ADAPTATIVE SIGNAL SAMPLING AND SAMPLE QUANTIZATION

FOR RESOURCE-CONSTRAINED STREAM PROCESSING

Deepak S. Turaga, Olivier Verscheure, Daby M. Sow and Lisa Amini

IBM T.J. Watson Research Center, 19 Skyline Drive, Hawthorne, NY, USA

Keywords:

Remote Health Monitoring, ECG Compression, Low-complexity, Non-Uniform Sampling, Quantization.

Abstract:

We propose a low-complexity encoding strategy for efﬁcient compression of biomedical signals. At the heart

of our approach is the combination of non-uniform signal sampling together with sample quantization to

improve the source coding efﬁciency. We propose to jointly extract and quantize information (data samples)

most relevant to the application processing the incoming data in the backend unit. The proposed joint sampling

and quantization method maximizes a user-deﬁned utility metric under system resource constraints such as

maximum transmission rate or encoding computational complexity. We illustrate this optimization problem

on electrocardiogram (ECG) signals, using the Percentage Root-mean-square Difference (PRD) metric as the

utility function measuring the distortion between the original signal and its reconstructed (inverse quantization

and linear interpolation) version. Experiments conducted on the MIT-BIH ECG corpus using the well-accepted

FAN algorithm as the non-uniform sampling method show the effectiveness of our joint strategy: Same PRD

as ’FAN alone’ at half the data rate for less than three times the (low) computational complexity of FAN alone.

1 INTRODUCTION

Remote Health Monitoring is an emerging technol-

ogy allowing medical practitioners to extend their ser-

vices to patients outside of traditional hospital set-

tings. Common remote health monitoring systems are

leveraging pervasive devices such as cellular phones

to collect biomedical readings on patients and relay

the data to servers while being non-intrusive and not

restricting the mobility of patients (Mohomed et al.,

2006). This usage of pervasive devices differ sig-

niﬁcantly from traditional client server usage mod-

els where the pervasive device acts as a client receiv-

ing data from a more powerful server. In the cur-

rent model, the roles are reversed. Pervasive devices

are used to stream data to back-end servers. Their

resource scarceness creates interesting research chal-

lenges dictating the need for efﬁcient, low complex-

ity signal encoding schemes. This work proposes a

generic method for streaming continuous signals un-

der very strict resource constraints while minimizing

the loss in valuable information the original signals

carry.

While our method is applicable to a wide vari-

ety of signals, we describe it in the context of efﬁ-

cient, low complexity compression of electrocardio-

gram (ECG) signals. An ECG signal provides es-

sential information to the cardiologist and is used for

both monitoring and diagnostic purposes. An ECG

monitoring device essentially measures the electrical

impulses that stimulate the heart to contract. Be-

tween 125 and 500 sample points are collected every

second, each coded on 8 or 12 bits (Nygaard et al.,

2001). Thus, a single-lead uncompressed ECG sig-

nal requires between 1 kbps and 6 kbps of sustained

wireless bandwidth. Any application based on wire-

less transmission of even moderate amounts of data

must deal with the reality that usage of wireless spec-

trum will always incur some monetary cost. Efﬁcient,

low complexity compression is thus crucial to make

remote health monitoring via low-end pervasive de-

vices a reality.

The main goal of any compression technique is

to achieve maximum data volume reduction while

preserving the signiﬁcant signal morphology fea-

tures upon reconstruction (Jalaleddine et al., 1990).

In ECG signal compression algorithms the goal is

to achieve a minimum information rate, while re-

taining the relevant diagnostic information in the

reconstructed signal. Compression techniques for

ECG waveforms can be broadly classiﬁed into two

main groups: direct time-domain techniques (Barr,

1988; Cox et al., 1968), and transform-domain tech-

niques (Bradie, 1996; Hilton, 1997; Addison, 2005).

Transform-based methods (e.g., wavelet-based) usu-

ally outperform time-domain techniques but require

S. Turaga D., Verscheure O., M. Sow D. and Amini L. (2008).

ADAPTATIVE SIGNAL SAMPLING AND SAMPLE QUANTIZATION FOR RESOURCE-CONSTRAINED STREAM PROCESSING.

