![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](https://image.slidesharecdn.com/htupload-140206135949-phpapp01/85/Ht-upload-4-320.jpg)
![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
+
+
+
+](https://image.slidesharecdn.com/htupload-140206135949-phpapp01/85/Ht-upload-5-320.jpg)
![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
+
+
+
+](https://image.slidesharecdn.com/htupload-140206135949-phpapp01/85/Ht-upload-6-320.jpg)
![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
+
+
+
+](https://image.slidesharecdn.com/htupload-140206135949-phpapp01/85/Ht-upload-7-320.jpg)
![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
+
+
+
+
+](https://image.slidesharecdn.com/htupload-140206135949-phpapp01/85/Ht-upload-8-320.jpg)
![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
+
+
+
+
+](https://image.slidesharecdn.com/htupload-140206135949-phpapp01/85/Ht-upload-9-320.jpg)
Ht upload
- 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
- 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 + + + + +
- 12. 3-D Image DHT Compression M=3
- 13. Experiments and Results XA Image MR Brain Image
- 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