Making

nonlinear

manifold

learning

models

interpretable:

The

manifold

grand

tour

Making

nonlinear

manifold

learning

models

interpretable:

The

manifold

grand

tour

Article

LJMU

has

developed

LJMU

Research

Online

for

users

to

access

the

research

output

of

the

University

more

effectively.

Lisboa,

P,

Martín-Guerrero,

JD

and

Vellido,

A

Making

nonlinear

manifold

learning

models

interpretable:

The

manifold

grand

tour.

Expert

Systems

with

Applications,

reduction

is

required

to

produce

visualizations

of

high

dimensional

data.

In

this

framework,

one

of

the

most

straightforward

approaches

to

visualising

high

dimensional

data

is

based

on

reducing

complexity

and

applying

linear

projections

while

tumbling

the

projection

axes

in

a

defined

sequence

which

generates

a

Grand

Tour

of

the

data.

We

propose

using

smooth

nonlinear

topographic

maps

of

the

data

distribution

to

guide

the

Grand

Tour,

increasing

the

effectiveness

of

this

approach

by

prioritising

the

linear

views

of

the

data

that

are

most

consistent

with

global

data

structure

in

these

maps.

A

further

consequence

of

this

approach

is

to

enable

direct

visualisation

of

the

topographic

map

onto

projective

spaces

that

discern

structure

in

the

data.

The

experimental

results

on

standard

databases

reported

in

this

paper,

using

Self-Organising

Maps

and

Generative

Topographic

Mapping,

illustrate

the

practical

value

of

the

proposed

approach.

It

must

be

remarked

the

novelty

of

the

proposed

method

that

improves

some

of

the

aspects

of

previous

approaces

based

on

the

Grand

Tour.

When

exploring

any

environment,

it

would

be

unusual,

or

even

counterintuitive,

not

to

try

to

visualize

it

first.

The

same

applies

to

the

exploration

of

data

Visualisation

processes

are

straightforward

when

data

sets

comprise

a

handful

of

attributes,

since

data

visualisation

can

be

readily

implemented,

for

instance

with

multiple

scatter-plots..

However,

high

dimensional

data

require

the

application

of

more

advanced

methods.

This

may

involve

the

application

of

projective

or

mapping

algorithms

and

becomes

an

important,

or

even

necessary,

stage

of

data

analysis.

This

is

specially

true

when

the

interpretability

of

the

results

is

a

requirement

of

the

analysis

Vellido

et

al.

Such

techniques

are

data

visualisation-oriented

instances

of

the

more

general

family

of

Dimensionality

Reduction

(DR)

methods..

Some

of

the

most

frequently

used

DR

methods

involve

only

linear

combinations

of

the

covariates.

These

methods

have

the

advantage

over

their

non-linear

counterparts

that

when

a

gap

in

the

observed

data

is

seen

from

a

particular

projection

(revealing

data

grouping

structure),

then

that

gap

is

known

to

be

present

and

cannot

close

when

the

dimensionality

of

the

projection

is

increased.

A

popular

such

method

is

Principal

Component

Analysis

(PCA),

which

is

typically

applied

in

practice

using

biplots

Jolliffe

This

approach

compensates

for

one

of

its

limitations,

in

particular

sensitivity

to

noise

and

the

lack

of

a

robust

criterion

for

choosing

the

adequate

number

of

PCs,

by

the

straightforward

interpretability

of

the

resulting

projections..

Alternatively,

Non-linear

Dimensionality

Reduction

(NLDR)

(Lee

and

Verleysen,

methods

are

potentially

more

powerful

to

model

complex

highdimensional

data.

These

methods

are

well-suited

to

map

the

topological

structure

of

the

data,

especially

when

the

regions

of

interest

cannot

be

well-separated

using

linear

discrimination

functions,

or,

equivalently,

whenever

mean

values

are

not

representative

of

density

functions

due

to

deviations

from

normality..

Manifold

learning

methods

are

part

of

the

NLDR

family

of

techniques

that

attempt

to

represent

multivariate

data

by

assuming

they

can

be

closely

approxi-mated

using

low-dimensional

manifolds,

typically

chosen

to

be

for

visualisation

purposes.

However,

these

methods

can

generate

complex

surfaces

with

possible

occurrence

of

folds,

which

are

often

the

result

of

overfitting.

Moreover,

the

projection

of

data

onto

the

visualization

maps

is

heavily

conditioned

by

the

assumed

structure

of

the

map

and

so

does

not

necessarily

provide

a

clear

picture

of

the

empirical

data

density.

In

order

to

increase

the

interpretability

of

manifold

learning

techniques,

it

is

of

interest

to

combine

generative

models

and

NLDR

with

linear

projective

methods..

An

alternative

approach

is

to

produce

different

views

of

the

data

arising

from

a

succession

of

linear

projections.

A

framework

to

generate

a

comprehensive

range

of

low-dimensional

projections

is

the

Grand

Tour

proposed

by

Diane

Cook

and

colleagues

(Buja

et

al.,

Cook

and

Swayne,

In

this

approach,

the

data

are

effectively

tumbled

in

a

systematic

way

and

viewed

through

the

prism

of

low-dimensional

linear

projections,

looking

for

indicators

of

structure,

typically

gaps

between

sub-population

cohorts.

Due

to

the

usefulness

of

the

visualisations

obtained,

and

the

need

of

an

easy

and

straightforward

way

to

obtain

them,

this

method

has

been

recently

implemented

in

an

R

package

(Wickham

et

al.,

with

a

user-friendly

graphical

user

interface

(GUI)

(Huang

et

al.,

While

this

approach

is

powerful

in

principle,

we

reckon

that

the

search

procedure

may

be

expedited

by

prioritising

the

most

informative

views

of

the

data;

this

is

the

goal

and

main

novelty

of

the

proposed

approach

compared

to

(Buja

et

al.,

Cook

and

Swayne,

In

(Lecerf

and

Bouchard,

a

method

based

on

selecting

candidate

projections

from

the

space

of

all

projections

was

proposed..

Our

proposed

method

also

pursues

that

goal

but

from

a

different

and

more

complete

perspective,

since

it

does

not

require

any

user-interaction

and

a

twodimensional

track

is

employed

to

guide

three-dimensional

projections

of

the

data

without

the

need

for

space-filling

patterns..

Our

conjecture

is

hence

that

prioritising,

we

are

effectively

introducing

an

implicit

narrative

in

the

process

of

visual

data

analysis.

This

added

contextual

information

becomes

a

way

of

storytelling

that

should

potentially

provide

more

actionable

knowledge

(Segel

and

Heer,

The

prioritisation

of

the

most

informative

views

of

the

data

can

effectively

be

provided

by

the

NLDR

methods.

Therefore,

in

this

study,

we

propose

using

NLDR

manifolds

as

the

base

surface

over

which

we

roll-out

sequences

of

linear

projections,

knowing

that

they

will

cross

regions

of

high

data

density.

This

approach

is

intended

to

use

the

power

of

linear

projections

and

leverage

it

on

the

data

coverage

generated

by

NLDR

methods,

in

particular

Self-Organising

Map

(SOM)

networks

(Kohonen,

and

Generative

Topographic

Mapping

(GTM)

(Bishop

et

al.,

In

particular,

this

method

aims

to

quickly

discover

gaps

in

the

data

distribution,

which

may

be

consistent

with

a

hierarchical

structure

that

may

not

be

explicitly

available

even

with

prior

clustering.

The

proposed

methodology

is

the

Manifold

Grand

Tour..