In Proceedings of the First International Conference on Bio-inspired Systems and Signal Processing, pages 96-103

DOI: 10.5220/0001067400960103

 SciTePress

a computational power beyond what a mainstream

pervasive device can handle. Instead, well-accepted

time-domain techniques, such as FAN (Barr, 1988)

and AZTEC (Cox et al., 1968), rely on simple heuris-

tics so as to non-uniformly sample the original wave-

form and retain only those data samples that con-

tribute the most to the quality of the reconstructed (in-

terpolated) signal.

Another well-known compression strategy is

quantization. There are two types of quantization.

Vector quantization, where the input symbols are

gathered together in groups called vectors and pro-

cessed to give the output, and scalar quantization,

where each input symbol is treated separately in pro-

ducing the output. Scalar quantization has a low com-

putational complexity, is easy to implement and can

achieve reasonably good compression performance if

applied properly. There has been recent interests in

the scientiﬁc community to design schemes perform-

ing jointly both quantization and uniform sampling in

order to match the underlying system resource con-

straints (Derpich et al., 2006). Uniform sampling in-

volves discarding samples of the data regularly to re-

duce the data rate. While uniform sampling can re-

duce the stream rate appropriately it does not guaran-

tee the retention of all samples of interest (features),

especially when the frequency characteristics of the

signal are not well-behaved, which is clearly the case

for ECG waveforms.

This work investigates the beneﬁt of jointly

performing non-uniform sampling (e.g., FAN or

AZTEC) and quantization operations in the context

of remote health monitoring. The paper is organized

as follows: Section 2 introduces some notations and

describes, in generic terms, the concept of joint non-

uniform sampling and quantization. This concept ap-

plied to signal compression is the subject of Section 3,

while Section 4 formulates the problem speciﬁcally

for ECG compression under resource constraints us-

ing FAN (Barr, 1988) as the non-uniform sampling

technique. The problem is posed as an optimization

problem. The optimization problem is solved in Sec-

tion 5. Finally, our strategy is validated in Section 6.

And, Section 7 gives concluding remarks.

2 SIGNAL COMPRESSION: NON

UNIFORM SAMPLING AND

QUANTIZATION

Let x[k], 0 ≤ k < N denote a discrete time signal

represented with b

bits per sample.

2.1 Non Uniform Sampling

Non uniform sampling of x[k] extracts N

SOI

≤ N sam-

ples of interests (SOI) from x[k]. We denote such

sampling by the operator S : x[k]→x[k

] where k

cor-

responds to the location of the retained samples of

interest. The operator S is often lossy, and only an

approximation to the original signal x

[k] may be re-

covered by interpolating x[k

] appropriately. If, af-

ter sampling, we retain N

SOI

out of N samples, the

achieved compression ratio is is

SOI

, correspond-

ing to a rate

SOI

bits per sample. Additionally, in

the compressed rate we also need to include the bits

required to encode the locations of the retained sam-

ples, i.e. an additional b

loc

bits per sample. The se-

lectivity of the sampling operator S is controlled by a

sampling sensitivity parameter ε, with low values of

ε corresponding to low selectivity, i.e. most samples

from x[k] are retained. To explicitly indicate the de-

pendence of S on ε, we present it as S

2.2 Quantization

Quantization is another well known lossy technique

used to reduce the signal rate, when applications can

tolerate the resultant distortion. We denote the quan-

tization operator as Q : x[k]→ ˆx[k] where ˆx[k] uses

< b

bits per sample, thereby reducing the average

data rate of the stream by a factor

Given a periodic signal such as ECG, with rela-

tively stationary probability density function (under

known context, i.e. physical activity, health state

etc.) the quantizer sensitivity is controlled only by

the number of desired reconstruction levels

L = 2

As before, to explicitly indicate the dependence of Q

on L, we represent it as Q

2.3 Joint Non-uniform Sampling and

Quantization

Quantization, when used in conjunction with non-

uniform sampling can further reduces the rate of the

stream. When quantization is applied prior to sam-

pling we have the resultant signal S

(x[k])) and

when the signal is sub-sampled before quantization,

the resultant signal is Q

(x[k])). Note that these

operators are not commutative, and the two cases are

likely to achieve different compression factors. The

compression gain is multiplicative, i.e. the corre-

sponding rate of signal Q

(x[k])) is

SOI

+ b

loc

bits per sample.

