Computer and Robot Vision I

Computer and Robot Vision I

Chapter 7Conditioning and Labeling

Presented by: 傅楸善 & 江祖榮 r02922085@ntu.edu.tw指導教授 : 傅楸善 博士

Recognition Methodology

Conditioning Labeling Grouping Extracting Matching

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dDC & CV Lab.CSIE NTU

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7-1 Introduction

Conditioning: noise removal, background normalization,…

Labeling: thresholding, edge detection, corner finding,…

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7-2 Noise Cleaning

noise cleaning: • uses neighborhood spatial coherence• uses neighborhood pixel value homogeneity

7-2 Filter

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f(x-1,y-1) f(x,y-1) f(x+1,y-1)f(x-1,y) f(x,y) f(x+1,y)

f(x-1,y+1) f(x,y+1) f(x+1,y+1)

W(-1,-1) W(0,-1) W(1,-1)W(-1,0) W(0,0) W(1,0)W(-1,1) W(0,1) W(1,1)

Y axis

X axis

image Filter

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box filter: computes equally weighted averagebox filter: separablebox filter: recursive implementation with “two+”, “two-”, “one/” per pixel

7.2 Noise Cleaning

filter: separable

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9 multiplications 3 multiplications

* = *

3 multiplications

* = *

9 multiplications 3 multiplications 3 multiplications

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filter: separable

box filter: recursive implementation

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7.2 Noise Cleaning

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=* = *

7.2 Noise Cleaning

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=* = *

7.2 Noise Cleaning

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7.2 Noise Cleaning

Gaussian filter: linear smoother)(

21

2

22

),( cr

kecrw

for all where,),( Wcr

Wcr

cr

e

k

),(

)(21

2

22

1

linear noise-cleaning filters: defocusing images, edges blurred

size of W: two or three from center

weight matrix:

Take a Break 0

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A Statistical Framework for Noise Removal

Idealization assumption: if there were no noise, the pixel values in each image neighborhood would be the same constant

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Outlier or Peak Noise

outlier: peak noise: pixel value replaced by random noise value

neighborhood size: larger than noise, smaller than preserved detail

center-deleted: neighborhood pixel values in neighborhood except center

Outlier or Peak Noise

X1 X2 X3

X4 y X5

X6 X7 X8

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Decide whether y is an outlier or not

:y center pixel value

N

nnx

N 1

1

:,...,1 Nxx center-deleted neighborhood

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Outlier or Peak Noise

:,...,1 Nxx

:y

:

N

nnx

N 1

1

center-deleted neighborhood

center pixel value

mean of center-deleted neighborhood

2

1

)ˆ-(

N

nnxminimizes

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Outlier or Peak Noise

:_ removaloutlierzˆif | |

_ ˆ otherwise{y y

outlier removalz

output value of neighborhood outlier removal

:y

::

not an outlier value if reasonably close toy use mean value when outlierthreshold for outlier valuetoo small: edges blurredtoo large: noise cleaning will not be good

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Outlier or Peak Noise

:ˆ 2

N

nnx

N 1

22 )ˆ(1

1ˆ

ˆif | |

ˆˆ otherwise

{yy

contrast dependent outlier removalz

center-deleted neighborhood variance

: use neighborhood mean if pixel value significantly far from mean

7-2-3 Outlier or Peak Noise

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7-2-3 Outlier or Peak Noise

smooth replacement: instead of complete replacement or not at all

:treplacemensmoothz convex combination of input and mean

:K weighting parameter

yKy

K

Ky

y

z treplacemensmooth

|

ˆˆ

|ˆ

|ˆ

ˆ|

|ˆ

ˆ|

use neighborhood mean

use input pixel value

:0K

:K y

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7-2-4 K-Nearest Neighbor

K-nearest neighbor: average equally weighted average of k-nearest neighbors

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7-2-5 Gradient Inverse weighted

gradient inverse weighted: reduces sum-of-squares error within regions

' '

' '