The

remaining

of

the

paper

starts

with

a

summary

description

of

manifold

learning

models

such

as

SOM

and

GTM,

together

with

an

overview

of

cohortbased

linear

visualisation,

which

improves

on

PCA

by

using

data

labels

from

cluster

or

class

membership,

whenever

this

information

is

available.

This

is

followed

by

a

detailed

description

of

the

Manifold

Grand

Tour

procedure.

Empirical

results

for

two

public

domain

data

sets

illustrate

the

application

of

the

method.

This

paper

has

presented

a

new

approach

for

NLDR

methods

oriented

to

multivariate

exploratory

data

visualisation

that

combines

the

modelling

flex-ibility

of

one

such

method,

namely

GTM,

with

the

interpretability

of

linear

models

for

data

visualisation.

In

the

reported

experiments,

GTM

has

been

used

to

guide

a

Grand

Tour

of

some

real

data

sets

that

uses

a

recently

proposed

linear

DR

method

for

data

visualisation,

which

is

based

on

a

clustering

approach..

The

achieved

results

illustrate

the

suitability

of

the

proposed

method

to

produce

useful

representations

that

intuitively

reveal

the

internal

data

structure.

This

paper

involves

a

relevant

theoretical

advance

with

respect

to

the

standard

Grand

Tour

since

views

are

not

random

but

selected

according

a

smart

guide,

such

us

GTM.

This

work

also

improves

and

completes

the

approach

presented

in

(Lecerf

and

Bouchard,

since

neither

user

interaction

nor

a

two-dimensional

track

is

required

to

guide

three

dimensional

projections.

The

main

limitation

of

the

study

is

related

to

the

number

of

class

structures

since

a

high

number

of

classes

might

difficult

the

visualization;

this

is

however

a

common

problem

in

this

kind

of

visualizations.

It

is

finally

remarkable

that

an

appealing

advantage

of

the

proposed

method

is

that

it

could

straightforwardly

be

extended

to

alternative

manifold

learning

algorithms.

