1. Pyramid Vector Quantization

2. What is Pyramid Vector Quantization?
A Vector Quantizer
That has a simple algebraic structure
To perform gain-shape quantization

3. Motivation

4. Why Vector Quantization?
3 classic advantages (Lookabaugh et al. 1989):
Space filling advantage: VQ codepoints tile space more efficiently
Example: 2-D, squares vs. hexagons
Maximum possible gain for large dimension: dB
Shape advantage: VQ can use more points where PDF is higher
1.14 dB gain for 2-D Gaussian, 2.81 for high dimension
Memory advantage: exploit statistical dependence between vector components

5. Why Vector Quantization?
3 classic advantages (Lookabaugh et al. 1989):
Space filling advantage: VQ codepoints tile space more efficiently
Example: 2-D, squares vs. hexagons
Maximum possible gain for large dimension: dB
Shape advantage: VQ can use more points where PDF is higher
Can be mitigated with entropy coding
Memory advantage: exploit statistical dependence between vector components
Transform coefficients are not strongly correlated

6. Why Vector Quantization
Important: Space advantage applies even when values are totally uncorrelated
Another important advantage
Can have codebooks with less than 1 bit per dimension

7. Why Algebraic VQ?
Trained VQ impractical for high rates, large dimensions
High dimension → large LUTs, lots of memory
Exponential in bitrate
No codebook structure → slow search
“Algebraic” VQ solves these problems
Structured codebook: no LUTs, fast search
Space-filling lattice for arbitrary dimension unknown: have to approximate
PVQ asymptotically optimal for Laplacian sources

8. Why Gain-Shape Quantization?
Separate “gain” (energy) from “shape” (spectrum)
Vector = Magnitude × Unit Vector (point on sphere)
Potential advantages
Can give each piece different rate allocations
Preserve energy (contrast) instead of low-passing
Scalar can only add energy by coding ±1’s
Implicit activity masking
Can derive quantization resolution from the explicitly coded energy
Better representation of coefficients

9. How it Works (High-Level)

10. Simple Case: PVQ without a Predictor
Scalar quantize gain
Place K unit pulses in N dimensions
Up to N = 1024 dimensions for large blocks
Only has N-1 degrees of freedom
Normalize to unit norm
K is derived implicitly from the gain
Can also code K and derive gain

11. Codebook for N=3 and different K

12. PVQ vs. Scalar Quantization

13. PVQ with a Predictor Video provides us with useful predictors
We want to treat vectors in the direction of the prediction as “special”
They are much more likely!
Subtracting and coding the residual would lose energy preservation
Solution: align the codebook axes with the prediction, and treat one dimension differently

14. 2-D Projection Example
Input
Input

15. 2-D Projection Example
Input + Prediction
Prediction
Input

16. 2-D Projection Example Input + Prediction
Compute Householder Reflection
Prediction
Input

17. 2-D Projection Example Input + Prediction
Compute Householder Reflection
Apply Reflection
Prediction
Input

18. 2-D Projection Example Input + Prediction
Compute Householder Reflection
Apply Reflection
Compute & code angle
Prediction θ
Input

19. 2-D Projection Example Input + Prediction
Compute Householder Reflection
Apply Reflection
Compute & code angle
Code other dimensions
Prediction θ
Input

20. What does this accomplish?
Creates another “intuitive” parameter, θ
“How much like the predictor are we?”
θ = 0 → use predictor exactly
θ determines how many pulses go in the “prediction” direction
K (and thus bitrate) for remaining N-1 dimensions adjusted down
Remaining N-1 dimensions have N-2 degrees of freedom (no redundancy)
Can repeat for more predictors

21. Details...

22. Band Structure DC coded separately with scalar quantization
AC coefficients grouped into bands
Gain, theta, etc., signaled separately for each band
Layout ad-hoc for now
Scan order in each band optimized for decreasing average variance

23. Band Structure
4x4
8x8
16x16
Scan order is possibly over-fit...

24. To Predict or Not to Predict...
θ ≥ π/2 → Prediction not helping
Could code large θ’s, but doesn’t seem that useful
Need to handle zero predictors anyway
Current approach: code a “noref” flag
Currently jointly code up to 4 flags at once, with fixed order-0 probability per band (5% of KF rate)
Patches in review cut this down this a lot
Force noref=1 when predictor is zero in keyframes
Separate probabilities for each block size
Adapt the probabilities

25. Quantization Matrix Simple approach (what we’re doing now)
Separate quantization resolution for each band
Keep flat quantization within bands
Advanced approach?
Scaling after normalization complicated
Unit pulses no longer “unit” (how to sum to K?)
Householder reflection scrambles things further
Better(?): Pre-scale vector by quantization factors
Effects on energy preservation?

26. Quantization Matrix Example
Flat Quantizer (base Q=35)
Adjusted Per-Band (base Q=23)

27. Quantization Matrix Example
Flat Quantizer (base Q=35)
Adjusted Per-Band (base Q=23)
Metrics: +15% PSNR, +12% SSIM, -18% PSNR-HVS

28. Activity Masking Goal: Use better resolution in flat areas
Low contrast → low energy (gain)
Derivations in doc/video_pvq.lyx, doc/theoretical_results.lyx
Currently wrong/incomplete, will fix
Step 1: Compand gain (g)
Goal: Q ∝ g2α (x264 uses α = 0.173)
Quantize ĝ = (Qgĥ)β, encode ĥ
β = 1/(1-2α)
Qg = (Q/β)β

29. Activity Masking cotd. Step 2: Choose θ resolution
D = (g - ĝ)2 + gĝ(Dθ + sin θ sin ϑ Dpvq)
Dθ = 2 – 2cos(θ – ϑ) = distortion due to θ quant.
Dpvq = distortion due to PVQ
Assume g = ĝ, ignore Dpvq...
Qθ = (dĝ/dĥ)/ĝ = β/ĥ