computer and robot vision i chapter 7 conditioning and labeling presented by: 傅楸善 &...
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![Page 1: Computer and Robot Vision I Chapter 7 Conditioning and Labeling Presented by: 傅楸善 & 江祖榮 r02922085@ntu.edu.tw 指導教授 : 傅楸善 博士](https://reader038.vdocuments.net/reader038/viewer/2022102718/56649e0b5503460f94af2836/html5/thumbnails/1.jpg)
Computer and Robot Vision I
Chapter 7Conditioning and Labeling
Presented by: 傅楸善 & 江祖榮 [email protected]指導教授 : 傅楸善 博士
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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
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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
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filter: separable
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9 multiplications 3 multiplications
* = *
3 multiplications
* = *
9 multiplications 3 multiplications 3 multiplications
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filter: separable
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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
)(2
12
22
),( cr
kecrw
for all where,),( Wcr
Wcr
cr
e
k
),(
)(2
12
22
1
linear noise-cleaning filters: defocusing images, edges blurred
size of W: two or three from center
weight matrix:
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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
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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 outlier
threshold for outlier value
too small: edges blurred
too 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 | |
ˆ
ˆ otherwise{
yy
contrast dependent outlier removalz
center-deleted neighborhood variance
: use neighborhood mean if pixel value significantly far from mean
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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
yK
yK
Ky
y
z treplacemensmooth
|
ˆ
ˆ|
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( , ) ( , )' ' 2
otherwise1
max{ , | ( , ) ( , )|}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
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7-2-6 Order Statistic Neighborhood Operators
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:,, 91 xx
:,..., )9()1( xx
x1 x2 x3 x4 x5 x6 x7 x8 x9
16 128 109 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
z
yz Q
median
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
][2
1)()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, DOWN
applied 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
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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
Y
Z
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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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S/N ratio (signal to noise ratio):
VS: image gray level variance VN: noise variance
𝑆𝑁𝑅=20∗ log10√𝑉𝑆√𝑉𝑁
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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
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Uniform
S & P Gaussiannoise
Original
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default peak noise removal
2.0,99;{|ˆ|
ˆ_
yify
otherwiseremovaloutlierz
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Uniform
S & P Gaussian
outlier removal
Uniform
S & P Gaussian
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contrast-dependent noise removal
1.0,99;{|
ˆ
ˆ|
ˆ
yify
otherwiseremovaloutlierdependentcontrastz
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Uniform
S & P Gaussian
contrast-dependent outlier removal
Uniform
S & P Gaussian
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2.0,4.0,99;ˆ|
ˆˆ
|
|ˆ
ˆ|
|ˆ
ˆ|
KK
y
y
yK
yK
z treplacemensmooth
smooth replacement
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Uniform
S & P Gaussian
contrast-dependent outlier removal with smooth
replacement
Uniform
S & P Gaussian
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hysteresis smoothing
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Uniform
S & P Gaussian
hysteresis smoothing
Uniform
S & P Gaussian
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inter-quartile mean filter
5.0,77;{||
Q
zyify
otherwisez
median
median
z
)4
2()
4
23(
NN xxQ
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Uniform
S & P Gaussian
inter-quartile mean filter
Uniform
S & P Gaussian
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neighborhood midrange filter
77;][2
1)()1( Nmidrange xxz
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Uniform
S & P Gaussian
neighborhood midrange filter
Uniform
S & P Gaussian
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neighborhood running-mean filter
)77,( filterbox
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Uniform
S & P Gaussian
neighborhood running-mean filter
Uniform
S & P Gaussian
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sigma filter
Mn
nxM
y#
1ˆ
2.0,99;}22|{ yxynMwhere n
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Uniform
S & P Gaussian
sigma filter
Uniform
S & P Gaussian
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neighborhood weighted median filter
(7X7)
)77(
1111111
1222221
1233321
1234321
1233321
1222221
1111111
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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
3
2,
5
1,:
:
:
betweenreasonablefractionk
ntconstascalings
valuepixelcentery
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
Wcrccrrfz
Wcrccrrfz
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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()( ''
xfxf
xfxfxfxfxfxf
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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
11
11
11
11
121
000
121
11
11
11
11
101
202
101
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Gradient Edge Detectors
11
11
11
11
101
202
101
000
011
011
11
11
11
11
000
101
101
000
1111
1111
11
11
11
11
011
112
101
01010
11011
101
11
11
11
11
101
202
101
10111
11112
101
11
11
11
11
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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
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Gradient Edge Detectors orthogonal
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1
0
1-
2
0
2-
1
0
1-
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
nnn
kg7,...,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 only
2,1,0
done by only four masks since negation of each mask is also a mask gradient magnitude and direction same as Kirsch operator
nnn
rg7,...,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
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Robinson and Kirsch compass operator: detect lineal edges
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90
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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
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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)
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Zero-Crossing Edge Detectors
2
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Zero-Crossing Edge Detectors
)(k)-kk(k)k-k(k)kk(k)k-k(k 1421421631631 41111
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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
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Zero-Crossing Edge Detectors
3X3 mask for computing minimum-variance digital Laplacian
First type
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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
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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(),,(),,(
N
NamplitudejiimIjinimI 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
![Page 147: Computer and Robot Vision I Chapter 7 Conditioning and Labeling Presented by: 傅楸善 & 江祖榮 r02922085@ntu.edu.tw 指導教授 : 傅楸善 博士](https://reader038.vdocuments.net/reader038/viewer/2022102718/56649e0b5503460f94af2836/html5/thumbnails/147.jpg)
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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)