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


On spurious behavior of CFD simulations Spurious behavior in underresolved grids and/or semi-implicit temporal discretizations for four computational fluid dynamics (CFD) simulations are studied. The numerical simulations consist of (a) a 1-D chemically relaxed nonequilibrium model, (b) the direct numerical simulation (DNS) of 2-D incompressible flow over a backward facing step, (c) a loosely-coupled approach for a 2-D fluid-structure interaction, and (d) a 3-D compressible unsteady flow simulation of vortex breakdown in delta wings. Using knowledge from dynamical systems theory, various types of spurious behaviors that are numerical artifacts were systematically identified. These studies revealed the various possible dangers of misinterpreting numerical simulation of realistic complex flows that are constrained by the available computing power. In large scale computations underresolved grids, semi-implicit procedures, loosely-coupled implicit procedures, and insufficiently long time integration in DNS are most often unavoidable. Consequently, care must be taken in both computation and in interpretation of the numerical data. The results presented confirm the important role that dynamical systems theory can play in the understanding of the nonlinear behavior of numerical algorithms and in aiding the identification of the sources of numerical uncertainties in CFD. In a series of papers (Yee et al. 1991 , 1995a ,b, Lafon & Yee 1996a ,b and Griffiths et al. 1992a , the dynamics of finite discretizations for constant time steps was studied using tools from dynamical systems theory. The approach was to take continuum nonlinear model ordinary differential equations (ODEs) and partial differential equations (PDEs) with known explicit solutions, discretize them according to various standard numerical methods, and apply techniques from discrete dynamical systems theory to analyze the nonlinear behavior of the resulting nonlinear difference equations. Particular attention was paid to the isolation of the different nonlinear behavior and spurious dynamics due to some of the numerical uncertainties that were observed in practical computational fluid dynamics (CFD) computations. Here, spurious solutions refer to numerical artifacts or physically irrelevant solutions. They are true solutions of the resulting nonlinear difference equations but are not solutions of the governing equations. The numerical schemes considered for these nonlinear model ODEs and PDEs were selected to illustrate the following different nonlinear behavior of numerical methods:. 0 Occurrence of stable and unstable spurious asymptotes above the linearized stability limit 0 Occurrence of stable and unstable spurious steady states below the linearized stability limit 0 Stabilization of unstable steady states by implicit and semi-implicit methods 0 Interplay of initial data and time steps on the permissibility of spurious asymptotes 0 Interference with the dynamics of the underlying implicit scheme by procedures in solving the nonlinear algebraic equations (resulting from implicit discretization of the continuum) 0 Dynamics of the linearized implicit Euler scheme solving the time-dependent equations vs. Newton's method solving the steady equation 0 Spurious dynamics independently introduced by spatial and time discretizations 0 Convergence rate and spurious behavior of high-resolution shock-capturing methods 0 Numerically induced & suppressed chaos, and numerically induced chaotic transients 0 Spurious dynamics generated by grid adaptations of the scheme (for constant time steps) of the scheme (for constant time steps) For time-accurate computations, variable time steps based on local error control (Aves et al., 1995 , Gresho 1997 or adaptive error control (Estep & William 1996 , Johnson 1995 are more suitable than constant time steps and grid spacings. However, these variable step size approaches are not at a point where they are either efficient or practical for CFD computations. The rationale for the study of the dynamics of constant step sizes is that the majority of time-accurate CFD computations are still based on constant time steps that satisfy the CFL condition.. The type of study performed in Yee et al. (1991) , Yee & Sweby (1994, 1995a,b) , Sweby & Yee (1994) , Lafon & Yee (1996a,b) and Griffiths et al. (1992a,b) is often viewed in the CFD community as of academic interest, and the results found are often dismissed as irrelevant to practical CFD computations. The general consensus is that except for special situations, an analysis similar to that in Yee et al. (1991) , Yee & Sweby (1994, 1995a,b) , Lafon & Yee (1996a,b) and Griffiths et al. (1992a,b) , is not possible for more complicated models. Nevertheless, they believe phenomena observed in simple model nonlinear problems can serve as guidelines in practical computations. To support their beliefs, Yee & Sweby (1996a,b) took the initiative to illustrate this connection. To correlate the knowledge gained from the aforementioned studies with practical computations, Yee & Sweby (1996a) identified some examples in time-marching to the steady state CFD computations that exhibit behavior similar to that of the simple model case. Yee & Sweby (1996b) gave an overview of the subject with some unsteady CFD examples. These later studies exemplify the importance of the knowledge gained in the nonlinear behavior of numerical schemes if the physics of the problem under study are not well understood and numerical simulations are the only viable means of solutions.. The present study continues the same line of research but concentrates on spurious behavior in underresolved grids and/or semi-implicit time discretizations for four realistic time-accurate CFD simulations. The numerical simulations consist of (a) a 1-D chemically relaxed nonequilibrium model, (b) the direct numerical simulation (DNS) of 2-D incompressible flow over a backward facing step, (c) a loosely-coupled approach for a 2-D fluid-structure interaction, and (d) a 3-D compressible unsteady flow simulation of vortex breakdown in delta wings. These examples were chosen based on their non-apparent spurious behavior that is difficult to detect without extensive refinement study and some knowledge from dynamical systems theory. The main purpose of the present study is to illustrate that inaccuracy associated with underresolved grids (e.g., disappearance of the fine details of the flow due to large numerical dissipation and/or low order of the scheme) and linear instability associated with time discretizations are easier to detect than spurious numerical solutions.. The allowable upstream initial inputs strongly depend on the grid spacing for the 1-D chemically relaxed nonequilibrium model. There is a drastic reduction in the basins of the initial upstream inputs with a slight increase in grid spacing. The semi-implicit time-like discretizations behave in a manner similar to standard explicit Runge-Kutta methods except the allowable upstream initial inputs are slightly less sensitive to the grid spacing. Spurious time-periodic solutions exist for grid spacings that are near the stability limit. Long time transients or chaotic-like transients followed by divergent or spurious time-periodic solutions were observed for the backward facing step simulations using a spectral element method if sufficiently long time integrations were performed. Insufficiently long time integration in conjunction with underresolved grids can lead to the erroneous conclusion of chaotic flow. Time-asymptotic numerical solutions are sensitive to initial conditions. Spurious time-periodic solutions and numerically induced chaotic-like behavior were observed for a 2-D fluid-structure interaction using a loosely-coupled (time lag) implicit temporal procedure of solving the fluid and structure equations. The spurious behavior ceased to exist if a fully-coupled procedure is used. Nonunique numerical solutions, some of which are spurious, were observed as a function of initial angles of attack in vortex breakdown of delta wing simulations using a fully-implicit procedure. A grid refinement study confirmed that the nonunique solutions are numerical artifacts and are independent of the studied initial angles of attack.. The present study serves to point out the various possible dangers of misinterpreting numerical simulations of realistic complex flows that are constrained by the available computing power. In large scale computations, underresolved grids, semi-implicit procedures, loosely-coupled implicit procedures, and insufficiently long time integration in DNS are most often unavoidable. Consequently, care must be taken in both computation and in interpretation of the numerical data. A good knowledge of the nonlinear dynamical behavior of numerical schemes is shown to be very useful. Details of the four CFD simulations will be presented in the next four sections. An analysis similar to that in Yee & Sweby (1994, 1995a,b) is only possible for the 1 -D chemically relaxed nonequilibrium model. Note that the background material presented in Section 2.1 on the interplay of initial data and time steps on the occurrence of spurious numerical solutions is relevant for Sections I1 -V. Refinement studies were performed on four CFD computations to illustrate spurious behavior in underresolved grids and/or semi-implicit temporal discretizations. These examples were chosen based on their non-apparent spurious behaviors that were difficult to detect without extensive refinement study and some knowledge from dynamical systems theory. Each example illustrates different spurious behaviors as discussed in Sections I1 -V.. Among the four examples, the DNS of the backward facing step concerning the transition point Reynolds number was by far the most controversial (in the early nineties). As illustrated, long time transients and numerically induced chaotic transients exhibited in large scale DNS or time-accurate computations are not easy to detect due to the CPU intensive nature of the simulations. The study also revealed a nonstandard guideline in grid clustering and grid adaptation. Fine grids are needed in regions of the smooth flow in order to avoid spurious solutions. The fine grid requirement of the smooth solution defies the standard logic in grid clustering and grid adaptation methods.. In the case of the 3-D vortex breakdown in delta wings, there already exists uncertainties in experimental studies by different investigators on multiple time-asymptotic solutions for a given static angle of attack. The spurious behaviors observed in numerical simulations contribute to additional uncertainties if systematic resolution assessments are not performed.. In all of the underresolved grid cases, the grids actually are finer than grids commonly used in realistic complex CFD simulations. The three semi-implicit methods considered are typical numerical procedures employed in active research areas such as chemical reacting flows, combustions, fluid-structure interactions, DNS and large eddy simulations. The present study serves to point out the various possible dangers of misinterpreting numerical simuIations of realistic complex flows that are constrained by available computing power. Table 3 .5. Initial data study for Re = 800 and M grid at t 5 2000. Fig. 2.1 . Bifurcation diagrams and basins of attraction of asymptotes of 6 explicit meL,,ods the two-species reacting flow. According to the experiments of Magness (1991) , vortex breakdown moves upstream over the delta wing when the angle of attack is slowly increased (i.e. in a quasistatic manner) beyond a critical value a,, M 30.7'. In order to study computationally this quasistatic onset of vortex breakdown, calculations were performed initially near a,, using a 1" increment in angle of attack and employing two of the grids previously described. Although smaller steps in incidence would had been desirable, this was not computationally feasible. The solutions obtained prior to breakdown onset (a = 30") using Grid I (Fig. 5.2a) and Grid 2 (Fig. 5.2d ) are found to be in reasonable agreement with each other. The computed onset of vortex breakdown occurred for Grid I when a was increased from 31" to 32", whereas for Grid 2, it took place between 30" and 31", in closer agreement with experiment (Magness, 1991) . The process by which breakdown appeared in the near-wake and moved over the wing was also found to be qualitatively similar for both grids. Based on these comparisons, one would conclude that the effect of numerical resolution on the quasistatic breakdown onset is small. However, as described below, the non-linear dynamic behavior near acr was found to be affected significantly by numerical grid resolution when large increments in the initial angle of attack are imposed.. Initial Angles of Attack: As discussed in Visbal (1995a) , some experimental studies have reported the existence of vortex breakdown static hysteresis on delta wings (i.e. multiple, time-asymptotic solutions for a given static angle of attack). Motivated by these findings, the existence of non-unique solutions for different initial conditions was investigated numerically. However, instead of observing static hysteresis phenomena similar to that in some of the experimental studies, spurious behavior due to numerics were encountered. Computations were performed on Grids I and 2 for a = 30" using three different initial angles of attack (YO. With the exception of the initial conditions and grids, all remaining numerical parameters (i.e., time step, damping coefficient, time-marching procedure and boundary conditions) were kept the same. In all of the computations, the fixed non-dimensional At = 0.005. This non-dimensional Fig. 5.2b . The frequency spectra of the streamwise velocity fluctuations at a point within the reverse flow region in the vortex core is given in Fig. 5.4 . It indicates the nearly periodic character of this spiral breakdown. For Case c, a third distinct solution was obtained for a large jump in the initial angle of attack from a0 = 17" to the desired a = 30". The lift coefficient time history appears to be chaotic-like. This flow field exhibits breakdown well upstream of the trailing edge, as shown in Fig. 5 . 2~ . Also, the corresponding velocity fluctuations associated with this stronger breakdown display multiple frequencies (Fig. 5.5 ). Both Cases b and c were run till tS = tU,/C M 80, during which time no tendency was observed for the breakdown to leave the wing. For the purpose of comparison with breakdown computations in tubes or isolated vortices, t ' = 80.0 corresponds in the present case to 2400 characteristic times based on the vortex core radius at X/C = 0.5,. Comparison of the three solutions in terms of the X-component of vorticity in the vortex core at a location upstream of breakdown (X/C = 0.4) is shown in Fig. 5 .6. All flow solutions are seen to be essentially the same at this upstream location, indicating that the different cases are not simply due to lags in the development of the vortex following a change in angle of attack.

