Rate Control Simulator

Action scene, CRF 23, a 4-frame buffer. CBR's worst frame scores 60 out of the 85 this model allows it, and the rules are all on the page: CRF gives each frame a share of the bits proportional to its complexity, CBR pays a flat 2000 bits and tops that up from a buffer holding 4 frames' worth, and CBR's quality is 85 times the square root of supply over demand. Derive the 60.

1. The CRF allocation is written relative to the scene's mean complexity, so averaging it hands the mean straight back. Whatever the scene and whatever the CRF, the average comes out at the 2000-bit target — that readout is a definition, not a measurement.

   $$\bar{b}=\frac{1}{120}\sum _{i=1}^{120}B\frac{c_i}{\bar{c}}=B=2000bits/frame$$

2. The peak is the informative one. 4078 against 2000 says the busiest frame carries 2.039 times the average complexity, and the generator caps complexity at 0.95, so this scene's average complexity has to be 0.466.

   $$\frac{b_{\max }}{\bar{b}}=\frac{c_{\max }}{\bar{c}}=\frac{4078}{2000}=2.039⟹\bar{c}=\frac{0.95}{2.039}=0.466$$

3. CBR faces the same curve. Its demand per frame is the same expression as the CRF allocation, so the frame CRF pays 4078 bits for is the frame CBR is asked 4078 bits for and hands over 2000.

   $$D_t=B\frac{c_t}{\bar{c}}=b_t⟹D_{\max }=4078,B=2000$$

4. Covering that is the buffer's job. Full, it holds 8000 bits and lifts supply to 5040, comfortably past the 4078 the hardest frame wants — a full buffer would hide the peak entirely and quality would never leave 85.

   $$S=B+0.38\beta ,{\beta }_{\max }=4\times 2000=8000⟹S_{\max }=5040$$

5. Quality does leave 85, so the buffer cannot be full. Set it empty instead: supply is the bare 2000, the ratio is 0.4904, its square root is 0.7003, and 85 times that is 59.5, which the panel rounds to 60. The printed minimum is the empty-buffer floor exactly.

   $$\beta =0:\frac{S}{D}=\frac{2000}{4078}=0.4904,Q=85\sqrt{0.4904}=59.5\to 60$$

6. The buffer drains at whatever demand exceeds the budget, which is 2078 bits on the peak frame. 8000 bits of buffer is 3.85 frames of that. This scene's busy stretch runs 40.

   $${\beta }_{t+1}-{\beta }_t=B-D_t=-2078⟹\frac{8000}{2078}=3.85\mathrm{frames}$$

Both modes spend the same 2000 bits per frame on this scene. CRF's quality line sits at 89.8, CBR averages 79. Work out what that 10.8-point gap is bought with, and what it would cost in CRF steps.

1. The CRF score comes from the slider alone: 45 points spread over the 22 steps from CRF 18 to CRF 40. CRF 23 is 5 steps down, so 89.8.

   $$Q^{\mathrm{CRF}}=100-\frac{23-18}{22}\times 45=100-10.227=89.8$$

2. One step is therefore 2.045 points anywhere on the scale — this model makes CRF quality linear, which is what lets you price the gap in steps later.

   $$\frac{\Delta Q}{\Delta \mathrm{CRF}}=\frac{45}{22}=2.045$$

3. The bitrates are not merely close, they are identical. CRF averages the target by construction, and CBR pays the target on every single frame, so both produce 240,000 bits over the 120 frames.

   $${\bar{b}}^{\mathrm{CRF}}=2000,b_i^{\mathrm{CBR}}=2000\forall i⟹120\times 2000=240000\mathrm{bits}$$

4. Nothing in the 10.8 points between 89.8 and 79, then, is a bitrate difference. Same bits, same file size, 10.8 points apart.

   $$\Delta Q=89.8-79=10.8$$

5. The gap splits in two. CBR is capped at 85 even on frames it could sail through, which costs 4.8 points against CRF's 89.8, and the remaining 6.0 points are lost on the frames where the buffer had nothing left.

   $$89.8-85=4.8,85-79=6.0,4.8+6.0=10.8$$