Edge detection Edge Detection - The Blavatnik School of Computer
Image Convolution and Edge Detection
Transcript of Image Convolution and Edge Detection
![Page 1: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/1.jpg)
Image Convolution and
Edge Detection
![Page 2: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/2.jpg)
Edge Formation Factors
Depth discontinuity
Surface color discontinuity
Surface normal discontinuity
Illumination discontinuity
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What’s in front?
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What it is?
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http://www.cfar.umd.edu/~fer/optical/index.html
Cornelia Fermüller
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121 121 118 111 . . . 21
134 136 137 132 . . . 23
133 131 136 136 . . . 25
136 145 148 151 . . . 34
137 140 147 149 . . . 54
. . . . . . . . . . . . . . . . . .
231 233 243 244 . . . 179
i
j
Any 2D matrix can be seen as an image
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200 130 20
255 100 10
200 100 30
Image I
0.5 0 0
0 1.5 0.5
0 -0.5 0
Kernel f
205
Image filtering: Replacing each pixel value by
weighted average of its neighbors
Linear Filtering
![Page 8: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/8.jpg)
Simple Neural Network
521 5
22 2
I(x)=
f(x)=
161
0
g(x)=0
This leads to parallel computation
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Anatomy of eye
The macula is the center of vision (and retina) and
the fovea (FAZ) is the focal point approximately
only 0.4mm in diameter. Reading, driving, etc. is all
performed here.
http://www.retinatexas.com/Anatomy_eng.html
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Photo-sensors
1) Light pass through retina cells, excites Rod and Cone
2) Cone: color(spectral) sensitive, R,G,B, 6 Million
3) Rod: more photo sensitive, peak at 580nm(yellow), 120 M
4) What happens if you miss one type of Cone cells?
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100 0
10.3 0.6
I(x)=
f(x)=
0.60
0
g(x)=
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100 0
10.3 0.6
I(x)=
f(x)=
10.60
10.3 0.6
0
g(x)=
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100 0
10.3 0.6
I(x)=
f(x)=
10.60 0.3
10.3 0.6
10.3 0.6
0
g(x)=
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1
0 0 0
0
0 0
0
0
Image
Kernel
Linear Filter
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1
0 0 0
0
0 0
0
0
Image
Kernel
Linear Filter
Image no change
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1
0 0 0
0
0 0
0
0
Image
Kernel
Linear Filter
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Image
Kernel
Linear Filter
Image Shifted
1
0 0 0
0
0 0
0
0
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0.3
0 0 0
0 0 0
Image
Kernel
Linear Filter
0.30.3
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0.3
0 0 0
0 0 0
Image
Kernel
Linear Filter
0.30.3
Blurred
(horizontal)
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0.3
0
0
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0
0
0
Image
Kernel
Linear Filter
0.3
0.3
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0.3
0
0
0
0
0
0
Image
Kernel
Linear Filter
0.3
0.3
Blurred
(vertical)
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Image
Filter
Kernel
Linear Filter
0.3
0 0 0
0 0 0
0.30.32
0 0 0
0
0 0
0
0-
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Image
Kernel Image
Linear Filter
0.3
0 0 0
0 0 0
0.30.32
0 0 0
0
0 0
0
0-
-
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Image
Kernel Image
Linear Filter
0.3
0 0 0
0 0 0
0.30.32
0 0 0
0
0 0
0
0-
-
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Image filtering
𝑔 𝑚, 𝑛 =
𝑘,𝑙
𝐼 𝑚 + 𝑘, 𝑛 + 𝑙 ∗ 𝑓(𝑘, 𝑙)
Output
Image
Input
Image
Kernal
Image
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0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
Image filtering
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
Image I
1 2 3
4 5 6
7 8 9
Kernel f Output g8x83x3
Loop over every pixel
a b c
d e f
g h i
Register
Calculate result = a*1+b*2+…+i*9
Same position
𝑔 𝑚, 𝑛 =
𝑘,𝑙
𝐼 𝑚 + 𝑘, 𝑛 + 𝑙 ∗ 𝑓(𝑘, 𝑙)
![Page 27: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/27.jpg)
Special case: impulse function
1 2 3
4 5 6
7 8 9
Kernel fImage I
0 0 0
0 1 0
0 0 0
0 1 0
0 0 0
0 0 0
Register
9 8 7
6 5 4
3 2
Output g
Calculate result = 0*9+…+1*2+0*1
![Page 28: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/28.jpg)
Special case: impulse function
1 2 3
4 5 6
7 8 9
Kernel fImage I
0 0 0
0 1 0
0 0 0
1 0 0
0 0 0
0 0 0
Register
9 8 7
6 5 4
3 2 1
Output g
Calculate result = 0*9+…0*2+1*1
<Note> The output is the kernel flipped left-right, up-down!