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


Designing an Information System for Updating Land Records in Bangladesh: Action Design Ethnographic Research (ADER) Abstract Information Systems (IS) has developed through adapting, generating and applying diverse methodologies, methods, and techniques from reference disciplines. Further, Action Design Research (ADR) has recently developed as a broad research method that focuses on designing and redesigning IT and IS in organizational contexts. This paper reflects on applying ADR in a complex organizational context in a developing country. It shows that ADR requires additional lens for designing IS in such a complex organizational context. Through conducting ADR, it is seen that an ethnographic framework has potential complementarities for understanding complex contexts thereby enhancing the ADR processes. This paper argues that conducting ADR with an ethnographic approach enhances design of IS and organizational contexts. Finally, this paper aims presents a broader methodological framework, Action Design Ethnographic Research (ADER), for designing artefacts as well as IS. This is illustrated through the case of a land records updating service in Bangladesh. Given the wide view and diverse nature of the information systems (IS) discipline, there is a need for rigorous research methodologies. However, there are currently few rigorous methodologies in this discipline (Avgerou and Madon 2004) . One approach is to draw methodological contributions from reference disciplines with a view to investigating complex issues (Thompson 2012) . At the same time, IS scholars have been relentlessly striving to develop innovative methodologies, for example, action research in IS (Baskerville 1999b) , canonical action research (Davison et al. 2004) , design research (Hevner et al. 2004) , action design research (ADR) (Sein et al. 2011) , participatory action design research (Bilandzic and Venable 2011) and action design ethnographic research (ADER) (Alam, Brooks and Khan 2012) . Among these, ADR has emerged as a significant approach, for example by its publication in MIS Quarterly (one of the top journals of the discipline (Sein et al. 2011) ). This paper employs ADR to design an IS for service delivery of updating land records in Bangladesh. Through conducting ADR it has been found that ethnographic methodological approaches and tools enhance the processes of ADR for complex organizational contexts, such as exist in this service delivery process. Thus, the ethnographic methodology offers a significant complement to ADR. This paper applies ADR together with ethnographic approaches and tools. Consistent with this, it demonstrates how ethnographic approaches can enhance action design research (ADR) and examines whether they can work together as a novel and rigorous methodology for the IS discipline.. Ethnography is a paradigm, a methodology and a basket of research approaches and techniques which shows potential contributions for qualitative research within the IS discipline (Myers 2004; Nandhakumar and Jones 2002; Myers 1999; Nandhakumar and Jones 1997; Geertz 1973) . The ethnographic methodology has been used in IS research for over 20 years and played a substantial role in studying complex contexts and solving practitioners' problems (Walsham 1995; Orlikowski 1992) .. The nature of ethnographic research is empirical, longitudinal, inductive, interpretive, participant observation oriented and interventional. It seeks insights into everyday practices through applying its dual perspectives i.e. emic view (insider) and etic view (outsider) (Geertz 1983) . Further, ethnographic research can provide a deeper understanding in IS research (Walsham 1995) and shows complementarities with interpretive, applied and interventional research e.g., action research (Davison et al. 2004; Nilsson 2000; Baskerville 1999a ), design research (Gregor and Hevner 2013; Barab et al. 2004; Hevner et al. 2004) and action design research (Sein et al. 2011) .. On the other hand, ADR is a prescriptive research method that aims to conduct design research through action research steps and processes, without separating context-design-intervention-evaluation (Sein et al. 2011 ). This paper chose ADR with a view to designing an IS for a land records updating service from a public section organization and then evaluating its organizational context. Remarkably, ADR sees organizational contexts as inseparable from the design and redesign of artifacts and IS. However, since ADR has been derived from design research, it suffers from a lack of tools and approaches to understand the contexts. In addition, ADR aims to identify problems and build processes from the users' perspective alongside researchers' skills and knowledge. The ethnographic research is a hallmark for understanding the context and users' perspectives (Geertz 1983) . Thus, both ethnographic approaches and ADR are potentially complimentary, enabling better understanding of the context of users' perspectives and designing IS in organizational contexts. This paper seeks to frame a rigorous and broader methodology that can accommodate both ADR and ethnographic approaches with a view to understanding, intervening and evaluating the complex organizational contexts, designing and redesigning systems, processes and artifacts in the contexts in which learning and knowledge develop. ADR and ethnographic approaches have the potential to complement each other. Thus applying them together provides a better understanding of IS phenomena in the complex context of developing countries. This paper, therefore, aims to develop 'Action Design Ethnographic Research' as a methodological framework through using ethnographic approaches throughout the stages and processes of ADR, illustrated by the design of an IS for updating land records service in Bangladesh. The case of land records updating service has shown how the complex organizational contexts have continuously reshaped the designed IS. The complex organizational contexts include the inherent networks between the staff and the middlemen, the 'insiders' as well as the staff's intention of reshaping the designed IS and the continual insiders' behavior. The designed IS and the organizational contexts are iterative. In order to trace every iteration, requires methods, approaches, tools and techniques for capturing users' intentions, behaviors and organizational contexts. Interventional methodologies without appropriate approaches and tools cannot reveal the underlying contexts and hidden intentions of the users.. ADR as an interventional methodology has the ambition to formulate problems in the context and design of artifact or IS and to evaluate those contexts. However, it has not explicitly specified how to understand the contexts and what the approaches and tools are required for tracing, designing and evaluating the contexts. Implicitly ADR refers to mutual reciprocity between the users and researchers which is derived from the insider and outsider views within the ethnographic methodology. Moreover, the ethnographic methodology includes: participant observation (engaging and intervening with the contexts), interpreting the contexts through insider and outsider views and generating multiple layers of interpretations through thick description.. This study, thus, has applied ADR together with ethnographic approaches as a methodological framework, or Action Design Ethnographic Research (ADER), which has elicited complex organizational contexts around the land records updating service in Bangladesh. It has revealed how the staff and the middlemen are connected and how they reshaped every iteration of the designed and redesigned IS in this case. ADER offers iterative processes, alongside different lenses and roles, from problem formulation to design (BIE) to evaluation to learning, all intimately tied together.. ADER can play a key role in better understanding complex contexts, where informal practices dominate over the organizational rules and the designed IS and processes. ADER gives the opportunity to formulate problems and interact with BIE through multiple cycles such that the contextual problems can be formulated from the data i.e. rather than relying on the researchers' knowledge and perceptions, the problem can be formulated through empirical evidence arising from engagement and observation.. Finally, understanding on-going use and continuous reshaping of designed IS by users and contexts is challenging where practices are diverse, hidden and deliberately deceptive. Thus this ADER study has provided the researchers' engaged with the context as insider with the opportunity to trace the continuous reshaping of the IS by the users. Moreover, an objective evaluation is often helpful in the case of interventional research, thus applying an outsider viewpoint helps to conduct context free evaluation. However, the insider view also brings forth the researchers' reflection that could be seen as subjective evaluation. Finally, ADER has the potential to identify learning and transfer it to the practitioners through model building. Although conducting ADER is challenging because it requires long term involvement, commitment and skills, it has the potential to provide valuable ethnographic analysis i.e. 'thick description', for both IS practitioners and professionals. Bangladesh is a populous country with a total of 160 million people, and an average land per person of only 0.22 acre. However, it relies on an agro-economy because agriculture contributes 60 % of total GDP. Therefore, land is the only capital and source of livelihood for the majority of people. About 80 % of people depend on agriculture and the rate of land ownership transfer is very high. Thus, updating of land records is a significant service for citizens.. Updating land records, a core public service in the public sector, refers to changes in existing records. It is a process of updating the name of the owner of land in record registers after land registration (purchase) or inheritance (either loosing or gaining). Due to change of ownership, updating of existing records is essential to legalize land ownership.. Updating of land records service has been identified as problematic and outdated, a source of corruption and civil and criminal litigations by the government itself, the development partners, practitioners and civil society. Rampant corruption in this service delivery is a barrier to economic growth of the country. In addition to fraud, forgery and physical assaults, including murders took place due to land records related litigations. About 80 % of civil cases and 70 % of criminal cases are involved with land records related matters (Barakat and Prasanta 2004) .. This service is delivered from a public sector organization, namely the Upazila Land Office (ULO). Upazila refers to a sub-district, the bottom tier of public administration in the country. There are a total of 500 ULOs in the country. The ULO is headed by an Assistant Commissioner of Land (ACL). A ULO comprises of several local union Land Offices (LO) that is headed by a Union Assistant Land Officer (UALO).. There are two main provisions for updating land records: ULO's own initiatives and citizens' applications to the ULO for updating land records. Firstly, the ULO can update land records on the basis of two information networks: one, receiving a Land Transfer Notice after land registration is completed by the land Registration Office and the other is a report on updating land records from a LO. According to the legal provisions, following land registration a Land Transfer Notice is issued to the ULO for updating the appropriate records. On the other hand, where there is death of a landowner or any change takes place in the type of land, the LO reports to the ULO for the purpose of updating land records. However, functionally sending LO's reports to the ULO for updating land records as well as ULO's own initiative on the basis of Land Transfer Notice rarely takes place.. On the other hand, citizens can apply to the ULO to update land records. When citizens apply for a land records update, a number of complications can occur: Usually a middleman submits a citizens' application with the requisite fees and required documents: land registration deeds, certified copies of existing records, sketch map of the land plots, and inheritance certificate from local union council. Thereafter the ULO conducts an initial scrutiny of the fees and supporting documents and asks the concerned LO to send a field inspection report on the application. The ULO hears the applicant or first party (new owner) and the second party (the seller and related individuals) and consults the LO's report and scrutinizes the registers of the ULO. Finally, the ULO either approves or rejects the application i.e., updates the land record or no change made.