The optimal values of these reconstruction levels are

known for a standard MSE quantizer

ADAPTATIVE SIGNAL SAMPLING AND SAMPLE QUANTIZATION FOR RESOURCE-CONSTRAINED STREAM

PROCESSING

3 DESIGN OF JOINT

NON-UNIFORM SAMPLING

AND QUANTIZATION BASED

COMPRESSION

We can exploit the multiplicative gain in compres-

sion achieved by joint sampling and quantization to

design better signal compression schemes. However,

different types of signals and applications can toler-

ate different levels of quantization noise and require

different numbers of samples of interest. Hence the

joint design of quantization and non-uniform sam-

pling needs to be performed carefully. Consider the

two different operator options S

) and Q

and let the corresponding rates be

)

SOI

)

loc

and

)

SOI

)

+ b

)

loc

. In order to de-

sign a good compression scheme, we also need

to formally deﬁne a distortion metric. Let x

[k]

represent the reconstructed signal, after decompres-

sion, i.e. x

[k] = S

−1

(x[k])))) or x

[k] =

−1

(x[k])))). Then the utility associated

with the compression may be deﬁned in terms of x[k]

and x

[k] as U (x[k], x

[k]). The goal of designing the

right compression scheme is to maximize this utility

under a rate constraint. If the desired rate constraint

is b

con

(in bits per sample), the optimal compression

scheme may be designed by solving the following

constrained optimizations:

opt

, S

opt

} = argmax

}

[U (x[k], x

[k])]

subject to

Q (S

)

SOI

)

+ b

)

loc

≤b

con

(1)

and

opt

, Q

opt

} = argmax

}

[U (x[k], x

[k])]

subject to

)

SOI

)

+ b

)

loc

≤b

con

(2)

As mentioned earlier, designing the quantizer Q

re-

quires determining the number of quantization lev-

els L and designing the non-uniform sampling S

epsilon

strategy requires determining the optimal value for ε

for a givennon uniform sampling scheme. We thus re-

duce the problem of ﬁnding Q

opt

and S

opt

to the iden-

tiﬁcation of the values of L and ε that maximizes the

utility. Consequently, since b

epsilon

)

= ⌈log

L⌉,1

and 2 can be rewritten as:

{ε

opt

, L

opt

} = argmax

{L,ε}

[U (x[k], x

[k])]

subject to

Q (S

)

SOI

(⌈log

L⌉)

+ b

)

loc

≤b

con

(3)

and

{ε

opt

, L

opt

} = argmax

{ε,L}

[U (x[k], x

[k])]

subject to

)

SOI

(⌈log

L⌉)

+ b

)

loc

≤r

con

(4)

If the order of the quantization and non-uniform

sampling also needs to be determined, we may com-

pare the optimal utilities in the two cases to deter-

mine the best order. Solving the joint optimization

presented in equations 4 and 3 is non-trivial. This

optimization is heavily dependent on the relationships

between U and N

SOI

and the pair (ε, L). For a generic

sampling algorithm, for a signal with arbitrary char-

acteristics, it is likely to be very difﬁcult to determine

the optimal solution without some form of computa-

tionally complex exhaustive search. In some cases,

however, for sampling algorithms such as FAN, and

for well-behaved signals such as ECG, we show that

these relationships can be estimated experimentally,

and modeled using simple parametric functions. This

enables tractable, and low complexity algorithms to

solve the optimization in real time. In the following

sections, we present several parametric model based

approaches to trade-off computational complexity for

accuracy, while solving this optimization for the FAN

algorithm with MSE quantization for the ECG signal.

4 ENCODING ECG SIGNALS FOR

REMOTE HEALTH

MONITORING

We illustrate our approach to jointly quantize and

sample non uniformly waveforms by focusing on the

representation of electrocardiogram (ECG) signals.

This proposed technique implements adaptive sam-

pling before quantization (i.e. S before Q ).

4.1 Brief Background on ECG Signals

A typical electrocardiogram monitoring device gen-

erates large volumes of digital data. Depending on

the intended application, the sampling rate may range

from 125 to 1000 Hz, with each data sample digitized

to a 8-16 bit value. This translates to a minimum data

rate of 15 KB per minute. Transmitting this signal

over a low-bandwidth channel, especially when ag-

gregating data from multiple sensors, requires com-

pression. The data needs to also be recorded over long

periods, often as much as 24 hours, and doctors may

wish to build a database of ECG recordings for their

patients. Minimizing the storage resources also re-

quires data compression.

