1. Image Compression System Megan Fuller and Ezzeldin Hamed 1

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

3. The 2D Discrete Fourier Transform 3

4. The 2D Discrete Cosine Transform 4

5. High Level Architecture Separable, in- place 2D DFT/DCT Input Memory Coefficient > Threshold? Output Module (sending data to PC) 5 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 6

7. Number of FFTs 7

8. Reduction for the DFT case 8 S 00 S 01 S 02 S 03 S 10 S 11 S 12 S 13 S 20 S 21 S 22 S 23 S 30 S 31 S 32 S 33 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 S 31 S 11 RealImag

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 9 S 10 S 11 S 12 S 13 S 00 S 01 S 02 S 03 S 30 S 31 S 32 S 33 S 20 S 21 S 22 S 23 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 RealImag

10. In-Place Radix-4 FFT 10

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 11 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 YesNo DFTDCTDFT 8 12 16 DCT 8 12 16 12 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 13 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) 14 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 15 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. 16

17. Physical Considerations Transform# of BitsDynamic Scaling? Critical PathSlice Registers Slice LUTs BRAMDSP48Es DFT16No11.458ns16%23%29%7 DFT16Yes11.763ns17%24%29%7 DFT12No11.273ns15%22%24%7 DFT12Yes11.464ns16%23%24%7 DFT8Yes11.287ns15%22%18%6 DCT16Yes11.458ns19%26%29%10 DCT12Yes11.273ns18%25%24%10 DCT8Yes11.066ns17%23%18%8 17 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 ComponentLatency (clock cycles)Potential Frame Rate with 50MHz Clock Initialization870,000- DCT263,900189 images/second DFT262,200191 images/second 18

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 19

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. 20

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) 21

22. Relationship Between the DFT and the DCT (continued) 22

23. Rounding DesignMSE Decrease with Rounding 12 bits, no dynamic scaling, DFT20 16 bits, no dynamic scaling, DFT0 12 bits, dynamic scaling, DFT2 16 bits, no dynamic scaling, DCT0 12 bits, dynamic scaling, DCT2 16 bits, dynamic scaling, DCT0 Conclusion: Never hurt, often helped. Free in hardware (just a register initialization), so always use it. All subsequent results will be using rounding. 23

24. Dynamic Scaling of DCT 24

25. Dynamic Scaling of DCT (continued) 25

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 26

27. Performance of 8 Bit Systems 27

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 28