Quality Assessment for Hyperspectral Airborne Systems

282 Pages·2015·12.32 MB·English

Quality Assessment For Hyperspectral Airborne Systems
From Data Up To Land-Products
DISSERTATION
zur Erlangung des akademischen Grades
Doctor Rerum Naturalium
(Dr. rer. nat.)
im Fach Informatik
eingereicht an der
Mathematisch-Naturwissenschaftliche Fakultät
der Humboldt-Universität zu Berlin
von
Ingénieur Généraliste Supaéro,
Diplômé de l’Institut Supérieur de l’Aéronautique et de l’Espace:
Grégoire Henry Gérard Kerr
Präsident der Humboldt-Universität zu Berlin:
Prof. Dr. Jan-Hendrik Olbertz
Dekan der Mathematisch-Naturwissenschaftliche Fakultät:
Prof. Dr. Elmar Kulke
Gutarter/in:
1. Prof. Dr. Ralf Reulke
2. Prof. Dr. Thomas Udelhoven
3. Dr. René Preusker
Tag der Verteidigung: 30.07.2015

Short Abstract
This work proposes a methodology for performing a quality assessment on the complete air-
borne hyperspectral system, thus ranging from data acquisition up to land-product generation.
It is compliant with other quality assessment initiatives, such as the European Facility for Air-
borne Research (EUFAR), the Quality Assessment for Earth observation work-group (QA4EO)
and the Joint Committee for Guides in Metrology (JCGM). These are extended into a generic
framework allowing for a flexible but reliable quality assessment strategy. Since airborne hyper-
spectral imagery is usually acquired in several partially overlapping flight-lines, it is proposed to
use this information redundancy to retrieve the imagery’s internal variability. The underlying
method is generic and can be easily introduced in the German Aerospace Center DLR’s hyper-
spectral processing chain. The comparison of two overlapping flight-lines is not straightforward,
should it only be because the presence of geo-location errors present in the data. A first step
consists in retrieving the relative variability of the pixel’s geo-locations, hence providing pairs
of pixels imaging the same areas. Subsequently, these pairs of pixels are used to obtain quality
indicators accounting for the reproducibility of mapping-products, thus extending the EUFAR’s
quality layers up to land-products. The third stage of the analysis consists of using these reli-
ability results to improve the mapping-products: it is proposed to maximise the reliability over
the mapping-methods’ parameters. Finally, the repeatability assessment is back propagated to
the hyperspectral data itself. As a result, an estimator of the reflectance variability (including
model-, and scene-induced uncertainties) is proposed by means of a blind-deconvolution ap-
proach. Altogether, this complements and extends the EUFAR quality layers with estimates
of the data and products repeatability while providing confidence intervals as recommended by
JCGM and QA4EO.
Keywords— Airborne Remote Sensing, Hyperspectral Processing Chain, Imaging Spectro-
scopy, Quality Assessment, Signal Processing, EUFAR, QA4EO
i
Zusammenfassung
Im Rahmen dieser Arbeit wird ein Konzept entwickelt und umgesetzt, welches eine um-
fassende Bewertung von Daten flugzeuggetragener hyperspektraler Systeme ermöglicht. Es baut
auf mehreren aktuellen Initiativen zur Erfassung der Datenqualität flugzeuggetragener Sensoren
auf: Der ’European Facility for Airborne Reseach’, der ’Quality Assessment for Earth Obser-
vation Workgroup’ und dem ’Joint Committee for Guides in Metrology’. Bei der Befliegung
eines Gebietes mit hyperspektralen Sensorsystemen werden mehrere, teilweise sich überlap-
pende, Flugstreifen aufgenommen. Es wird vorgeschlagen, die Bildinformationen dieser Über-
lappungsbereiche als redundant anzusehen und so die innere Variabilität der Daten zu erfassen.
Die jeweils zwischen zwei Flugstreifen auftretende Variabilität kann (aufgrund unterschiedlicher
Blickrichtungen) als ungünstigster anzunehmender Fall (’worst-case’) betrachtet werden und er-
gänztdaherexistierendeAnsätze,diesichaufdieAuswertunghomogenerFlächenkonzentrieren.