1 ( , ) ( , )' ' 2otherwise

1max{ , | ( , ) ( , )|}2

( , , , ) { r c r c

k

f r c f r c

w r c r c

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Take a Break 1

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7-2-6 Order Statistic Neighborhood Operators

order statistic: linear combination of neighborhood sorted values

neighborhood pixel values

sorted neighborhood values from smallest to largest

:,,1 Nxx

:,..., )()1( Nxx

7-2-6 Order Statistic Neighborhood Operators

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:,, 91 xx

:,..., )9()1( xx

x1 x2 x3 x4 x5 x6 x7 x8 x916 128 109 66 4 6 96 84 53

x(1) x(2) x(3) x(4) x(5) x(6) x(7) x(8) x(9)4 6 16 53 66 84 96 109 128

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7-2-6 Order Statistic Neighborhood Operators

Median Operator

median: most common order statistic operator

median root: fixed-point result of a median filter

median roots: comprise only constant-valued neighborhoods, sloped edges

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Original Image

Median Root Image

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7-2-6 Order Statistic Neighborhood Operators

median: effective for impulsive noise (salt and pepper)

median: distorts or loses fine detail such as thin lines

)2

1( Nmedian xz

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7-2-6 Order Statistic Neighborhood Operators

inter-quartile distance:Q

)4

2()4

23( NN xxQ

otherwise

||if zy

zy

z Qmedian

mediandianrunning me

Running-median Operator

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7-2-6 Order Statistic Neighborhood Operators

Trimmed-Mean Operator

trimmed-mean: first k and last k order statistics not used

trimmed-mean: equal weighted average of central N-2k order statistics

kN

knnmeantrimmed x

kNz

1)(2

1

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7-2-6 Order Statistic Neighborhood Operators

Midrange operator

midrange: noise distribution with light and smooth tails

][21

)()1( Nmidrange xxz

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7-2-7 Hysteresis Smoothing

hysteresis smoothing: removes minor fluctuations, preserves major transients

hysteresis smoothing: finite state machine with two states: UP, DOWNapplied row-by-row and then column-by-column

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7-2-7 Hysteresis Smoothing

if state DOWN and next one larger, if next local maximum does not exceed threshold then stays current value i.e. small peak cuts flat

otherwise state changes from DOWN to UP and preserves major transients

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7-2-7 Hysteresis Smoothing

if state UP and next one smaller, if next local minimum does not exceed threshold then stays current value i.e. small valley filled flat

otherwise state changes from UP to DOWN and preserves major transients

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7-2-7 Hysteresis Smoothing

Take a Break 2

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Sigma Filter

sigma filter: average only with values within two-sigma interval

Mn

nxM

y#

1ˆ

}22|{ yxynMwhere n

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Selected-Neighborhood Averaging

selected-neighborhood averaging: assumes pixel a part of homogeneous region (not required to be squared, others can be diagonal, rectangle, three

pixels vertical and horizontal neighborhood)

noise-filtered value: mean value from lowest variance neighborhood

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Minimum Mean Square Noise Smoothing

minimum mean square noise smoothing: additive or multiplicative noise

each pixel in true image: regarded as a random variable

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Minimum Mean Square Noise Smoothing

YZ

ZY

])[(

error squaremean minimize to and choose :Goal

22 YYE

variables:,valuepixelestimated:

noise:valuepixel:

valuepixelobserved:

Y

YZ

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Noise-Removal Techniques-Experiments

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Noise-Removal Techniques-Experiments

uniform Gaussian salt and pepper varying noise (the noise energy varies across the image)

types of noise

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Noise-Removal Techniques-Experiments

salt and pepper

:maxmini

gray value at given pixel in output image

:p:u:),( crI:),( crO

),( cr

{ if),(

otherwise)( minmaxmin

),(pucrI

uiiicrO

minimum/ maximum gray value for noise pixels

fraction of image to be corrupted with noise

uniform random variable in [0,1]

gray value at given pixel in input image

),( cr

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Generate salt-and-pepper noise

Noise-Removal Techniques-Experiments

I(nim, i , j) = 0 if uniform(0,1) < 0.05I(nim, i , j) = 255 if uniform(0,1) > 1- 0.05I(nim, i , j) = I(im, i ,j) otherwiseuniform(0,1) : random variable uniformly distributed over [0,1]

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Noise-Removal Techniques-Experiments

S/N ratio (signal to noise ratio):

