Download pdf - Optimal Delayed Decisions in Decoding of Predictively ......Speech coding via adaptive differential pulse code modulation (G.726, G.722) Continuously variable slope delta modulation

Optimal Delayed Decisions

in Decoding of Predictively

Encoded Sources

Vinay Melkote and Kenneth Rose

Signal Compression Lab

Department of Electrical and Computer Engineering

University of California, Santa Barbara

Signal Compression Lab, ECE, UCSB 2

Introduction

� Decoders simply reconstruct data, no parameter choices to make

� Can decoder delay, and thus accrued future coded data, improve current reconstruction?

� Feasible if adequate correlation exists between coded

data units

� Predictive coding systems provide the right setting:

assume an underlying correlation model for the source


Introduction

� Predictive coding widely employed in signal compression standards:

� Motion-compensated video coding (H.264)

� Speech coding via adaptive differential pulse code

modulation (G.726, G.722)

� Continuously variable slope delta modulation (Bluetooth hands-free profile)

� Attractive for low-delay/low-complexity applications


Introduction

� Assume a scalar first-order autoregressive (AR) source: a sequence of zero-mean random variables

that evolve as

1 1, ,n n nx x x− +⋯ ⋯

1n n nx x zρ −= +

( )Zp zi.i.d innovations withzero-mean pdf

nz nx

1zρ −

correlated source samples with inter-sample correlation coefficient ρ


Introduction

� Consider coding with a differential pulse code modulation scheme (DPCM)

� The prediction here is

� Generally, , i.e., predictor matched to source

1 1, ,n n nx x x− +⋯ ⋯

nx ne

nxɶnxɶ

ni

n̂e

n̂x

+

-

++

Q

nxɶ

n̂en̂x

+1

az−

1az

−

DPCM Encoder

DPCM Decoder

1ˆ

n nx ax −=ɶ

a ρ=


� DPCM encoder and decoder operate at zero delay

� At asymptotically high bit-rates:

�

� Matched predictor is optimal

�

� Hence indices are approximately i.i.d

� DPCM encoder and decoder operate at zero delay

� At asymptotically high bit-rates:

�

� Matched predictor is optimal

�

� Hence indices are approximately i.i.d

� Future indices provide no information on

� Zero-delay decoder optimal for the given encoder

Introduction

1 1ˆ

n n nx x xρ ρ− −= ≈ɶ

1 1, ,n n ni i i− +⋯ ⋯

1 2, ,n ni i+ + ⋯

1 1ˆ

n n n n n ne x x x x zρ ρ− −≈∴ = − − =

nx

1 1ˆ

n nx x− −≈


Introduction

� At low bit-rates, prediction errors are correlated, and the indices as well

� Future indices contain information on

� Can this be exploited, by appropriate decoding delay, to

improve the reconstruction of ?

nx

nx

1 1, ,n n ni i i− +⋯ ⋯

1 2, ,n ni i+ + ⋯


Prior work

� Interpolative DPCM (IDPCM) [Sethia & Anderson, ‘78] and

Smoothed DPCM (SDPCM) [Chang & Gibson, ‘91]

� Apply a non-causal post-filter to smooth the zero-delay

reconstructions: non-causality implemented by delay

Regular zero-

delay DPCM

reconstructions

ˆ ˆ or idpcm sdpcmn nx x

1ˆ

nx + 2ˆnx +1ˆnx − ˆn Lx +2ˆnx − ˆnx

+

n Lb +2nb +1nb +1nb −2nb − n

b

Delayed reconstructions

after filtering


Prior work

� IDPCM and SDPCM differ in the design of the non-causal

filter

� The IDPCM design:

� Filter taps determined by minimization of an expectedmean squared error that involves statistics of

unquantized samples

� Process autocorrelation determines filter taps

� Ignores bit-rate and innovation densities

� No gains by increasing look-ahead beyond process order


Prior work

� The SDPCM design:

� Employs a Kalman fixed-lag smoother

� The AR process provides the ‘plant’ model with source samples viewed as the ‘plant state’.