BIOSIGNALS 2008 - International Conference on Bio-inspired Systems and Signal Processing

4.2 Adaptive Sampling

FAN (Barr, 1988) is a standard sampling technique

for ECG signal compression and was reported in 1964

by (Gardenhire, 1964). It extracts samples of inter-

est by approximating the signal using a piecewise lin-

ear representation, and discards all but the terminal

points along these line segments. More precisely,

the FAN algorithm replaces the signal with straight

line segments such that none of the original points

lies further from the line segment than some prede-

termined maximum deviation threshold τ. Figure 1

visually describes the algorithm. The ﬁrst point x[k

]

is accepted as non-redundant (permanent sample).

Two slopes {L

} are drawn between x[k

] and

{x[k

] − τ, x[k

] + τ}. The third sample point x[k

)]

falls within the area bounded by the two slopes. Thus

new slopes {L

} are calculated between x[k

] and

x[k

] ± τ respectively. Then the two pairs of slopes

are compared and the most restrictive are retained:

= min(U

) and L

= max(L

, L

). Since sam-

ple x[k

] lies inside the range it is thus discarded;

while x[k

] is accepted as a permanent sample and the

procedure above is repeated, comparing future sam-

ple values to the most restrictive lines. During signal

reconstruction, the discarded samples are linearly in-

terpolated from their neighboring retained samples.

Discard

Keep

Discard

Figure 1: FAN algorithm for non-uniform sampling.

The deviation threshold τ determines the quality

of the approximation with large τ leading to more

samples being discarded, and coarser signal approx-

imation. In our setting, this threshold τ maps directly

to the sampling sensitivity ε, and we use the two inter-

changeably. The FAN algorithm has been used widely

for ECG signal compression as it is extremely com-

putationally lightweight (O(N) for N samples), and

performs reasonably well in practice, in terms of re-

taining samples and features of interest. However, for

small target bit-rates (under 2 bits per sample), the

FAN algorithm often underperforms computationally

more complex (O(N

)) algorithms such as Cardinal-

ity Constrained Shortest Path (CCSP). In this bit-rate

range, we wish to improve the performance of FAN

by combining it with quantization. Combination with

a simple quantization can retain the low-complexity

nature of FAN, while improving its compression qual-

ity.

4.3 Joint FAN Sampling and

Quantization

The reconstruction quality of compressed ECG sig-

nals is often captured using the percentage root-mean-

square difference (PRD) between the original signal

and its reconstructed (inverse quantization and linear

interpolation) version. The reconstructed signal x

[k]

is determined from the sampled and quantized signal

by inverse quantization and linear interpolation.

Hence, the utility function is deﬁned as:

U (x[k], x

[k]) = −100∗

∑

j=1

(x[ j] − x

[ j])

∑

j=1

x[ j]

(5)

Finally the joint sampling and quantization prob-

lem, given a rate constraint b

con

(in bits per sample),

may be written as the following optimization:

{ε

opt

, L

opt

} = argmax

{ε,L}

[U (x[k], x

[k])]

subject to

)

SOI

)

+ b

)

loc

≤b

con

(6)

The search complexity for a naive implementa-

tion of the solution to this problem is O(|Ω

| × |Ω

where Ω

is the set of possible values for ε, Ω

the set of possible values for L and | • | is the car-

dinality operator. This is a constant factor that mul-

tiplies the complexity of the FAN algorithm (thereby

linearly increasing the complexity). However, this is a

worst case metric as it assumes no apriori knowledge

of the underlying ECG signal. Due to the periodic na-

ture of the ECG signal, the designed answer is likely

to change slowly with time (across consecutive win-

dows of N samples each), and hence we can distribute

this complexity over several windows. This may be

done by either solving the optimization once every Z

windows, thereby reducing the overhead complexity

to O(

|Ω

|×|Ω

) or by reducing the space of possible

search values, i.e. the number of elements in each set

(allowing only for small variations in the previously

designed values).