VS: image gray level variance VN: noise variance

𝑆𝑁𝑅=20∗ log10√𝑉𝑆√𝑉𝑁

Noise-Removal Techniques-Experiments

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𝑆𝑁𝑅=20∗ log10√𝑉𝑆√𝑉𝑁

𝑉𝑆=∑∀ 𝑛

( (𝐼 (𝑖 , 𝑗 )−𝜇𝑠 ))2

‖𝑛‖

𝜇𝑠=∑∀𝑛

𝐼 (𝑖 , 𝑗 )

‖𝑛‖

𝜇𝑁𝑜𝑖𝑠𝑒=∑∀𝑛

𝐼𝑁𝑜𝑖𝑠𝑒 (𝑖 , 𝑗 )− 𝐼 (𝑖 , 𝑗)

‖𝑛‖

𝑉𝑁=∑∀𝑛

( ( 𝐼𝑁𝑜𝑖𝑠𝑒 (𝑖 , 𝑗 )− 𝐼 (𝑖 , 𝑗)−𝜇𝑁𝑜𝑖𝑠𝑒 ) )2

‖𝑛‖

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Take a Break 3

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Noise-Removal Techniques-Experiments

uniform noise

variessalt and pepper noise

Gaussian noise

Uniform

S & P Gaussiannoise

Original

default peak noise removal

2.0,99;{ |ˆ|

ˆ_

yify

otherwiseremovaloutlierz

Uniform

S & P Gaussian

outlier removal

Uniform

S & P Gaussian

contrast-dependent noise removal

1.0,99;{ |ˆ

ˆ|

ˆ

yify

otherwiseremovaloutlierdependentcontrastz

Uniform

S & P Gaussian

contrast-dependent outlier removal

Uniform

S & P Gaussian

2.0,4.0,99;ˆ|

ˆˆ

|

|ˆ

ˆ|

|ˆ

ˆ|

KKy

y

yKy

Kz treplacemensmooth

smooth replacement

Uniform

S & P Gaussian

contrast-dependent outlier removal with smooth

replacement

Uniform

S & P Gaussian

hysteresis smoothing

Uniform

S & P Gaussian

hysteresis smoothing

Uniform

S & P Gaussian

inter-quartile mean filter

5.0,77;{ ||

Qzy

ify

otherwisez

median

median

z

)4

2()4

23( NN xxQ

Uniform

S & P Gaussian

inter-quartile mean filter

Uniform

S & P Gaussian

neighborhood midrange filter

77;][21

)()1( Nmidrange xxz

Uniform

S & P Gaussian

neighborhood midrange filter

Uniform

S & P Gaussian

neighborhood running-mean filter

)77,( filterbox

Uniform

S & P Gaussian

neighborhood running-mean filter

Uniform

S & P Gaussian

sigma filter

Mn

nxM

y#

1ˆ

2.0,99;}22|{ yxynMwhere n

Uniform

S & P Gaussian

sigma filter

Uniform

S & P Gaussian

neighborhood weighted median filter

(7X7)

)77(

1111111122222112333211234321123332112222211111111

Uniform

S & P Gaussian

neighborhood weighted median filter

Uniform

S & P Gaussian

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Uniform Gaussian_30 S & P 0.1Contrast-dependent outlier removal 26.408 25.758 20.477Smooth replacement 29.344 26.821 26.792Outlier removal 26.407 25.768 20.406Hysteresis 18.278 13.804 1.9040Interquartile mean filter 29.193 24.517 35.720Midrange filter 22.171 19.629 -0.1658Running-mean filter 29.522 28.278 21.194Sigma filter 34.717 18.831 -2.9151Weighted-median filter 33.964 30.720 36.033

Noise-Removal Techniques-Experiments with Lena

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7.3 Sharpening

unsharp masking: subtract fraction of neighborhood mean and scale result

N

nn meanodneighborhoisx

Nwhere

1

1

32,

51,:

