Quantization

1
Quantization

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

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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
reconstructed signal
Information theory VLC Huffman Arithmetic Run-leng
th
Quantization
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Quantization

• Entropy coding techniques
• 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

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Typical Quantizer
Forward Quantizer
x
y
Q
input
output
y
6
Typical Inverse Quantizer

• Typical reconstruction
• Quantization error

y
111
110
101
100
011
010
001
x
000
clipping, overflow
decision boundaries
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Mid-rise versus Mid-tread
y
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

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Quantization Errors

• Approximation 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

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Quantization Error Model

• Assumptions

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Quantization Error Variance
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Uniform Quantization Bounded Input
y
x
b-bit Quantizer
12
Uniform Quantization Bounded Input
b-bit quantizer
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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
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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

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Example of Popular Quantization

• Round
• Floor
• Ceiling

y
x
Uniform midtread quantizer from Round and Floor
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Quantization from Rounding
x
6
10
14
14
6
10
2
2
Uniform Quantizer, step-size4
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Dead-zone Scalar Quantization

• The bin size around zero is doubled
• Other bins are still uniform
• Create more zeros
• Useful for image/video

2?
?
-?
-2?
0
x
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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
0
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Optimal Scalar Quantization

• Problem Statement

• Notes
• Non-uniform quantizer under consideration
• Reconstruction can be anywhere, not necessarily
  the center of the interval

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Optimal Scalar Quantization
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Optimal Scalar Quantization

• Optimal Decoder for a Given Encoder

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Lloyd-Max Quantizer

• Main idea Lloyd 1957 Max 1960
• solving these 2 equation iteratively until D
  converges
• Assumptions
• Input PDF is known and stationary
• Entropy has not been taken into account

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Example
y
0
x
a
b
a
b
x
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Example
y
0
x
a
b
a
b
x
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Embedded Quantization

x
x
y
-1
Q
Q
x

• Also called bit-plane quantization, progressive
  quantization
• Most significant information is transmitted first
• JPEG2000 quantization strategy

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Embedded Quantization
R 1
R 2
R 3
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Embedded Forward Quantization
x
Embedded Quantizer, N2
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Embedded Inverse Quantization
Original symbol x 22
Range16, 32)
Range16, 24)
Range20, 24)



x 24
x 20 24 4
x 22 20 2
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Vector Quantization

• n-dimensional generalization of scalar quantizer
• Nearest neighbor and centroid rule still apply

codebook, containing code-vectors or codewords
n-dimensional input vectors
Separable Scalar Q