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Image Compression System

Megan Fuller and Ezzeldin Hamed

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Transforms of Images

Original Image

Image Reconstructed from 25% of DFT coefficients

Magnitude of DFT of Image-128 (otherwise DC component = ~8e6)

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The 2D Discrete Fourier Transform

Where This can be computed separably by rearranging:

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The 2D Discrete Cosine Transform

• Computed separably• Computed as a DFT + 1 multiply• Generally gives better energy compaction than DFT

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High Level Architecture

Separable, in-place 2D DFT/DCTInput

Memory

Coefficient > Threshold?

Output Module (sending data to PC)

• The choice between DFT and DCT is provided at compile time• Threshold is provided by the user at run time

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What’s Interesting?

• Reducing the computation required

• Sharing resources in the DCT case

• Some memory organization tricks

• Reducing bit width

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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?

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Reduction for the DFT case• 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

S31S11

Real Imag

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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

S10 S11 S12 S13

S00 S01 S02 S03

S30 S31 S32 S33

S20 S21 S22 S23

• 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

Real Imag

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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?

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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

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Design Space ExploredDynamic Scaling

Yes No

DFT DCTDFT

8 12 16

DCT

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

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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

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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

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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).

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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.

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Physical ConsiderationsTransform # of Bits Dynamic

Scaling?Critical Path Slice

RegistersSlice LUTs

BRAM DSP48Es

DFT 16 No 11.458ns 16% 23% 29% 7

DFT 16 Yes 11.763ns 17% 24% 29% 7

DFT 12 No 11.273ns 15% 22% 24% 7

DFT 12 Yes 11.464ns 16% 23% 24% 7

DFT 8 Yes 11.287ns 15% 22% 18% 6

DCT 16 Yes 11.458ns 19% 26% 29% 10

DCT 12 Yes 11.273ns 18% 25% 24% 10

DCT 8 Yes 11.066ns 17% 23% 18% 8

• 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

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LatencyComponent Latency (clock cycles) Potential Frame Rate

with 50MHz ClockInitialization 870,000 -

DCT 263,900 189 images/second

DFT 262,200 191 images/second

N 𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐶𝑦𝑐𝑙𝑒𝑠≅𝑁 2log 4𝑁

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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

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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.

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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)

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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)

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Rounding

Design MSE Decrease with Rounding

12 bits, no dynamic scaling, DFT 20

16 bits, no dynamic scaling, DFT 0

12 bits, dynamic scaling, DFT 2

16 bits, no dynamic scaling, DCT 0

12 bits, dynamic scaling, DCT 2

16 bits, dynamic scaling, DCT 0

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

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Dynamic Scaling of DCT

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Dynamic Scaling of DCT (continued)

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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

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Performance of 8 Bit Systems

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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