::

betweenreasonablefractionk

ntconstascalingsvaluepixelcentery

possible to replace neighborhood mean with neighborhood median

]ˆ[ kyszsharpened

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Extremum Sharpening

extremum-sharpening: output closer of neighborhood maximum or minimum

{ ),(),(

max

min

maxminmin

max

}),(|),(max{}),(|),(min{

crfzzcrfifz

otherwisezsharpenedextremumz

WcrccrrfzWcrccrrfz

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Take a Break

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7.4 Edge Detection

digital edge: boundary where brightness values are significantly different

edge: brightness value appears to jump

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Gradient Edge Detectors

gradient magnitude: where are values from first, second masks

respectively

22

21 rr

Roberts operators: two 2X2 masks to calculate gradient

21, rr

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Roberts operators with threshold = 30

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Gradient Edge Detectors

22

21 ppg

21, pp)arctan( 21 pp

Prewitt edge detector: two 3X3 masks in row column direction

gradient magnitude: gradient direction: clockwise w.r.t. column axis where are values from first, second masks respectively

P1

P2

)arctan( 21 pp

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Gradient Edge Detectors

)()1()(' xfxfxf

)1()()1(' xfxfxf

)1()1()1()()()1()1()( ''

xfxfxfxfxfxfxfxf

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Prewitt edge detector with threshold = 30

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Gradient Edge Detectors Sobel edge detector: two 3X3 masks in row column direction

gradient magnitude: gradient direction: clockwise w.r.t. column axis where are values from first, second masks respectively

22

21 ssg

)arctan( 21 ss21, ss

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Gradient Edge Detectors Sobel edge detector: 2X2 smoothing followed by 2X2 gradient

1111

1111

121000121

1111

1111

101202101

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Gradient Edge Detectors

1111

1111

101202101

000011011

1111

1111

000101101

00011111111

1111

1111

011112101

0101011011101

1111

1111

101202101

1011111112

101

1111

1111

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Sobel edge detector with threshold = 30

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Gradient Edge Detectors Frei and Chen edge detector: two in a set of nine orthogonal masks (3X3)

gradient magnitude: gradient direction: clockwise w.r.t. column axis where are values from first, second masks respectively

22

21 ffg

)arctan( 21 ff21, ff

Gradient Edge Detectors orthogonal

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101-2

02-

101-

1 2 1 0 0 0 1- 2- 1-

0 1 1- 1 - 1 1 1 (-1) 1 1(-1) (-1) (-1)

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Gradient Edge Detectors Frei and Chen edge detector: nine orthogonal masks (3X3)

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Frei and Chen edge detector with threshold = 30

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Gradient Edge Detectors

nnnkg

7,...,0,max

Kirsch: set of eight compass template edge masks

gradient magnitude:

gradient direction: nkmaxarg45

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Kirsch edge detector with threshold = 30

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Gradient Edge Detectors Robinson: compass template mask set with

only2,1,0

done by only four masks since negation of each mask is also a mask gradient magnitude and direction same as Kirsch operator

nnnrg

7,...,0,max

nrmaxarg45

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Robinson edge detector with threshold = 30

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Gradient Edge Detectors Nevatia and Babu: set of six 5X5 compass template masks

參考軸： y 軸

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Nevatia and Babu with threshold = 30

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Gradient Edge Detectors

edge contour direction: along edge, right side bright, left side dark

edge contour direction: more than gradient direction90

90

contour direction

gradient direction

Robinson and Kirsch compass operator: detect lineal edges

90

90 90 90

90

9090 90

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Gradient Edge Detectors

four important properties an edge operator might have

accuracy in estimating gradient magnitude accuracy in estimating gradient direction accuracy in estimating step edge contrast accuracy in estimating step edge direction

gradient direction: used for edge organization, selection, linking

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7.4.1 Gradient Edge Detectors

參考軸： X 軸 , 順時針方向

contour

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Take a Break

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Zero-Crossing Edge Detectors

first derivative maximum: exactly where second derivative zero crossing

first derivative maximum: exactly where second derivative zero crossing

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Zero-Crossing Edge Detectors

2

2

2

2

2

2

2

22 )(

cI

rII

crI

),( crILaplacian of a function

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Zero-Crossing Edge Detectors

two common 3X3 masks to calculate digital Laplacian

First type Second type

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Zero-Crossing Edge Detectors the 3X3 neighborhood values of an image function

