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Asymptotically Scale-invariant Multi-resolution Quantization Cheuk Ting Li Dept. of Information Engineering, Chinese University of Hong Kong Email: ctli@ie.cuhk.edu.hk 2020 IEEE International Symposium on Information Theory Multi-resolution

Asymptotically Scale-invariant Multi-resolution Quantization Cheuk Ting Li Dept. of Information Engineering, Chinese University of Hong Kong Email: ctli@ie.cuhk.edu.hk 2020 IEEE International Symposium on Information Theory
Multi-resolution Quantizer (MRQ) 0 2 4 8 12 Q 3 0 2 4 6 8 10 12 Q 2 0 1 2 3 4 5 6 7 8 10 12 Q 1 Studied by, e.g. Equitz and Cover [1991], Brunk and Farvardin [1996], Jafarkhani et al. [1997], Wu and Dumitrescu [2002], Dumitrescu and Wu [2004], Effros and Dugatkin [2004] A sequence of quantizers Coarser quantization can be obtained from finer quantization by discarding information Finer quantization can be obtained from coarser quantization by adding additional information (successive refinement)
Multi-resolution Quantizer (MRQ) 0 2 4 8 12 Q 3 0 2 4 6 8 10 12 Q 2 0 1 2 3 4 5 6 7 8 10 12 Q 1 Studied by, e.g. Equitz and Cover [1991], Brunk and Farvardin [1996], Jafarkhani et al. [1997], Wu and Dumitrescu [2002], Dumitrescu and Wu [2004], Effros and Dugatkin [2004] A sequence of quantizers Coarser quantization can be obtained from finer quantization by discarding information Finer quantization can be obtained from coarser quantization by adding additional information (successive refinement)
Multi-resolution Quantizer (MRQ) 0 2 4 8 12 Q 3 0 2 4 6 8 10 12 Q 2 0 1 2 3 4 5 6 7 8 10 12 Q 1 Studied by, e.g. Equitz and Cover [1991], Brunk and Farvardin [1996], Jafarkhani et al. [1997], Wu and Dumitrescu [2002], Dumitrescu and Wu [2004], Effros and Dugatkin [2004] A sequence of quantizers Coarser quantization can be obtained from finer quantization by discarding information Finer quantization can be obtained from coarser quantization by adding additional information (successive refinement)
Multi-resolution Quantizer (MRQ) 0 2 4 8 12 Q 3 0 2 4 6 8 10 12 Q 2 0 1 2 3 4 5 6 7 8 10 12 Q 1 Studied by, e.g. Equitz and Cover [1991], Brunk and Farvardin [1996], Jafarkhani et al. [1997], Wu and Dumitrescu [2002], Dumitrescu and Wu [2004], Effros and Dugatkin [2004] A sequence of quantizers Coarser quantization can be obtained from finer quantization by discarding information Finer quantization can be obtained from coarser quantization by adding additional information (successive refinement)
Multi-resolution Quantizer (MRQ) 10 0 10 0 10 1 10 1 10 2 10 2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 A quantizer is a function Q : R → R where the range Q ( R ) is finite/countable A MRQ is a set of quantizers { Q s } s with parameter s > 0 ( ≈ step size), such that output of coarser quantizer can be deduced from output of finer quantizer, without knowledge of the original data: Q s 2 ( Q s 1 ( x )) = Q s 2 ( x ) for s 2 ≥ s 1 > 0