Additional improvement in performance may be

obtained by actually designing the complete quantizer

(including the design of the optimal resonstruction

levels) dynamically. This however comes at a cost of

increased complexity. In the worst case, a standard k-

means based implementation of quantizer design has

complexity O(N

). Of course, this cost may also be

distributed across several windows (due to the nature

ADAPTATIVE SIGNAL SAMPLING AND SAMPLE QUANTIZATION FOR RESOURCE-CONSTRAINED STREAM

PROCESSING

of the ECG signal) to reduce the computational com-

plexity. The design of optimal low-cost quantizers in

conjunction with the sampling is an interesting direc-

tion of future research.

5 MODEL BASED SEARCH

STRATEGY

A model based search strategy is enabled by the rea-

sonably stationary characteristics of the ECG signal,

and the somewhat predictable behavior of the FAN

algorithm. Speciﬁcally, we observe, that in a partic-

ular operating region (deﬁned by the rate constraint,

e.g. number of bits per sample, and the correspond-

ing quality metric, i.e. PRD) we may develop simple

parametric models that capture the effect of L and ε

on the utility (PRD) and the rate (bits per sample). As

an example, we run the FAN algorithm several times

on a real ECG signal with different values of ε and

plot the resulting number N

SOI

of samples retained,

and the corresponding distortion PRD in Figure 2.

0 0.05 0.1 0.15 0.2

200

400

600

800

1000

1200

SOI

0 0.05 0.1 0.15 0.2

PRD

Figure 2: N

SOI

and PRD as functions of ε.

As is clear, N

SOI

has an almost exponentially de-

caying relationship with ε, while the PRD has a near-

linear relationship with ε, and we can capture these

relationships very simply as follows

SOI

(ε) = µe

−νε

(7)

and

PRD(ε) = α+ βε (8)

where µ, ν, α, and β are the model parameters. If we

now combine this sampling with quantization using

L levels, we can derive the resulting bit-rate for the

compressed signal (in bits per sample) as

)

SOI

log

(L)

(9)

Using Equation 7, we may rewrite this as

)

µe

−νε

log

(L)

. (10)

In order to build a similar model for the PRD as a joint

function of ε and L, we plot the resulting PRD (after

FAN followed by quantization) in Figure 3. Clearly,

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

PRD

L=4

L=8

L=16

L=32

L=64

L=128

Figure 3: PRD as function of ε and L.

the slope and intercept of the line relating PRD to ε

change with L. After further investigation, we ﬁnd

that this relationship may be captured as

α(L) = γe

−ρL

(11)

and

β(L) = η×log(ξL

). (12)

Combining these equations, we may rewrite the

model for PRD after joint sampling followed by quan-

tization as

PRD = γe

−ρL

+ ηεlog(ξL

) (13)

These models for PRD and b

)

are validated for

a real ECG signal in Figure 4 and Figure 5. While

the models tend to underestimate the real values (es-

pecially for small ε), the shapes of the curves re-

main similar allowing for a search strategy using this

model.

5.1 Three Times FAN Strategy

In order to compress the ECG signal under a rate

constraint, we ﬁrst partition it into ﬁxed size windows

(each with W samples

). Then per window, we run

Note that since we process the window independently,

we have N = W

BIOSIGNALS 2008 - International Conference on Bio-inspired Systems and Signal Processing

100

0.02

0.04

0.06

0.08

PRD

Real

Model

Figure 4: PRD: Real value versus model prediction.

0.02

0.04

0.06

0.08

)

Real

Model

Figure 5: b

: Real value versus model prediction.

the FAN algorithm for two different values of ε (a

high value and a low value), to determine two values

for N

SOI

followed by quantization with two different

numbers of levels L (a high value and a low value)

to determine four values of PRD. The four values

of PRD provide us with four equations to solve for

parameters γ, ρ, η and ξ. Similarly, the two values

of N

SOI

provide us with two equations to solve for

parameters µ and ν. Once we determine the model

for a given window, it is straightforward to determine

the optimal parameter settings for ε and L under any

speciﬁed rate constraint. Once we determine the

optimal parameter settings, we then need to run FAN

once with the selected ε

opt

followed by quantization

with L

opt

levels. Hence, per window we run the FAN

algorithm three times.