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Zero-Crossing Edge Detectors

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

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

( 1,-1) ( 1, 0) ( 1, 1)2

652

4321),( ckrckrkckrkkcrI

( r, c)

654

321

kkkkkk

631 kkk

654

321

kkkkkk

1k

421 kkk

421 kkk

654

321

kkkkkk

631 kkk

654

321

kkkkkk

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

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

( 1,-1) ( 1, 0) ( 1, 1)2

652

4321 )1()1)(1()1()1()1()1,1( kkkkkkI

654

321

kkkkkk

631 kkk

654

321

kkkkkk

1k

421 kkk

421 kkk

654

321

kkkkkk

631 kkk

654

321

kkkkkk

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

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

( 1,-1) ( 1, 0) ( 1, 1)2

652

4321 )0()0)(1()1()0()1()0,1( kkkkkkI

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

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

( 1,-1) ( 1, 0) ( 1, 1)2

652

4321 )1()1)(1()1()1()1()1,1( kkkkkkI

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

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

( 1,-1) ( 1, 0) ( 1, 1)2

652

4321 )1()1)(0()0()1()0()1,0( kkkkkkI

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

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

( 1,-1) ( 1, 0) ( 1, 1)2

652

4321 )0()0)(0()0()0()0()0,0( kkkkkkI

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

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

( 1,-1) ( 1, 0) ( 1, 1)2

652

4321 )1()1)(0()0()1()0()1,0( kkkkkkI

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

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

( 1,-1) ( 1, 0) ( 1, 1)2

652

4321 )1()1)(1()1()1()1()1,1( kkkkkkI

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

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

( 1,-1) ( 1, 0) ( 1, 1)2

652

4321 )0()0)(1()1()0()1()0,1( kkkkkkI

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

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

( 1,-1) ( 1, 0) ( 1, 1)2

652

4321 )1()1)(1()1()1()1()1,1( kkkkkkI

I(r,c) = k1+k2r+k3c+k4r2+k5rc+k6c2

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Zero-Crossing Edge Detectors

2

2

2

2

2

2

2

22 )(

cI

rII

crI

265

24321),( ckrckrkckrkkcrI

542 2 ckrkkrI

42

2

2krI

653 2ckrkkcI

62

2

2kcI

642

2

2

2

2

2

2

22 22)( kk

cI

rII

crI

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Zero-Crossing Edge Detectors

)(k)-kk(k)k-k(k)kk(k)k-k(k 1421421631631 41111

64 22 kk

] 111181

111 [31

654321654321421

6311631

654321421654321

)kkkkk(k)k-kk-kk(k)kk(k)kk(k)(k)-k-k(k

)k-kkk-k(k)k-k(k)kkk-k-k(k

6464 22 66 31 kkkk

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Zero-Crossing Edge Detectors 3X3 mask computing a digital Laplacian 12),44( babae

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Zero-Crossing Edge Detectors

)24()0(

)24()0()0(

)44(

6

5

4

3

2

1

bakk

bakkk

ebak

64 22 kk { )44(

12

ba-e

ba

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Zero-Crossing Edge Detectors

3X3 mask for computing minimum-variance digital Laplacian

First type

)(21

22

22

21:

cr

eGaussian

2

2

2

2

:cI

rILaplacian

2

2

2

2

),(cG

rGcrLOG

)1(2

1

)(2

1

)(2

1

))2

1(()(

2

2)(21

4

)(21

2

2)(21

4

)(21

22

)(21

22

2

2

22

2

22

2

22

2

22

2

22

re

ere

err

errr

Grr

G

cr

crcr

cr

cr

)1(2

12

2)(21

42

22

22

ce

cG cr

)1(2

12

2)(21

42

22

22

re

rG cr

)(21

2

22

4

2

2)(21

42

2)(21

4

2

22

2

22

2

22

)2(2

1

)1(2

1)1(2

1),(

cr

crcr

ecr

cerecrLOG

Laplacian of the Gaussian kernel

-2 -9 -23 -1 103 178 103 -1 -23 -9 -2

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Zero-Crossing Edge Detectors

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Zero-Crossing Edge Detectors

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Zero-Crossing Edge Detectors

