8d040 basis beeldverwerking feature extraction anna vilanova i bartrolí biomedical image analysis...
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8D040 Basis beeldverwerking
Feature Extraction
Anna Vilanova i Bartrolí
Biomedical Image Analysis Groupbmia.bmt.tue.nl
N=M=30
What is an image?
Image is a 2D rectilinear array of pixels (picture element)
N=M=256
L=15(4 bits)L=255 (8 bits)
What is an image?No continuous values - Quantization
255
170
15
8
Binary Image L=1 (1 bit)L=3 (2 bits)
An image is just 2D?
No! – It can be in any dimension
Example 3D:
Voxel-Volume Element
Segmentation
Reduction of dimensionality
Why feature extraction ?
Pixel levelImage of 256x256 and
8 bits
256 65536 ~ 10 157826
possible images
• Incorporation of cues from human perception• Transcendence of the limits of human perception
• The need for invariance
Why feature extraction ?
Apple detection …
Transformation (Rotation)
cos( ) sin( ) 0sin( ) cos( ) 0
0 0 1
2a
1a
1b2b
P
( )T P
1 2( , ,1)P x x
1
1 2 2
cos( ) sin( ) 0( ) ( , ,1) sin( ) cos( ) 0
0 0 1 A
aT P x x a
O
1 1 2cos( ) sin( )b a a
2 1 2sin( ) cos( ) b a a
cos( )
sin( )
sin( )
cos( )
How do we transform an image?
• We transform a point P
• How do we transform an image f(P)?
1
1 2 2( ) ( , ,1)
A
aT P x x a
OT ( ) .T P P T
( ) ( )newf Q f Pnewff
QPHow do we know
which Q belongs to P?
1b
2
a
1
a
2
b
.Q P T
How do we transform an image?
• How do we transform an image f(P)?
1b
1. Q PT
We know T which is the transformation we want to achieve.
1( ) ( . )newf Q f Q T
( ) ( )newf Q f Pnewff
QP2
a
1
a
2
bHow do we know
which Q belongs to P?
Apple detection …
Feature Characteristics
• Invariance (e.g., Rotation, Translation)• Robust (minimum dependence on)• Noise, artifacts, intrinsic variations• User parameter settings
• Quantitative measures
We extract features from…
Region of InterestSegmented Objects
Classification
Features
Texture Based(Image & ROI)
Shape(Segmented objects)
Shape Based Features
• Object based• Topology based (Euler Number)• Effective Diameter
(similarity to a circle to a box)• Circularity• Compactness• Projections• Moments (derived by Hu 1962)
• …
4-neighbourhood of
8-neighbourhood of
Adjacency and Connectivity – 2D
pp
p
*( ) ( ) { }k kN p N p p
Notation: k-Neighbourhood of is p ( )kN p
[ , ]p i j
Adjacency and Connectivity – 3D
6-neighbourhood 18-neighbourhood 26-neighbourhood
Objects or Components (Jordan Theorem)
• In 2D – (8,4) or (4,8)-connectivity• In 3D – (6,26)-,(26,6)-,(18,6)- or
(6,18)-connectivity
Connected Components Labeling
Each object gets a different label
Connected Components Labeling
A
B
CRaster Scan
Note: We want to label A. Assuming objects are 4-connected B, C are already labeled.
Cortesy of S. Narasimhan
Connected Components Labeling
1
0
0 label(A) = new label
0
X
X label(A) = “background”
1
0
C label(A) = label(C)
1
B
0 label(A) = label(B)
1
B
C If
label(B) = label(C)then, label(A) = label(B)
Cortesy of S. Narasimhan
2
4 2 2 2 3
4 4 4 4 ?
2
2 2 2 2 3
2 2 2 2 2
2
2 2 2 2 3
2 2 2 2 2 2 2 3
2 2 2 2 2 2 2
2 2 2
What if label(B) not equal to label(C)?
Connected Components Labeling
1
B
C
Connected Components Labeling
Each object gets a different label
Classification
Features
Texture Based(Image & ROI)
Shape(Segmented objects)
Topology based – Euler Number
E C H
Euler Number E describes topology.
C is # connected components
H is # of holes.
