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					ρ-domain Modeling
for Rate Estimation

Carri Chan and Yuki Konda
EE398 Project Presentation
 ρ-domain       Model
     Overview of Model
     Our Estimation Results
     Observations

   Rate Control
     Transmission   Model
     Optimization
     Results

   Conclusion
  Explanation of ρ-domain model
                                                       Quantized matrix

         Original matrix
                               Transformed matrix

     ρ = % of zero coefficients
     Different quantization levels
      correspond to different ρ’s

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.
    Using ρ values to calculate RD
   Relationship between transform coefficients and ρ very
    similar for different images

                              q domain               ρ domain

   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
Our RD results for SPIHT codec

         Dots indicate estimated RD
         Solid line indicates empirical RD
Benefits and limitations of ρ-domain
   Simple – allows for accurate RD model based on
    easy to calculate image/frame statistics
   Fast – encoding at many rates is very time
   Model improves if training set has similar
    statistics to the actual images to estimate
   Best estimates at low rates– high ρ
Variable Bit Rate Channel

     Video Frames         Buffer

   Lagrangian optimization gives best performance
   The buffer constraint may not make this policy
   Let’s optimize given the buffer constraint and RD
    estimation of each frame
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
Dynamic Programming Optimization

                minimum Cost-to-Go
                                      Future cost
   Then:

            Immediate cost

   Terminal Costs:

   Based on Training Data
Use Estimation!
                   Identical: Expected Cost-to-Go from n+1

 Estimated Distortion using ρ-domain modeling
                   Blue dots
Frame Estimation   Estimated (R,D)

                   Red line
                   Empirical (R,D)
Summary: Results
 We get much better performance than no-
 For large buffer sizes we approach
  Lagrangian optimal
 For estimation to help more, we need
  video frames that vary more
 ρ-domain model allows fast/effective
  Rate-Distortion Estimation
 We can use this estimate to perform
  fast/effective Rate Control
Thank you!
Calculation of θ(reference)
   Qnz : pseudo bit rate to describe non-zero
        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
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 ∑ ρi 2 – (∑ ρi )2)
         n: number of estimate points (6 in above example)
CBR Channel
   We also looked at a Constant Bit Rate Channel
   No room for improvement over Lagrangian
Using Empirical Data to Calculate
   Interpolated Values and Actual Values are very