A pixel is declared to have a zero crossing if it is less than –t and one of its eight neighbors is greater than t or if it is greater than t and one of its eight neighbors is less than –t for some fixed threshold t

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Zero-Crossing Edge Detectors

t

-t

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Take a Break

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Edge Operator Performance

edge detector performance characteristics: misdetection/false-alarm rate

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7.5 Line Detection

A line segment on an image can be characterized as an elongated rectangular region having a homogeneous gray level bounded on both its longer sides by homogeneous regions of a different gray level

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7.5 Line Detection

one-pixel-wide lines can be detected by compass line detectors

detecting lines of one to three pixels in width

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semilinear line detector by step edge on either side of the line

𝑠=max {𝑎𝑖+𝑏𝑖|𝑎𝑖>𝑡∧𝑏𝑖>𝑡 }

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Project due Nov. 26

Write the following programs

1. Generate additive white Gaussian noise

2. Generate salt-and-pepper noise

3. Run box filter (3X3, 5X5) on all noisy images

4. Run median filter (3X3, 5X5) on all noisy images

5. Run opening followed by closing or closing followed by opening

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Project due Nov. 26

box filter on white Gaussian noise with amplitude = 10Gaussian noise After 5x5 box filterAfter 3x3 box filter

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Project due Nov. 26

box filter on salt-and-pepper noise with threshold = 0.05salt-and-pepper noise After 5x5 box filterAfter 3x3 box filter

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Project due Nov. 26

median filter on white Gaussian noise with amplitude = 10Gaussian noise After 5x5 median filterAfter 3x3 median filter

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Project due Nov. 26

median filter on salt-and-pepper noise with threshold = 0.05salt-and-pepper noise After 5x5 median filterAfter 3x3 median filter

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Project due Nov. 26

Generate additive white Gaussian noise

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

NNamplitudejiimIjinimI

Gaussian random variable with zero mean and st. dev. 1

amplitude determines signal-to-noise ratio, try 10, 30

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Project due Nov. 26 Generate additive white Gaussian noise with amplitude = 10

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Project due Nov. 26 Generate additive white Gaussian noise with amplitude = 30

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Project due Nov. 26

Generate salt-and-pepper noise

I( noiseImage,i ,j) = , if 𝑢𝑛𝑖𝑓𝑜𝑟𝑚 (0,1 )<0.05, if 𝑢𝑛𝑖𝑓𝑜𝑟𝑚 (0,1 )>1 −0.05, otherwise

1.0and05.0bothtry]1,0[overddistributeuniformlyriablevarandom:)1,0(uniform

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Project due Nov. 26 Generate salt-and-pepper noise with threshold = 0.05

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Project due Nov. 26 Generate salt-and-pepper noise with threshold = 0.1

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Project due Dec. 3

Roberts operator Prewitt edge detector Sobel edge detector Frei and Chen gradient operator Kirsch compass operator Robinson compass operator Nevatia-Babu 5X5 operator

Write programs to generate the following gradient magnitude images and choose proper thresholds to get the binary edge images:

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Project due Dec. 3

Roberts operator with threshold = 30

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Project due Dec. 3

Prewitt edge detector with threshold = 30

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Project due Dec. 3

Sobel edge detector with threshold = 30

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Project due Dec. 3

Frei and Chen gradient operator with threshold = 30

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Project due Dec. 3

Kirsch compass operator with threshold = 30

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Project due Dec. 3

Robinson compass operator with threshold = 30

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Project due Dec. 3

Nevatia-Babu 5X5 operator threshold = 30

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Project due Dec. 10

Write the following programs to detect edge:

Zero-crossing on the following four types of images to get edge images (choose proper thresholds), p. 349 Laplacian minimum-variance Laplacian Laplacian of Gaussian Difference of Gaussian, (use tk to generate D.O.G.) dog (inhibitory , excitatory , kernel size=11)

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Difference of Gaussian

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Difference of Gaussian

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Project due Dec. 10

Zero-crossing on the following four types of images to get edge images (choose proper thresholds t=1)

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Laplacian

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minimum-variance Laplacian

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Laplacian of Gaussian

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Difference of Gaussian (inhibitory , excitatory , kernel size=11)