Image Processing Reference
In-Depth Information
For odd indices with a support of (15) we get:
B 7
S 2 B 4
S 6 B 4
S 4 B 5
S 2 B 6
S 6 B 6
=
S 1 X 1
S 7 X 7
S 3 X 3
S 5 X 5
101110
10
11
10
4
(22)
1
10
101
110
10
1
A direct approach from the classical definition requires: a single multiplication for even
indices (20) and 5 multiplications for odd indices (22). The scaled coefficients S 1,7,3,5 X 1,7,3,5
in (22) can be expressed in an equivalent way introduced by Arai, Agui, Nakajima (AAN,
1988)., which allows reducing an amount of multiplications from 5 to 4 only.
=
B 7
S 4 B 5
S 1 X 1
S 7 X 7
S 3 X 3
S 5 X 5
1110
1
11
101
(
+
)
4
S 2
S 6
B 6
(23)
1
10
11
(
)
S 2
S 6
B 4
1
101
1
(
)
S 6
B 6
B 4
Fig. 4. A fast DCT algorithm developed in 1988 by Arai, Agui and Nakajima
A minimization of multiplications amounts is one of a fundamental goal in long-term
numerical calculations. Reduction of product terms significantly speed up sophisticated
calculations, because a single multiplication requires several clock cycles of processor.
Multiplications in powerful FPGA chips can be however performed in very fast dedicated
DSP blocks in a single clock cycle. Signals processed in parallel threads in a hardware
implementation of a pipeline design have to be synchronized to each other. Pipeline approach
requires additional shift registers for synchronization also for signal currently not being
processed. However, such synchronization needs additional resources. Fig. 5 shows the part
of pipeline chain corresponding to odd indices of DCT coefficients (lower part in Fig. 4).
 
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