� Quantizer operation provides the ‘observation’ model, with quantized source samples ( ) perceived as ‘observations’

� The model assumes that the quantization noise is white

and uncorrelated with the source

� Kalman filter optimal for linear Gaussian model: ignores the true innovation pdf

ˆnx


� Decoder has more information: unused by mere averaging of the zero-delay reconstructions

� For instance, decoder has information

� Smoothed reconstructions need not lie in

which is known to the decoder

0

Q

( )na i ( )nb i

( )n n

x a i+ɶ ( )n nx b i+ɶnxɶ

nx+ ɶ lay in this interval ne

lies in this interval= + n n nx x eɶ

Sub-optimalities

( , , )n nx i Qɶ

[ )( ) ( )n n n n nI x a i x b i= + +ɶ ɶ

12

Proposed method

� Estimation-theoretic approach that optimally combines the

information to obtain the - sample

delayed reconstruction of

� Recursively calculates the pdf of conditioned on all

available information

nx1 1, , , , ,n n n n Li i i i− + +⋯ ⋯

nx

L

ni n̂x ˆsdpcm

nxˆidpcm

nx

Regular DPCM

Decoder

Optimal Delayed

Decoderni n̂x

*

n̂x

IDPCM or SDPCM

Proposed method


� Distortion criterion - mean squared error (MSE)

� The optimal estimate of at the decoder, with delay

� Intervals are an equivalent

representation of information available to the decoder

� Expectation over the conditional pdf

� Distortion criterion - mean squared error (MSE)

� The optimal estimate of at the decoder, with delay :

Optimal Delayed Decoder

nx L*

1ˆ [ | , , , , ]n n n n n Lx E x i i i− += ⋯ ⋯

1[ | , , , , ]n n n n LE x I I I− += ⋯ ⋯

1( | , , , , )n n n n Lp x I I I− +⋯ ⋯

[ )( ) ( )n n n n nI x a i x b i= + +ɶ ɶ

[Gibson & Fischer, ‘82]


� By application of Bayes’ rule and Markov property of the process

� is the zero-delay pdf – combines all

information up to time

� weighs the zero-delay pdf to incorporate future information

({ } | )k n k n L np I x< ≤ +

( |{ } )n k k np x I ≤

( |{ } ) ({ } | )( |{ } )

( |{ } ) ({ } | )

n k k n k n k n L nn k k n L

n k k n k n k n L n n

p x I p I xp x I

p x I p I x dx

≤ < ≤ +≤ +

≤ < ≤ +

=

∫

Optimal Delayed Decoder

n

15

Forward recursion

� Recursion for the zero-delay pdf: update from time to

Say, zero-

delay pdf at

time is

known

1n −

n1n −

1nI −

1 1( |{ } )n k k np x I− ≤ −

1nx −

16

Time

n-1

1nI −

1 1( |{ } )n k k np x I− ≤ −

1nx −

n

nx

1( )Z n np x xρ −−

Forward recursion

� Recursion for the zero-delay pdf: update from time to n1n −

17

Time

n-1

1nI −

1 1( |{ } )n k k np x I− ≤ −

1nx −

n

nx

1 1 1( |{ } ) ( )n k k n Z n np x I p x xρ− ≤ − −−

Forward recursion


18

1nI −

1 1( |{ } )n k k np x I− ≤ −

1nx −

nx

1 1 1 1 1( |{ } ) ( |{ } ) ( )n k k n n k k n Z n n np x I p x I p x x dxρ≤ − − ≤ − − −= −∫

Forward recursion


19

1nI −

1 1( |{ } )n k k np x I− ≤ −

1nx −

nxnI

1( |{ } )

0

n k k n n np x I x I

otherwise

≤ − ∈

Forward recursion


20

Time

n-1

1nI −

1 1( |{ } )n k k np x I− ≤ −

1nx −

n

nx

( |{ } )n k k np x I ≤

nI

1

1

( |{ } )

( |{ } )( |{ } )