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


Input-state-output representations and constructions of finite support 2D convolutional codes Abstract Two-dimensional convolutional codes are considered, with codewords having compact support indexed in N 2 and taking values in F n , where F is a finite field. Input-state-output representations of these codes are introduced and several aspects of such representations are discussed. Constructive procedures of such codes with a designed distance are also presented. Multidimensional (mD) convolutional codes are the higher dimensional (nontrivial) generalizations of one-dimensional (1D) convolutional codes. These codes may prove useful in transmission of mdimensional data, such a 2D pictures, 3D animation, etc. (see [4, 13, 17] ), or for the storage of digital information, [14] . While 1D convolutional codes have been thoroughly understood, the literature about mD convolutional codes is quite limited. Moreover, most of the existing research is focused on algebraic aspects and fundamental issues. Fornasini and Valcher [2] introduced the general theory for the study of two-dimensional (2D) convolutional codes constituted by sequences indexed on Z 2 , and discussed issues such as the characterization of such codes in terms of their internal properties and input-output representations. In [15] the same authors considered 2D convolutional codes in which the codewords have compact support in Z 2 , and presented several properties of their encoders and syndrome formers (parity-check matrices) under different hypotheses on the code structure. They also introduced the dual codes of such codes. Multidimensional (mD) convolutional codes having finite support in N m were first studied by Weiner in [16] , where he explored some connections of mD convolutional codes with commutative algebra and algebraic geometry. In [3] , GluesingLuerssen, Rosenthal and Weiner analyzed the relation between multidimensional convolutional codes and systems. In the same line, a different approach is given by Kitchens in [6] , where he set a concept of mD convolutional code from the symbolic dynamical point of view, establishing five different equivalent notions. More recently, for the purpose of studying mD convolutional codes from a more practical point of view, R. Lobo, in his dissertation [8] introduced the concept of locally invertible encoders which lead to a particular class of basic convolutional codes. However crucial aspects of the theory of convolutional codes such as the construction of good convolutional codes and of minimal representations of a code have not been deeply studied by these authors. In particular, only Weiner studied related problems in [16] for a very particular and simple class of multidimensional codes, the unit memory codes, which are the ones whose encoders are constituted by polynomials of either first or zero degree.. In this paper, we consider two-dimensional (2D) convolutional codes over a finite field F, constituted by polynomials in two indeterminates with coefficients in These codes are called 2D finite support convolutional codes. We study such codes by means of state space representations. Rosenthal and collaborators [10, 12] defined input-state-output (ISO). representations for 1D finite support convolutional codes and used these representations in the construction of good codes. We introduce such representations for 2D finite support convolutional codes, considering the Fornasini-Marchesini state space model for 2D linear systems [1] . Minimality of such representations is very important not only for an efficient implementation of the code but also for the construction of good codes. However, minimality of the Fornasini-Marchesini state space models is not completely characterized as it happens for 1D state space models [1, 5] . In this paper, we define the complexity of a 2D finite support convolutional code, similarly to the 1D case, and show that it is a lower bound on the dimension of such representations. In addition, we give a sufficient condition to construct minimal ISO representations of 2D finite support convolutional codes. The advantages of this approach is that ISO representations show how the encoding algorithm proceeds by explicitly displaying the corresponding evolution of the state space (the memory of the code). In particular, we take advantage of the nature of such representations to construct 2D finite support convolutional codes with a designed distance. The structure of the paper is as follows. We begin by introducing, in Section 2, some necessary background on polynomial matrices in two indeterminates and on the Fornasini-Marchesini state space models, centering around concepts such as reachability and observability. Section 3 is devoted to provide an overview of the theory of 2D finite support convolutional codes already available in the literature. In Section 4 we introduce the ISO representations for 2D finite support convolutional codes, study minimality and characterize the properties of such representations to obtain non-catastrophic 2D finite support convolutional codes. We then consider, in Section 5, the restriction of 2D finite support convolutional codes to the semi-axes N × {0} and {0} × N, and show that they constitute 1D finite support convolutional codes. In Section 6 we relate the distance of a type of 2D codes with the distance of its restriction to the axes. This allow us to present a construction of 2D finite support convolutional codes with a designed distance. In this paper we have established a framework to represent 2D finite support convolutional codes by means of linear systems. To that purpose we have introduced input-state-output representations, which have several advantages with respect to the representations used by Fornasini and Valcher in [2] . In particular, we have overcome the important issue of minimality, namely, our input-stateoutput representations allowed us to construct minimal representations of 2D convolutional codes, whereas such constructions are not known for the representations used in [2] . We have characterized catrastrophicity of these representations and show how it is possible to make constructions with a designed distance. We think that these representations are a good tool for the construction of optimal 2D finite support convolutional codes. It is a topic of future research to obtain an upper bound for the maximal distance of 2D convolutional codes and construct codes that reach such a bound. Moreover, we believe that such an approach will lead to a future algebraic decoding algorithm.

