machine vision - miun.se

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Copyright (c) Benny Thörnberg 1:44

Machine visionImage analysis I

W

V

Benny Thörnberg

Assistant professor

in electronics


Outline

•Uniform sampling and quantization

•Spatial and frequency domains

•Image filtering in frequency domain

•Spatial image processing

•Point processing

•Sliding window

•Geometric transformations

•Affine transformations

•Pixel interpolation

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Outline

•Region based image segmentation

•Global thresholding

•Local adaptive thresholding

•Contour based image segmentation

•Sobel and Laplacian operators

•Subpixel precise image segmentation

•Connected components and connectivity

•Component labeling

•Feature extraction


Spatial domain - Uniform sampling and

Quantization

Reference: R.C. Gonzales and R.E. Woods, Digital Image Processing, Addison-Wesley

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Spatial domain - Uniform sampling and

Quantization


Gray-level and spatially

continuous imageUniformly sampled image with

digitized gray-levels


Uniform sampling and Quantization

−−−−

−

−

=

)1,1()1,1()0,1(

)1,1()1,1()0,1(

)1,0()1,0()0,0(

),(

MNfNfNf

Mfff

Mfff

yxf

K

M

L

L

•Digitization of spatial coordinates (x,y) is called image sampling and

amplitude digitization is called gray-level quantization

{ } +∈∧−∈∧∈ ZRIIyxf R 12..0),( 2

Fundamental data structures for an image

•Image processing that operates directly on pixels in one single image is

called spatial image processing.

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Frequency domain – DFT - IDFT

{ } ∑∑−

=

−

=

+−

⋅≡=1

0

1

0

)(2

],[],[],[M

x

N

y

N

vy

M

uxj

eyxfvuFyxfFπ

2-dimensional discrete Fourier transform DFT

{ } ∑−

=

−

⋅≡=1

0

2

][][][N

x

N

uxj

exfuFxfF

π1-dimensional discrete Fourier transform DFT

{ } ∑∑−

=

−

=

+− ⋅≡=

1

0

1

0

)(21 ],[],[],[

M

u

N

v

N

vy

M

uxj

evuFyxfvuFFπ

2-dimensional inverse discrete Fourier transform IDFT


Frequency domain - Examples

Spatial domainAmplitude spectrum

of Frequency domain

F

F

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Frequency domain - Examples

Spatial domain

F

F

Amplitude spectrum

of Frequency domain


Image filtering in Frequency domain


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Image smoothing in Frequency domain



Image smoothing in Frequency domain


Original Radii=5

Radii=30

Radii=230Radii=80

Radii=15

2:nd order Butterworth LPF

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Spatial image enhancement – point processing

( )( )12,0,5.0maxmin)( −++= bbaggf

Linear gray level transformation and quantization

12 −b

12 −b

g

f(g)

a

b

a: contrastb: brightness



Histogram equalization

•The purpose with applying histogram equalization is to improve contrast

and normalize the image histogram

•We want to apply a gray level transform such that an arbitrary image will

have a uniform Probability Distribution Function.

•The PDF is usually not defined for an image but we can estimate the PDF

from calculation of the histogram.

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)(rTs =

Histogram equalization

Assume a monotonically increasing gray level transformation function

Hence, there exist a reverse function 1010)(1 ≤≤∧≤≤∧= −rssTr

)(1

)()(sTr

rsds

drrpsp

−=

⋅=(*) Relation between pdf of r and s is given by

)()()(0

rcdfdwwprTs

r

r === ∫Consider the following transformation

Then )(rpdr

dsr=

1)(

1)()(

)(1

=

⋅=

−= sTrr

rsrp

rpspWith (*), we have



Histogram equalization - Example

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Histogram equalization - Example

Global histogram equalization Local histogram equalization

using 7x7 neighborhood




Image averaging

•The purpose of image averaging is to reduce temporal noise by calculating

a mean value image out of a sequence of images

),(),(),( yxyxfyxg η+=We consider additive Gaussian noise

∑ ==

N

i i yxgN

yxg1

),(1

),(We calculate the mean value image

{ } 2

),(

2

),(

1),(),( yxyxg

NyxfyxgE ησσ =∧=It follows that

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Image averaging

Averaging of 8 images Averaging of 128 imagesSingle noisy image


Spatial image enhancement – Sliding Window


+

−= −=

∈∧

−−⋅=

∗=

∑ ∑

ZS

byaxfbacyxg

cfg

S

Sa

S

Sb

),(),(),(

A filtered output image g(x,y) is calculated

through convolution with the input image f(x,y)

and a mask c(x,y).

Describes a square sized filter mask c of size 2S+1 x 2S+1

Filter mask c is often symmetric and has odd number of

coefficients in both dimensions.

Describes a FIR filter in 2 dimensions

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1/25 1/25 1/25 1/25 1/25

1/25 1/25 1/25 1/25 1/25

1/25 1/25 1/25 1/25 1/25

1/25 1/25 1/25 1/25 1/25

1/25 1/25 1/25 1/25 1/25

∗



1/25 1/25 1/25 1/25 1/25

1/25 1/25 1/25 1/25 1/25

1/25 1/25 1/25 1/25 1/25

1/25 1/25 1/25 1/25 1/25

1/25 1/25 1/25 1/25 1/25

Frequency response of a 5x5 mean filter

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Geometric transformations – Affine

transformations

Translation

+

+=

⋅

=

0

0

0

0

*

*

110

01

YY

XXY

X

Y

X

Y

X

New coordinates (X*,Y*) are calculated for all previous coordinates (X,Y)

such that a translational movement (X0,Y0) becomes visual.