0

n

n k k nn n

n k k n nn k k n

I

p x Ix I

p x I dxp x I

otherwise

≤ −

≤ −≤

∈

=

∫

Forward recursion


Zero-delay pdf

at time n

21

1 1( | ) ( )

n L

n L n L Z n L n L n L

I

p I x p x x dxρ

+

+ + − + + − += −∫

n LI +

1n Lx + −

n Lx +

Backward recursion

� Recursion for the probability of future outcomes: step back from

time to nn L+

22

1 1( | ) ( )

n L


I

p I x p x x dxρ

+

+ + − + + − += −∫

n LI +

1n Lx + −

n Lx +Time

n+L-1

n+L

n LI +

1n Lx + −

n Lx +

1 1( | ) ( )

n L


I

p I x p x x dxρ

+

+ + − + + − += −∫

Backward recursion


time to nn L+

23

n LI +

1n Lx + −

n Lx +

1 1( | ) ( )

n L


I

p I x p x x dxρ

+

+ + − + + − += −∫

Backward recursion


time to nn L+

24

n LI +

1n Lx + −

n Lx +

1 1( | ) ( )

n L


I

p I x p x x dxρ

+

+ + − + + − += −∫

Backward recursion


time to nn L+

25

n LI +

1n Lx + −

n Lx +

1( | )n L n Lp I x+ + −

Backward recursion


time to nn L+

26

n LI +

1n Lx + −

n Lx +

1n LI + −

1 1 1( | )

0

n L n L n L n Lp I x x I

otherwise

+ + − + − + −∈

Backward recursion


time to nn L+

27

Time

n+L-1

n+L

n LI +

1n Lx + −

n Lx +

1n LI + −

n+L-2

2n Lx + −

1

1 2 1 1 2 1( , | ) ( | ) ( )

n L

n L n L n L n L n L Z n L n L n L

I

p I I x p I x p x x dxρ

+ −

+ + − + − + + − + − + − + −= −∫

Backward recursion


time to nn L+

28

Time

n+L-1

n+L

n LI +

1n Lx + −

n Lx +

1n LI + −

n+L-2

2n Lx + −

1 2( , | )n L n L n Lp I I x+ + − + −

Backward recursion


time to nn L+


n Li +

time

n1n−2n− 1n+ n L+1n L+ −

n Lx +ɶ

n LI +

Summary

� At time n L+


n1n−2n− 1n+ n L+1n L+ −

nI 1nI + 1n LI + − n LI +

1 1( |{ } )n l l np x I− ≤ −

Summary

� At time n L+


n1n−2n− 1n+ n L+1n L+ −

nI 1nI + 1n LI + − n LI +

1 1( |{ } )n l l np x I− ≤ −

( |{ } )n l l np x I ≤

Summary

� At time n L+


n1n−2n− 1n+ n L+1n L+ −

1nI + 1n LI + − n LI +

( |{ } )n l l np x I ≤

({ } | )l n l n L np I x< ≤ +

Summary

� At time n L+


n1n−2n− 1n+ n L+1n L+ −

1nI + 1n LI + − n LI +

( |{ } )n l l np x I ≤

({ } | )l n l n L np I x< ≤ +

( |{ } )n l l n Lp x I ≤ +

Summary

� At time n L+


n1n−2n− 1n+ n L+1n L+ −

1nI + 1n LI + − n LI +

( |{ } )n l l np x I ≤

( |{ } )n l l n Lp x I ≤ +

*ˆnx

Summary

� At time n L+


Special case: matched predictor

� The L-step recursion for future probabilities can be simplified

� There exists function such that,

� A codebook of the functions can be

constructed

� Recursion can be replaced by codebook access with

, and translation of the function by

1ˆ

n nx xρ −=ɶ

1 , ,( )

Li ixΛ

⋯

1 , ,ˆ({ } | ) ( )

n n Lk n k n L n i i n np I x x x

+ +< ≤ += Λ −

⋯

1, ,n n Li i+ +⋯ ˆnx

1 , ,( )

Li ixΛ

⋯


Table look-up via ?2 1, , ,n n ni i i− −⋯

Codebook-based Delayed Decoder

� Henceforth, we exclusively consider the matched predictor

� Optimal delayed estimate:

1ˆ

n nx xρ −=ɶ

*

1ˆ [ | , , , , ]n n n n n Lx E x I I I− += ⋯ ⋯

({ } | )k n k n L np I x< ≤ +( |{ } )n k k np x I ≤

Table look-up via 1, ,n n Li i+ +⋯

1 , ,ˆ( )

n n Li i n nx x

+ +Λ −

⋯


Table look-up via ?2 1, , ,n n ni i i− −⋯




1ˆ

n nx xρ −=ɶ

*

1ˆ [ | , , , , ]n n n n n Lx E x I I I− += ⋯ ⋯


Table look-up via 1, ,n n Li i+ +⋯Growing history of indices precludes an optimal

look-up table for the zero-delay pdf

1 , ,ˆ( )

n n Li i n nx x

+ +Λ −

⋯


A good approximation is still feasible !




