ρ -domain Modeling for Rate Estimation

1. ρ-domain Modeling for Rate Estimation Carri Chan and Yuki Konda EE398 Project Presentation 3/14/06

2. Outline • ρ-domain Model • Overview of Model • Our Estimation Results • Observations • Rate Control • Transmission Model • Optimization • Results • Conclusion

3. Explanation of ρ-domain model Quantized matrix • ρ = % of zero coefficients • Different quantization levels correspond to different ρ’s Original matrix Transformed matrix Zhihai He, Sanjit Mitra, "A Unified Rate-Distortion Analysis Framework for Transform Coding," IEEE Trans. on Circuits and Systems for Video Technology, vol. 11, no. 12, pp. 1221-1236, December 2001.

4. Relationship between transform coefficients and ρ very similar for different images Linear model to approximate R(ρ) = θ(1- ρ) R hits 0 at ρ = 1. Calculate θ by linear regression of observed behavior in ρ domain estimate D based on transform coefficients to obtain RD curve Using ρ values to calculate RD q domain ρ domain

5. Our RD results for SPIHT codec • Dots indicate estimated RD • Solid line indicates empirical RD

6. Benefits and limitations of ρ-domain model • Simple – allows for accurate RD model based on easy to calculate image/frame statistics • Fast – encoding at many rates is very time consuming • Model improves if training set has similar statistics to the actual images to estimate • Best estimates at low rates– high ρ

7. Tx i Enc Buffer Video Frames Variable Bit Rate Channel • Lagrangian optimization gives best performance • The buffer constraint may not make this policy possible • Let’s optimize given the buffer constraint and RD estimation of each frame

8. Dynamic Programming Optimization (1) : State • i = channel state: Discrete Markov Chain, transition probability qij • Rc(i) = channel rate given channel state • b = amount of bits in the buffer • T = total amount of bits available—necessary to maintain average bit constraint

9. Dynamic Programming Optimization • minimum Cost-to-Go • Then: • Terminal Costs: • Based on Training Data Future cost Immediate cost

10. Use Estimation! Identical: Expected Cost-to-Go from n+1 Estimated Distortion using ρ-domain modeling

11. Frame Estimation Blue dots Estimated (R,D) Red line Empirical (R,D)

12. Results

13. Summary: Results • We get much better performance than no-control • For large buffer sizes we approach Lagrangian optimal • For estimation to help more, we need video frames that vary more

14. Conclusion • ρ-domain model allows fast/effective Rate-Distortion Estimation • We can use this estimate to perform fast/effective Rate Control

15. Thank you!

16. Calculation of θ(reference) • Qnz : pseudo bit rate to describe non-zero coefficients Qnz = (1/M) ∑ S(x): S(x) = floor(log2|x|) + 2 M coefficients in matrix, x is value of coefficient • Qz : pseudo bit rate to describe zero coefficients Qz = Aiκ + Bi κ = Qnz(qo)/(1- ρ(qo)) A and B are obtained from linear regression

17. Calculation of θ cont’d (reference) • R (ρi) = A(ρi) • Qnz(ρi) + B(ρi) • Qz(ρi) + C(ρi) A: [1.1018 0.8825 0.5780 0.6078 1.0325 0.4176] B: [1.2431 1.0448 0.9718 1.2732 1.2802 0.6390] C: [0.0503 0.0469 0.1398 0.0111 -0.1167 4.9123e-005] ρ: [0.7207 0.8047 0.8957 0.9550 0.9791 0.9985] • θ = (∑ρi ∑ R(ρi) - n ∑ ρiR(ρi) ) / (n ∑ ρi2 – (∑ ρi )2) n: number of estimate points (6 in above example)

18. CBR Channel • We also looked at a Constant Bit Rate Channel • No room for improvement over Lagrangian Optimal!

19. Using Empirical Data to Calculate PSNR • Interpolated Values and Actual Values are very close