![Page 29: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/29.jpg)
Convolution
• Let I be an Signal(image), Convolution kernel f,
𝑔 𝑚, 𝑛 =
𝑘,𝑙
𝐼 𝑚 − 𝑘, 𝑛 − 𝑙 ∗ 𝑓(𝑘, 𝑙)
𝐼⨂𝑓
Output
Image
Input
Image
Kernal
Image
![Page 30: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/30.jpg)
Convolution
• Convolution is filtering with kernel flipped
9 8 7
6 5 4
3 2 1
Kernel f
flipped
Image I
0 0 0
0 1 0
0 0 0
1 0 0
0 0 0
0 0 0
Register
1 2 3
4 5 6
7 8 9
Output g
=Kernel
Calculate result = 1*9+0*8+…+0*1
![Page 31: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/31.jpg)
Impulse functions shift images
a b c
d e f
g h i
Image I
1 0 0
0 0 0
0 0 0
Kernel f
0 0 0
0 0 0
0 0 1
Kernel f ’
e f 0
h i 0
0 0 0
Result 𝐼⨂𝑓
• In this case the resulting image shifted to the upper left
![Page 32: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/32.jpg)
Convolution is associative
a b c
d e f
g h i
I
1 0 0
0 0 0
0 0 0
f
e f 0
h i 0
0 0 0
𝐼⨂𝑓
a b c
d e f
g h i
1 0 0
0 0 0
0 0 0
e f 0
h i 0
0 0 0
f I f⨂𝐼
![Page 33: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/33.jpg)
9 8 7
6 5 4
3 2 1
Kernel f
flipped
Image I
2 0 0
0 3 0
0 0 0
37 32 21
22 17 12
9 6 3
Output g
1 0 0
0 0 0
0 0 0
0 0 0
0 1 0
0 0 0
Decompose
*2
*3
5 4 0
2 1 0
0 0 0
*2
9 8 7
6 5 4
3 2 1
*3
Intermediate
Add together
Linear independence
![Page 34: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/34.jpg)
Linear independence
0 1 0 1 0
0 1 0 1 0
0 1 0 1 0
0 1 0 1 0
0 1 0 1 0
Image I
1 0 0
0 1 0
0 0 0
Kernel f
0 1 0 1 0
0 1 0 1 0
0 1 0 1 0
0 1 0 1 0
0 1 0 1 0
Output g2
5x5
1 0 0
0 0 0
0 0 0
Kernel f1
0 0 0
0 1 0
0 0 0
Kernel f2
1 0 1 0 0
1 0 1 0 0
1 0 1 0 0
1 0 1 0 0
0 0 0 0 0
Output g1
1 1 1 1 0
1 1 1 1 0
1 1 1 1 0
1 1 1 1 0
0 1 0 1 0
Output g=g1+g2
![Page 35: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/35.jpg)
• Convolution has commutative property
1 0 0
0 0 0
0 0 0
0 0 0
1 0 0
0 0 0
0 0 0
0 0 0
1 0 0
0 0 0
0 0 0
0 1 0
a b c
d e f
g h i
Image I
a b c
d e f
g h i
Image I
𝐼⨂𝑓
Decompose
1 0 0
1 0 0
1 1 0
Kernel f
⨂
b c 0
e f 0
h i 0
0 0 0
b c 0
e f 0
0 0 0
a b c
d e f
e f 0
h i 0
0 0 0
![Page 36: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/36.jpg)
• Convolution has commutative property
1 0 0
1 0 0
1 1 0
Kernel f
1 0 0
0 0 0
0 0 0
0 0 0
1 0 0
0 0 0
0 0 0
0 0 0
1 0 0
0 0 0
0 0 0
0 1 0
Decompose
b c 0
e f 0
h i 0
0 0 0
b c 0
e f 0
0 0 0
a b c
d e f
e f 0
h i 0
0 0 0
a b c
d e f
g h i
Image I
𝑓⨂𝐼
⨂
a b c
d e f
g h i
![Page 37: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/37.jpg)
Proof of Commutative property
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Output Size of Image
Convolutionf g
f
g
f
g
f
g
Full Same Valid
filter2(g, f, shape) in MATLAB
Full: output_size = f_size + g_size – 1
Same: output_size = f_size
Valid: output_size = f_size – (g_size – 1)
![Page 39: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/39.jpg)
2D visualization of convolution (full)
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
Image I
1 2 3
4 5 6
7 8 9
Kernel f
1 3 6 6 6 6 6 6 5 3
5 1
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79
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
Output g
8x8
3x3
10x10
![Page 40: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/40.jpg)
g : 10 x 10 Gaussian kernel
f : 640 x 360 resolution
Output Size of Image
Convolution
Full
filter2(g, f, shape) in MATLAB
Full: output_size = f_size + g_size – 1
>> full = filter2(g, im, 'full');
>> size(full)
ans =
369 649
![Page 41: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/41.jpg)
2D visualization of convolution
(same)
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
Image I
1 2 3
4 5 6
7 8 9
Kernel f
1
2
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4
2
4
2
4
1
7
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
Output g
8x8
3x3
8x8
![Page 42: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/42.jpg)
g : 10 x 10 Gaussian kernel
f : 640 x 360 resolution
Output Size of Image
Convolution
Same
filter2(g, f, shape) in MATLAB
Full: output_size = f_size + g_size – 1
>>same = filter2(g, im, 'same');