1ˆ

n nx xρ −=ɶ

*

1ˆ [ | , , , , ]n n n n n Lx E x I I I− += ⋯ ⋯



1 , ,ˆ( )

n n Li i n nx x

+ +Λ −

⋯


A good approximation is still feasible !




1ˆ

n nx xρ −=ɶ

*

1ˆ [ | , , , , ]n n n n n Lx E x I I I− += ⋯ ⋯



A codebook-based approximation for the optimal delayed estimate

1 , ,ˆ( )

n n Li i n nx x

+ +Λ −

⋯


� Approximation for the zero-delay pdf:

� Let denote the stationary marginal prediction error pdf - a fixed (time invariant) pdf [Farvardin & Modestino, ’85]

� The pdf of conditioned on past indices is approximated as:

� Thus the zero-delay pdf is just:


1ˆ

n n ne x xρ −= −∵

( )Ep e

2 1 1 1ˆ ˆ( | , , ) ( | ) ( )n n n n n E n np x i i p x x p x xρ ρ− − − −≈ = −⋯

nx

1

1

ˆ( )

ˆ( )( |{ } )

0

n

E n nn n

E n n nn k k n

I

p x xx I

p x x dxp x I

otherwise

ρ

ρ

−

−≤

− ∈ −

≈

∫


� Approximate delayed estimate:


*

1ˆ [ | , , ,

({ } | )

({ } |

( |{ } )

( |{ }]

),

)

k nn n

n n n n n

n k k n

n k k n

k n

L

n

L n

k n k n L n

p I xx dxx E x I I I

p x I

xx I p dp I x−

< ≤ +

< ≤

≤

+

+≤

= =∫∫

⋯ ⋯

1 , ,ˆ( )

n n Li i n nx x

+ +Λ −

⋯

1

1

ˆ( )

ˆ( )

0

n

E n nn n

E n n n

I

p x xx I

p x x dx

otherwise

ρ

ρ

−

−

− ∈ −

∫

[ )1 1ˆ ˆ( ) ( )n n n n nI x a i x b iρ ρ− −= + +1

ˆ ˆ ˆ ( )n n n nx x e iρ −= +

*

1ˆ ˆ ( , , )n n n n Lx x c i iρ − +≈ + ⋯ Look-up table/codebook


� Numerical evaluation via and

� Alternative - a training-set based design:

� => is the estimate of

the prediction error at time given the window of indices

� Encoder is fixed: run it on a long enough training set of the

source, and obtain prediction error training set and indices

� Train delayed decoding codebook

Codebook design

( )Ep e

*

1ˆ ˆ ( , , )n n n n Lx x c i iρ − +≈ + ⋯ ( , , )n n Lc i i +⋯

n, ,n n Li i +⋯

1 , ,ˆ( )

n n Li i n nx x

+ +Λ −

⋯


Results

� Source is first order AR

� DPCM Encoder:

� Rate: first order entropy of output indices

� Employs uniform threshold quantizer: scaled suitably to achieve different rates

� Thresholds fixed by scale-factor, reconstructions optimized iteratively similar to [Farvardin & Modestino, ’85]

� Iterative optimization also provides for codebook approach

� Predictor matched to source

( )Ep e


0.25 0.5 0.75 1 1.25 1.5 1.75 2-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

1.5

Rate (bits/sample)

SN

R g

ain

ov

er

reg

ula

r D

PC

M (

dB

)

