Medical Image Compression
using 3-D Hartley Transform
Authors: Sunder, R. S., Eswaran, C. and Sriraam, N.
Source: to appea...
3-D Image Compression
3-D Image Compression
DFT
Run Length
10110…

DCT
Huffman
Coding
DHT
DFT: Discrete Fourier Transform
DCT: Discrete Cosine T...
Discrete Hartley Transform
f(x)

8

5

1

4

F(x)
23

3

0
f(x)

1

2

3

4

8

5

1

4

3

0
0
F(0) = 8 × [cos(2 × π × × ...
Discrete Hartley Transform
f(x)

8

F(x)

5

1

4

3

23

0

1

2

3

4

8

5

1

4

3

5.3

0
0
F(1) = 8 × [cos(2 × π × ×...
Inverse Discrete Hartley Transform
(IDHT)
f(x)

F(x)

8

23
0
F(x)

23

1

2

5.3

4.31

5.3

3

4

-1.4

4.31 -1.4 8.81

...
Inverse Discrete Hartley Transform
(IDHT)
f(x)

8

F(x)
5

1
0

23
f(1) =

4

3

1

2

23
3

5.3

4.31 -1.4 8.81

4

5.3 4...
Discrete Hartley Transform – 2D
0

f(x,y) 0
1

1

2

8
3

5
4

1
5

F(x,y) 0

0
26

1

2

1

0
0
0
0
F(0,0) = 8 × [cos(2π ...
Discrete Hartley Transform – 2D
f(x,y)

