presentation

Quantization Design
Jie Ren
jr843@drexel.edu
Adaptive Signal Processing and Information Theory Group
Department of Electrical and Computer Engineering
Drexel University, Philadelphia, PA 19104
July 28th and 30th, 2014
Jie Ren (Drexel ASPITRG) QD July 28th
and 30th
, 2014 1 / 35

References
S. P. Lloyd, “Least squares quantization in pcm,” IEEE Transactions on Acoustics, Speech
and Signal Processing, vol. 28, no. 2, pp. 129–137, March 1982.
J. Max, “Quantizing for minimum distortion,” IEEE Transactions on Acoustics, Speech
and Signal Processing, vol. 6, no. 1, pp. 7–12, March 1960.
N. Farvardin and J. W. Modestino, “Optimum quantizer performance for a class of
non-gaussian memoryless sources,” IEEE Trans. Inform. Theory, vol. 30, no. 3, pp.
485–497, May 1984.
D. K. Sharma, “Design of absolutely optimal quantizers for a wide class of distortion
measures,” IEEE Trans. Inform. Theory, vol. 24, no. 6, pp. 693–702, November 1978.
P. A. Chou, T. Lookabaugh, and R. M. Gray, “Entropy-constrained vector quantization,”
IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 37, no. 1, pp. 31–42,
Juanuary 1989.
L. R. Varshney, “Unreliable and resource-constrained decoding,” Ph.D. dissertation,
Massachusetts Institute of Technology, 2010.
and 30th
, 2014 2 / 35

Problem Setup
Outline
1 Problem Setup
2 Lloyd-max Quantizer Design
Local Optimality Conditions
By Alternating Optimization
By Dynamic Programming
3 Variable Rate Optimum Quantizer Design
Problem Setup
Analysis
Generalized Lloyd-Max Algorithm
and 30th
, 2014 3 / 35

Problem Setup
Quantization
Map a Large Set Θ of Input Values to a Smaller set A
• Discrete Quantization : Countable Θ, Countable A
• Continuous Quantization : Uncountable Θ, Countable A
• Scalar Quantization
Q : Θ → A (1)
• Vector Quantization
Q : Θ × Θ × · · · × Θ → A (2)
and 30th
, 2014 4 / 35

Problem Setup
Problem Setup
N-level Continuous scalar Quantizer
• Source Θ with normalized support [0, 1] and pdf p(θ)
• N-level quantizer QN(·)
• N Reconstruction levels A = {a1, . . . , aN}
• Thresholds b1, . . . , bN−1 partitioning [0, 1] with b0 = 0 and bN = 1
and 30th
, 2014 5 / 35

Problem Setup
N-level Continuous scalar Quantizer
• Huﬀman encode/decode QN(Θ) with rate
R = −
N
n=1
Pn log2 Pn (3)
where
Pn
bn
bn−1
p(θ)dθ (4)
• Distortion Metric
d : [0, 1] × A → R+
(5)
with
d(a, a) = 0 (6)
• Average Distortion
DN(b, a) =
N
n=1
bn
bn−1
d(θ, an)p(θ)dθ (7)
and 30th
, 2014 6 / 35

Problem Setup
Fixed Rate Quantizer Design and Variable Rate Quantizer Design
• N-level Lloyd-Max quantizer : minimize the average distortion for a
ﬁxed number of levels N.[1][2]
• N-level Optimum Quantizer : minimize the average distortion for a
ﬁxed number of levels N subject to an entropy constraint of
rate.[3]
and 30th
, 2014 7 / 35

Lloyd-max Quantizer Design
Outline
1 Problem Setup
Problem Setup
Analysis
and 30th
, 2014 8 / 35

Lloyd-max Quantizer Design Local Optimality Conditions
Outline
1 Problem Setup
Problem Setup
Analysis
and 30th
, 2014 9 / 35

• Nearest Neighbor Condition : For ﬁxed reconstruction levels {ak},
Given any θ ∈ [ak, ak+1],
Q(θ) = d(θ, ak) ≤ d(θ, ak+1) ? ak : ak+1 (8)
• Centroid Condition : For ﬁxed regions {Rk} with thresholds {bk},
ak = arg min
a
bk
bk−1
d(θ, a)p(θ)dθ (9)
• Zero Probability Boundary Condition, for all bk, k = 1, . . . , K − 1
P(θ = bk) = 0 (10)
and 30th
, 2014 10 / 35

Necessary and Sufficient Conditions
Theorem
The nearest neighbor condition, the centroid condition, and the zero
probability of boundary condition are necessary for a Lloyd-Max quantizer
to be optimal.
Theorem
If the following conditions hold for a source Θ and distortion function
d(θ, a) :
1 p(θ) is positive and continuous in (0, 1)
2
1
0 d(θ, a)p(θ)dθ is finite for all a
3 d(θ, a) is zero only for θ = a, is continuous in θ for all a, and is
continuous and convex in a
then the nearest neighbor condition, centroid condition, and zero
probability of boundary conditions are sufficient to guarantee local
optimality of a quantizer.
and 30th
, 2014 11 / 35