IDPCM

SDPCMProposed Optimal Decoder

Proposed Codebook Decoder

L=3

L=1

L=3

L=1

Results

Performance comparison of competing delayed decoders for

a Gaussian source with 0.95ρ =

Performance of

zero-delay DPCM

at different bit-rates


0.25 0.5 0.75 1 1.25 1.5 1.75 2-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

1.5

Rate (bits/sample)

SN

R g

ain

ov

er

reg

ula

r D

PC

M (

dB

)

IDPCM



L=3

L=1

L=3

L=1

Results



SDPCM with lag of 3

samples, worse at

lower delays


0.25 0.5 0.75 1 1.25 1.5 1.75 2-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

1.5

Rate (bits/sample)

SN

R g

ain

ov

er

reg

ula

r D

PC

M (

dB

)

IDPCM



L=3

L=1

L=3

L=1

Results



IDPCM, delay

limited to 1 sample

automatically


0.25 0.5 0.75 1 1.25 1.5 1.75 2-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

1.5

Rate (bits/sample)

SN

R g

ain

ov

er

reg

ula

r D

PC

M (

dB

)

IDPCM



L=3

L=1

L=3

L=1

Results



Codebook-

based approach

using 1 and 3

future indices

Performance curves

for the optimal

delayed decoder

hidden beneath plots

for the codebook

approach


0.25 0.5 0.75 1 1.25 1.5 1.75 2-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

1.5

Rate (bits/sample)

SN

R g

ain

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er

reg

ula

r D

PC

M (

dB

)

IDPCM



L=3

L=1

L=3

L=1

Zoom in to see the

performance gap

between optimal

and codebook

approaches

Results




0.35 0.45 0.55 0.65 0.75 0.851.35

1.4

1.45

1.5

Rate (bits/sample)

SN

R g

ain

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er

reg

ula

r D

PC

M (

dB

)

Proposed Optimal Decoder


Results




0.25 0.5 0.75 1 1.25 1.5 1.75 2-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

1.5

Rate (bits/sample)

SN

R g

ain

ov

er

reg

ula

r D

PC

M (

dB

)

IDPCM



L=3

L=1

L=3

L=1

� Performance of SDPCM and IDPCM not guaranteed to be better than zero-delay DPCM

� Proposed approaches at 1 sample delay outperform SDPCM at higher delay (3) : indices contain a lot of information

� At low bit-rates increasing delay provides more gains

Results




Results

0.8ρ =

0.25 0.5 0.75 1 1.25 1.5 1.75 2-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

1.5

Rate (bits/sample)

SN

R g

ain

ov

er

reg

ula

r D

PC

M (

dB

)

IDPCM

SDPCM

Proposed Optimal DecoderL=3

L=1L=3

L=1

Performance comparison of competing delayed decoders

for a Gaussian source with

� Lower correlation naturally implies lesser to be gained from looking into the future


Results


a source with Laplacian innovations with 0.95ρ =

0.25 0.5 0.75 1 1.25 1.5 1.75 2-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

1.5

Rate (bits/sample)

SN

R g

ain

ov

er

reg

ula

r D

PC

M (

dB

)

IDPCM

SDPCM

Proposed Optimal Decoder

L=3

L=1

L=1

L=3


Other contributions

� Codebook approach trades computational complexity for memory

� Proposed an approach for codebook-size reduction via an index mapping technique with very minimal performance loss

� Optimal and codebook approaches readily extended to higher ordersources (equivalence via an appropriate first-order vector AR process)

� Index window employed in the codebook can be extended to includea few past indices: useful in the case of higher order sources

� Training-set based design, and codebook-based operation, particularly attractive for higher order sources (due to to the higher dimensionality involved)


Summary

� Proposed an estimation-theoretic approach for optimal delayed decoding in predictive coding systems

� Combines all known information at the decoder in a recursively calculated conditional pdf

� Motivates a codebook-based delayed decoder that is nearly optimal even for modest dimensions

� Substantial performance gains compared to prior smoothing/filtering techniques


Future directions

� Encoder optimization based on the proposed delayed decoder

� Employ delayed reconstructions for prediction via local

decoder

� Delayed decoding in adaptive predictive coding scenarios

� Application for speech/audio coding in Bluetooth systems

� Delayed decoding codebook adaptation techniques