0
0
1

1

2

8
3

5
4

1
5

F(x,y) 0
1

0

1

2

26 6.098 0.902
2

10.83 2.17

0
0...
100

120

134

256

100

100

125

97

210

175

85

97

145

33

45

61

33

50

78

100

37

125

37

89

97

80

78

99...
72
1

36

39

3

-10

6

13

-54

23

6

-9

16

2

-2

11

-35

49

2

-13

1

0

-16

35

38

25

5

-7

1

-2

2

-3

0...
3-D Image DHT Compression

M=3
Experiments and Results

XA Image

MR Brain Image
XA Image

M=2

M=8

M=4

M=16
MR Brain Image

M=2

M=8

M=4

M=16
Conclusions
• Discrete hartley transform (DHT)
• 3D-image DHT compression

• XA image – DHT with M = 8
• MR brain image – ...
DHT
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  1. 1. Medical Image Compression using 3-D Hartley Transform Authors: Sunder, R. S., Eswaran, C. and Sriraam, N. Source: to appear in Computers in Biology and Medicine Reporter: Tzu-Chuen Lu Date: Aug. 18, 2005
  2. 2. 3-D Image Compression
  3. 3. 3-D Image Compression DFT Run Length 10110… DCT Huffman Coding DHT DFT: Discrete Fourier Transform DCT: Discrete Cosine Transform DHT: Discrete Hartley Transform
  4. 4. Discrete Hartley Transform f(x) 8 5 1 4 F(x) 23 3 0 f(x) 1 2 3 4 8 5 1 4 3 0 0 F(0) = 8 × [cos(2 × π × × 0) + sin( 2 × π × × 0)] 5 5 1 1 5 × [cos(2 × π × × 0) + sin( 2 × π × × 0)] 5 5 2 2 1× [cos(2 × π × × 0) + sin( 2 × π × × 0)] 5 5 3 3 4 × [cos(2 × π × × 0) + sin( 2 × π × × 0)] 5 5 4 4 3 × [cos(2 × π × × 0) + sin( 2 × π × × 0)] 5 5 + + + + cos
  5. 5. Discrete Hartley Transform f(x) 8 F(x) 5 1 4 3 23 0 1 2 3 4 8 5 1 4 3 5.3 0 0 F(1) = 8 × [cos(2 × π × × 1) + sin( 2 × π × × 1)] 5 5 1 1 5 × [cos(2 × π × × 1) + sin( 2 × π × ×1)] 5 5 2 2 1× [cos(2 × π × × 1) + sin( 2 × π × ×1)] 5 5 3 3 4 × [cos(2 × π × × 1) + sin(2 × π × ×1)] 5 5 4 4 3 × [cos(2 × π × ×1) + sin( 2 × π × × 1)] 5 5 4.31 -1.4 8.81 + + + +
  6. 6. Inverse Discrete Hartley Transform (IDHT) f(x) F(x) 8 23 0 F(x) 23 1 2 5.3 4.31 5.3 3 4 -1.4 4.31 -1.4 8.81 8.81 1 0 0 × 23 × [cos(2 × π × × 0) + sin( 2 × π × × 0)] f(0) = 5 5 5 1 1 1 × 5.3 × [cos(2 × π × × 0) + sin( 2 × π × × 0)] 5 5 5 1 2 2 × 4.31× [cos(2 × π × × 0) + sin( 2 × π × × 0)] 5 5 5 1 3 3 × −1.4 × [cos(2 × π × × 0) + sin( 2 × π × × 0)] 5 5 5 1 4 4 × 8.81× [cos(2 × π × × 0) + sin( 2 × π × × 0)] 5 5 5 + + + +
  7. 7. Inverse Discrete Hartley Transform (IDHT) f(x) 8 F(x) 5 1 0 23 f(1) = 4 3 1 2 23 3 5.3 4.31 -1.4 8.81 4 5.3 4.3 -1.4 8.8 1 1 0 0 1 × 23 × [cos(2 × π × ×1) + sin( 2 × π × ×1)] 5 5 5 1 1 1 × 5.3 × [cos(2 × π × ×1) + sin( 2 × π × ×1)] 5 5 5 1 2 2 × 4.31× [cos(2 × π × ×1) + sin( 2 × π × × 1)] 5 5 5 1 3 3 × −1.4 × [cos(2 × π × ×1) + sin(2 × π × × 1)] 5 5 5 1 4 4 × 8.81× [cos(2 × π × × 1) + sin( 2 × π × ×1)] 5 5 5 + + + +
  8. 8. Discrete Hartley Transform – 2D 0 f(x,y) 0 1 1 2 8 3 5 4 1 5 F(x,y) 0 0 26 1 2 1 0 0 0 0 F(0,0) = 8 × [cos(2π × × 0 + 2π × × 0) + sin( 2π × × 0 + 2π × × 0)] 3 2 3 2 1 0 1 0 5 × [cos(2π × × 0 + 2π × × 0) + sin( 2π × × 0 + 2π × × 0)] 3 2 3 2 2 0 2 0 1× [cos(2π × × 0 + 2π × × 0) + sin( 2π × × 0 + 2π × × 0)] 3 2 3 2 0 1 0 1 3 × [cos(2π × × 0 + 2π × × 0) + sin( 2π × × 0 + 2π × × 0)] 3 2 3 2 1 1 1 1 4 × [cos(2π × × 0 + 2π × × 0) + sin( 2π × × 0 + 2π × × 0)] 3 2 3 2 2 1 2 1 5 × [cos(2π × × 0 + 2π × × 0) + sin( 2π × × 0 + 2π × × 0)] 3 2 3 2 + + + + +
  9. 9. Discrete Hartley Transform – 2D f(x,y) 0 0 1 1 2 8 3 5 4 1 5 F(x,y) 0 1 0 1 2 26 6.098 0.902 2 10.83 2.17 0 0 0 0 F(0,1) = 8 × [cos(2π × × 0 + 2π × ×1) + sin( 2π × × 0 + 2π × ×1)] 3 2 3 2 1 0 1 0 5 × [cos(2π × × 0 + 2π × ×1) + sin( 2π × × 0 + 2π × × 1)] 3 2 3 2 2 0 2 0 1× [cos(2π × × 0 + 2π × ×1) + sin( 2π × × 0 + 2π × ×1)] 3 2 3 2 0 1 0 1 3 × [cos(2π × × 0 + 2π × × 1) + sin( 2π × × 0 + 2π × × 1)] 3 2 3 2 1 1 1 1 4 × [cos(2π × × 0 + 2π × × 1) + sin( 2π × × 0 + 2π × ×1)] 3 2 3 2 2 1 2 1 5 × [cos(2π × × 0 + 2π × ×1) + sin( 2π × × 0 + 2π × × 1)] 3 2 3 2 + + + + +
  10. 10. 100 120 134 256 100 100 125 97 210 175 85 97 145 33 45 61 33 50 78 100 37 125 37 89 97 80 78 99 89 117 57 125 85 97 33 45 100 120 45 61 134 256 125 37 85 97 37 89 85 97 117 57 78 100 33 45 97 33 45 100 120 45 61 37 DHT Quantization 5770 584 710 96 -310 100 210 -432 370 110 -167 583 60 -33 194 -553 888 41 -266 37 4 -321 666 687 797 168 -296 35 -155 63 -90 13 -212 -593 -560 105 -368 349 -296 -305 396 151 -551 -5 -554 44 -372 -265 26 -121 220 523 -22 -183 -300 -807 221 -41 -626 353 433 -174 580 -209 72 1 36 39 3 -10 6 13 -54 23 6 -9 16 2 -2 11 -35 49 2 -13 1 0 -16 35 38 25 5 -7 1 -2 2 -3 0 -7 -16 -14 2 -6 9 -8 -10 22 8 -28 0 -14 2 -20 -15 2 -7 12 15 -1 -10 -18 -50 28 -3 -35 11 14 -10 36 -26
  11. 11. 72 1 36 39 3 -10 6 13 -54 23 6 -9 16 2 -2 11 -35 49 2 -13 1 0 -16 35 38 25 5 -7 1 -2 2 -3 0 -7 -16 -14 2 -6 9 -8 -10 22 8 -28 0 -14 2 -20 -15 2 -7 12 15 -1 -10 -18 -50 28 -3 -35 11 14 -10 36 -26 721 36 0 0 -10 6 13 -54 23 6 0 0 2 -2 11 -35 0 0 0 0 0 0 0 38 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -20 0 2 -7 0 0 0 -10 -18 -50 28 -3 0 0 14 -10 36 -26
  12. 12. 3-D Image DHT Compression M=3
  13. 13. Experiments and Results XA Image MR Brain Image
  14. 14. XA Image M=2 M=8 M=4 M=16
  15. 15. MR Brain Image M=2 M=8 M=4 M=16
  16. 16. Conclusions • Discrete hartley transform (DHT) • 3D-image DHT compression • XA image – DHT with M = 8 • MR brain image – DCT with M = 2
  17. 17. DHT

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