Automatic Image Segmentation by Dynamic Region Merging Abstract: This paper addresses the automatic image segmentation problem in a region merging style. With an initially over-segmented image, in which the many regions (or super-pixels) with homogeneous color are detected, image segmentation is performed by iteratively merging the regions according to a statistical test.There are two essential issues in a region merging algorithm: order of merging and the stopping criterion. In the proposed algorithm, these two issues are solved by a novel predicate, which is defined by the sequential probability ratio test (SPRT) and the maximum likelihood criterion. Starting from an over-segmented image, neighboring regions are progressively merged if there is an evidence for merging according to this predicate.We show that the merging order follows the principle of dynamic programming. This formulates image segmentation as an inference problem, where the final segmentation is established based on the observed image. We also prove that the produced segmentation satisfies certain global properties. In addition, a faster algorithm is developed to accelerate the region merging process, which maintains a nearest neighbor graph in each iteration. Experiments on real natural images are conducted to demonstrate the performance of the proposed dynamic region merging algorithm. Image segmentation is a fundamental yet still challenging problem in computer vision and image processing.. In particular, it is an essential process for many applications such as object recognition, target tracking, content-based image retrieval and medical image processing, etc. Generally speaking, the goal of image segmentation is to partition an image into a certain number of pieces which have coherent features (color, texture, etc.) and in the meanwhile to group the meaningful pieces together for the convenience of perceiving [1] . In many practical applications, as a large number of images are needed to be handled, human interactions involved in the segmentation process should be as less as possible. This makes automatic image segmentation techniques more appealing. Moreover, the success of many high-level segmentation techniques (e.g. class-based object segmentation [2, 3] ) also demands sophisticated automatic segmentation techniques.. Dating back over decades, there is a large amount of literature on automatic image segmentation. For example, some classical algorithms are based on the abrupt changes in image intensity or color, thus salient edges can be detected. Well-known edge detection algorithms include the Robert edge detector, the Sobel edge detector, the Laplacian operator [4] and the Canny edge detector [5] [6] [7] . However, due to the resulting edges are often discontinuous or over-detected, they can only provide candidates for the object boundaries.. As the image segmentation problems can be easily translated into graph-related problems, some algorithms were proposed based on the criterion of minimizing the cut on a graph. The most widely used cut criteria include normalized cut [8] , ratio cut [9] and minimum cut [10] , in which the cost functions are minimized by different optimization strategies. Modeling the problem as an energy minimization process, these methods aim to produce a desirable segmentation by achieving global optimization. However, these methods only provide a characterization of each cut rather than the whole regions and they might be computationally inefficient for many practical applications. In most cases, it is very difficult to find the globally optimal solution for cut based energy functions. Since a single optimal partition of an image is not easy to obtain, it makes a possibility of finding different level-based explanations of an image. From this aspect, there are works [11] [12] tackling the image segmentation as a hierarchical bottom-up problem.. Another classical category of segmentation algorithms is based on the similarity among the pixels within a region, namely region-based segmentation. A lot of work has investigated the use of primitive regions as a preprocessing step for image segmentation [13] [14] [15] . The advantages are twofold: first, regions carry on more information in describing the nature of objects. Second, the number of primitive regions is much fewer than that of pixels in an image and thus largely speeds up the region merging process. Starting from a set of primitive regions, the segmentation is conducted by progressively merging the similar neighboring regions according to a certain predicate. Although the segmentation is obtained by making local decisions, some techniques have been proved to be very efficient [16] [17] [18] [19] [20] [21] . In region merging techniques, the goal is to merge regions that satisfy a certain homogeneity criterion. In previous works, there are region merging algorithms based on statistical properties [16, 22, 23] , graph properties [17] [18] [19] [24] [25] or spatio-temporal similarity [21] .. However, most region merging algorithms are hardly proven to preserve the global properties.. In this paper, we implement the segmentation algorithm in a region merging style for its merit of efficiency, where similar neighboring regions are iteratively merged according to a novel merging predicate.. The proposed predicate can be interpreted as a combination of the consistency measure and the similarity measure. As stated above, homogeneity criteria (cues) are essential to the region merging process. However, in this paper we do not focus on how to find good cues. Instead, we model the cues as a function of random variables. Evaluation of the homogeneity of regions follows the principle of Gestalt theory [26] [27] , which suggests a preference to having consistent elements in the same data set. Unlike some traditional methods which measure the data consistency with a certain threshold [20] , the function we use is associated to a binary value which is defined by the Sequential Probability Ratio Test (SPRT) [28] . More specifically, the extent of consistency is measured by two hypotheses, i.e. null hypothesis H 0 "the tested data are inconsistent" and an alternative hypothesis H 1 "the tested data are consistent". The similarity measure describes the affinity between two neighboring regions. In each iteration, a region finds its closest neighbor at the lowest value according to some cost function. This casts as that of finding in a spatial-time search space an optimal path that most satisfies the data consistency. If taking the image segmentation as a labeling problem [1] , the label of a region will transit to the one of its closest neighbor as iteration goes on. As a result, a merging will happen when two neighboring regions exchange their labels. We can prove that this merging scheme satisfies the maximum likelihood criterion, by which a primitive region will be merged into its most similar group at the lowest cost in the final segmentation.. A good segmentation algorithm should preserve certain global properties according to the perceptual cues [16, [22] [23] . This leads to another essential problem in a region merging algorithm: the order that is followed to perform the region merging. Since the merging process is inherently local, most existing algorithms have difficulties to possess some global optimality. However, we demonstrate that the proposed algorithm holds certain global properties, i.e. being neither over-merged nor under-merged, using the defined merging predicate. In addition, to speed up the region merging process, we introduce the structure of Nearest Neighbor Graph to accelerate the proposed algorithm in searching the merging candidates. The experimental results indicate the efficiency of the acceleration algorithm.. The rest of this paper is organized as follows. In Section II, a novel region merging predicate is defined.. The consistency test of neighboring regions is defined in Section III by using SPRT. In Section IV, we present the dynamic region merging algorithm and prove some global properties it holds for the segmentation. In Section V, an accelerated algorithm for the merging process is presented and analyzed. In Section VI, we show the segmentation results on real images. Section VII concludes the paper.

O B I O O B I O O O O O O B I O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Impact

of

variance

components

on

reliability

of

absolute

quantification

using

digital

PCR

Background:

Digital

polymerase

chain

reaction

(dPCR)

is

an

increasingly

popular

technology

for

detecting

and

quantifying

target

nucleic

acids.

Its

advertised

strength

is

high

precision

absolute

quantification

without

needing

reference

curves.

The

standard

data

analytic

approach

follows

a

seemingly

straightforward

theoretical

framework

but

ignores

sources

of

variation

in

the

data

generating

process.

These

stem

from

both

technical

and

biological

factors,

where

we

distinguish

features

that

are

hard-wired

in

the

equipment,

user-dependent

and

provided

by

manufacturers

but

may

be

adapted

by

the

user.

The

impact

of

the

corresponding

variance

components

on

the

accuracy

and

precision

of

target

concentration

estimators

presented

in

the

literature

is

studied

through

simulation.

Results:

We

reveal

how

system-specific

technical

factors

influence

accuracy

as

well

as

precision

of

concentration

estimates.

