Image processing
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• Connected components Extraction:
– Start from p belong to desired region.
0
1
1
1,2,3,
Until:
k k
k k
X p
X X B A k
X X
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Morphological Image processing
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• Example (1):
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Morphological Image processing
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Original
Thr
Erosion (5×5)
Connected Components with size
• Example (2)
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Digital Image Processing
Morphological Image processing
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• Convex Hull:
– Convex Set:
– Convex Hull: Smallest Convex set H, containing S
• Define Four basic structural elements, Bi, i=1,2,3,4
0
1
4
1
1,2,3,4 and 1, 2,3,
,
i
i i i
k k
i i i
i
X A
X X B A i k
C A D D X
converged
, , 0,1 : 1s t s t S S
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Morphological Image processing
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C: Converged
C C
C C
• Example:
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Morphological Image processing
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• Shortcoming of previous algorithm:
– Grow more than minimum required convex size.
– Limit to vertical-horizontal expansion.
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Digital Image Processing
Morphological Image processing
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• Thinning:
1 2
1 2
1
2
Another approach:
, , ,
One Pass with
One Pass with
...
One Pass with
Repeat until convergence
c
n
n
n
A B A A B A A B
B B B
A A B B B
B
B
B
B
B
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Digital Image Processing
Morphological Image processing
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Convert to m-connection
• Example:
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Morphological Image processing
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• Thickening:
– Structural elements are as before (1 and 0 interchange).
– Usually thin the background, then complement the results.
1 2 n
A B A A B
A A B B B
B
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Digital Image Processing
Morphological Image processing
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• Skeletonization A with notation S(A):
– For z belong to S(A) and (D)z, the largest disk centered at z and contained in A, one can not find a larger disk containing (D)z and included in A.
– Disk (D)z touches the boundary of A at two or more different points.
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Morphological Image processing
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• Skeleton by example:
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Morphological Image processing
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• Formulation:
0
k=0
, : Opening
: times
K=max k
Reconstruction:
A=
: times
k
k
k
k
k k
S A S A
S A A kB A kB B
A kB A B B B k
A kB
S A kB
S A kB S A B B B k
K
K
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Morphological Image processing
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• Example:
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Morphological Image processing
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• Pruning:
– Remove Parasitic components (clean-up)
1
8
2 1
1
3 2
4 1 3
k
k
X A
X X B
X X H A
X X X
B
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Morphological Image processing
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• Example:
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Morphological Image processing
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• Morphological Reconstruction:
– Until now: An Image and a structuring elements
– Now: Two images and one structuring elements • Marker Image: Starting points for transformation.
• Mask Image: Constrains the transformation
• Structuring element: Define connectivity
– Two main operator:
• Geodesic Dilation
• Geodesic Erosion
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Morphological Image processing
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• Geodesic Dilation:
– F: Marker Image
– G: Mask Image
– B: Structuring Elements
– Geodesic Dilation of size 1:
– Geodesic Dilation of size n:
– G: Limit the growing nature of dilation by B
1
GD F F B G
1 1 0,
n n
G G G GD F D F D F D F F
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Morphological Image processing
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• Illustration of Geodesic Dilation:
1
GD F F B G
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Morphological Image processing
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• Geodesic Erosion:
– F: Marker Image
– G: Mask Image
– B: Structuring Elements
– Geodesic Erosion of size 1:
– Geodesic Erosion of size n:
– G: Results is greater or equal to mask
1
GE F F B G
1 1 0,
n n
G G G GE F E F E F E F F
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Morphological Image processing
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• Illustration of Geodesic Erosion:
1
GE F F B G
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Morphological Image processing
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• Morphological Reconstruction:
– By Dilation
1
, :k k kD
G G G GR F D F k D F D F
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Morphological Image processing
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• Morphological Reconstruction:
– By Erosion
1
, :k k kE
G G G GR F E F k E F E F
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Digital Image Processing
Morphological Image processing
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• Sample Application:
– Opening/Closing by Reconstruction: • Classical Opening:
– Erosion: Remove small objects
– Subsequent Dilation: Try to restore the shape of objects that remains.