Das entwickelte Konzept ist auf unterschiedliche Sensorsysteme anwendbar, somit generisch und
kannproblemlosindieaktuelleDatenprozessierungskettedesDeutschenZentrumsfürLuft-und
Raumfahrt e.V. integriert werden. Im ersten Abschnitt der Arbeit wird dargelegt, wie korres-
pondierende Pixelpaare, die in den jeweiligen Streifen an gleicher Geolokation liegen, ermittelt
werdenkönnen. DaraufaufbauenderfolgteinePlausibilitätsüberprüfungdererfaßtenPixelpaare
unter Verwendung von Zuverlässigkeitsmetriken, die auf Basis höherwertigerer Datenprodukte
berechnetwerden. IneinemweiterenSchrittwerdendieErgebnissegenutzt,umdienotwendigen
Parameter für eine optimierte Bildauswertung - hier im Sinne der Zuverlässigkeit - abzuleiten.
Abschließend werden die Pixelpaare benutzt, um die globale Variabilität der Reflektanzwerte
abzuschätzen. Insgesamt werden durch diese Arbeit die existierenden Methoden zur Qualität-
skontrolle optischer Bilddaten umfassend ergänzt.
Schlagwörtern — flugzeugetragene Sensorsysteme, Prozessierungskette für hyperspektrale
Bilddaten, spektroskopische Bildaufzeichnung, Qualitätssicherung, Signalverarbeitung, EUFAR,
QA4EO
ii
Long Abstract
In order to be able to use results derived from hyperspectral imagery, it is necessary to first
validatethecorrespondingdataandproducts: ’Are they reliable enough for achieving this goal?’,
’What is their quality?’...Although some initiatives are currently aiming at addressing these
questions, several issues remain open. The first one regards the exact definition of ’quality’. In
thefollowingthetermisrestrictedtothenotionofrepeatability. Inotherwords,thiscorresponds
to answering the question: ’What are the discrepancies between two similar measurements?’
A second issue regards the feasibility of such an assessment. Hyperspectral data allows for
theretrievalof awide rangeof -possiblysubtle -physical processes. Althoughthis isusually an
advantage, it also implies that any measurement is influenced by many unwanted sources. As a
result, hyperspectral imagery requires a complex pre-processing chain for translating the ’raw’
measurements into valuable information. This involves, in turn, further complex considerations.
Tostartwith,thesepre-treatmentsarebasedonmodels-orapproximationsoftheactualprocess
- they are therefore introducing their own uncertainties into the data, which would in turn also
need to be assessed. Besides these limitations, more classical considerations should be made:
the sensor itself is not perfect and need to be characterised and calibrated. This stage is also
associatedwithanuncertaintywhichaffectsanyfurtherstep. Inordertoevaluatethequalityof
aproduct-oreventocheckwetheradatasetissuitableforaspecifictask-thecompletesystem
should be investigated: the sensor and the associated characterisation & calibration, the data
acquisition, the data pre-processing and the products generation steps. As a result any direct
uncertainty propagation within the complete system is - in practice - illusory.
In order to bypass these limitations, the European Facility for Airborne Research (EUFAR)
has already produced several sets of quality tags. They do not account for the reliability as
such, but only indicate whether such or such aspect of the data is susceptible to affect the
quality. Although useful, this information has only been developed for the data itself - up to
reflectance: due to the wide variety of products (e.g. classification, quantitative analysis...),
mappingapplicationswereleftaside. TheGermanAerospaceCenter, beingpartoftheEUFAR,
is however interested in extending the existing quality tags toward land-products. This work
aims at addressing this issue.
Itsfirststepsregardsthedataqualityitself: althoughofinterest,theEUFARqualitytagsare
not detailed enough for obtaining any quantification of the imagery’s reliability. For practical
applications, this is theoretically by passed by so-called vicarious (or cross-) validation. The
sensor system images a large homogeneous area whose properties are known and compared to
the output of the processing chain. However, the huge majority of the areas of interest are
heterogeneous, hence bringing further uncertainties (e.g. adjacency effects, influence of differing
pixel’s footprints). Homogeneous areas are therefore corresponding to an over-optimistic case.
Thus, another solution had to be introduced in order to account for more plausible cases.