We

find

that

a

well-chosen

sample

dilution

level

and

modifiable

settings

such

as

the

fluorescence

cut-off

for

target

copy

detection

have

a

substantial

impact

on

reliability

and

can

be

adapted

to

the

sample

analysed

in

ways

that

matter.

User-dependent

technical

variation,

including

pipette

inaccuracy

and

specific

sources

of

sample

heterogeneity,

leads

to

a

steep

increase

in

uncertainty

of

estimated

concentrations.

Users

can

discover

this

through

replicate

experiments

and

derived

variance

estimation.

Finally,

the

detection

performance

can

be

improved

by

optimizing

the

fluorescence

intensity

cut

point

as

suboptimal

thresholds

reduce

the

accuracy

of

concentration

estimates

considerably.

Conclusions:

Like

any

other

technology,

dPCR

is

subject

to

variation

induced

by

natural

perturbations,

systematic

settings

as

well

as

user-dependent

protocols.

Corresponding

uncertainty

may

be

controlled

with

an

adapted

experimental

design.

Our

findings

point

to

modifiable

key

sources

of

uncertainty

that

form

an

important

starting

point

for

the

development

of

guidelines

on

dPCR

design

and

data

analysis

with

correct

precision

bounds.

Besides

clever

choices

of

sample

dilution

levels,

experiment-specific

tuning

of

machine

settings

can

greatly

improve

results.

Well-chosen

data-driven

fluorescence

intensity

thresholds

in

particular

result

in

major

improvements

in

target

presence

detection.

We

call

on

manufacturers

to

provide

sufficiently

detailed

output

data

that

allows

users

to

maximize

the

potential

of

the

method

in

their

setting

and

obtain

high

precision

and

accuracy

for

their

experiments.

We

studied

the

influence

of

several

sources

of

variation

on

estimators

produced

by

digital

PCR.

We

showed

how

some

have

higher

impact

than

others

and

found

certain

background

conditions

to

be

more

vulnerable

to

this

than

others.

This

impact

may

stay

hidden

to

the

naive

user

who

could

take

away

suboptimal

results

with

a

false

sense

of

precision,

accuracy

and

reliability..

A

first

source

of

variance

is

technical

variation,

which

includes

pipette

error.

Although

careful

sample

preparation

can

keep

this

error

relatively

small,

it

is

unavoidable

and

reduces

precision.

This

can

not

be

captured

with

a

single

replicate

and

previously

published

asymptotic

confidence

intervals.

Replicates

allow

this

source

of

variation

to

be

included

in

the

data-analytic

process

and

provide

correct

precision

estimates..

Unequal

partition

size

reduces

the

accuracy

for

highly

concentrated

samples.

Since

this

source

of

variation

is

dependent

on

the

machine

itself,

it

is

one

of

the

priorities

for

manufacturers

to

optimize

it

and

keep

the

variation

between

partitions

small.

We

have

shown

that

the

bias

is

small

for

realistic

limited

deviations

and

can

be

neglected

when

the

concentration

is

not

close

to

the

upper

limit.

Users

are

advised

to

avoid

strongly

concentrated

samples

and

dilute

samples

when

necessary..

The

fluorescence

threshold

chosen

for

target

detection

drives

the

misclassification

rates

and

has

a

high

impact

on

the

results,

reducing

accuracy.

Samples

with

few

target

copies

and

experiments

with

a

very

high

concentration

of

target

nucleic

acid

are

especially

vulnerable.

In

the

former

case,

the

focus

may

usefully

shift

to

detection

rather

than

quantification

while

in

the

latter,

dilutions

or

qPCR

may

be

advised.

Misclassification

to

some

extent

is

unavoidable,

but

the

informed

user

can

do

a

lot

to

reduce

it..

The

underlying

continuous

distribution

of

the

fluorescence

is

a

mixture

distribution

composed

of

output

from

both

positive

and

negative

partitions.

These

two

parts

may

be

partially

overlapping

as

a

result

of

biological

factors

such

as

inhibition,

contamination

or

primer

depletion.

The

choice

of

the

threshold

results

in

corresponding

false

positive

and

false

negative

rates.

The

optimal

trade-off

naturally

depends

on

the

concentration

of

target

copies

in

the

sample.

When

the

software

allows

it,

users

can

calibrate

the

threshold

to

reflect

expected

misclassification

rates

of

their

application

and

get

more

accurate

results..

Additionally,

dilution

series

may

help

to

determine

the

concentration

where

the

variance-bias

trade-off

is

lowest

and

the

measurement

reflects

the

best

combination

of

accuracy

and

precision.

This

is

especially

useful

at

high

concentrations

when

the

focus

is

on

accurate

quantification.

Users

can

achieve

this

by

comparing

precision

and

linearity

plots

to

the

patterns

in

Figure

to

picture

the

bias,

while

an

estimate

of

the

standard

deviation

follows

from

correctly

analysed

replicate

experiments..

Since

we

identified

misclassification

as

the

major

bottleneck

that

induces

the

largest

accuracy

drop,

methods

to

optimise

classification

and

accuracy

are

promising

topics

of

future

research..

Finally,

it

is

worth

emphasizing

that

our

results

have

focussed

on

technical

replicates

involving

variation

in

results

generated

by

machine

settings

and

human

handling

of

a

given

biological

sample.

It

is

essential

to

acknowledge

sources

of

variation

such

as

systematic

pipette

error

and

correct

for

them

when

necessary.

As

for

any

biological

measurement,

additional

sampling

variation

may

be

present

in

many

experiments

at

several

levels.

This

happens

quite

independently

of

the

technology

and

is

discussed

widely

in

the

literature

A

thoughtful

protocol

to

correct

also

for

this

source

of

variation

should

be

generally

considered

in

addition

to

the

specific

digital

PCR

protocol..

Digital

PCR

is

a

promising

tool

for

high

precision

estimation.

We

showed

how

several

sources

of

variation

can

influence

results

and

can

be

accommodated

with

the

correct

knowledge

such

that

accurate

and

precise

concentration

estimates

remain

possible.

Our

findings

indicate

that

reliability

can

be

increased

by

well-chosen

sample

preparation

and

machine

settings.

Machine

calibration

in

theory

allows

the

researcher

to

adapt

the

technology

to

yield

results

optimized

for

each

specific

setting.

While

it

is

of

course

essential

to

provide

default

settings

to

simplify

the

process

for

the

users,

it

is

at

least

as

important

that

manufacturers

provide

detailed

output

to

facilitate

personalized

treatment

and

thus

enhance

the

quality

of

their

results.

Below,

we

detail

our

simulation

set-up

presenting

plausible

departures

from

the

theoretical

model

described

above.

From

each

generated

dataset,

we

estimate

the

number

of

target

copies

in

the

sample

and

derive

the

concentration

of

the

original

dilution

relying

on

the

assumptions

of

the

simple

working

model,

which

may

ignore

components

of

variation

in

the

data-generating

model.

For

each

set-up

simulations

are

run

to

get

stable

results.