>> size(same)
ans =
360 640
Same: output_size = f_size
![Page 43: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/43.jpg)
2D visualization of convolution
(valid)
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
Image I
1 2 3
4 5 6
7 8 9
Kernel f
4
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0 0 0 0 0 0
0 0 0 0 0 0
Output g
8x8
3x3
6x6
![Page 44: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/44.jpg)
g : 10 x 10 Gaussian kernel
f : 640 x 360 resolution
Output Size of Image
Convolution
Valid
filter2(g, f, shape) in MATLAB
Full: output_size = f_size + g_size – 1
>> valid = filter2(g, im, 'valid');
>> size(valid)
ans =
351 631
Same: output_size = f_size
Valid: output_size = f_size – (g_size – 1)
![Page 45: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/45.jpg)
Image Boundary Effect
The filter window falls off at the edge of image.
![Page 46: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/46.jpg)
![Page 47: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/47.jpg)
Image Extrapolation
(Mirroring)
J = imread(‘image.bmp’);
figure; imshow(J);
Code
J
![Page 48: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/48.jpg)
Image Extrapolation
(Mirroring)Code
boarder = 40;
[nr,nc,nb] = size(J);
J_big = zeros(nr+2*boarder, nc + 2*boarder,nb);
J_big(boarder+1:boarder+nr,boarder+1:boarder+nc,:) = J;
40 nc
nr
J_big
![Page 49: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/49.jpg)
Image Extrapolation
(Mirroring)Code
for i=1:border,
for j=1:border,
J_big(i,j,:) = J(border-i+1,border-j+1,:);
end
end40 nc
nr
J_big
Mirroring with respect to the boarder
(i,j)
(border-i+1,border-j+1)
![Page 50: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/50.jpg)
Image Extrapolation
(Mirroring)Code
for i=1:boarder,
for j=border+1:border+nc,
J_big(i,j,:) = J(border-i+1,j-border,:);
end
end40 nc
nr
J_big
Mirroring with respect to the boarder
(i,j)
(border-i+1,j-border)
![Page 51: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/51.jpg)
Image Extrapolation
(Mirroring)Code
for i=nr+border+1:border*2+nr,
for j=border+1:border+nc,
J_big(i,j,:) = J(2*nr-i+border+1,j-border,:);
end
end40 nc
nr
J_big
Mirroring with respect to the boarder
![Page 52: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/52.jpg)
Image Extrapolation
(Mirroring)Code
for i=border+1:border+nr;
for j=1:border,
J_big(i,j,:) = J(i-border,border-j+1,:);
end
end40 nc
nr
J_big
Mirroring with respect to the boarder
![Page 53: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/53.jpg)
Examples of image operation as
convolution
![Page 54: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/54.jpg)
Gaussian Averaging
• Rotationally symmetric.
• Weights nearby pixels more than distant ones.
– This makes sense as probabalistic inference. • A Gaussian gives a good
model of a fuzzy blob
![Page 55: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/55.jpg)
2D filters, more on this later…
• is the Laplacian operator:
Laplacian of Gaussian
Gaussian derivative of Gaussian
![Page 56: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/56.jpg)
An Isotropic Gaussian
• The picture shows a
smoothing kernel
proportional to
• (which is a reasonable
model of a circularly
symmetric fuzzy blob)
![Page 57: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/57.jpg)
Smoothing with a Gaussian
![Page 58: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/58.jpg)
The effects of smoothing
Each row shows smoothing
with gaussians of different
width; each column shows
different realizations of
an image of gaussian noise.
![Page 59: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/59.jpg)
![Page 60: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/60.jpg)
![Page 61: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/61.jpg)
Image smoothing can remove noise, and also …
![Page 62: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/62.jpg)
![Page 63: Image Convolution and Edge Detection](https://reader033.fdocuments.in/reader033/viewer/2022043006/626b4c107d98d471372910c5/html5/thumbnails/63.jpg)