A pyramid is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilateral, or of any polygon shape. As such, a pyramid has at least three outer triangular surfaces. Wikipedia पिरामिड जैसे ज्यामितीय आकार से मिलती जुलती संरचनाओं को पिरामिड कहते हैं। विश्व में बहुत सी संरचनाएँ पिरैमिड के आकार की हैं जिनमें मिस्र के पिरामिड बहुत प्रसिद्ध हैं। पिरामिड आकार की संरचनाओं की सबसे बड़ी विशेषता यह है कि इसके भार का अधिकांश भाग जमीन के पास होता हैA pyramid is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilateral, or of any polygon shape. As such, a pyramid has at least three outer triangular surfaces. Wikipedia पिरामिड जैसे ज्यामितीय आकार से मिलती जुलती संरचनाओं को पिरामिड कहते हैं। विश्व में बहुत सी संरचनाएँ पिरैमिड के आकार की हैं जिनमें मिस्र के पिरामिड बहुत प्रसिद्ध हैं। पिरामिड आकार की संरचनाओं की सबसे बड़ी विशेषता यह है कि इसके भार का अधिकांश भाग जमीन के पास होता है

1. Pyramids
Lecture‐5

4. 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.

5. 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.

6. Image Pyramid

7. Pyramid

8. Gaussian pyramid
Source: Forsyth

9. Laplacian pyramid
Source: Forsyth

10. Deformable Part Model (DPM)
Slide Credit: Pedro F. Felzenszwalb

11. Contents
• Gaussian and Laplacian Pyramids
– Reduce
– Expand
• Applications of Laplacian pyramids
– Image compression
– Image composting

12. Cited by 5067

14. Gaussian Pyramids (reduce)
)
2
,
2
(
)
,
(
)
,
(
2
2
2
2
1 n
j
m
i
g
n
m
w
j
i
g
m n
l
l 

  
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[ 1

 l
l g
REDUCE
g
Level l

15. Convolution

16. Reduce (1D)

17. Reduce
0 1 2 3 4 5 6 7 8
0 1 2 3 4
0 1 2

18. Gaussian Pyramids (expand)

19. Expand (1D)

20. Expand (1D)

21. Expand
0 1 2 3 4 5 6 7 8
0 1 2 3 4

22. Expand
0 1 2 3 4 5 6 7 8
0 1 2 3 4

23. Convolution Mask

24. Convolution Mask
• Separable
•Symmetric

25. 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.

26. Convolution Mask
c
c
b
b
a
b

27. Convolution Mask
a=.4 GAUSSIAN, a=.5 TRINGULAR

28. Triangular
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
c b a b c
Triangular
a=.4 GAUSSIAN, a=.5 TRINGULAR

29. 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

30. Gaussian

31. Gaussian
-3 -2 -1 0 1 2 3
.011 .13 .6 1 .6 .13 .011
x
g(x)

33. Separability

34. 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.

35. Gaussian Pyramid

36. Laplacian Pyramids
.

37. 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.

38. Laplacian Pyramids
• Similar to edge detected images.
• Most pixels are zero.
• Can be used for image compression.

40. Coding using Laplacian Pyramid
•Compute Gaussian pyramid
•Compute Laplacian pyramid
•Code Laplacian pyramid

41. Decoding using Laplacian pyramid
• Decode Laplacian pyramid.
• Compute Gaussian pyramid from Laplacian
pyramid.
• is reconstructed image.

42. Laplacian Pyramid

43. Image Compression (Entropy)
7.6
4.4
5.0
5.6
6.2
.77
1.9
3.3
4.2
Bits per pixel

44. Image Compression (Entropy)
7.6
4.4
5.0
5.6
6.2
.77
1.9
3.3
4.2
Bits per pixel

45. Image Compression
1.58
.73

46. Combining Apple & Orange

47. Combining Apple & Orange

48. 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.

49. 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