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Medical Image Compression using 3-D
Hartley TransformAuthors: Sunder, R. S., Eswaran, C. and Sriraam, N.
Source: to appear in Computers in Biology and Medicine
Reporter: Tzu-Chuen LuDate: Aug. 18, 2005
3-D Image Compression
3-D Image Compression
Run Length
Huffman Coding
10110…
DFT
DCT
DHT
DFT: Discrete Fourier TransformDCT: Discrete Cosine TransformDHT: Discrete Hartley Transform
Discrete Hartley Transform
8 5 1 4 3f(x) F(x)
)]05
02sin()0
5
02[cos(8
)]05
12sin()0
5
12[cos(5
)]05
22sin()0
5
22[cos(1
)]05
32sin()0
5
32[cos(4
)]05
42sin()0
5
42[cos(3
F(0) = +
+
+
+
34158
0 1 2 3 4
f(x)
23
cos
Discrete Hartley Transform
8 5 1 4 3
8 5 1 4 3
f(x) F(x)
23
0 1 2 3 4
)]15
02sin()1
5
02[cos(8
)]15
12sin()1
5
12[cos(5
)]15
22sin()1
5
22[cos(1
)]15
32sin()1
5
32[cos(4
)]15
42sin()1
5
42[cos(3
F(1) = +
+
+
+
5.3 4.31 -1.4 8.81
Inverse Discrete Hartley Transform (IDHT)
f(x) F(x)
23
)]05
02sin()0
5
02[cos(23
5
1
)]05
12sin()0
5
12[cos(3.5
5
1
)]05
22sin()0
5
22[cos(31.4
5
1
)]05
32sin()0
5
32[cos(4.1
5
1
)]05
42sin()0
5
42[cos(81.8
5
1
f(0) = +
+
+
+
5.3 4.31 -1.4 8.818
8.81-1.44.315.323
0 1 2 3 4
F(x)
8
23 5.3 4.31
-1.4 8.81
f(x) F(x)
23
0 1 2 3 4
)]15
02sin()1
5
02[cos(23
5
1
)]15
12sin()1
5
12[cos(3.5
5
1
)]15
22sin()1
5
22[cos(31.4
5
1
)]15
32sin()1
5
32[cos(4.1
5
1
)]15
42sin()1
5
42[cos(81.8
5
1
f(1) = +
+
+
+
5.3 4.31 -1.4 8.815 1 4 3
Inverse Discrete Hartley Transform (IDHT)
Discrete Hartley Transform – 2D
8 5 1
3 4 5
f(x,y) F(x,y)
)]02
020
3
02sin()0
2
020
3
02[cos(8
)]02
020
3
12sin()0
2
020
3
12[cos(5
)]02
020
3
22sin()0
2
020
3
22[cos(1
)]02
120
3
02sin()0
2
120
3
02[cos(3
)]02
120
3
12sin()0
2
120
3
12[cos(4
F(0,0) =
)]02
120
3
22sin()0
2
120
3
22[cos(5
+
+
+
+
+
260 1 2
0
1
0 1 2
0
1
Discrete Hartley Transform – 2D
f(x,y) F(x,y)
)]12
020
3
02sin()1
2
020
3
02[cos(8
)]12
020
3
12sin()1
2
020
3
12[cos(5
)]12
020
3
22sin()1
2
020
3
22[cos(1
)]12
120
3
02sin()1
2
120
3
02[cos(3
)]12
120
3
12sin()1
2
120
3
12[cos(4
F(0,1) =
)]12
120
3
22sin()1
2
120
3
22[cos(5
+
+
+
+
+
26 6.098 0.902
2 10.83 2.17
8 5 1
3 4 5
0 1 2
0
1
0 1 20
1
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
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
721
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
DHT Quantization
-26 36 -10 -3 28
-50 -18 2
-35 11 -2 6 23
-54 13 6
14 00
-10 0 00-7
0-20 00 0 0 0 0
00 000000
0 0 0000 00
38 0 0 0 00 00
2 00
-10 0 0 36 721
721
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
3-D Image DHT Compression
M=3
Experiments and Results
XA Image MR Brain Image
XA Image
M=2 M=4
M=8 M=16
MR Brain Image
M=2 M=4
M=8 M=16
Conclusions
• Discrete hartley transform (DHT)
• 3D-image DHT compression
• XA image – DHT with M = 8
• MR brain image – DCT with M = 2
DHT