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 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 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 O O O O O O O O O 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 O O O O O O O O


Bacteria Hunt Evaluating multi-paradigm BCI interaction The multimodal, multi-paradigm brain-computer interfacing (BCI) game Bacteria Hunt was used to evaluate two aspects of BCI interaction in a gaming context. One goal was to examine the effect of feedback on the ability of the user to manipulate his mental state of relaxation. This was done by having one condition in which the subject played the game with real feedback, and another with sham feedback. The feedback did not seem to affect the game experience (such as sense of control and tension) or the objective indicators of relaxation, alpha activity and heart rate. The results are discussed with regard to clinical neurofeedback studies. The second goal was to look into possible interactions between the two BCI paradigms used in the game: steady-state visually-evoked potentials (SSVEP) as an indicator of concentration, and alpha activity as a measure of relaxation. SSVEP stimulation activates the cortex and can thus block the alpha rhythm. Despite this effect, subjects were able to keep their alpha power up, in compliance with the instructed relaxation task. In addition to the main goals, a new SSVEP detection algorithm was developed and evaluated. In this study a game was used to investigate the interaction with BCIs in a natural environment. The goal of the study was twofold.. Firstly, the aim was to show the efficacy of alpha-based neurofeedback. Sham feedback was used to examine the effect that the perceived control over the BCI has on the game experience. It was expected that real feedback would subjectively lead to a more relaxed state and greater feeling of confidence. Furthermore, differences in objective indicators of relaxation were expected. The analysis showed no difference in subjective and objective indicators of relaxation or control. Several possible methodological and technical factors for the inefficacy of the neurofeedback were identified.. Future work should try to evaluate their role to enable a successful alpha neurofeedback-based BCI. Secondly, the interaction between different, simultaneously used BCI paradigms, namely alpha neurofeedback and SSVEP, were studied. Due to the antagonistic nature of these two mental tasks, relaxation and focused attention, and due to the blockage of the alpha rhythm during visual stimulation, a decrease of alpha was expected during SSVEP. However, an increase in alpha power was found. Further analysis suggested that the increase was due to the increase of the power in the first harmonic of the SSVEP stimulation (7.5 Hz), which is close to the alpha band (8) (9) (10) (11) (12) . Careful examination of possible influences of multiple BCI paradigms on each other is advised.. Furthermore, a novel SSVEP detection algorithm was developed and compared against a standard method. The advantage over the standard method is an independence of prior baseline recordings. The viability of the new approach was shown by evaluation results comparable to the standard method. A multimodally controlled multi-paradigm BCI game was designed and implemented [30] to test the simultaneous use of several BCI paradigms in addition to traditional input in one game. One goal of the current study was to examine the effect of neurofeedback on the ability of the user to manipulate his mental state (i.e. relaxation) to control an aspect of the game world (i.e. the bacteria's flight speed). A second goal was the identification of possible interactions between the two BCI paradigms employed within the game. Additionally, the novel approach to SSVEP detection is discussed. In the previous section the game and system design were described. In this section we will outline the methods used to study the application of neurofeedback and SSVEP BCI in the context of multimodal gaming.

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


Nonparametric Discriminant Analysis for Face Recognition Abstract-In this paper, we develop a new framework for face recognition based on nonparametric discriminant analysis (NDA) and multiclassifier integration. Traditional LDA-based methods suffer a fundamental limitation originating from the parametric nature of scatter matrices, which are based on the Gaussian distribution assumption. The performance of these methods notably degrades when the actual distribution is non-Gaussian. To address this problem, we propose a new formulation of scatter matrices to extend the two-class NDA to multiclass cases. Then, in order to exploit the discriminant information in both the principal space and the null space of the intraclass scatter matrix, we develop two improved multiclass NDA-based algorithms (NSA and NFA) with each one having two complementary methods that are based on the principal space and the null space of the intraclass scatter matrix, respectively. Comparing to the NSA, the NFA is more effective in the utilization of the classification boundary information. In order to exploit the complementary nature of the two kinds of NFA (PNFA and NNFA), we finally develop a dual NFA-based multiclassifier fusion framework by employing the overcomplete Gabor representation for face images to boost the recognition performance. We show the improvements of the developed new algorithms over the traditional subspace methods through comparative experiments on two challenging face databases, the Purdue AR database and the XM2VTS database. LINEAR Discriminant Analysis (LDA) [6] is a representative subspace analysis method which has been extensively studied for face recognition. It aims to find the most discriminative features by maximizing the ratio of the determinant of the between-class scatter matrix to that of the within-class scatter matrix. To enhance the stability and performance of LDA, a number of improved LDA-based methods [1] , [2] , [4] , [5] , [11] , [15] , [18] , [19] , [20] , [21] , [24] , [25] , [26] , [27] , [28] have been proposed. However, most of the existing LDA-based methods inherit the parametric nature from the traditional LDA approach: The construction of the scatter matrices relies on the underlying assumption that the samples in each class satisfy the Gaussian distribution. Thus, they suffer performance degradation in cases of non-Gaussian distribution. In [7] , a nonparametric technique is developed to overcome the problem by introducing a new definition for the between-class scatter matrix, which explicitly emphasizes the samples near boundary. Under the new formulation, by utilizing the whole training set, instead of merely the class centers, and weighting the sample pairs according to their contributions to discrimination, the learning algorithm generates a model more adapted to the sample space, especially in the non-Gaussian cases. However, this nonparametric definition is restricted to the two-class cases and cannot be applied in the multiclass classification such as face recognition. In this paper, we propose a new formulation of between-class scatter matrix by extending the definition of the original nonparametric between-class scatter matrix to the multiclass problem [9] . Based on this new formulation, a new method called multiclass nonparametric discriminant analysis (NDA) is proposed.. The well-known small sample size problem often arises when applying LDA in face recognition due to the high dimensionality of the sample space and the limited training set. In these cases, the within-class scatter matrix becomes nearly singular, which would incur serious instability and overfitting. In order to address this problem, we further propose a new method called principal nonparametric subspace analysis (PNSA) to extract nonparametric discriminating features within the principal subspace of withinclass scatter matrix. This helps stabilize the transformation and thus improves the generalization performance.. A limitation of the PNSA method is that it only utilizes the principal subspace of the intrapersonal scatter with the whole null space discarded. It has been shown that the null space of withinclass scatter also contains important discriminative information [3] , [19] , so we develop another method called null-space nonparametric subspace analysis (NNSA) to make use of the null space of the within-class scatter matrix.. However, the within-class scatter matrix in nonparametric subspace analysis (NSA) still has the parametric form, which may cause recognition performance degradation. In order to address this problem, we further propose a new formulation of scatter matrices in which both the within-class and between-class scatter matrices are redefined in nonparametric form to better exploit the discriminant information in training data. Based on this new formulation, an enhanced NSA algorithm called nonparametric feature analysis (NFA) is derived accordingly. Similar to NSA, we also derive two additional methods for the principal space and the null space: The principal space NFA (PNFA) is based on the principal space of the within-class scatter matrix and the null-space NFA (NNFA) is based on the null space of the within-class scatter matrix. Inspired by the dual-space LDA in [19] , we can see that the two NFA-based approaches, PNFA and NNFA, are inherently complementary. Thus, it is desirable to combine the two types of classifiers.. We apply the developed NFA methods on Gabor features for face recognition. Gabor wavelets have been shown to outperform original appearance features [10] , [22] . However, previous methods either downsample the Gabor responses [10] or use only the Gabor responses at certain fiducial points [22] . To fully utilize all the information embedded in the overcomplete Gabor representations without creating an extremely high-dimensional Gabor space, we use multiple classifiers in a dual NFA framework to handle the high dimensionality of Gabor features. Significant improvement over conventional subspace methods are achieved as demonstrated on two challenging face databases, the Purdue AR database and the XM2VTS database. We chose these two data sets because of their large variation of face samples over a reasonable data size. In this section, we conduct experiments on two standard face data sets, the AR database [12] and the XM2VTS database [13] . Comparing to other standard data sets, these two data sets have large within-class variations for a relatively large number of people, thus showing a higher degree of non-Gaussian distribution. To better evaluate the recognition performance with geometric and photometric interferences filtered out, we preprocess the face images through the following steps:. 1. Rotate the face images to align the vertical face orientation. 2. Scale the face images so that the distances between the two eyes are the same for all images. 3. Crop the face images to remove the background and the hair region. 4. Apply histogram equalization to the face images for photometric normalization.

