1. Image Compression System Megan Fuller and EzzeldinHamed

2. Transforms of Images Original Image Magnitude of DFT of Image-128 (otherwise DC component = ~8e6) Image Reconstructed from 25% of DFT coefficients

3. The 2D Discrete Fourier Transform Where This can be computed separably by rearranging:

4. The 2D Discrete Cosine Transform • Computed separably • Computed as a DFT + 1 multiply • Generally gives better energy compaction than DFT

5. High Level Architecture Output Module (sending data to PC) Separable, in-place 2D DFT/DCT Input Memory Coefficient > Threshold? • The choice between DFT and DCT is provided at compile time • Threshold is provided by the user at run time

6. What’s Interesting? • Reducing the computation required • Sharing resources in the DCT case • Some memory organization tricks • Reducing bit width

7. Number of FFTs • Using FFT to calculate the 1D-DFT • We need FFTs to calculate the 2D-DFT • Can we reduce the number of FFTs?

8. Reduction for the DFT case Real Imag • Using the DFT properties • Input is real • Output is symmetric • Combining rows • Even/Odd decomposition S00 S01 S02 S03 S10 S11 S12 S13 S20 S21 S22 S23 S30 S31 S32 S33 • N/2 FFTs of the rows, followed by Even/Odd decomposition • Output is symmetric (discard half the columns) • N/2 FFTs of the columns • Total of N FFT computations S11 S31

9. Reduction in the DCT case • Again combining the rows in the same way as in DFT (N/2 FFTs) • Even/Odd decomposition then extra multiplication to calculate the DCT S00 S20 S01 S21 S02 S22 S03 S23 Real Imag • Results are not symmetric • But the DCT is real • We can combine the columns the same way we combined the rows (N/2 FFT) • The same multiplier inside the FFT is used • Another Even/Odd decomposition is required here with an extra complex multiplier • Total of N FFT computations + few extra multiplications S10 S30 S11 S31 S12 S32 S13 S33

10. In-Place Radix-4 FFT • Critical path • Fixed point arithmetic • Bit Width? • Quantization noise • Rounding instead of Truncation • Avoid any overflow • additions • Needs extra bits • Can we do better?

11. Static Scaling Vs. Dynamic Scaling • Shift when you expect an overflow • Shift after each addition • The location of the fraction point is fixed at each computation step • Almost no overhead compared to fixed point • Higher effective bit width only in the first computation steps • No effect on the critical path • Shift only when overflow occurs • Track overflows and account for them • The location of the fraction point is the same for each 1D-FFT frame • Needs simple circuitry to track the overflow and shift when required • Effective bit width depend on the data. • No effect on the critical path

12. Design Space Explored Dynamic Scaling Yes No DFT DCT DFT DCT 8 12 16 8 12 16 8 12 16 8 12 16 • 8 bits with dynamic scaling considered later • 8 bits without dynamic scaling (and 12 for DCT) perform too poorly to be considered • 12 does as good as 16 bits with dynamic scaling in the DFT

13. Dynamic Scaling of DFT • 50% of coefficients is sufficient for perfect reconstruction because of the symmetry of the DFT • 16 bits without dynamic scaling does as well as floating point • 12 bits with dynamic scaling also does nearly as well as floating point

14. Dynamic Scaling of DFT(continued) • Improvement in performance when dynamic scaling is used more than makes up for reduced compression because the scaling bits have to be saved • 12 bits with dynamic scaling does nearly as well as 16 bits

15. DCT Vs. DFT • All cases are using dynamic scaling • DCT provides better energy compaction • For DCT, 12 bits gives a lower MSE for a given compression ratio (this was not the case for the DFT).

16. 8 Bits Image reconstructed from 50% of the DFT coefficients, computed with 8 bits, using dynamic scaling. MSE = 452. Image reconstructed from 6% of the DFT coefficients, computed with 16 bits, MSE = 129.

17. Physical Considerations • Critical path about the same for all designs, could probably be improved with tighter synthesis constraints • Resource usage increases with bitwidth, addition of dynamic scaling, and DCT, but overall doesn’t change much • DCT uses extra DSP blocks because of the extra multiplication

18. Latency

19. Future Work • Use of DRAM to allow compression of larger images • Support for color images • Support for rectangular images of arbitrary edge length • Combining the DCT and DFT into a single core that could compute either transform, as selected by the user at runtime

20. Relationship Between the DFT and the DCT The N-point DFT of a sequence is the Fourier Series coefficients for that sequence made periodic with period N.

21. Relationship Between the DFT and the DCT (continued) The N-point DCT of a sequence is a twiddle factor multiplied by the first N Fourier Series coefficients of the 2N point sequence y(n) made periodic with period 2N. y(n) = x(x) + x(2N-1-n) x(n)

22. Relationship Between the DFT and the DCT (continued) The DCT can be computed from the DFT as follows: • Define the sequences y(n) = x(n) + x(2N-1-n) v(n) = y(2n) • Compute the N-point DFT of v(n), V(k)

23. Rounding Conclusion: Never hurt, often helped. Free in hardware (just a register initialization), so always use it. All subsequent results will be using rounding.

24. Dynamic Scaling of DCT

25. Dynamic Scaling of DCT (continued)

26. Limitations of MSE Image reconstructed from 5.7% of the DCT coefficients, computed with dynamic scaling. MSE = 193 Image reconstructed from 6.1% of the DCT coefficients, computed without dynamic scaling. MSE = 338

27. Performance of 8 Bit Systems

28. More Limitations of MSE (Left) 8 bit DFT coefficients, computed with rounding. Compression ratio = 2.3, MSE = 869. (Right) 8 bit DFT coefficients, computed without rounding. Compression ratio = 2.1, MSE = 664 (Left) 8 bit DCT coefficients, computed with rounding. Compression ratio = 2.2, MSE = 517. (Right) 8 bit DCT coefficients, computed without rounding. Compression ratio = 2.4, MSE = 563