View
213
Download
0
Category
Preview:
Citation preview
Pyramids
Lecture‐5
Egyptian Pyramids• There are 138 pyramids discovered in Egypt as of 2008.• Most were built as tombs for the country's Pharaohs. • The earliest known Egyptian pyramids are found at Saqqara, northwest of
Memphis. – The earliest among these is the Pyramid of Djoser (constructed 2630 BCE–2611 BCE) .
– This pyramid and its surrounding complex were designed by the architect Imhotep.– The estimate of the number of workers to build the pyramids range from a few
thousands to up to 100,000.
• The most famous Egyptian pyramids are those found at Giza, on the outskirts of Cairo. Several of the Giza pyramids are counted among the largest structures ever built.
• The Pyramid of Khufu at Giza is the largest Egyptian pyramid. It is the only one of the Seven Wonders of the Ancient World still in existence.
Pyramids
• Very useful for representing images.• Pyramid is built by using multiple copies of image.
• Each level in the pyramid is 1/4 of the size of previous level.
• The lowest level is of the highest resolution.• The highest level is of the lowest resolution.
Image Pyramid
Pyramid
Gaussian pyramid
Source: Forsyth
Laplacian pyramid
Source: Forsyth
Deformable Part Model (DPM)
Slide Credit: Pedro F. Felzenszwalb
Contents
• Gaussian and Laplacian Pyramids– Reduce– Expand
• Applications of Laplacian pyramids– Image compression– Image composting
Cited by 5067
Gaussian Pyramids (reduce)
)2,2(),(),(2
2
2
21 njmignmwjig
m nll
][ 1 ll gREDUCEg
Level l
Convolution
Reduce (1D)
Reduce
0 1 2 3 4 5 6 7 8
0 1 2 3 4
0 1 2
Gaussian Pyramids (expand)
Expand (1D)
Expand (1D)
Expand
0 1 2 3 4 5 6 7 8
0 1 2 3 4
Expand
0 1 2 3 4 5 6 7 8
0 1 2 3 4
Convolution Mask
Convolution Mask
• Separable
•Symmetric
Convolution Mask
• 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.
Convolution Mask
cc
bba
b
Convolution Mask
a=.4 GAUSSIAN, a=.5 TRINGULAR
Triangular
00.050.1
0.150.2
0.250.3
0.350.4
0.450.5
c b a b c
Triangular
a=.4 GAUSSIAN, a=.5 TRINGULAR
Approximate Gaussian
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
c b a b c
Gaussian
a=.4 GAUSSIAN, a=.5 TRINGULAR
Gaussian
Gaussian
-3 -2 -1 0 1 2 3
.011 .13 .6 1 .6 .13 .011
x
g(x)
Separability
Algorithm
• Apply 1‐D mask to alternate pixels along each row of image.
• Apply 1‐D mask to each pixel along alternate columns of resultant image from previous step.
Gaussian Pyramid
Laplacian Pyramids
.
Pierre‐Simon Laplace (1749–1827)
• Mathematician and astronomer • French Newton
• Laplacian• Laplace Transform• Celestial mechanics• Spherical harmonics
• On 15 March 1788 at the age of thirty-nine, Laplace married Marie-Charlotte de Courty de Romanges, a pretty eighteen-and-a-half-year-old girl from a good family in Besançon.
Laplacian Pyramids
• Similar to edge detected images.• Most pixels are zero.• Can be used for image compression.
Coding using Laplacian Pyramid
•Compute Gaussian pyramid
•Compute Laplacian pyramid
•Code Laplacian pyramid
Decoding using Laplacian pyramid
• Decode Laplacian pyramid.• Compute Gaussian pyramid from Laplacian pyramid.
• is reconstructed image.
Laplacian Pyramid
Image Compression (Entropy)
7.6
4.4
5.0
5.6
6.2
.77
1.9
3.3
4.2
Bits per pixel
Image Compression (Entropy)
7.6
4.4
5.0
5.6
6.2
.77
1.9
3.3
4.2
Bits per pixel
Image Compression
1.58
.73
Combining Apple & Orange
Combining Apple & Orange
Algorithm• Generate Laplacian pyramid Lo of orange image.
• Generate Laplacian pyramid La of apple image.
• 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.
Reading Material
• http://ww‐bcs.mit.edu/people/adelson/papers.html
– The Laplacian Pyramid as a compact code, Burt and Adelson, IEEE Trans on Communication, 1983.
• Fundamental of Computer Vision, Section 4.5. http://www.cs.ucf.edu/courses/cap6411/book.pdf
Recommended