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


Super-Resolution in Respiratory Synchronized Positron Emission Tomography Abstract-Respiratory motion is a major source of reduced quality in positron emission tomography (PET). In order to minimize its effects, the use of respiratory synchronized acquisitions, leading to gated frames, has been suggested. Such frames, however, are of low signal-to-noise ratio (SNR) as they contain reduced statistics. Super-resolution (SR) techniques make use of the motion in a sequence of images in order to improve their quality. They aim at enhancing a low-resolution image belonging to a sequence of images representing different views of the same scene. In this work, a maximum a posteriori (MAP) super-resolution algorithm has been implemented and applied to respiratory gated PET images for motion compensation. An edge preserving Huber regularization term was used to ensure convergence. Motion fields were recovered using a B-spline based elastic registration algorithm. The performance of the SR algorithm was evaluated through the use of both simulated and clinical datasets by assessing image SNR, as well as the contrast, position and extent of the different lesions. Results were compared to summing the registered synchronized frames on both simulated and clinical datasets. The super-resolution image had higher SNR (by a factor of over 4 on average) and lesion contrast (by a factor of 2) than the single respiratory synchronized frame using the same reconstruction matrix size. In comparison to the motion corrected or the motion free images a similar SNR was obtained, while improvements of up to 20% in the recovered lesion size and contrast were measured. Finally, the recovered lesion locations on the SR images were systematically closer to the true simulated lesion positions. These observations concerning the SNR, lesion contrast and size were confirmed on two clinical datasets included in the study. In conclusion, the use of SR techniques applied to respiratory motion synchronized images lead to motion compensation combined with improved image SNR and contrast, without any increase in the overall acquisition times. A MAP based super-resolution algorithm has been implemented and applied to respiratory gated images. Results on simulated datasets including realistic respiratory motion and on clinical data show that this method leads to motion compensation combined with globally improved image SNR and locally improved contrast and resolution enhancement. Respiratory motion in PET/CT imaging is responsible for a reduction in overall image quality associated with a loss of both quantitative and qualitative accuracy. Proposed solutions are based on the use of respiratory synchronized acquisitions which reduce the effects of respiratory motion. However, such gated acquisitions are also associated with an increase in PET image statistical noise since only part of the data available throughout a respiratory motion average acquisition is used. In order to account for these effects the synchronized gated datasets can be combined using deformable image registration applied either during or after the reconstruction of the respiratory synchronized datasets.. An alternative approach is proposed in this work based on the use of a MAP super-resolution algorithm. The principle of super-resolution is to use motion information derived from an image sequence representing different views of the same scene to enhance the quality of the image sequence by taking advantage of the additional spatio-temporal information available. In the proposed methodology motion information inherent in the respiratory synchronized PET image frames is combined with a MAP super-resolution algorithm. The proposed algorithm was assessed using both simple and patient specific anthropomorphic simulated datasets and was tested on a couple of clinical PET studies. As expected in all of the simulated datasets used in the comparative evaluation the respiratory synchronized images led to the worst signal to noise ratio as a result of the reduced count statistics. Despite this, an improvement in the size of the detected lesions and the associated lesion contrast relative to the respiratory motion average images was measured. This improvement was however smaller than the one achieved by the same synchronized frame image reconstructed using the same number of counts as the respiratory motion average image. These results are in agreement with previous studies by others [10] .. A respiratory motion corrected image was obtained based on previously proposed approaches by summing the individually reconstructed synchronized frame images following deformable image registration [5] , [6] . As expected the performance of this approach in terms of SNR as well as in determining lesion location and/or size was similar to that of the single respiratory synchronized frame using the same count statistics as the motion average image. However, one thing worth noting was the larger variability in determining the lesion size as a function of the lesion location in the lung field which was also reported by Lamare et al. [10] . This variability can be explained by a combination of two factors. The first is the limited precision of the deformable model used to align the individual synchronized frame images. Although spline-based motion models are well adapted to modeling respiratory motion [9] , [10] , they can fail on small structures or noisy images. The second factor is the nonuniformity of respiratory motion in the lungs. Lesions located in the lower part of the lung fields are generally associated with a larger magnitude motion than lesions located in the upper part. The super-resolution images calculated with and without PSF had higher SNR (by a factor of over 4 on average) and lesion contrast (by a factor of 2) than the single respiratory synchronized frame using the same reconstruction matrix size. In comparison to the motion corrected or the motion free images a similar SNR was obtained. However, the SR image without PSF modeling achieved a better performance in terms of recovered lesion contrast (up to 40%) and size (up to 10%), particularly in the case of the smaller lesions (7-11 mm). Incorporating resolution modeling in the SR model further improved contrast and resolution (FWHM was lower). In addition, although both SR images use the same motion fields as the image obtained by summing the frames previously registered to a common position, a smaller variability in the measured lesion size with respect to the lesion location in the lung fields was seen in the standard NCAT SR images. This can be explained by the function minimized during the super-resolution process (see (9) , Section II-A), where we seek to minimize the sum of differences between each synchronized frame and the super-resolution image registered to this frame. By not treating each registered frame independently, this process encourages a consistency of the registration. Our super-resolution approach should be, therefore, less dependent on the registration process accuracy than the register-and-sum technique.. The proposed approach was implemented on a standard personal computer (Intel Xeon 3 GHz dual core). The super-resolution process considering six synchronized frames of dimension voxels took min. For comparison the reconstruction of each individual respiratory synchronized image on the same computer took min. Three previously proposed super-resolution PET enhancing methods based on the use of the IBP and POCS super-resolution algorithms involved either wobbling the detector arrays, moving the patient bed during the acquisition, or shifting the reconstruction matrix. The obvious advantage of the proposed approach is the exploitation of the respiratory motion information, which is an inherent part of the acquisition process, to improve overall image quality. In addition, this methodology does not require longer acquisition periods as is the case with the implementation requiring successive shifts of the patient bed. Furthermore, and unlike the approach consisting in shifting the reconstruction grid, the noise content in each of our frames is independent, which allows a better noise recovery in the final image.. Furthermore, previously proposed approaches for the use of super-resolution in PET concentrated on the use of the IBP and the POCS algorithms. The advantages of the MAP approach over these algorithms are threefold. First, the algorithm converges towards a unique solution. The result is thus independent of the initial image or of the number of iterations used. Second, unlike the one used in IBP, the cost function to be minimized incorporates an a priori term which can lead to a solution having some desirable properties chosen by the user. Finally, the blurring introduced by the finite resolution of the PET detectors can be modeled in the algorithm, which allows better resolution recovery.. There are, however, two potential pitfalls when using superresolution to correct respiratory motion instead of using one of the previously proposed approaches. The first stems from the nonuniformity of the respiratory motion. The previous approaches led to a resolution enhancement uniform throughout the image. In our approach, however, the resolution enhancement is nonuniform and takes place only in moving parts of the image. In parts of the anatomy where there is no respiratory motion (such as for example in the spine), the sample points are the same for all gated frames. Super-resolution in these regions thus performs a mere averaging of all frames, enhancing SNR but not resolution. Thus, in this sense the resulting images have a nonuniform resolution recovery.. The second potential pitfall of the proposed approach is its dependency on the deformable registration step, which is performed independently of the super-resolution process. Inaccuracies in the registration parameters can lead to errors in the high resolution