O O O O O B O O O O B I O O B I I O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


State-space mixing cell model of unsteady solute transport in unsaturated soil Abstract The purpose of this model is to enable implementation of the theory of linear systems control in operational management of waste and fertiliser applications onto land, so that the underlying groundwater is protected from pollution by leachate. The state-space form of the model enables use of the extensive theory and available software on stochastic linear S)lstCms. In particular, the Kalman filter is relevant to the imperfectly understood and highly variable processes of solute transport and transformation in field soils. The series of mixing cells was selected as a linear system model of one-dimensional, vertical, advective-dispersive transport, and based on cumulative soil water drainage as the index variable for application to unsteady flow in unsaturated soil. For each cell, solute transfer between mobile and immobile soil water, as well as equilibrium and nonequilibrium linear adsorption, are represented as lumped processes by two fractions linked by rate-limited transfer. The resident solute concentrations in the cell fractions are the states of the system. The complete model of solute transport and transformation for a unifonn soil has four parameters, and can be described in MATLAB® with about ten lines of code. The software library can then be used to produce the discrete form of the model, which is unconditionally stable for any drainage interval. as well as to implement state estimation and control algorithms. A demonstration of the model is reported for 35S-labelled sulphate leached from five replicated lysimeters (800 mm diameter, lIDO mm depth) of an undisturbed field soil (a free-draining silt loam) under pasture receiving rainfall and irrigation, The results show satisfactory one-step-ahead forecasts with the Kalman filter for the period of record, and a forecast is given of the complete response to the solute pulse application beyond the data record.Application of fertilisers or wastes to the land surface can degrade the quality of underlying groundwater if these substances are in excess of crop requirements or the assimilative capacity of the soil. The model presented in this paper is designed to support management of the amount and timing of susbstance applications to the land so that groundwater quality is protected. The proposed management process involves monitoring leachate quality by means of lysimeters and using these data for operational decision making.Each lysimcter is an undisturbed column of soil, surrounded by a sealed vertical casing, from which freely draining soil water is collected for analysis (Cameron et al., 1992) . The collected leachate is a ;00% sample of drainage from the horizontal area of the lysimeter and therefore substance mass balances are measured for the soil volume. A reliable estimate of water and solute mass flux in the field is obtained with appropriate replication of Iysimeter units.For reasons explained in the following section, cumulative drainage is the index variable rather than time. For practical monitoring, leachate data are collected after significant rainfall or irrigation events. and the drainage interval since the previous collection is unknown. Therefore, the leachate concentration data represent the mean values during these irregular intervals of cumulative drainage.2, MOIJELTransport and transformation of substances in soil involves processes which are highly variable and have properties which are difficult to measure accurately. Management under these circumstances is facilitated by application of the theory of stochastic control (e,g., Bertsekas, 1976; Davis and Vinter, 1985) which enables formulation of rules for monitoring and control with optimal performance. This theory requires the processes of interest be described by a mathematical model with the "state-space" structure which relates the measured outputs Y(t) of a system to the internal states X(t) in response to inputs U(t) in the form:(1)The coefficient matrices A, B, C, and D for the present paper are assumed to be constant with respect to the index variable 1. A model in the form of (1) can easily be coded in MA TLAB® software for access to a library of algorithms for linear system control (Grace et al., 1992) , Application of fertilisers or wastes to the land surface can degrade the quality of underlying groundwater if these substances are in excess of crop requirements or the assimilative capacity of the soil. The model presented in this paper is designed to support management of the amount and timing of susbstance applications to the land so that groundwater quality is protected. The proposed management process involves monitoring leachate quality by means of lysimeters and using these data for operational decision making.. Each lysimcter is an undisturbed column of soil, surrounded by a sealed vertical casing, from which freely draining soil water is collected for analysis (Cameron et al., 1992) . The collected leachate is a ;00% sample of drainage from the horizontal area of the lysimeter and therefore substance mass balances are measured for the soil volume. A reliable estimate of water and solute mass flux in the field is obtained with appropriate replication of Iysimeter units.. For reasons explained in the following section, cumulative drainage is the index variable rather than time. For practical monitoring, leachate data are collected after significant rainfall or irrigation events. and the drainage interval since the previous collection is unknown. Therefore, the leachate concentration data represent the mean values during these irregular intervals of cumulative drainage. The purpose of the model development was to enable access to control theory and supporting software so that this technology could be applied to managing the impact on groundwater of applying wastes and fertilisers to the land, The model is designed for application to leachate data from undisturbed field soils where uncertainty of process parameters and measurement is high. Processes of advective-dispersive transport, mobile-immobile soil waler, and linear adsorption (equilibrium and non equilibrium) were explicitly included in the model development, but have been shown to be represented by a smaller set of dynamic characteristics. Sources and sinks, such as plant uptake, are implicitly included in the concept of this model being part of a controlled system which is tracking these assimilative processes.. Demonstration of the model was centred on its role as a component in an application of the Kalman filter, a pivotal theory in estimation and controi under uncertainty. The results (Figure 4 ) display satisfactory dynamic tracking in response to new data but the error between forecast and measurement shows some persistence, which indicates imperfections in specification of the model and uncertainty profile (Figure 3 ). The forecast of the complete response. at 1100 mm depth, to a solute application at the land surface ( Figure 5 ) demonstrates again how the mood can be used to incorporate available data and subjective judgements about uncertainty into making an optimal estimate of dynamic behaviour.. 5.

B B I O O O B I O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Paley type partial difference sets in non p-groups By modifying a construction for Hadamard (Menon) difference sets we construct two infinite families of negative Latin square type partial difference sets in groups of the form Z 3 2 × Z p 4t where p is any odd prime. One of these families has the wellknown Paley parameters, which had previously only been constructed in p-groups. This provides new constructions of Hadamard matrices and implies the existence of many new strongly regular graphs including some that are conference graphs. As a corollary, we are able to construct Paley-Hadamard difference sets of the StantonSprott family in groups of the form Z 3 2 × Z p 4t × Z 9p 4t ±2 when 9p 4t ± 2 is a prime power. These are new parameters for such difference sets. Let G be a finite group of order v, and let D be a subset of G with cardinality k. Then D is a (v, k, λ)-difference set (DS) if the list of differences d 1 d Hadamard (Menon) difference sets, having parameters (4u 2 , 2u 2 − u, u 2 − u), are of particular interest due to the fact that their ±1 incidence matrices form Hadamard matrices. The text of Beth, Jungnickel, and Lenz [1] and the survey of Jungnickel [7] are excellent references for DSs.. Similarly, suppose G is a finite group of order v with a subset D of order k such that the differences d [9] . A partial difference set having parameters (n 2 , r(n + 1), −n + r 2 + 3r, r 2 + r) is called a negative Latin square type PDS.. Another important family of difference sets is the Paley-Hadamard difference sets having parameters (v, [10] along with the Paley-Hadamard difference sets in the context of Hadamard matrices, and in fact both Paley PDSs and Paley-Hadamard difference sets can be used to construct both Hadamard designs and matrices. In the survey of Ma [9] were the following questions: "Questions 13.4. Suppose G is an abelian group of order v ≡ 1(mod 4). If v is not a prime power, does there exist a Paley PDS in G? If v is a prime power, does G need to be elementary abelian?". The articles of Davis as well as Leung and Ma give various constructions of Paley partial difference sets in p-groups ( [3] , [8] ) that are not elementary abelian and thus answer the second question. In this paper, we will give the first construction of Paley PDSs in groups having an order which is not a prime power and therefore provide the first positive answer to the first of the questions. Using these Paley PDSs, we will also construct Paley-Hadamard DSs with new parameters.. There have also been relatively few constructions of negative Latin square type partial difference sets. Originally most of these were in nonelementary abelian groups, [9] . Recently, there have been constructions given in nonelementary abelian p-groups such as Davis and Xiang in [5] and Polhill [11] . Jørgensen and Klin [6] constructed negative Latin square type PDSs in groups of order 100. In this paper, we will not only construct the Paley PDSs but another pair of families of negative Latin square type PDSs in non p-groups as well.. Regular partial difference sets generate Cayley graphs which are strongly regular, and in particular the Paley-type and negative Latin square type PDSs give rise to conference graphs and negative Latin square type strongly regular graphs respectively.. Often DSs and PDSs are studied within the context of the group ring. . This is abuse of notation which is widely accepted; so that depending on the context D will represent the difference set D or the element d∈D d in the group ring Z[G]. Character theory is frequently used to simplify calculations with difference sets and partial difference sets in abelian groups. Turyn [13] and separately Yamamoto [18] first used character theory to study abelian difference sets. A character on an abelian group G is a homomorphism from the group to the complex numbers with modulus 1. The principal character is 1 on all the elements of G; any other character is called nonprincipal. One can naturally extend a character on G to a homomorphism of the group ring Z[G] as follows: if χ is a character on G, then for an element A = g∈G a g g let χ(A) = g∈G a g χ(g) so that if S is a subset of G, then χ(S) = s∈S χ(s). See [13] for a proof of similar results to the following. Theorem 1.1 (a) Let G be an abelian group of order v with a subset D of cardinality k and let λ be a positive integer satisfying. Let G be an abelian group of order v with a subset P of cardinality k such that 0 ∈ P with k 2 = k + λk + µ(v − k − 1) for positive integers λ and µ. Then P is a (v, k, λ, µ)-partial difference set in G if and only if for every nonprincipal character χ on

