The document discusses medical image compression using the 3D Discrete Hartley Transform (DHT). It provides examples of applying the DHT to compress 3D X-ray angiography (XA) images and magnetic resonance (MR) brain images. Experiments showed that an XA image could be compressed with little loss of quality using a DHT with a quantization value of M=8. An MR brain image was effectively compressed using a DHT with M=2. The DHT provides an alternative transform to the Discrete Cosine Transform for lossy 3D medical image compression.

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. 3-D Image Compression

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. 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. 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. 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. 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. 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. 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. 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. 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. 3-D Image DHT Compression
M=3

13. Experiments and Results
XA Image
MR Brain Image

14. XA Image
M=2
M=8
M=4
M=16

15. MR Brain Image
M=2
M=8
M=4
M=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. DHT