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Dr. J. [email protected]
M.HosseinKord
Science and Research Branch of Islamic Azad University
Lecture 07 – Pyramids
Machine Vision
1/49 slides
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Table of Contents2
7-1) Gaussian Pyramids
7-2) Laplacian Pyramids
7-3) Applications of Laplacian pyramids
7-5) Optical flow using Pyramids
7-1-1) Reduce 7-1-2) Expand
7-3-1) Image compression 7-3-2) Image composting
7-4) Interpolation
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3
Original ImgHighest Resolution
Lowest Resolution
.
.
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7-1)Gaussian Pyramids
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 4: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/4.jpg)
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7-1-1) Reduce4
Level l
Level l-1
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 5: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/5.jpg)
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7-1-1) Reduce- Convolution
5
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
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7-1-1) Reduce (1D):Example
6
i = 2
Convolution Mask:[w(-2), w(-1), w(0), w(1), w(2)]
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 7: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/7.jpg)
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7-1-1) Reduce (1D)
7
gl = REDUCE (gl-
1)
Convolution Mask:[w(-2), w(-1), w(0), w(1), w(2)][ c , b , a , b , c ]
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
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7-1-2) Expand8
n=1n=2
Notice:
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 9: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/9.jpg)
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7-1-2) Expand(1D)9
i = 4
[w(-2), w(-1), w(0), w(1), w(2)][ c , b , a , b , c ]
Involved weights [c , a , c]
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
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7-1-2) Expand(1D)10
i = 3
[w(-2), w(-1), w(0), w(1), w(2)][ c , b , a , b , c ]
Involved weights [b , b]
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
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Expand11
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 12: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/12.jpg)
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Convolution Mask12
•Separable
•Symmetric
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 13: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/13.jpg)
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Convolution Mask13
• The sum of mask should be 1.
•All nodes at a given level must contribute the same total weight to the nodes at the next higher level.
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 14: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/14.jpg)
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Convolution Mask14
a + 2b + 2c = 1
a + 2c = 2b
b= ¼
cacb b
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
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Convolution Mask 15
a= 0.5 TRINGULAR
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
a= 0.4 GAUSSIAN
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Gaussian Mask16
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
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Gaussian Pyramid17
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 18: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/18.jpg)
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Gaussian Pyramid18
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
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Laplacian Pyramids
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Laplacian Pyramids20
• Similar to edge detected images.• Most pixels are zero.• Can be used for image compression.
L1 = g1 – EXPAND[g2]L2 = g2 – EXPAND[g3]L3 = g3 – EXPAND[g4]L4 = g4
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 21: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/21.jpg)
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Laplacian Pyramids21
Lower in size and resolution
Gaussian Pyramid
Laplacian Pyramid
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
L1 = g1 – EXPAND[g2]L2 = g2 – EXPAND[g3]L3 = g3 – EXPAND[g4]L4 = g4
![Page 22: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/22.jpg)
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Applications of Laplacian pyramids
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Image compression23
•Compute Gaussian pyramid
•Compute Laplacian pyramid
•Code Laplacian pyramid
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 24: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/24.jpg)
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24
• Decode Laplacian pyramid.• Compute Gaussian pyramid from Laplacian pyramid.
• g1 is reconstructed image.
Image compression
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 25: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/25.jpg)
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Image Compression (Entropy)
25
7.6
4.4
5.0
5.6
6.2
0.77
1.9
3.3
4.2
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 26: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/26.jpg)
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Image Compression26
1.58
0.73
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 27: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/27.jpg)
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Combining Apple & Orange27
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
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Algorithm28
• Generate Laplacian pyramid Lo of orange image.
• Generate Laplacian pyramid La of appleimage.
• Generate Laplacian pyramid Lc by– copying left half of nodes at each level from apple
and– right half of nodes from orange pyramids.• Reconstruct combined image from Lc.
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 29: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/29.jpg)
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Interpolation
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Interpolation30
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 31: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/31.jpg)
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1‐D Interpolation31
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
1=< x =<2
![Page 32: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/32.jpg)
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2‐D Interpolation32
Bilinear
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 33: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/33.jpg)
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Bi‐linear Interpolation33
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 34: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/34.jpg)
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Optical flow using Laplacian Pyramid
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Why Lucas Kanade with Pyramids?35
• Horn-Schunck and Lucas-Kanade optical method works only for small motion.
• If object moves faster, the brightness changes rapidly, 2x2 or 3x3 masks fail to estimate spatiotemporal derivatives.
• Pyramids can be used to compute large optical flow vectors.
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 36: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/36.jpg)
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Lucas Kanade with Pyramids
36
LK for highest level of Laplacian pyramid ui , viDo the interpolation u*i-1 , v*i-1Multiply by 2 u*i-1 , v*i-1Calculate ft according to displacement of u*i-1 , v*i-1 LK for level l-1 of Laplacian pyramid u’i-1 , v’i-1Accurate value of Optical flow is ui-1 = u*i-1 + u’i-1
vi-1 = v*i-1 + v’i-1
Lucas Kanade
Lucas Kanade
Interpolation
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 37: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/37.jpg)
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Laplacian Pyramid37
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 38: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/38.jpg)
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Laplacian Pyramid38
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 39: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/39.jpg)
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Laplacian Pyramid39
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 40: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/40.jpg)
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Laplacian Pyramid40
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids
![Page 41: Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord](https://reader036.vdocuments.mx/reader036/viewer/2022062305/568150c7550346895dbeeb2e/html5/thumbnails/41.jpg)
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Laplacian Pyramid41
7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask
7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting
7-4) interpolation
7-5) Optical flow using Pyramids