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O


Combining precomputation and on-demand routing for multicast QoS routing Abstract-In QoS routing, two computational approaches have been studied: precomputation and on-demand computation. The precomputation approaches have the advantage of extreme efficiency in connection requests processing, but suffers from large storage overhead for maintenance of precomputed QoS routes. In contrast, the on-demand computation approaches have higher connection acceptance probability, but incur large perrequest computation cost and long join latency. In this paper, we try to combine the two approaches in the design of a multicast routing protocol PPMRP (Probabilistic Precomputation Multicast Routing Protocol). PPMRP is the first attempt to include the precomputation technique in the design of multicast QoS routing protocols. It employs two innovative ideas to achieve a nice balance of cost and performance: scope-limited adverting of multicast tree information and probabilistic selection of precomputation routers. Through extensive simulations, we demonstrate that PPMRP exhibits the low join latency characteristic and superior connection acceptance probability without excessive computation and control message overhead. With real-time streaming applications becoming more and more popular on the Internet, finding effective QoS routes has become an increasingly important issue. QoS routing is the process of finding feasible paths subject to connection QoS constraints while achieving efficient network resource utilization.. A reference model for a linked state based router supporting QoS routing is presented in [2] , and reproduced in Fig.1 . QoS-aware routers exchange routing information to create a topology database representing the router's view of the current network connectivity state, from which a Routing Information Base (RIB), also called QoS routing table [2, 3, 4] , is constructed. Upon a connection request arrival, a path satisfying connection QoS requirement is extracted from RIB and inserted into Forwarding Information Base (FIB), which is structured for efficient look-up when data packets arrive. Two main computational approaches in QoS routing are illustrated through the model. For the on-demand computation approach shown on the left side of Fig.1 , complete computation from the topology database to the forwarding state is performed for each connection request. The right side of Fig.1 depicts the precomputation approach, which constructs the RIB containing partial or complete QoS routing paths before connection arrival time. The forwarding state is extracted from the QoS routing table only when connection requests arrive later.. The on-demand computation approaches [2, 5] use the most up-to-date database information to calculate an routing path for each connection. However, it incurs overhead for computing QoS paths for each request. The computation overhead is costly if requests arrival rate is very high [2] . The precomputation-based QoS routing approach [1] [2] [3] [4] [5] [6] tries to reduce the computation cost by having routers precompute all possible QoS paths, in a manner that can potentially account for all possible connection resource requirements, to all possible destinations. The QoS extension to OSPF scheme -QOSPF [3, 4] is a representative precomputation proposal in the unicast setting. Its join latency is extremely low since the results are already computed and only a path extraction procedure needs to be executed.. In the multicast setting, no precomputation based multicast extensions have been proposed for QoS routing since it is impossible for a router to maintain complete information of all possible multicast trees of all possible QoS grades in its RIB database. Therefore, the predominant proposals belong to greedy, hop-by-hop, on-demand computation approach where, for each join request, the route computation involves searching for the existing multicast tree via various probing strategies. Representative examples include YAM [7] , QoSMIC [8] , and QMRP [9] . In YAM [7] , the search initiator uses a spanning ring search to probe for on-tree nodes. Since YAM's spanning ring search is extremely costly in terms of protocol control overhead, QoSMIC [8] improves upon YAM by including a centralized manager in the search procedure to reduce the control overhead. Both of the protocols rely on a timer to wait for responses from on-tree routers before selecting a best path segment that satisfies the QoS requirements. These approaches are greedy and cost-sensitive, which may construct lower cost tree. But they suffer from high control overhead and long join latency resulting from the use of a timer to collect responses. QMRP [9] first performs a single-path shortest path search. In case of heavy network load, the more costly multi-path search is performed if the single-path search fails. Since no timer scheme is used to wait for responses, this strategy has superior join latency performance when network load is light. The problem with the approach is that QMRP constructs the shortest-path tree when network load is light, which almost always has higher tree cost than other cost-sensitive protocols. Besides, its join latency performance will exacerbate in case of heavy network load.. The goal of our research is to propose a protocol, called PPMRP, to improve the join latency performance of current on-demand multicast QoS routing strategies using the precomputation technique. The major problem is to alleviate the control and storage overhead for route precomputation. We highlight below the major issues addressed in this paper.. • We propose a novel precomputation-based multicast QoS routing protocol, which requires only minimal storage overhead via two innovative ideas: scope-limited advertising and probabilistic selection of precomputation router.. • The protocol is the first attempt on a multicast extension of QOSPF in the literature.. • Our proposal incorporates additive delay constraints ∆ d in addition to non-additive bandwidth constraints ∆ b when precomputing QoS routes.. • We demonstrate how our design requires reduced join latency compared with on-demand approaches. The rest of this paper is organized as follows. The mechanisms of our protocol: scope-limited advertising and probabilistic router selection will be described in the next section. Our proposal, the PPMRP protocol, is described in section III. Section IV compares the routing performance of PPMRP to representative on-demand approaches through extensive simulations. A conclusion is given in section V. This paper proposes to combine precomputation-based techniques with traditional on-demand approaches for multicast QoS routing. We present a concrete protocol PPMRP that adopts elements from the QOSPF protocol to support multipath QoS-aware routing. PPMRP employs two important techniques: scope-limited advertisement, and probabilistic router participation. Through extensive simulations, it is shown to improve the long join latency characteristic of previous ondemand protocols based on the precomputed QoS search table while maintaining good routing performance with acceptable control message and storage overhead.