image. However, as we explained before in this section, the super-resolution step can, to some extent, compensate for inconsistencies on the registration between individual frames.. Other limitations of the current implementation of the algorithm include the performance dependency of the super-resolution image on the regularization parameters, which are currently chosen empirically, as well as the modeling of the scanner PSF as a spatially invariant Gaussian blur. Although, this simplified model was shown to yield better results in terms of contrast and resolution enhancement than the absence of blur modeling, a more accurate spatially variant PSF model might yield further resolution enhancement. Finally, consistent differences were observed in the performance of the different reconstructions in terms of the parameters considered (contrast, SNR, FWHM, and position error), demonstrating a better performance for the super-resolution reconstructed images for both simulated and clinical datasets. However, these differences were not statistically significant given the small number of both simulated and clinical datasets and the potential clinical impact of the proposed approach needs to be demonstrated in larger patient image series.. Future developments will address these issues by developing an automatic framework for the selection of optimal regularization parameters for each image sequence, and estimating the motion parameters within the super-resolution process. Within this context the alternative that will be pursued is the integration of the super-resolution process within the reconstruction of the 4-D datasets compared to its current application following the independent reconstruction of the individual respiratory synchronized frame images. The influence of the regularization parameters on convergence and image quality was assessed using the results presented in Fig. 3 . Since contrast and FWHM results were similar for all spheres, only the ones concerning the 22 mm sphere were presented in this figure. All problems with a nonzero value of were found to converge to a unique solution. Globally, background SNR, contrast and FWHM were more sensitive on than on . As expected, background SNR was higher for higher values of (+64.5% to +90.6% when is multiplied by 10). However, as increases, background SNR stays within 5% of the same value. Conversely to background SNR, lesion contrast increases with between 0.01 and 0.1 (an average increase of 17.9%), while lesion contrast decreases for (-1.4% to -16.0%). Once again, variations of lead only to moderate variations of contrast (-1% to 5.9%). Concerning FWHM, increases in both and lead to increased estimated lesion size, although variations with respect to are of larger amplitude than variations with respect to ( versus ). The super-resolution images were computed with an upsampling set to 2 in every direction. The regularization parameters and for each of the simulated anthropomorphic and clinical datasets were chosen in order to achieve both high SNR and high contrast. Convergence was not a criterion since all problems with a nonzero were found to converge to a unique solution. For all simulated and clinical datasets and were used. The second simulated dataset, being reconstructed with smaller voxels and similar statistics as the other simulated datasets resulted in noisier frames and hence required more regularization ( and ). Fig. 4 presents a comparison of the seven different reconstructed images of the simulated NCAT phantom with the spherical lung lesions considered in the evaluation (see Section II-D). For these same NCAT simulated datasets, Table I contains the results of the quantitative analysis on the SNR, contrast, and FWHM as a function of lesion size.. The use of only part of the statistics associated with the synchronized images (Frame 1) leads to a lower SNR by a factor of 1.5-3 and 1.1-2 relative to the image corresponding to the same frame reconstructed with the entire respiratory cycle count statistics (All Counts image) and the Motion Average image respectively. A higher resolution synchronized frame image (Large Frame 1) leads to the worst SNR with a reduction by relative to the standard reconstruction matrix (Frame 1). Finally the super-resolution frames, computed with or without PSF modeling (SR and SR without PSF modeling) lead to an SNR equivalent to that of the images derived either from the sum of the images realigned to the same frame in the respiratory cycle (Sum of Registered Frames) or from the same frame reconstructed with the entire respiratory cycle count statistics (All Counts image).. In terms of lesion contrast, the reduced SNR associated with the Large Frame 1 leads to the worst lesion contrast performance, with a reduction of 15%-30% (depending on the lesion size) relative to the standard synchronized image Frame 1 . On the other hand, the Frame 1 image exhibits similar contrast compared to the Motion Average image ( differences throughout the range of lesion sizes considered), demonstrating that the gain of reducing respiratory motion effects is practically cancelled by the reduced statistics associated with a single synchronized frame image. The lesion contrast performance of the Sum of Registered Frames image was very close to that of the All Counts images. Finally, the highest contrast was associated with the SR images, particularly when the modeling of the scanner PSF was included. Lesion contrast in the SR and SR without PSF modeling images was respectively up to 15% and 10% higher than in the All Counts image.. The motion average reconstructed image systematically yielded the largest recovered lesion size, with a FWHM larger than that of the All Counts images by on average 44%, 26%, 37%, and 31% for the 7, 11, 15, and 21 mm lesions, respectively. In addition, this same image was associated, as a result of the respiratory motion effects, with the largest variability in the recovered lesion size as a function of their placement in the different parts of the lung field. On the other hand, differences in the recovered lesion sizes were on average between 7% and 12% between the respiratory synchronized image (Frame 1) and the All Counts image, increasing to 14%-24% for the high resolution respiratory synchronized image (Large Frame 1). More importantly, this image showed an increase by over a factor of 2 in the variability of the recovered size throughout the lung fields when compared to the standard Frame 1. This increased variability was most probably caused by uncertainties associated with the measured decrease in the SNR highlighted above in this same section.. Considering the respiratory motion corrected images, the recovered lesion sizes from the Sum of Registered Frames image were within 5% of the ones obtained using the All Counts image for the entire range of lesion sizes considered. The associated variability was similar to that obtained for the Frame 1 or the All Counts images. Finally, the smallest FWHM and associated variability as a function of lesion location was measured with the SR images, particularly noticeable for the smaller size lesions (7-11 mm), with average differences of up to -18% relative to the All Counts image. The advantage of PSF modeling within the super-resolution process was mostly seen for the smaller lesion sizes, with an 8% larger FWHM measured in the SR without PSF modeling image. On the other hand, the FWHM of the larger lesions was similar (differences ) in both images. is presented in Tables II and III In terms of image contrast smaller differences were seen in performance between the different reconstructed images for each of the two simulated datasets. The worst contrast (between -10% and -20% relative to the All Counts image) was the one obtained from the Motion Average and the Large Frame 1 images. Globally similar levels of contrast were seen in the images accounting for the effects of the respiratory motion (Sum of Registered Frames, SR and SR without PSF modeling). Concerning the lesion size and location the SR images allowed the best accuracy, being closer to the real simulated lesion sizes (26 mm and 18 mm for the tumor slice location considered on the first and second simulated dataset, see Figs. 5 and 6) and localizations as well as to the results obtained using the All Counts image. Although globally a similar performance was obtained using the Sum of Registered Frames images, there was a larger variability with lesion location in the accuracy of the results concerning both the recovered lesion size and position. Finally, although the use of respiratory gating led to better results than the Motion Average image, the increased noise levels associated with the lower statistical quality for Frame 1 and Large Frame 1 images compromised the accuracy of determining both the lesion size and associated location, explaining the higher FWHM and position errors observed.. The images obtained for the two clinical datasets are presented in Figs. 7 and 8 . Corresponding quantitative results of SNR, contrast, and FWHM assessing lesion size are presented in Table IV . As a result of the reduced statistics the respiratory synchronized images led to the worst SNR ( to ) relative to the respiratory average or respiratory motion corrected (Sum of Registered Frames, SR and SR without PSF modeling) images. The SR images led systematically to the highest SNR compared to the Motion Average andSum of Registered Frames images. In terms of both contrast and FWHM there were only small differences between the respiratory synchronized Frame 1 and the Motion Average image. The highest lesion contrast was obtained using the SR image with PSF modeling compared to the Motion Average and Sum of Registered Frames images. Similar trend but with smaller magnitude differences ( and compared to the Motion Average and the Sum of Registered Frames images, respectively) were obtained for the lesion size measurements.

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