5.2 Two Times FAN Strategy

We exploit the near stationarity of the ECG signal

characteristics to reduce the complexity of the Three

Times FAN strategy. Speciﬁcally, while for the ﬁrst

window we employ the same approach (with two

times FAN followed by two times quantization) for

every subsequent window we run the FAN algorithm

for only one additional value of ε followed by quan-

tization with two different values of L. This provides

us with two values of PRD and one value of N

SOI

In order to compute the model parameters, we then

combine this with two values of PRD and one value

of N

SOI

computed from the previous window. We al-

ternate between recomputing the PRD and N

SOI

for

the high ε, and the PRD and N

SOI

for the low ε (corre-

spondingly reusing these for the low ε and high ε, re-

spectively, from the previous window), for every suc-

cessive window. Note that it is possible to easily ex-

tend this approach to recompute the model parameters

only once every Z windows, to further reduce com-

plexity. We examine some of the tradeoffs between

complexity and accuracy by comparing the perfor-

mance of these algorithms, and using that to identify

trends for other extensions.

6 EXPERIMENTAL RESULTS

We evaluate the performance of these algorithms on

ECG signals from the MIT-BIH database. Specif-

ically, we use a subset of this database, consisting

of 10 different ECG signals, of duration 8000 sam-

ples each. For these signals we evaluate the different

strategies described in Table 1. We compare FANEX,

FANQEX, FANQMS-3 and FANQMS-2 in terms of

their distortion (PRD)-rate (bits per sample) curves,

and also in terms of their computational complex-

ity. We present results for different processing win-

dow sizes (W) to identify the general performance

trend variations. Each window consists of W sam-

ples of the signal, and is analyzed and processed in-

dependently by the different algorithms, speciﬁcally

in terms of computing the optimal parameters ε and

L, and using the FAN algorithm with these param-

eters. We limit the search space for the exhaustive

search strategies by considering a ﬁnite small set of

possible values that ε and L can take. For our experi-

ments we have ε∈{0.002, 0.004, ··· , 0.04, 0.05, 0.06}

and L∈{4, 8, 16, 32, 64}. We ﬁrst consider a process-

ing window of size W = 1000 samples, and present

the distortion-rate (D-R) curve averaged across these

signals (across the 8 windows per signal) for the four

different algorithms in Figure 6.

In Figure 6 we observe that the schemes with joint

ADAPTATIVE SIGNAL SAMPLING AND SAMPLE QUANTIZATION FOR RESOURCE-CONSTRAINED STREAM

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101

Table 1: Algorithms Considered.

Name Uses Quantization ε only or (ε, L) search strategy

FANEX No Exhaustive

FANQEX Yes Exhaustive

FANQMS-3 Yes Model Based - Three Times FAN

FANQMS-2 Yes Model Based - Two Times FAN

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Bits per sample

Resultant PRD

W=1000 Samples − 8 Windows

FANQMS−3

FANQMS−2

FANQEX

FANEX

Figure 6: D-R Curves: W = 1000.

quantization and FAN signiﬁcantly outperform the

FAN only scheme, a compression factor of 2 for the

same PRD. This makes the performance of the FAN

algorithm comparable to the state-of-the art compres-

sion algorithms (with signiﬁcantly higher complex-

ity). Furthermore, we ﬁnd that the model based

searches FANQMS-3 and FANQMS-2 have perfor-

mance very close to that achieved by the exhaustive

search for target bit-rates less than 2 bits per sample.

As the target bit-rate starts to approach 2 bits per sam-

ple, the model based search strategies underperform

the FANEX strategy, as the models are inaccurate

for

this range. However, note that for this higher bit-rate

range, the performance of the FAN algorithm by it-

self is comparable to the best ECG compression algo-

rithms presented, thereby limiting any gains obtained

by additionally quantizing the signal. We also repeat

these experiments for a smaller window (W = 500)

and a larger window (W = 2000, and the results are

presented in Figure 7.

From Figure 7, the same performance trend is ob-

served as in Figure 6 for the four algorithms, how-

ever it is clear that the performance of FANQEX,

FANQMS-2, and FANQMS-3 are closer to each other

for largerW. This may be explained by the fact that a

Assumptions on linear, exponential and log-linear rela-

tionships are violated in this range.