O B O O O O B B I O B I O O O O O O B O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B B O O O O O O O O O O B O O O O O B B I O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O B O O O O O B I O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B I O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B B I O O O O O O O O O O O B I O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


User perspectives in public transport timetable optimisation The present paper deals with timetable optimisation from the perspective of minimising the waiting time experienced by passengers when transferring either to or from a bus. Due to its inherent complexity, this bi-level minimisation problem is extremely difficult to solve mathematically, since timetable optimisation is a non-linear non-convex mixed integer problem, with passenger flows defined by the route choice model, whereas the route choice model is a non-linear non-continuous mapping of the timetable. Therefore, a heuristic solution approach is developed in this paper, based on the idea of varying and optimising the offset of the bus lines. Varying the offset for a bus line impacts the waiting time passengers experience at any transfer stop on the bus line.In the bi-level timetable optimisation problem, the lower level is a transit assignment calculation yielding passengers' route choice. This is used as weight when minimising waiting time by applying a Tabu Search algorithm to adapt the offset values for bus lines. The updated timetable then serves as input in the following transit assignment calculation. The process continues until convergence.The heuristic solution approach was applied on the large-scale public transport network in Denmark. The timetable optimisation approach yielded a yearly reduction in weighted waiting time equivalent to approximately 45 million Danish kroner (9 million USD). In a report from the Capital Region of Denmark (RH, 2009) , it was estimated that 11.5 billion Danish kroner (DKK) will be lost due to travellers being delayed because of congestion in the Copenhagen Region in 2015. Furthermore, it was stated in the report that, to avoid the outlined scenario, people ought to start travelling by public transport rather than by car. The question is how this change in the market share between private and public transport is actually realised?. The present paper deals with timetable optimisation from the perspective of minimising the waiting time experienced when transferring either to or from a bus. passenger's trip purpose. The waiting time was the time that elapsed between the alighting of one run and the boarding of the next run.. The optimisation was performed with the view of finding the optimal departure time (offset) for each bus line to reduce passengers' waiting time when transferring. Instead of treating passengers' route choice as static and predetermined, the approach developed in this paper treated their route choice as pseudo-dynamic. This was done in an iterative process where the output of the timetable optimisation (i.e., the new timetable) served as input to the public assignment model. The output from this public assignment model (i.e., passengers' travel patterns) was then used as input in the timetable. Due to its inherent complexity, finding the optimal offset with this particular objective for the bus lines was extremely difficult analytically. Tabu Search was chosen because of its ability to avoid being trapped in local minima, and also because it had proven to be superior compared to other metaheuristics when bus timetables in Copenhagen were to be optimised (Jansen et al., 2002) .. The heuristic solution approach was applied to the highly complex bus network in Denmark. Applying the heuristic solution approach to the large-scale public transport network in Denmark should preferably reduce the waiting time passengers experience when transferring, while keeping their in-vehicle time and their total generalised travel cost at a constant level. The current study proposed a timetable optimisation approach that explicitly considered passengers' modified route choice as a reaction to changes in the timetable. The approach was applied to a real large-scale transit network. The optimisation yielded a significant reduction in weighted transfer waiting time, while only affecting the in-vehicle travel time and the generalised travel journey cost to a lesser extent.. Overall, the study contributes to the literature by proposing a new optimisation tool, which has its strengths with respect to both the timetable optimisation and the reliability perspective, since changes in the travellers' route choice decisions are considered explicitly whereby demand effects are taken into account.. A topic for future research could be to use stochastic rather than deterministic travel times. Another additional feature could be to consider the problem as multi-objective, e.g., minimise fleet size and transfer waiting time. In this way, convincing bus companies of the applicability of the results might be a smoother process. Finally, the order in which offset changes were imposed could be changed from the ''greedy'' approach introduced in this paper, where the feasible offset change with the largest optimisation potential was chosen. Instead, a knapsack-inspired approach could be applied. Combining the bus lines (for which the offset should be changed) in a way that yields the largest total potential improvement, while still obeying the idea of sub-networks, could be an idea for future research. The current study applied a bi-level timetable optimisation approach, where the objective was to minimise the weighted transfer waiting time. Timetable optimisation (upper level) was integrated with a public assignment model (lower level) to assess how travellers change their behaviour according to the changes imposed in the timetable. This is not a constructive heuristic. Therefore, to make the developed approach work properly, it is necessary to have an existing transit network as an initial solution and being able to run a schedule-based transit assignment, i.e., having a timetable explicitly stating when every transit vehicle departs from its initial stop and when it arrives and departs from the sub sequent stops all the way to its destination.