Multi-resolution Quantizer (MRQ) 10 0 10 0 10 1 10 1 10 2 10 2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 A quantizer is a function Q : R → R where the range Q ( R ) is finite/countable A MRQ is a set of quantizers { Q s } s with parameter s > 0 ( ≈ step size), such that output of coarser quantizer can be deduced from output of finer quantizer, without knowledge of the original data: Q s 2 ( Q s 1 ( x )) = Q s 2 ( x ) for s 2 ≥ s 1 > 0
Multi-resolution Quantizer (MRQ) 2.5 0 1 2 3 4 5 6 7 8 10 12 Q 1 0 2 4 6 8 10 12 Q 2 0 1 2 3 4 5 6 7 8 10 12 Q 1 0 2 4 8 12 Q 3 Consider a piece of data being sent and recompressed multiple times between multiple users Let the data be x ∈ R , compressed by the first user to Q s 1 ( x ) , then by the second user to Q s 2 ( Q s 1 ( x )) , ... If Q s is MRQ, final output y = Q s n ( · · · Q s 1 ( x ) · · · ) is as good as the worst quantizer Q max j s j If Q s is not MRQ, final output can be far from x
Multi-resolution Quantizer (MRQ) 2.5 0 1 2 3 4 5 6 7 8 10 12 Q 1 0 2 4 6 8 10 12 Q 2 0 1 2 3 4 5 6 7 8 10 12 Q 1 0 2 4 8 12 Q 3 Consider a piece of data being sent and recompressed multiple times between multiple users Let the data be x ∈ R , compressed by the first user to Q s 1 ( x ) , then by the second user to Q s 2 ( Q s 1 ( x )) , ... If Q s is MRQ, final output y = Q s n ( · · · Q s 1 ( x ) · · · ) is as good as the worst quantizer Q max j s j If Q s is not MRQ, final output can be far from x
Multi-resolution Quantizer (MRQ) 2.5 0 1 2 3 4 5 6 7 8 10 12 Q 1 0 2 4 6 8 10 12 Q 2 0 1 2 3 4 5 6 7 8 10 12 Q 1 0 2 4 8 12 Q 3 Consider a piece of data being sent and recompressed multiple times between multiple users Let the data be x ∈ R , compressed by the first user to Q s 1 ( x ) , then by the second user to Q s 2 ( Q s 1 ( x )) , ... If Q s is MRQ, final output y = Q s n ( · · · Q s 1 ( x ) · · · ) is as good as the worst quantizer Q max j s j If Q s is not MRQ, final output can be far from x
Multi-resolution Quantizer (MRQ) 2.5 0 1 2 3 4 5 6 7 8 10 12 Q 1 0 5 10 11 12 Q 2 0 1 2 3 4 5 6 7 8 10 12 Q 1 0 2 4 8 12 Q 3 Consider a piece of data being sent and recompressed multiple times between multiple users Let the data be x ∈ R , compressed by the first user to Q s 1 ( x ) , then by the second user to Q s 2 ( Q s 1 ( x )) , ... If Q s is MRQ, final output y = Q s n ( · · · Q s 1 ( x ) · · · ) is as good as the worst quantizer Q max j s j If Q s is not MRQ, final output can be far from x
Binary Multi-resolution Quantizer (BMRQ) 10 0 10 1 10 2 0.0 0.2 0.4 0.6 0.8 1.0 Q ❜✐♥ ( x ) = 2 ⌊ log 2 s ⌋ ( ⌊ 2 −⌊ log 2 s ⌋ x ⌋ + 1 / 2 ) s Uniform quantizers with step sizes that are powers of 2 Q ❜✐♥ changes abruptly rather than smoothly with s s If we require step size ≤ 3 . 9, then we have to choose step size 2, and use almost twice as many quantization cells as needed Not “scale-invariant”