– Restoration in strongly depend on similarity between SE and that objects.
• Restore exactly the shapes of the remained objects after erosion.!
• Opening Reconstruction of size n, we use F as mask
n D
R F
n E
R F
O F R F nB
C F R F nB
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Morphological Image processing
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• Example:
– Opening Reconstruction:
• Goal: Extract Characters with long vertical strokes
• B: 51×1
Original Erosion
Opening 1
FO F
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Morphological Image processing
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• Fully Automated Filling Hole Procedure:
– I(x,y): Binary Image
– F(x,y): Marker image that is 0 everywhere except borders
1 , , is on the border of ,
0 otherwise
c
cD
I
I x y x y IF x y
H R F
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Morphological Image processing
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• Example For Illustration:
– B: A full 3×3 mask
1
1, :
c
c c c c
c
c
I
k k kD
I I I I
cD
I
D F F B I
R F D F k D F D F
H R F
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Morphological Image processing
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• Practical Example:
I Ic
F H
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Morphological Image processing
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• Border Clearing:
– Goal: Remove objects that touch (connected to) the border
1 , , is on the border of ,
0 otherwise
D
I
I x y x y IF x y
X I R F
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Morphological Image processing
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• Example For Illustration:
– B: A full 3×3 mask
1
1, :
I
k k kD
I I I I
D
I
D F F B I
R F D F k D F D F
X I R F
D
IR F
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Morphological Image processing
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• Summary of Morphological Tools:
– Figure 9.33 and Table 9.1 (Pages 662-664)
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Morphological Image processing
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• Gray-Scale Morphoogy
– f(x,y): The input gray-scale image
– b(x,y): a structuring element (a subimage function)
– (x,y): Integers.
– f and b are functions that assign a gray-level value (real number or real integer) to each distinct pair of coordinate (x,y)
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Morphological Image processing
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• Two Types of Structuring Elements:
– Non-flat
– Flat
• Mostly used:
– Flat SE
– Symmetrical
Horizontal Intensity Profile Through the center
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Morphological Image processing
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• Erosion and Dilation:
– Erosion • Flat SE with origin in centered at (x,y)
• Similar to correlation procedure
– Dilation • Flat reflected SE with origin in centered at (x,y),
• Similar to Convolution procedure
,
, min ,s t b
f b x y f x s y t
,
, ,s t b
f b x y Max f x s y t
ˆ , ,b x y b x y
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Morphological Image processing
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• Example:
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Morphological Image processing
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• Erosion and Dilation:
– Erosion • Non-Flat SE with origin in centered at (x,y)
• Similar to correlation procedure
– Dilation • Non-Flat reflected SE with origin in centered at (x,y),
• Similar to Convolution procedure
,
, min , ,N
N Ns t b
f b x y f x s y t b s t
,
, max , ,N
N Ns t b
f b x y f x s y t b s t
ˆ , ,b x y b x y
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• Dilation Similarity with Convolution: – f (x-s,y-t): is simply mirrored version of f (x,y) with respect
to the original of the x-y axis. the function f (x-s,y-t) moves to the right for positive s-t, and to the left for negative s-t.
– Max operation replaces the sums of convolution – Addition operation replaces with the products of
convolution.
• General effect – If all the values of the structuring element are positive, the
output image tends to be brighter than the input – Dark details either are reduced or eliminated, depending
on how their values and shapes relate to the structuring element used for dilation.
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• 1D example:
max ( ) ( )N
Ns b
f b x f x s b x
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• Erosion similarity to 2D correlation
– f (x+s,y+t) moves to the left for positive s-t and to the right for negative s-t.
• General effect – If all the elements of the structuring element are positive,
the output image tends to be darker than the input
– The effect of bright details in the input image that are smaller in area than the structuring element is reduced, with the degree of reduction being determined by the gray-level values surrounding the bright detail and by the shape and amplitude values of the structuring element itself.
