1. The Johns Hopkins University
Quantization
Trac D. Tran
ECE Department
The Johns Hopkins University
Baltimore, MD 21218

2. Outline Review Quantization Nonlinear mapping
Forward and inverse quantization
Quantization errors
Clipping error
Approximation error
Error model
Optimal scalar quantization
Examples

3. Reminder original signal compressed bit-stream reconstructed signal
Information theory
VLC
Huffman
Arithmetic
Run-length
Quantization

4. Quantization Entropy coding techniques Quantization
Perform lossless coding
No flexibility or trade-off in bit-rate versus distortion
Quantization
Lossy non-linear mapping operation: a range of amplitude is mapped to a unique level or codeword
Approximation of a signal source using a finite collection of discrete amplitudes
Controls the rate-distortion trade-off
Applications
A/D conversion
Compression

5. Typical Quantizer Q Forward Quantizer x y input output y
111
110
101
000
001
010
011
100
quantization level
or codeword
clipping/overflow
quantization cell/bin/interval x
decision boundaries
clipping/overflow

6. Typical Inverse Quantizer
Typical reconstruction
Quantization error y x Q
Inverse Quantizer ^ -1 y
111
110
101
100
011
010
001
000 x
clipping, overflow
decision boundaries

7. Mid-rise versus Mid-tready y x x
Uniform Midrise Quantizer
Uniform Midtread Quantizer
Popular in ADC
For a b-bit midrise
Popular in compression
For a b-bit midtread

8. Quantization Errors Approximation error Clipping error
Lack of quantization resolution, too few quantization levels, too large quantization step-size
Causes staircase effect
Solution: increases the number of quantization levels, and hence, increase the bit-rate
Clipping error
Inadequate quantizer range limits, also known as overflow
Solution
Requires knowledge of the input signal
Typical practical range for a zero-mean signal

9. Quantization: Error Model
Assumptions: x Q ^ -1 + x ^
Quantization error:
Mean-squared distortion measure:

10. Quantization Error Variance+ x ^
would like to minimize

11. Uniform Quantization – Bounded Inputy x
b-bit Quantizer x q q
high bit-rate
assumption

12. Uniform Quantization – Bounded Inputx q q
b-bit quantizer
high bit-rate
assumption

13. Signal-to-Noise Ratio
Definition of SNR in decibel (dB)
power of the signal
power of the noise
For quantization noise
Suppose that we now add 1 more bit to our Q resolution:

14. Example
Design a 3-bit uniform quantizer for a signal with range [0,128]
Maximum possible number of levels:
Quantization stepsize:
Quantization levels:
Reconstruction levels:
Maximum quantization error:

15. Example of Popular Quantization
Round
Floor
Ceiling y x
Uniform midtread quantizer from Round and Floor

16. Quantization from Roundingy
–14
–10 –6 –2 2 6 10 14 x
Uniform Quantizer, step-size=4

17. Dead-zone Scalar Quantization
The bin size around zero is doubled
Other bins are still uniform
Create more zeros
Useful for image/video
Output
∆ 2∆ 3∆ Input
-3∆ -2∆ -∆
3.5∆
2.5∆
1.5∆
-1.5∆
-2.5∆
-3.5∆
Quantization mapping:
De-quantization mapping:
-2∆ -∆ 2∆ ∆ x

18. Non-Uniform Quantization
Uniform quantizer is not optimal if source is not uniformly distributed
For given M, to reduce MSE, we want narrow bin when f(x) is high and wide bin when f(x) is low
f(x) x

19. Optimal Scalar Quantization
Problem Statement:
Optimal Encoder for a Given Decoder:
Notes:
Non-uniform quantizer under consideration
Reconstruction can be anywhere, not necessarily the center of the interval

20. Optimal Scalar Quantization
Fundamental Theorem of Calculus
Nearest
Neighbor
Rule

21. Optimal Scalar Quantization
Optimal Decoder for a Given Encoder:
Centroid
Rule

22. Lloyd-Max Quantizer Main idea [Lloyd 1957] [Max 1960] Assumptions
solving these 2 equation iteratively until D converges
Assumptions
Input PDF is known and stationary
Entropy has not been taken into account
Nearest
Neighbor
Partitioning
Centroid
Computation
Input Codebook
Updated Codebook
index of m-th iteration

23. Example y –b –a x a b x

24. Example y –b –a x a b x

25. Embedded Quantization^ x y x -1 Q Q x S
MSB
LSB F1 F2 y
Discard N integer bit planes
+ all fractional bit planes
Also called bit-plane quantization, progressive quantization
Most significant information is transmitted first
JPEG2000 quantization strategy

26. Embedded Quantization
R = 1
R = 2
R = 3

27. Embedded Forward Quantizationy 16 12 8
–16
–12 –8 4 –4 x
Dead Zone
Embedded Quantizer, N=2

28. Embedded Inverse Quantization1 X 1
Truncate
4 bit planes X 1
Receive 1
refinement bit X 1
Receive 2
refinement bits
Original symbol
x = 22
Range=[16, 32)
Range=[16, 24)
Range=[20, 24) ^ ^ ^
x = 24
x = 20
= 24 – 4
x = 22 =
N-bit-plane truncation = scalar quantization with

29. Vector Quantization n-dimensional generalization of scalar quantizer
Nearest neighbor and centroid rule still apply
n-dimensional
input vectors
codebook, containing
code-vectors or codewords
Vector Q
Separable
Scalar Q