Binary Multi-resolution Quantizer (BMRQ) 10 0 10 1 10 2 0.0 0.2 0.4 0.6 0.8 1.0 Q ❜✐♥ ( x ) = 2 ⌊ log 2 s ⌋ ( ⌊ 2 −⌊ log 2 s ⌋ x ⌋ + 1 / 2 ) s Uniform quantizers with step sizes that are powers of 2 Q ❜✐♥ changes abruptly rather than smoothly with s s If we require step size ≤ 3 . 9, then we have to choose step size 2, and use almost twice as many quantization cells as needed Not “scale-invariant”
Binary Multi-resolution Quantizer (BMRQ) 10 0 10 1 10 2 0.0 0.2 0.4 0.6 0.8 1.0 Q ❜✐♥ ( x ) = 2 ⌊ log 2 s ⌋ ( ⌊ 2 −⌊ log 2 s ⌋ x ⌋ + 1 / 2 ) s Uniform quantizers with step sizes that are powers of 2 Q ❜✐♥ changes abruptly rather than smoothly with s s If we require step size ≤ 3 . 9, then we have to choose step size 2, and use almost twice as many quantization cells as needed Not “scale-invariant”
Binary Multi-resolution Quantizer (BMRQ) 10 0 10 1 10 2 0.0 0.2 0.4 0.6 0.8 1.0 Q ❜✐♥ ( x ) = 2 ⌊ log 2 s ⌋ ( ⌊ 2 −⌊ log 2 s ⌋ x ⌋ + 1 / 2 ) s Uniform quantizers with step sizes that are powers of 2 Q ❜✐♥ changes abruptly rather than smoothly with s s If we require step size ≤ 3 . 9, then we have to choose step size 2, and use almost twice as many quantization cells as needed Not “scale-invariant”
Dithered Binary Multi-resolution Quantizer (DBMRQ) 10 0 10 1 10 2 0.0 0.2 0.4 0.6 0.8 1.0  2 ⌊ log 2 s ⌋ + 1 ( ⌊ 2 −⌊ log 2 s ⌋− 1 x ⌋ + 1 / 2 )   Q ❞✐ ✐❢ ❢r❛❝ ( φ ⌊ 2 −⌊ log 2 s ⌋− 1 x ⌋ ) < 2 − 2 ⌊ log 2 s ⌋ + 1 / s s ( x ) :=  2 ⌊ log 2 s ⌋ ( ⌊ 2 −⌊ log 2 s ⌋ x ⌋ + 1 / 2 ) ♦t❤❡r✇✐s❡ ,  Same as BMRQ when s = 2 k (uniform quantizer w/ step 2 k ) When s decreases from 2 k to 2 k − 1 , more and more cells with size 2 k are bisected into cells with size 2 k − 1 Not “scale-invariant”
Dithered Binary Multi-resolution Quantizer (DBMRQ) 10 0 10 1 10 2 0.0 0.2 0.4 0.6 0.8 1.0  2 ⌊ log 2 s ⌋ + 1 ( ⌊ 2 −⌊ log 2 s ⌋− 1 x ⌋ + 1 / 2 )   Q ❞✐ ✐❢ ❢r❛❝ ( φ ⌊ 2 −⌊ log 2 s ⌋− 1 x ⌋ ) < 2 − 2 ⌊ log 2 s ⌋ + 1 / s s ( x ) :=  2 ⌊ log 2 s ⌋ ( ⌊ 2 −⌊ log 2 s ⌋ x ⌋ + 1 / 2 ) ♦t❤❡r✇✐s❡ ,  Same as BMRQ when s = 2 k (uniform quantizer w/ step 2 k ) When s decreases from 2 k to 2 k − 1 , more and more cells with size 2 k are bisected into cells with size 2 k − 1 Not “scale-invariant”
Dithered Binary Multi-resolution Quantizer (DBMRQ) 10 0 10 1 10 2 0.0 0.2 0.4 0.6 0.8 1.0  2 ⌊ log 2 s ⌋ + 1 ( ⌊ 2 −⌊ log 2 s ⌋− 1 x ⌋ + 1 / 2 )   Q ❞✐ ✐❢ ❢r❛❝ ( φ ⌊ 2 −⌊ log 2 s ⌋− 1 x ⌋ ) < 2 − 2 ⌊ log 2 s ⌋ + 1 / s s ( x ) :=  2 ⌊ log 2 s ⌋ ( ⌊ 2 −⌊ log 2 s ⌋ x ⌋ + 1 / 2 ) ♦t❤❡r✇✐s❡ ,  Same as BMRQ when s = 2 k (uniform quantizer w/ step 2 k ) When s decreases from 2 k to 2 k − 1 , more and more cells with size 2 k are bisected into cells with size 2 k − 1 Not “scale-invariant”
Dithered Binary Multi-resolution Quantizer (DBMRQ) 10 0 10 1 10 2 0.0 0.2 0.4 0.6 0.8 1.0  2 ⌊ log 2 s ⌋ + 1 ( ⌊ 2 −⌊ log 2 s ⌋− 1 x ⌋ + 1 / 2 )   Q ❞✐ ✐❢ ❢r❛❝ ( φ ⌊ 2 −⌊ log 2 s ⌋− 1 x ⌋ ) < 2 − 2 ⌊ log 2 s ⌋ + 1 / s s ( x ) :=  2 ⌊ log 2 s ⌋ ( ⌊ 2 −⌊ log 2 s ⌋ x ⌋ + 1 / 2 ) ♦t❤❡r✇✐s❡ ,  Same as BMRQ when s = 2 k (uniform quantizer w/ step 2 k ) When s decreases from 2 k to 2 k − 1 , more and more cells with size 2 k are bisected into cells with size 2 k − 1 Not “scale-invariant”

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