Optimum Uniform Quantizer

Transcription

Chapter 2 Quantization
1
How much can we compress this image losslessly?
How much can we compress this image with a loss?
CS4670/7670 Digital Image Compression
2
Outline
Quantization and Source encoder
Uniform quantization
• Basics
• Optimum Uniform quantizer
3
Quantization and Source
Encoder
Transmitter
Input
visual
information
A/D
Source
encoder
Channel
encoder
Modulation
Channel
Received
visual
information
D/A
Source
decoder
Channel
decoder
Demodulation
Receiver
Block diagram of a visual communication system4
Encoder
Storage
Input
visual
information
A/D
Source
encoder
D/A
Source
decoder
Retrieved
visual
information
Retrieval
Block diagram of a visual storage system
5
Encoder
Input
information
Transformation
Codeword
assignment
Quantization
Codeword
string
(a) source encoder
Codeword
string
Codeword
decoder
Inverse
transformation
Reconstructed
information
(b) source decoder
Block diagram of a source encoder and decoder
6
Encoder
Quantization:
• An irreversible process
• A source of information loss
• A critical stage in image and video
compression
It has significant impact on
• The distortion of reconstructed image
and video
• The bit rate of the compressed bitstream
7
Uniform Quantizer
Simplest
Most popular
Conceptually of great importance
8
Uniform Quantizer -- Basics
Definitions
• The input-output characteristic of the quantizer
Stair-case like
Non-linear
y i = Q ( x)
if
x ∈ (d i , d i +1 ),
Where yi and Q(x) is the output of the quantizer with respect to
the input x
9
Uniform Quantizer – Basics
(midtread quantizer)

y
Reconstructed levels
include 0
For odd # of reconstruction
y9
4
y8
3
y7
levels
2
1
y6
y5
-3.5
d2
d1=-∞
-2.5
d3
-1.5
d4
y3
y4
-0.5
d5
x
0 0.5
d6
-1
1.5
d7
2.5
d8
3.5
d9
d10=∞
-2
y2
-3
y1
-4
10
Decision levels
• The end points of the intervals, denoted by di
where i : index of intervals
Reconstruction level
• The output of the quantization, denoted by yi
Step size of the quantizer
• The length of the interval, denoted by ∆
11
Two features of a uniform quantizer
• Except possibly the right-most and left-most intervals, all
intervals along the x-axis are uniformly spaced
• Except possibly the outer intervals, the reconstruction
levels of the quantizer are also uniformly spaced
Furthermore, each inner reconstruction level is the
arithmetic average of the two decision levels of the
corresponding interval
12
Uniform Quantizer – Basics
(midrise quantizer)