Lloyd-max Quantizer Design By Alternating Optimization
Outline
1 Problem Setup
Problem Setup
Analysis
and 30th
, 2014 12 / 35

Lloyd-Max algorithm
Alternating Minimization
• For ﬁxed {ak}, minimize D w.r.t. {bk}
• For ﬁxed {bk}, minimize D w.r.t. {ak}
• D monotone decreasing
and 30th
, 2014 13 / 35

Lloyd-Max algorithm
Algorithm 1: Lloyd-Max
Result: Minimize the average distortion for a N-level Lloyd-Max quantizer
step 1) Choose an arbitrary set of initial reconstruction levels {an}
step 2) For each n = 1, . . . , N set Rn = {θ|d(θ, an) ≤ d(θ, aj ), j = n}
step 3) For each n = 1, . . . , N set an = arg mina E[d(Θ, a)|Θ ∈ Rn]
step 4) Repeat step 2 and 3 until change in average distortion is negligible
step 5) Revise {an} and {Rn} to satisfy the zero probability of boundary
condition
and 30th
, 2014 14 / 35

Analysis
• Local optimum guaranteed by Theorem. 2
• Monotonic Convergence in N
D∗
N(b∗
, a∗
) =
N
n=1
b∗
n
b∗
n−1
d(θ, a∗
n)p(θ)dθ (11)
D∗
= lim
N→∞
D∗
N (12)
and 30th
, 2014 15 / 35

Monotonic Convergence in N
The Lloyd-Max N-level quantizer is the solution of the following
problem:
D∗
N = min
N
n=1
bn
bn−1
d(θ, an)p(θ)dθ
s.t. b0 = 0
bN = 1
bn−1 ≤ bn, n = 1, . . . , N
an ≤ bn, n = 1, . . . , N
(13)
and 30th
, 2014 16 / 35

Monotonic Convergence in N
• Degenerate the N-level Lloyd-Max quantizer to N − 1
• By adding the additional constraint bN−1 = 1 to (13) and forcing
aN = 1, hence
D∗
N−1 ≥ D∗
N (14)
• D∗
N bounded below by 0
• D∗
N converges
and 30th
, 2014 17 / 35

Lloyd-max Quantizer Design By Dynamic Programming
Outline
1 Problem Setup
Problem Setup
Analysis
and 30th
, 2014 18 / 35

Lloyd-Max Quantizer Design Using Dynamic
Programming
• For discrete Θ
• One construction level in the interval (β1, β2) ⊆ [0, 1]
T1(β1, β2) = min
a
θ∈Θ∩(β1,β2)
d(θ, a)p(θ) (15)
• K construction levels in the interval (β1, β2)
TK (β1, β2) = min
a,b:β1<b1<···<bK−1<β2
K
k=1 θ∈Θ∩(bk−1,bk )
d(θ, a)p(θ) (16)
• Notice that
D∗
N = TN(0, 1) (17)
and 30th
, 2014 19 / 35

Programming
Theorem
Let b∗
1, . . . , b∗
K−1 be the optimizing boundary points for
TK (b∗
0 = 0, b∗
K = 1), then b∗
1, . . . , b∗
K−2 must be the optimizing boundary
points for TK−1(b∗
0, b∗
K−1), and
TK (b∗
0, b∗
K ) = min
bK−1:b∗
0 <bK−1<b∗
K
[TK−1(b∗
0, bK−1) + T1(bK−1, b∗
K )] (18)
and 30th
, 2014 20 / 35

Programming
• For any 1 < k ≤ K and any discrete β ∈ (b∗
0, b∗
K ]
Tk(b∗
0, β) = min
b:b∗
0 <b<β
[Tk−1(b∗
0, b) + T1(b, β)] (19)
• Optimizing threshold
b∗
k−1(b∗
0, β) = arg min
b:b∗
0 <b<β
[Tk−1(b∗
0, b) + T1(b, β)] (20)
and 30th
, 2014 21 / 35

Programming
Algorithm 2: DP algorithm for Lloyd-Max Quantizer Design
Result: Minimize the average distortion for a N-level Lloyd-Max quantizer
step 1) Compute the values of T1(β1, β2) for all discrete β1 and β2 in [0, 1]
step 2) For each n = 2, . . . , N compute Tn(0, β) and b∗
n−1(0, β) for all β
in (0, 1] using (19) and (20)
step 3) Let bN = 1, for each n = N, . . . , 2 set bn−1 = b∗
n−1(0, bn)
step 4) For each n = 1, . . . , N, set
ak = arg mina E[d(Θ, a)|Θ ∈ (bk−1, bk)]
and 30th
, 2014 22 / 35