+

+

=

⋅

=

11100

10

01

1

0

0

0

0

*

*

YY

XX

Y

X

Y

X

Y

X

The use of square matrixes simplifies sequential steps of several matrix operations

An affine transformation preserves lines, planes, parallel lines and parallel planes


Geometric transformations – Affine

transformations

+−

+

=

⋅

−=

1

cossin

sincos

1100

0cossin

0sincos

1

*

*

θθ

θθ

θθ

θθ

YX

YX

Y

X

Y

X

=

⋅

=

11100

00

00

1

*

*

YS

XS

Y

X

S

S

Y

X

Y

X

Y

X

Scaling

Rotation clockwise

X

Y

θ

(X,Y)

(X* ,Y*)

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Geometric transformations – Pixel

interpolation

Problem definition

Reference: C. Steger, M. Ulrich and C. Wiedermann, Machine Vision Algorithms and Applications


Geometric transformations – Pixel

interpolation


Bilinear interpolation

( ) ( )00100111 )1()1()1(~ gaagbgaagbg −+−+−+=

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Geometric transformations


Nearest neighbor versus Bilinear interpolation

Nearest neighbor

Bilinear interpolation




Aliasing effects at down scaling

Down scaled by a factor 3 using

bilinear interpolation

Same as above but preprocessed

with smoothing

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Polar transformation Perspective transformation


Image segmentation


•We want to extract regions in the image that correspond to the image

objects of interest.

•Image segmentation takes an image as input and returns one or more

regions or subpixel precise contours as output.

•Gray-levels, motion (in case of video), spectral analysis (colors) and texture

are examples of image properties that can be used as input to the

segmentation process.

•Image is decomposed into smaller pieces based on some kind of strategy

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Image segmentation

25590 maxmin =∧= gg


Global gray-level thresholding R

{ }maxmin ),(|),( gcrfgRcrS ≤≤∈=


Image segmentation


Local dynamic gray-level thresholding

{ }diffgcrgcrfRcrS ≥−∈= ),(),(|),(

Assume a gray-level image f(r,c) and a its smoothened correspondence g(r,c).

Then the dynamic thresholding of bright objects on a darker background is:

Smoothened

image g(r,c)

Original

image f(r,c)

Has ability to adapt to non homogeneous illumination, hence variations in background

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Image segmentation – Edge detection

•Illumination typically varies over large areas which makes large regions to grow or shrink if

we apply a global threshold.

•Another robust region segmentation can be achieved if we focus on finding the borders of

regions.

•Finding borders is often done as edge detection.



First order derivate

Second order derivate


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( )22

∂

∂+

∂

∂=∇

y

f

x

ffmag

The magnitude of the gradient vector is an obvious way to enhance edges in a picture

Z9Z8Z7

Z6Z5Z4

Z3Z2Z1

However, since this is a “computational demanding” operation,

we need to find approximations.

The Sobel operator is probably the most well known and used approximation

( ) ( ) ( ) ( ) ( )741963321987 2222 zzzzzzzzzzzzfmag ++−+++++−++≈∇

The Sobel operator can be implemented as two consecutive computational steps using the

following masks.

121

000

-1-2-1

10-1

20-2

10-1 The Sobel operators constitute both a

smoothing filter and a gradient

operator.


Image segmentation – Sobel operator

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Image segmentation – Laplacian operator

The Laplacian is a second order derivate and is defined as:

)(4 86425

2zzzzzf +++−≈∇Laplacian is approximated as :

2

2

2

22

y

f

x

ff

∂

∂+

∂

∂=∇

0-10

-14-1

0-10

The Laplacian is very sensitive to noise and is for that reason preferable combined with a

smoothing filter (Gaussian). Zero crossings should be used to detect the edges.


Subpixel-precise image segmentation


A continuous image function is estimated from

bilinear interpolation of gray levels between four

pixel centers.

Subpixel precise thresholding

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Subpixel-precise edge detection


The Laplacian operator can be combined with Gaussian smothing and subpixel precise zero

crossing in order to detect edges at subpixel precision.

Subpixel precise

thresholding on gray values

Laplacian and subpixel precise

zero crossing


Extraction of connected components

{ } ),()1,(),1,(),,1(),,1()(4 yxpyxyxyxyxpN =∧−+−+=

Four-connectivity

p

),()1,1(),1,1(),1,1(),1,1(

),1,(),1,(),,1(),,1()(8 yxp

yxyxyxyx

yxyxyxyxpN =∧

−−+−−+++

−+−+=

Eight-connectivity

p

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Extraction of 4-connected components

Component labeling

P

r

t

First pass:

If(P=1) then

If( t=1 and r=1 and different labels) then

assign P to one of the labels and note equivalence

else If( t=1 or r=1) then

assign P to label of r or t

else

assign new label to P

else nothing

Second pass:

Look for all equivalent labels and assign those to unique labels


Extraction of connected components

Component labeling

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Feature extraction

•Region features

- Compactness

- Anisometry

- Moments (can be normalized with respect to area

and/or position)

- Area, (p,q)=(0,0)

- Center of gravity

∑∈

=Ryx

qp

qp yxcrgm),(

, ),(

•Gray value features

- Max, min, mean, variance

- Moments

∑∈

=Ryx

qp

qp yxm),(

,

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