Reconstructed levels do not
include 0
For even # of reconstruction
y
y8
3.5
y7
levels
2.5
y6
1.5
y5
x
0.5
-4.0
d1=-∞
-3.0
d2
-2.0
d3
-1.0 y
4
d4
0
d5 -0.5
1.0
d6
2.0
d7
3.0
d8
4.0
d9 =∞
y3
-1.5
y2
-2.5
y1
-3.5
13
WLOG, assume both input-output
characteristics of the midtread and midrise
uniform quantizers are odd symmetric with
respect to the vertical axis x=0
• Subtraction of statistical mean of input x
• Addition of statistical mean back after quantization
• N: the total number of reconstruction levels of a
quantizer
14
Quantization Distortion
In terms of objective evaluation, we define quantization
error eq
eq = x − Q(x),
Quantization error is often referred to as quantization
noise
Mean square quantization error MSEq
d
N i+1
MSEq = ∑ ∫ ( x − Q( x))2 f ( x)dx
X
i=1 d
i
15
fx(x): probability density function (pdf)
• The outer decision levels may be -∞ or ∞
• When the pdf fx(x) remains unchanged, fewer
reconstruction levels (smaller N, coarser
quantization) result in more distortion.
• In general, the mean of eq is not zero. It is zero
when the input x has a uniform distribution. In
this case, MSEq is the variance of the
quantization noise eq.
16
y
granular
quantization
noise
overload
quantization
noise
0.5
x
-4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5
0 0.5 1
1.5 2.0 2.5 3.0 3.5 4.0 4.5
-0.5
overload
quantization
noise
17
Quantizer Design
The design of a quantizer (uniform/nonuniform)
• Choosing the # of reconstruction levels, N
• Selecting the values of decision levels and
reconstruction levels
The design of a quantizer is equivalent to
specifying its input-output characteristic
Optimum quantizer design
• For a given probability density function of the input
random variable, fX(x), design a quantizer such that the
mean square quantization error, MSEq, is minimized.
18
Quantizer Design
For uniform quantizer design
• N is usually given.
• According to the two features of uniform
quantizers
Only one parameter that needs to be decided: the
step size ∆
As to the optimum uniform quantizer design, a
different pdf leads to a different step size
19
y
y9
4
y8
3
y7
2
1
y6
y5
-4.5
d1
-3.5
d2
-2.5
d3
-1.5
d4 y
4
y3
-0.5
d5
x
0 0.5
d6
-1
1.5
d7
2.5
d8
3.5
d9
4.5
d10
-2
y2
-3
y1
-4
Uniform quantizer with uniformly distributed input
20
y
Zero
overload
quantization
noise
Granular
quantization
noise
0.5
x
-4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5
0 0.5 1
1.5 2.0 2.5 3.0 3.5 4.0 4.5
-0.5
Zero
overload
quantization
noise
Quantization distortion
21
The mean square quantization error
d
2
MSE q = N ∫ ( x − Q ( x)) 2 1 dx
N∆
d
1
2
∆
MSE q =
.
12
σ x2
SNRms = 10 log
= 10 log N 2 .
10
10 σ 2
q
If we assume
N = 2n,
we then have
SNRms = 20 log10 2n = 6.02n
dB.
22
The interpretation of the above result
• If use natural binary code to code the
reconstruction levels of a uniform quantizer with
a uniformly distributed input source, then every
increased bit in the coding brings out a 6.02 dB
increase in the SNRms
• Whenever the step size of the uniform quantizer
decreases by a half, the MSEq decreases four
times
23
Quantization Effects
1 bit quantizer
2 bit quantizer
3 bit quantizer
4 bit quantizer
24

Conditions of optimum quantization
• Derived by [Lloyd’57, 82; Max’60]
• Necessary conditions, for a given pdf fX(x)
x = −∞
1
xN +1 = +∞
d
i+1
∫ ( x − yi ) f X ( x)dx = 0 i = 1,2,L, N
d
i
d = 1(y + y )
i = 2,L, N
i 2 i −1 i
Centroid condition
Nearest neighbor
condition
25
•
•
•

First condition: for an input x whose range is
-∞ <x< ∞
Second: each reconstruction level is the centroid of
the area under the pdf and between the two adjacent
decision levels
Third: each decision level (except for the outer
intervals) is the arithmetic average of the two
neighboring reconstruction levels
These conditions are general in the sense that
there is no restriction imposed on the pdf.
26
Optimum uniform quantizer with different input
distributions
• A uniform quantizer is optimum when the input has
uniform distribution
• Normally, if the pdf is not uniform, the optimum
quantizer is not a uniform quantizer
• Due to the simplicity of uniform quantization, however, it
may sometimes be desirable to design an optimum
uniform quantizer for an input with a nonuniform
distribution.
27
Some Typical Distributions
Laplacian
Gamma
28
Optimal symmetric uniform quantizer for Gaussian, Laplacian and
Gamma distribution (with zero mean and unit variance).
[Max’60][Paez’72]. The numbers enclosed in rectangular are the
step sizes.
29
Under these circumstances, however, three
equations are not a set of simultaneous equations
one can hope to solve with any ease
Numerical procedures were suggested for design
of optimum uniform quantizers
• E.g., Newton method
Max derived uniform quantization step size for an
input with a Gaussian distribution [Max ’60]
30
Paez and Glisson found step size for
Laplacian and Gamma distributed input
signals [Paez’ 72]
Zero mean: if the mean is not zero, only a
shift in input is needed when applying these
results
Unit variance: if the standard deviation is not
unit, the tabulated step size needs to be
multiplied by the standard deviation.
31
Nonuniform Quantizer
In general, an optimal quantizer is a
nonuniform quantizer.
• Depends on statistic (pdf) of input source
Companding quantization
• Using uniform quantizer to realize non-uniform
quantization
• Reading: Section 2.3.2
Adaptive quantization
• Adapt to changing statistic of input source
• Reading: Section 2.4
HW #1: Ex. 2-2
32

Optimum Uniform Quantizer

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