O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Optimal sequential multiple hypothesis tests This work deals with a general problem of testing multiple hypotheses about the distribution of a discrete-time stochastic process. Both the Bayesian and the conditional settings are considered. The structure of optimal sequential tests is characterized. Let X 1 , X 2 , . . . , X n , . . . be a discrete-time stochastic process, whose distribution depends on an unknown "parameter" θ. We consider the classical problem of testing multiple hypotheses H 1 : θ = θ 1 , H 2 : θ = θ 2 , . . . ,. The main goal of this article is to characterize the structure of optimal sequential tests in this problem.. Let us suppose that for any n = 1, 2, . . . , the vector (X 1 , X 2 , . . . , X n ) has a probability "density" function f n θ (x 1 , x 2 , . . . , x n ). (1) (Radon-Nicodym derivative of its distribution) with respect to a product-measure. n times for some σ-finite measure µ on the respective space. We define a (randomized) sequential hypothesis test as a pair (ψ, φ) of a stopping rule ψ and a decision rule φ, with ψ = (ψ 1 , ψ 2 , . . . , ψ n , . . . ) , and φ = (φ 1 , φ 2 , . . . , φ n , . . . ) .. The functions ψ n = ψ n (x 1 , x 2 , . . . , x n ), n = 1, 2, . . . , are supposed to be some measurable functions with values in [0, 1]. The functions φ n = φ n (x 1 , x 2 , . . . , x n ), n = 1, 2, . . . are supposed to be measurable vector-functions with k non-negative components φ i n = φ i n (x 1 , . . . , x n ): φ n = (φ 1 n , . . . , φ k n ), such that k i=1 φ i n = 1 for any n = 1, 2, . . . . The interpretation of all these elements is as follows. The value of ψ n (x 1 , . . . , x n ) is interpreted as the conditional probability to stop and proceed to decision making, given that we came to stage n of the experiment and that the observations up to stage n were (x 1 , x 2 , . . . , x n ). If there is no stop, the experiments continues to the next stage and an additional observation x n+1 is taken. Then the rule ψ n+1 is applied to x 1 , x 2 , . . . , x n , x n+1 in the same way as as above, etc., until the experiment eventually stops.. It is supposed that when the experiment stops, a decision to accept some of H 1 , . . . , H k is to be made. The function φ i n (x 1 , . . . , x n ) is interpreted as the conditional probability to accept H i , i = 1, . . . , k, given that the experiment stops at stage n being (x 1 , . . . , x n ) the data vector observed.. The stopping rule ψ generates, by the above process, a random variable τ ψ (stopping time) whose distribution is given by P θ (τ ψ = n) = E θ (1 − ψ 1 )(1 − ψ 2 ) . . . (1 − ψ n−1 )ψ n .. Here, and throughout the paper, we interchangeably use ψ n both for ψ n (x 1 , x 1 , . . . , x n ) and for ψ n (X 1 , X 1 , . . . , X n ), and so do we for any other function of observations F n . This does not cause any problem if we adopt the following agreement: when F n is under probability or expectation sign, it is F n (X 1 , . . . , X n ), otherwise it is F n (x 1 , . . . , x n ).. For a sequential test (ψ, φ) let us define. and. i = 1, . . . , k, j = 1, . . . , k. The probabilities α ij (ψ, φ) for j = i can be considered "individual" error probabilities and β i (ψ, φ) "gross" error probability, under hypothesis H i , of the sequential test (ψ, φ).. Another important characteristic of a sequential test is the average sample number:. Let θ be any fixed (and known) value of the parameter (we do not suppose, generally, that θ is one of θ i , i = 1, . . . k).. In this article, we solve the two following problems:. Problem I. Minimize N (ψ) = N (θ; ψ) over all sequential tests (ψ, φ) subject to α ij (ψ, φ) ≤ α ij , for all i = 1, . . . k, and for all j = i,. where α ij ∈ (0, 1) (with i, j = 1, . . . k, j = i) are some constants.. Problem II. Minimize N (ψ) over all sequential tests (ψ, φ) subject to. with some constants β i ∈ (0, 1), i = 1, . . . , k.. More general problems of minimizing an average cost of type. with some cost function C n = C n (X 1 , X 2 , . . . X n ) can be treated in essentially the same manner. If k = 2 then Problems I and II are equivalent, because β 1 (ψ, φ) = α 12 (ψ, φ), and β 2 (ψ, φ) = α 21 (ψ, φ), by (2) and (3).. For independent and identically distributed (i.i.d.) observations and k = 2 the formulated problem, when θ = θ 1 and θ = θ 2 , is known as the modified Kiefer-Weiss problem (see [10] ), being the original Kiefer-Weiss problem minimizing sup θ N (ψ) under (5) (see [6] ).. For the latter problem, taking into account the usual relations between Bayesian and minimax procedures, it seems to be reasonable to generalize our problem of minimizing N (θ; ψ) to that of minimizing N (θ; ψ)dπ(θ), with some "weight" measure π. From what follows it is easily seen that, under natural measurability conditions, our method works as well for this latter problem.. In Section 2, we reduce Problems I and II to an unconstrained minimization problem. The new objective function is the Lagrange-multiplier function L(ψ; φ).. In Section 3, we find. where the infimum is taken over all decision rules. In Section 4, we minimize L(ψ) in the class of truncated stopping rules, i.e. such that ψ N ≡ 1 for some 0 < N < ∞.. In Section 5, we characterize the structure of optimal stopping rule ψ in the class of all stopping rules.. In Section 6, we apply the results obtained in Sections 2 -5 to the solution of Problems I and II.