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• 1D Example:
minN
N Ns b
f b x f x s b s
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• Erosion-Dilation Duality:
– For flat and non-flat SE:
ˆ , ,
ˆ , ,
ˆ( , ) and ( , )
c c
c c
c
f b x y f b x y
f b x y f b x y
f f x y b b x y
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Morphological Image processing
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• Dilation-Erosion by Example:
Original Dilation
Erosion
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• Opening-Closing:
– Same (binary) relation with Dilation-Erosion:
ˆ
ˆ
c c
c c
f b f b b
f b f b b
f b f b
f b f b
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• Opening/Closing 1D Example:
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• Opening-Closing Properties:
– Opening:
– Closing:
1 2 1 2
( )
( ) if , then ,
( )
i f b f
ii f f f b f b
iii f b b f b
1 2 1 2
( )
( ) if , then
( )
i f f b
ii f f f b f b
iii f b b f b
er indicates that the domain of e is a subset of the domain of r, and also that e(x,y) ≤ r(x,y) for any (x,y) in the domain of e
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• Opening – The structuring element is rolled underside the surface of f
– All the peaks that are narrow with respect to the diameter of the structuring element will be reduced in amplitude and sharpness
– So, opening is used to remove small light details, while leaving the overall gray levels and larger bright features relatively undisturbed.
– The initial erosion removes the details, but it also darkens the image.
– The subsequent dilation again increases the overall intensity of the image without reintroducing the details totally removed by erosion.
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Morphological Image processing
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• Opening Closing Example (1)
Opening: Decreased size of small bright details. No changes to dark region
Closing: Decreased size of small dark details. No changes to bright region
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• Opening and Closing Example (2):
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• Gray Level Morphological Examples:
– Smoothing:
– Gradient:
– Laplacian:
g f b f b
2g f b f b f
g f b b
Dilation Erosion Smoothing Gradient Laplacian
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Morphological Image processing
84
• Morphological Smoothing:
– Opening
– Closing
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E. Fatemizadeh, Sharif University of Technology, 2011
85
Digital Image Processing
Morphological Image processing
85
• Morphological Gradient:
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E. Fatemizadeh, Sharif University of Technology, 2011
86
Digital Image Processing
Morphological Image processing
86
• Top-hat and Bottom-hat Transforms:
hat
hat
T f f f b
B f f b f
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Morphological Image processing
87
• Example:
Original Thresholding
Opening
Top-hat
Thresholding
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E. Fatemizadeh, Sharif University of Technology, 2011
88
Digital Image Processing
Morphological Image processing
88
• Granulometry:
– Determine size distribution of particles in an image.
– Apply opening with SE’s on increasing size.
– Calculate sum of pixel values in the opening.
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E. Fatemizadeh, Sharif University of Technology, 2011
89
Digital Image Processing
Morphological Image processing
89
• Example:
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Morphological Image processing
90
• Example (Cont.):
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E. Fatemizadeh, Sharif University of Technology, 2011
91
Digital Image Processing
Morphological Image processing
91
• Textural Segmentation:
– Two different texture in an image
– Goal: Find Boundary between them.
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2011
92
Digital Image Processing
Morphological Image processing
92
• Gray-Scale Morphological Reconstruction:
– f: Marker Image
– g: Mask Image
– Same size and f ≤ g
– ˄: Point-wise minimum operator
– ˅: Point-wise maximum operator
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E. Fatemizadeh, Sharif University of Technology, 2011
93
Digital Image Processing
Morphological Image processing
93
• Geodesic Dilation of size 1:
• Geodesic Dilation of size n:
1
gD f f b g
1 1 0,
n n
g g g gD f D f D f D f f
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E. Fatemizadeh, Sharif University of Technology, 2011
94
Digital Image Processing
Morphological Image processing
94
• Geodesic Erosion of size 1:
• Geodesic Srosion of size n:
1
gE f f b g
1 1 0,
n n
g g g gE f E f E f E f f
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E. Fatemizadeh, Sharif University of Technology, 2011
95
Digital Image Processing
Morphological Image processing
95
• Morphological Reconstruction:
– By Erosion
– By Dilation
1, :
k k kE
g g g gR f E f k E f E f
1, :
k k kD
g g G GR f D f k D f D f
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Morphological Image processing
96
• Opening/Closing by Reconstruction:
n D
R f
n E
R f
O f R f nb
C f R f nb