Euler Number 3D
E C Cav G
Euler Number E describes topology. C is # connected components
Cav is # of cavities
G is # of genus
E=1+0-1=0 E=1+0-1=0 E=1+1-0=2
Euler Number 3D
E=2+0-0=2 E=1+1-0=2
Euler Number E describes topology. C is # connected components
Cav is # of cavities
G is # of genusE C Cav G
3D Euler Number
• The Euler Number in 3D can be computed with local operations• Counting number of vertices, edges and faces of the
surfaces of the objects
1
# # #
C Cav
i i ii
E vertices edges faces
Simple Shape Measurements
• 2D area - 3D volume • Summing elements
• 2D perimeter - 3D surface area• Selection of border elements • Sum of elemets with weights
• Error of precision
,
( , )x y
A f x y
1 22
where #ele. with 4c background ele.i
P N N
N i
Similarity to other Shape
• Effective Diameter
2
4 AC
P
2PComp
A
4
• Circularity (Circle C=1)
• Compactness – (Actually non-compactness)(Circle Comp= )
Ar
2A r
2P r
Moments
• Definition
• Order of a moment is• Moments identify an object uniquely
• ? is the Area• Centroid
pqr
, ,
[ , , ]p q rpqr
x y z
x y z f x y z p q r
000
100 010 001
000 000 000
( , , ) ( , , )
x y zc c c
[ ] : [1, ] {0,1}nf x N
Central Moments
• Moments invariant to position
• Invariant to scaling
, ,
( ) ( ) ( ) [ , , ]p q rpqr x y z
x y z
c x c y c z c f x y z
, ,
13
000
( ) ( ) ( ) [ , , ]p q rx y z
x y zpqr p q r
x c y c z c f x y z
Moments to Define Shape and Orientation
( , , )x y zc c c200
020
002
110
101
011
xx xy xz
yx yy yz
zx zy zz
xx
yy
zz
xy yx
xz zx
yz zy
I I I
I I I I
I I I
I
I
I
I I
I I
I I
Inertia Tensor
Eigenanalysis of a Matrix
• Given a matrix S , we solve the following equation
( ) =S I x 0 j j jSv = λ v
we find the eigenvectors and eigenvalues
• Eigenvectors and eigenvalues go in couples an usually are ordered as follows:
jjv
det( ) =S I 0
1 2 3
Eigenanalysis of the Inertia Matrix
Eigenanalysis
Sphere
Flatness
Elongated
( , , )x y zc c c
1 1v
2 2v
3 3v3
2
1
3
1
2
Eigenanalysis of the Inertia Matrix
Eigenanalysis
Sphere
Flatness
Elongated
( , , )x y zc c c
1 1v
2 2v
3 3v2
3
1
3
1
2
1 2 3
1 2 3
1 2 3
Orientation in 2D
• Using similar concepts than 3D
• Covariance or Inertia Matrix
• Eigenanalysis we obtain 2 eigenvalues and 2 eigenvectors of the ellipse
20 11
11 02IC
Moments Invariance
• Translation• Central moments are invariant
• Rotation• Eigenvalues of Inertia Matrix are invariant
• Scaling• If moment scaled by (3D) (2D) 1
3000
p q r
12
00
p q
Moments invariant rotation-translation-scaling
• For 3D three moments (Sadjadi 1980)
For 2D seven moments
1 200 020 002
2 2 22 200 020 200 002 020 002 101 110 011
23 200 020 002 002 110 110 101 011
2 2020 101 200 011
2
J
J
J
Classification
Features
Texture Based(Image & ROI)
Shape(Segmented objects)
Image Based Features
• Using all pixels individually• Histogram based features
− Statistical Moments (Mean, variance, smoothness)− Energy− Entropy− Max-Min of the histogram− Median
• Co-occurrance Matrix
Gonzalez & Woods – Digital Image ProcessingChapter 11 – 11.3.3 Texture
Histogram
0 1 2 3 4 5 6 7 8 9
Quantized , 0,...,
( ) # of voxels [ ]
total # of voxels
i
i i
b i L
N b f x b
N
L=9
( ) ( ) /i iP b N b N
bi
P(bi)
How do the histograms of this images look like?