O O O O O O O O O O O O O O B I O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Experimental signatures of extreme optical fluctuations in lumped Raman fiber amplifiers In this work, we experimentally investigate several temporal and spectral methods to highlight extreme fluctuations which can develop during the Raman amplification of an ultrashort pulse train. Forward and backward pumping schemes are compared to dual pass configurations.Keywords: Fiber optics; Raman amplifiers; extreme statistics 2The study of extreme statistics in nonlinear fiber optics is a topic that has recently stimulated much attention. Following the pioneering work of Solli and coworkers [1] that have reported the observation of statistics analogous to hydrodynamic rogue waves in the formation of optical supercontinuum, most of the initial works have focused on the passive propagation of picosecond pulses in photonic crystal fibers [2] . Those different studies carried out in the anomalous dispersion regime have highlighted the specific role played by modulation instability, Akhmediev breathers [3, 4] as well as Raman frequency shifted solitons that are also affected by third-order dispersion and resulting pulse-to-pulse interactions [5, 6] . However, observation of statistics that strongly deviate from a Gaussian distribution is absolutely not restricted to supercontinuum generation and can also be observed in telecommunication applications in the context of transmissions [7] [8] [9] , Raman fiber lasers [10, 11] as well as discrete amplification [12, 13] . More precisely, in this latest field, it has been shown that under certain conditions, statistical distribution of an amplified signal can be strongly reshaped during the amplification process.Examples of this deleterious degradation have been discussed for parametric amplifiers [12, 14, 15] and for Raman amplifiers [13, 16, 17] : in both cases, a quasi-instantaneous gain and a low walk-off between the signal and a partially coherent pump lead to a dramatic transfer of fluctuations from the pump to the signal in a co-propagating configuration. In a previous work,we have numerically and analytically described the evolution of a continuous signal in a copropagating amplifier in the presence of a low walk-off and pump depletion [17] . We have also Our experimental setup is sketched in Fig. 1a . An actively mode-locked erbium-doped fiber laser delivers a train of 2.9 ps pulses at a repetition rate of 10 GHz and with an initial average power of 10 mW (corresponding to a peak power of ~300 mW) at a wavelength of 1550 nm. Let us note that our previous works [13, 16] or [19] involved much lower repetition rates as the signal source was a MHz passively mode-locked fiber laser. The pump is delivered by a Raman continuous wave fiber laser centered at 1455 nm with an average power up to 2 W. An essential point is that this pump is a partially incoherent wave (~20 GHz of spectral width) that exhibits high contrast, fast intensity fluctuations [19] . Indeed, its intensity autocorrelation signal (plotted in Fig. 1b) presents a characteristic contrast of one half [13, 20] with a temporal width of 25 ps. This temporal duration is a characteristic time of the shortest pump fluctuations and is much larger than the picosecond signal to be amplified while being shorter than the temporal separation between two successive pulses.The amplification is performed in a 1-km long highly nonlinear fiber (HNLF respectively at 1550 nm). We have compared several pumping schemes: backward pumping (Fig. 1a1 ), forward pumping ( Fig. 1a3) and two bi-directional pumping (Figs 1a2 and 1a4) relying on the use of optical circulators. Moreover, several pump powers ranging from 500 to 1500 mW have been tested, leading to integrated gains from 5 to 15 dB in single pass configurations and gains from 10 to 22 dB in dual-pass configurations. Several characterizations of the optical properties of the amplified signal have been carried out. In the temporal domain, we have recorded the output pulse train and its associated statistics thanks to a high speed digital sampling oscilloscope (Tektronix TDS CSA 8200) combined with a photodiode having a 30 GHz bandwidth. We have complemented our study by using an optical intensity autocorrelator (Femtochrome, FR-103HS) to investigate pulse-to-pulse fluctuations.Radio frequency spectra as well as optical spectra were also recorded and further insights on the spectral stability were gained by mapping the spectral evolution into the temporal domain.The most intuitive and straightforward way to explore and evaluate signal variation at the output of the Raman amplifier is the direct observation on a high-speed oscilloscope. Indeed, the eyediagram enables us to visualize the variations of the signal level in the various experimental configurations. Given the finite bandwidth of the electronics under use (typically 30 GHz, much below the optical bandwidth of our pulses), we do not resolve the details of the temporal intensity profile of the picosecond pulses. Therefore, we do not monitor rigorously the peak power of the pulses, but rather the fluctuations of their energy. As a first approximation and given the fact that the amplified pulses do not exhibit any pulse splitting in our normally dispersive fiber, we can however consider that those energy fluctuations are directly proportional to peak power variations [21] .The choice of the pumping configuration has a strong impact on the amplifier performance as can be readily noticed in Fig. 2 . When the signal and the pump counter-propagate in the fiber, the output pulse train exhibits a low level of fluctuations (Fig. 2b) and maintains a quality comparable to the initial mode-locked signal (Fig. 2a) . On the contrary, in a copropagating 6 scheme, the stability of the output pulse train is drastically affected (Fig. 2c ) and striking fluctuations are observed. Similar conclusions can be drawn in bidirectional configurations (Fig 2d) . We took advantage of the statistical functions embedded in the oscilloscope to get more quantitative data on the statistical distribution of the pulse energy fluctuations (Fig. 3) . Results obtained for different pumping schemes (with a pump power of 1 W) are summarized in Fig. 3a and confirm the previous qualitative conclusions. In a backward pumping scheme, the statistical distribution of the pulse energy is quite narrow and can be compared to the initial pulse distribution. It heavily contrasts with the results recorded in a co-propagating scheme where the output statistical distribution presents heavy tails [13] . Events with pulse energy more than 30 times the median energy have indeed been recorded, illustrating the completely unusual amplitude of those fluctuations. Dual pass configurations exhibits slightly less degraded results, 7 the most extreme events being less frequent compared to the single pass co-propagating scheme.This can be mainly explained by the impact of gain saturation of the amplifier that ultimately limits the maximum energy that can be reached. We have also investigated in more details the influence of the pump power in a forward pumping scheme (Fig. 3b) . Increasing the pump power impacts the tail of the statistical distribution by increasing the level of fluctuations: measurements carried out for pumping values between 500 and 1500 mW show that the probability of observing an event between 2 and 12 times the median energy increases with pump power. However, for larger gains and given the high repetition rate of the signal to be amplified, pump depletion effects and gain saturation become important, which leads to an upper limit of the pump energy and rapidly falling 8 probability. Such a behavior is in full agreement with the numerical and analytical conclusions discussed in a previous article dealing with a continuous seed [17] .The electrical bandwidth of our oscilloscope is not high enough to resolve the details of the pulse structure and to get access to the duration of the picosecond pulses. Results obtained with a 10-GHz picosecond pulse train are plotted in Fig. 4 . For single-pass amplifiers, no major modification of the temporal intensity autocorrelation can be noticed compared to the initial pulses. Similar output durations can be recorded in co-and counterpropagating configurations. This has to be linked to the weak dispersion of the highly nonlinear fiber that leads to a dispersion length (defined as the distance for which the initial pulse broadens by a factor 2 due to dispersion [23]) above 5.5 km, which is much higher than the fiber length.However, let us note that the output duration obtained in the forward pumping scheme is slightly higher. Indeed, in this configuration, the pulse experiences more self-phase and cross-phase modulation [16, 24] (this point will be experimentally discussed in section 6 of this paper) so that the expanded spectrum leads to a slightly enhanced dispersion-induced temporal broadening.Differences are much more marked when comparing the results obtained in dual-pass configurations. More precisely, the scheme where forward pumping is followed by backward 9 pumping leads to significantly broadened output pulses. Such a broadening can once again be attributed to the large spectral expansion undergone in the first pass of the amplifier.Initial autocorrelation signal (blue solid line) is compared with autocorrelation for a counter-propagating configuration (dashed grey line) and for a counter-propagating pass followed by a co-propagating pass (dotted grey line). Results are compared to (b) autocorrelation for a co-propagating configuration (dashed grey line) and for a co-propagating pass followed by a counter-propagating pass (dotted grey line). Results obtained for an initial pump power of 1 W.We have also recorded the autocorrelation over a larger temporal span in order to observe the cross-correlation peaks between a pulse and its preceding pulse delayed by 100 ps. Such measurements were not possible in the previous published studies due to the low repetition rate of the passively modelocked source [12, 13, 16] (requiring therefore a delay of several tens of ns). Interestingly, the level of this cross-correlation peak can provide pertinent information on the amplitude jitter degradation that affects the pulse train [25] [26] [27] . Experimental results are summarized in Fig. 5 . The initial pulse train exhibits a level of the central peak identical to the level of the cross-correlation peak, confirming the regime of mode-locking of the source. Such The study of extreme statistics in nonlinear fiber optics is a topic that has recently stimulated much attention. Following the pioneering work of Solli and coworkers [1] that have reported the observation of statistics analogous to hydrodynamic rogue waves in the formation of optical supercontinuum, most of the initial works have focused on the passive propagation of picosecond pulses in photonic crystal fibers [2] . Those different studies carried out in the anomalous dispersion regime have highlighted the specific role played by modulation instability, Akhmediev breathers [3, 4] as well as Raman frequency shifted solitons that are also affected by third-order dispersion and resulting pulse-to-pulse interactions [5, 6] . However, observation of statistics that strongly deviate from a Gaussian distribution is absolutely not restricted to supercontinuum generation and can also be observed in telecommunication applications in the context of transmissions [7] [8] [9] , Raman fiber lasers [10, 11] as well as discrete amplification [12, 13] . More precisely, in this latest field, it has