0 0.5 1 1.5 2

Bits per sample

Resultant PRD

W=500 Samples − 16 Windows

FANQMS−3

FANQMS−2

FANQEX

FANEX

0 0.5 1 1.5 2

Bits per sample

Resultant PRD

W=2000 Samples − 4 Windows

FANQMS−3

FANQMS−2

FANQEX

FANEX

Figure 7: D-R Curves: W = 500(left), W = 2000(right) .

larger window size allows the model based algorithms

to ﬁt better parameterized curves, improving the per-

formance of the model based search schemes. This is

also evident from the fact that on average, the PRD

for the same target bit rate decreases with increasing

W. We also compare the computational complexity of

these algorithms in terms of the amount of CPU time

consumed per window. These CPU times are labeled

FANEX

, t

FANQEX

, t

FANQMS−2

and t

FANQMS−3

respec-

tively. We also label the time taken to run the FAN al-

gorithm on one window as t. Instead of presenting ab-

solute numbers, we present relative ratios of the com-

plexity of these algorithms to hide the dependency on

the underlying computer architecture, operating sys-

tem etc. These complexity ratios for the different al-

gorithms are presented in Table 2.

It is evident from Table 2 that FANQMS-3 has 29

(FANQMS-2 has 45 times) lower complexity than

FANQEX and 4 times (FANQMS-2 has 7 times)

lower complexity than FANEX. Further, as expected,

FANQMS-2 has lower complexity than FANQMS-3.

This observation holds across the two different win-

dow sizes considered. Furthermore, FANQMS-3 has

4 times the complexity of FAN, while FANQMS-2

has 3 times the complexity of running FAN one time.

This implies that the search for the optimal ε and L

has the complexity 1.5 times that of the FAN algo-

rithm. Note that by reusing the model parameters

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Table 2: Complexity Comparison.

FANQEX

FANQMS−3

FANEX

FANQMS−3

FANQEX

FANQMS−2

FANEX

FANQMS−2

500 29.18 4.50 4.18 47.75 7.35 2.80

1000 29.06 4.49 4.21 46.07 7.12 2.64

across more windows (updating model infrequently),

this overhead can also be signiﬁcantly reduced. This

is also indicated by the comparing the rowsof Table 2,

as the complexity gains for FANQMS-2 increase as

W increases from 500 to 1000 (the ratio

FANQMS−2

de-

creases from 2.80 to 2.64).

7 CONCLUSIONS

We present a low-complexity joint non-uniform sam-

pling and quantization based strategy for signal com-

pression. Speciﬁcally, we combine the FAN algo-

rithm with a minimum mean-squared error quantiza-

tion strategy to compress ECG signals. We ﬁrst for-

mulate the joint design of non-uniform sampling and

quantization for compression, as a constrained opti-

mization problem in terms of maximizing the relevant

distortion metric given the desired compression rate.

The solution of this optimization yields the optimal

sampling sensitivity, and the number of levels to be

used by the quantizer. In general, and for arbitrary

signals, it may not be possible to solve this optimiza-

tion efﬁciently. However, for ECG signals, we show

that we can develop simple parametric models to cap-

ture the impact of the FAN algorithm and quantiza-

tion on the resulting distortion (PRD) and rate, es-

pecially in very low bit-rate operating regions. Us-

ing these models we can efﬁciently determine the op-

timal FAN selectivity parameter ε and quantization

levels L to minimize the PRD for a given rate con-

straint. We design two model based algorithms, one

that re-estimates model parameters for every window

(W samples), and another that updates model parame-

ters every alternate window. We show that with these

strategies, we can achieve up to 2 times the compres-

sion rate of FAN (for the same PRD) with a com-

plexity less than 3 times that of FAN alone. We also

show that the performance of these algorithms ap-

proaches (within 10% in rate when ε < 1.8) an ex-

haustive search based strategy for different signals,

and window sizes. Given the low complexity of FAN

our algorithms still remain signiﬁcantly lower com-

plexity than state-of-the-art transform based compres-

sion schemes, while achieving comparable perfor-

mance. Directions for future research include design

of the optimal search strategy to re-estimate model pa-

rameters (how often, optimal window size etc.), the-

oretical analysis of the signal frequency and statis-

tical properties as well as algorithm complexity for

rate-distortion-complexityoptimal joint sampling and

quantization, and application of these ideas for other

multi-dimensional medical signals.

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