Bimodal Histogram
60 80 100 120
0.1
0.2
0.3
0.4
0.5
Trimodal Features
50 100 150 200 250
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Histogram Features
• Mean
• Central Moments0
( )
L
i ii
m b P b
0
( ) ( )
L
nn i i
i
c b m P b
Histogram Features
• Mean
• Variance
• Relative Smoothness
• Skewness
0
( )
L
i ii
m b P b
2 22
0
( ) ( )
L
i ii
b m P b c
2
11
1R
33
0
( ) ( )
L
i ii
c b m P b
Histogram Features
• Energy (Uniformity)
• Entropy
2
0
[ ( )]
L
ii
E P b
20
2
( ) log ( ( ))
if ( ) 0 then ( ) log ( ( )) 0
L
i ii
i i i
H P b P b
P b P b P b
Examples of Energy and Entropy
0
0,1
0,2
0,3
0 2 4 6 8 10 12 14 bi
P(bi)
0
0,02
0,04
0,06
0,08
0 2 4 6 8 10 12 14 bi
P(bi)
0
0,1
0,2
0,3
0,4
0 2 4 6 8 10 12 14 bi
P(bi)
0 2 4 6 8 10 12 14
00.10.20.30.40.50.60.70.80.9
11.1
bi
P(bi) Energy=1Entropy=0
Energy=0,111Entropy=3,327
Energy=0,255Entropy=2,018
Energy=0,0625Entropy= 4
Examples
Texture Mean std R 3rd moment Energy Entropy
1 82.64 11.79 0.002 -0.105 0.026 5.434
2 143.56 74.63 0.079 -0.151 0.005 7.783
3 99.72 33.73 0.017 0.750 0.013 6.374
1 2 3
The next slides were not given, during the lecture and will not be asked in the exam
Intensity Co-occurrance Matrix
• Operator Q defines the position between two pixels (e.g, pixel to the right)
• Co-occurance matrix G is (L+1) x (L+1) (6x6). Counts how often Q occurs
0 1 4
0 4 5
2 3 1
2 3 4
4
4
1
1
6
3
5
1
1 6 55 1
0 1 2 3 4 5 6
0 0 1 0 0 1 0 0
1 0 0 2 0 1 0 1
2 0 0 0 2 0 0 0
3 0 1 0 0 1 0 0
4 2 1 0 0 0 1 1
5 0 2 0 1 0 0 0
6 0 0 0 0 0 1 0
Image G
Example
• L=256• Q “one pixel immediately to the right”
Image
G - Matrix
Features based on the co-ocurrence Matrix
• The elements of G (gij) is converted to probability (pij) by dividing by the amount of pairs in G
• Based on the probability density function we can use• Maximum• Energy (uniformity)• Entropy
0 0
1
L L
iji j
p
Features based on the co-ocurrence Matrix
• Homogenity – closeness to a diagonal matrix
• Contrast
0 0 1 | | L L
ij
i j
p
i j
2
0 0
( )
L L
iji j
i j p
Features based on the co-ocurrence Matrix
• Correlation – measure of correlation with neighbours
0 0
( )( )L Lr c ij
i j r c
i m j m p
2 2
0 0 0
( ) ( ) ( ) ( )
L L L
i ij r i r r ii i i
P i p m iP i i m P i
2 2
0 0 0
( ) ( ) ( ) ( )L L L
j ij c j c c ij j i
P j p m jP j j m P i
Example
• L=256• Q “one pixel immediately to the right”
Image
G - Matrix
Example
Image
G - Matrix
Correlation Contrast Homogeneity
1 - 0.0005 10838 0.0366
2 0.9650 570 0.0824
3 0.8798 1356 0.2048
Moments
• Definition
• Order of a moment is• Moments identify an object uniquely
• Centroid
, ,
[ , , ]p q rpqr
x y z
x y z f x y z pqr p q r
100 010 001
000 000 000
( , , ) ( , , )x y zc c c
[ ] :[1, ] [1, ]nf x N L
Central Moments
• Moments invariant to position
• Normalized central moments, ,
( ) ( ) ( ) ( , , )p q rpqr x y z
x y z
c x c y c z c f x y z
, ,
13
000
( ) ( ) ( ) ( , , )p q rx y z
x y zpqr p q r
x c y c z c f x y z
Moments invariant rotation-translation-scaling• For 3D three moments (Sadjadi 1980)
• For 2D seven moments (Hu’s 1962)
1 200 020 002
2 2 22 200 020 200 002 020 002 101 110 011
23 200 020 002 002 110 110 101 011
2 2020 101 200 011
2
J
J
J
Moments invariant rotation-translation-scaling-mirroring (within minus sign)
1 2 3 4
5 6
, , , ,
,
are all
equal Mirroring
7 7
7 7
or
Projections
1
( ) ( , )
yN
xy
proj x f x y
1
( ) ( , )xN
yx
proj y f x y
x
y
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