been shown that under certain conditions, statistical distribution of an amplified signal can be strongly reshaped during the amplification process.. Examples of this deleterious degradation have been discussed for parametric amplifiers [12, 14, 15] and for Raman amplifiers [13, 16, 17] : in both cases, a quasi-instantaneous gain and a low walk-off between the signal and a partially coherent pump lead to a dramatic transfer of fluctuations from the pump to the signal in a co-propagating configuration. In a previous work,. we have numerically and analytically described the evolution of a continuous signal in a copropagating amplifier in the presence of a low walk-off and pump depletion [17] . We have also. shown that an adequate frequency shifted optical bandpass filter can isolate the most extreme structures [16] .. In this contribution, we focus our attention on experimental results outlining the drastic consequences of the transfer of intensity fluctuations from the pump to a pulsed signal in a lumped Raman amplifier. Indeed, due to their scarcity and their high amplitude, characterization of the impact of extreme statistics remains a tricky task. Contrary to widespread quantitative techniques such as the measurement of the optical noise signal ratio (OSNR) or the relative intensity noise (RIN) [18] , we have chosen to present here a set of alternative qualitative methods that show the emergence of extreme fluctuations in a 10 GHz pulse stream. More precisely, this paper will successively present statistical records using photodiodes, optical auto-/cross-correlations, RF spectrum and optical spectrum. Forward and backward pumping schemes will be compared to dual-pass configurations. We have presented in this paper several non-standard characterization methods which highlight qualitatively the specific nature of fluctuations observed in a lumped Raman amplifier pumped in a co-propagating way. Similar events may also be found in other devices like waveguides as it has been shown by Borlaug et al. [36] . All the various effects highlighted here are mainly related to the transfer of pump noise on the amplified signal. In other words, there are not linked to accumulation of amplified spontaneous emission in the amplifier nor by the beat noise-noise or noise-signal nor by multi-interference pattern. Therefore, several widely-spread methods used to perform optical performance monitoring and determine optical signal to noise ratio [37] may fail to efficiently take into account all those highly damaging impairments.. All the trends described in this work have been confirmed by intensive numerical simulations (not detailed here) based the integration of a set of two coupled nonlinear Schrödinger equation including the Raman amplification [13, 16, 17] with the assumption of a pump having a Gaussian statistics. Note however that this assumption fails in reproducing the results obtained by RF measurements, highlighting the complex nature of the output of the Raman fiber pump laser.. We expect our results to be expandable to other amplifiers such as parametric amplifiers.. The physical principle of parametric amplifiers may however lead to additional points to be discussed such as the consequences of the phase-sensitive nature of those devices. Our experimental setup is sketched in Fig. 1a . An actively mode-locked erbium-doped fiber laser delivers a train of 2.9 ps pulses at a repetition rate of 10 GHz and with an initial average power of 10 mW (corresponding to a peak power of ~300 mW) at a wavelength of 1550 nm. Let us note that our previous works [13, 16] or [19] involved much lower repetition rates as the signal source was a MHz passively mode-locked fiber laser. The pump is delivered by a Raman continuous wave fiber laser centered at 1455 nm with an average power up to 2 W. An essential point is that this pump is a partially incoherent wave (~20 GHz of spectral width) that exhibits high contrast, fast intensity fluctuations [19] . Indeed, its intensity autocorrelation signal (plotted in Fig. 1b) presents a characteristic contrast of one half [13, 20] with a temporal width of 25 ps. This temporal duration is a characteristic time of the shortest pump fluctuations and is much larger than the picosecond signal to be amplified while being shorter than the temporal separation between two successive pulses.. The amplification is performed in a 1-km long highly nonlinear fiber (HNLF respectively at 1550 nm). We have compared several pumping schemes: backward pumping (Fig. 1a1 ), forward pumping ( Fig. 1a3) and two bi-directional pumping (Figs 1a2 and 1a4) relying on the use of optical circulators. Moreover, several pump powers ranging from 500 to 1500 mW have been tested, leading to integrated gains from 5 to 15 dB in single pass configurations and gains from 10 to 22 dB in dual-pass configurations. Several characterizations of the optical properties of the amplified signal have been carried out. In the temporal domain, we have recorded the output pulse train and its associated statistics thanks to a high speed digital sampling oscilloscope (Tektronix TDS CSA 8200) combined with a photodiode having a 30 GHz bandwidth. We have complemented our study by using an optical intensity autocorrelator (Femtochrome, FR-103HS) to investigate pulse-to-pulse fluctuations.. Radio frequency spectra as well as optical spectra were also recorded and further insights on the spectral stability were gained by mapping the spectral evolution into the temporal domain.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I B I B I O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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A Covariance-Tying Technique for HMM-Based Speech Synthesis A technique for reducing the footprints of HMM-based speech synthesis systems by tying all covariance matrices of state distributions is described. HMM-based speech synthesis systems usually leave smaller footprints than unit-selection synthesis systems because they store statistics rather than speech waveforms. However, further reduction is essential to put them on embedded devices, which have limited memory. In accordance with the empirical knowledge that covariance matrices have a smaller impact on the quality of synthesized speech than mean vectors, we propose a technique for clustering mean vectors while tying all covariance matrices. Subjective listening test results showed that the proposed technique can shrink the footprints of an HMM-based speech synthesis system while retaining the quality of the synthesized speech. The most widely used speech synthesis technique is unit selection synthesis [1] - [3] , in which appropriate sub-word units are selected from large speech databases. Although this technique can synthesize high-quality speech, it requires large databases of recorded speech. Furthermore, it usually requires excessively large footprints to put it on embedded devices such as mobile phones, PDAs, car navigation systems, and game machines.. Statistical parametric speech synthesis based on HMMs [4] , [5] has grown in usage. Figure 1 gives an overview of a typical HMM-based speech synthesis system. In this system, the spectrum, excitation, and duration of speech are modeled simultaneously by contextdependent HMMs, and speech parameter trajectories are generated from the HMMs themselves under constraints between static and dynamic features [6] . One of the attractive points of HMM-based speech synthesis is its small footprint. HMM-based systems usually have smaller footprints than unit selection systems, because they store statistics rather than speech waveforms. However, further reduction is essential to put these systems on embedded devices that have little memory.. Speech parameters such as spectrum, excitation, and duration depend on a variety of contextual factors such as phoneme identities, accent types, and parts-of-speech. In the HMM-based speech synthesis system, contextdependent models are used to capture these contextual factors. If more combinations of these contextual factors are taken into account, we should be able to obtain more accurate models. However, as the number of contextual factors increases, the number of possible combinations also increases exponentially. As a result, it is difficult to robustly estimate model parameters due to the lack of training data. Furthermore, it is impossible for a finite set of training data to cover every possible combination of contextual factors. Various parameter-tying techniques have been developed [7] - [11] to avoid this problem. Among them, a decision tree-based context-clustering technique [12] has been widely used. In the HMM-based speech synthesis system, distributions of spectrum, excitation, and duration are clustered individually because they have their own contextual dependencies.. In this technique, top-down clustering is performed to maximize the likelihood of model parameters with respect to the training data by using questions about contexts. Then, HMM identifies which of those clustered into the same leaf node are tied. Unseen models can be generated by traversing the decision trees. Various criteria [13] - [17] have been proposed for selecting the questions to be used.. Conventionally, we construct an HMM stream-level tying structure in HMM-based speech synthesis, i.e., mean Copyright c 2010 The Institute of Electronics, Information and Communication Engineers vectors and covariance matrices have exactly the same parameter-tying structure ( Fig. 2 (a) ). However, we empirically know that covariance matrices have a smaller impact on the quality of synthesized speech than mean vectors. On the basis of this knowledge, a technique for contextclustering mean vectors while tying all covariance matrices ( Fig. 2 (b) ) should be tested in HMM-based speech synthesis. If each parameter is stored in a single-precision floating-point number and the dimensionality of Gaussian distributions is 120, approximately 938 kBytes are required to store 1,000 Gaussian distributions with diagonal covariance matrices (statistics associated to the leaf nodes). However, tying all covariance matrices reduced it almost by half (469 kBytes).. Semi-tied covariance [18] is one of the major covariance-tying techniques. This technique is a simple extension of the standard diagonal or full covariance matrices used with HMMs. Instead of having a distinct covariance matrix for every distribution, each covariance matrix consists of two elements, a component-specific diagonal covariance element and a tied transform. It is important to make the difference between the semi-tied covariance technique and the proposed technique clear.. The rest of this paper is organized as follows. Section 2 describes the decision-tree-based context clustering technique. Context clustering for semi-tied covariance matrices are presented in Sect. 3. Section 4 describes the proposed decision tree-based context-clustering technique for mean vectors while tying all covariance matrices. Subjective listening test results are shown in Sect. 5. Finally, concluding remarks and future plans are presented in Sect. 6. A technique for reducing the footprints of HMM-based speech synthesis systems by tying all covariance matrices is described. Experimental results showed that the proposed technique efficiently reduced the footprints of an HMMbased speech synthesis system to less than half of its original size while retaining the quality of the synthesized speech. Future work includes using a separated clustering technique for mean vectors and covariance matrices.

O O O O O B I O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O