Programming
Theorem
The boundaries {bk}K
k=0 and reconstruction levels {ak}K
k=1 returned by
Algorithm 2 represent the optimal quantizer.[4]
and 30th
, 2014 23 / 35

Variable Rate Optimum Quantizer Design
Outline
1 Problem Setup
Problem Setup
Analysis
and 30th
, 2014 24 / 35

Variable Rate Optimum Quantizer Design Problem Setup
Outline
1 Problem Setup
Problem Setup
Analysis
and 30th
, 2014 25 / 35

Fixed Rate Quantizer Design and Variable Rate Quantizer Design
• N-level Lloyd-Max quantizer : minimize the average distortion for a
ﬁxed number of levels N.[1][2]
• N-level Optimum Quantizer : minimize the average distortion for a
ﬁxed number of levels N subject to an entropy constraint of
rate.[3]
and 30th
, 2014 26 / 35

Problem Setup
• Huﬀman encode/decode QN(Θ) with rate
R = −
N
n=1
Pn log2 Pn (21)
where
Pn
bn
bn−1
p(θ)dθ (22)
• Distortion Metric
d : [0, 1] × A → R+
(23)
with
d(a, a) = 0 (24)
• Average Distortion
DN(b, a) =
N
n=1
bn
bn−1
d(θ, an)p(θ)dθ (25)
and 30th
, 2014 27 / 35

Problem Setup
Minimizing Distortion with a Rate Constraint
D∗
N = min
N
n=1
bn
bn−1
d(θ, an)p(θ)dθ
s.t. R = −
N
n=1
Pn log2 Pn ≤ H0
(26)
Form the Lagrangian
L = DN(b, a) + λ(R(b) − H0) (27)
and 30th
, 2014 28 / 35

Variable Rate Optimum Quantizer Design Analysis
Outline
1 Problem Setup
Problem Setup
Analysis
and 30th
, 2014 29 / 35

Analysis
Take Derivative w.r.t. an
∂
∂an
L =
∂
∂an
DN(b, a)
=
bn
bn−1
∂
∂an
d(θ, an) p(θ)dθ
(28)
Denote the optimum reconstruction levels by
a∗
n an(b) (29)
a∗
n can be obtained by (28), i.e. for the mean-square distortion [2]
a∗
n =
bn
bn−1
θp(θ)dθ
bn
bn−1
p(θ)dθ
(30)
and 30th
, 2014 30 / 35

Analysis
Take Derivative w.r.t. bn
∂
∂bn
L =
∂
∂bn
DN(b, a∗
) + λ
∂
∂bn
R(b) (31)
which leads to
λ(ln(Pn+1/Pn)) = d(bn, a∗
n+1) − d(bn, a∗
n) (32)
for all n ∈ {1, 2, . . . , N − 1}.
and 30th
, 2014 31 / 35

Variable Rate Optimum Quantizer Design Generalized Lloyd-Max Algorithm
Outline
1 Problem Setup
Problem Setup
Analysis
and 30th
, 2014 32 / 35

Generalized Max Algorithm
Algorithm 3: Generalized Max Algorithm
Result: Minimize the average distortion for a N-level variable rate
optimum quantizer
step 1) Given N and ﬁxed λ, set b0 = 0, choose an initial value for b1 and
set n = 1
step 2) For the present values of bn−1 and bn, use (28) and (32) to ﬁnd
bn+1. If n ≤ N − 1, replace n by n + 1 and go to step 2). Otherwise
continue
step 3) If bN obtained in step 2) is equal to 1, the initial guess for b1 is
good and the resulting b and a satisfy the necessary conditions for
optimality. Otherwise go to step 1) and change b1
and 30th
, 2014 33 / 35

Generalized Lloyd Algorithm
Apply the mean square distortion to (32)
λ(ln(Pn+1/Pn)) = (a∗
n+1 − a∗
n)(a∗
n+1 + a∗
n − 2bn) (33)
which can be written as
b∗
n =
a∗
n+1
a∗
n
−
λ
2(a∗
n+1 − a∗
n)
ln(Pn+1/Pn) (34)
and 30th
, 2014 34 / 35

Generalized Lloyd Algorithm
Algorithm 4: Generalized Lloyd Algorithm
Result: Minimize the average distortion for a N-level variable rate
optimum quantizer
step 1) Given N and ﬁxed λ, set b0 = 0, bN = 1, and choose an initial
value for b1, . . . , bN−1.
step 2) Compute {a1, . . . , aN} by (28)
step 3) Compute {b1, . . . , bN−1} by (34)
step 4) Run step 2) and step 3) times. If D∗
N converges, output b and a.
Otherwise go to step 1) and change initial guess for b
and 30th
, 2014 35 / 35

presentation

Recommandé

Recommandé

Contenu connexe

Tendances

Tendances (20)

Similaire à presentation

Similaire à presentation (20)

presentation