Sinceairbornehyperspectralimageryisusuallyacquiredinseveralpartiallyoverlappingflight-
lines, it is proposed to use this information redundancy to retrieve the imagery’s internal vari-
ability. This implies a data-driven quality assessment specific to each data-set. This specificity
is however not a limiting factor since this analysis can be performed on virtually every data-set.
A further consequence is however the fact that the repeatability assessment is only accounting
forrelativeerrors. Again, sincethegoalisheretoinvestigatetherepeatabilityofmeasurements,
absolute biases are of limited interest - they can actually be assessed by other means. Finally, it
is shown that this assessment is actually corresponding to a pessimistic - or worst case scenario
- quality assessment, hence complementing existing work on homogeneous areas.
iii
In practice, the comparison of two overlapping flight-lines is however not straight-forward,
should it only be because the presence of geo-location errors present in the data. A first step
consiststhereforeinretrievingtherelativevariabilityofthepixel’sgeo-locations,henceproviding
pairs of pixels imaging the same geo-locations. This stage is performed on atmospherically
corrected (reflectance) geo-rectified data. Subsequently, these pairs of pixels are used to obtain
’quality indicators’ accounting for the reproducibility of mapping-products. It is in particular
advocated to replace the classical Cohen’s κ coefficient used in remote sensing by the more
generic Krippendorff’s α. The latter is indeed more generic: it is not limited to analysis of two
classifications-itcanforexamplebeusedtoanalysemorethantwoquantitativemappingresults
(e.g. organic carbon content of the soil) - while being more stable with respect to configuration
changes(e.g. numberofinvestigatedpixels). Severalotherdedicatedindicatorsarealsoproposed,
as well as means to obtain their confidence intervals, hence extending EUFAR results over land-
products.
The third stage of this analysis consists in studying whether it is possible to use these reli-
ability results to improve the mapping-products. Once the reliability of a mapping-product has
been established, it is logical to to improve it. This work proposes to optimise the reliability
over the mapping-methods’ parameters. Since this approach is only depending on the reliability
retrieved by means of pairs of observations - which can be computed on virtually every data-set
- the approach is quite generic.
Finally, the repeatability assessment is back propagated to the hyperspectral data itself. As
a result, an estimator of the reflectance variability (including models, as well as scene induced
uncertainties) is proposed, hence both complementing the EUFAR quality layers with estimates
of the data repeatability and going beyond them by producing quantitative results. This is
achieved by means of a blind-deconvolution framework used in conjunction with the use of levy
alpha stable distributions.
Theproposedapproachesarealtogetherprovidinggenerictoolsthatcouldbeusedonalmost
any data-set with minimal human interaction, hence opening the way to their incorporation
within the German Aerospace Center processing chain for airborne hyperspectral data. They
are furthermore extensively tested against classical data-sets as well as against a specialised test
study elaborated within this work.
iv
Contents
1 Introduction & Objectives 1
1.1 Hyperspectral Airborne Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Proposed Approach and Outline of the Work . . . . . . . . . . . . . . . . . . . . 3
2 Remote Sensing & Spectroscopy 5
2.1 Introduction: Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Hyperspectral Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Basic Principles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Interactions with Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.3 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.4 Measurement Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 DLR Processing Chain & Associated Errors . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Processing Chain Description . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.2 Further Sources of Uncertainties . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Common Spectroscopy Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.1 Spectral Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.2 Band Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Quality Assessment 21
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Generic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.1 Quality Assessment: Uses and Interest . . . . . . . . . . . . . . . . . . . . 22
3.2.2 Quality for Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.3 Quality for Data Provider . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.4 Quality as Traceability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.5 Quality as Reproducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.6 Profane Requirements and Conclusions . . . . . . . . . . . . . . . . . . . 24
3.3 Generic Quality Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.1 Recommendations from JCGM . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.2 Quality Assessment for Earth Observation (QA4EO) . . . . . . . . . . . . 26
3.3.3 EUFAR Quality Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.4 Connex Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Quality Assessment in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4.1 Expert Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4.2 In-Situ Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.3 Cross Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4.4 Quality Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4.5 Bayesian Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4.6 Noise Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.7 Miscellaneous Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Conclusions and Retained Methodology . . . . . . . . . . . . . . . . . . . . . . . 35
v
4 Corresponding Pixels Automated Matcher 37
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.1 Generic Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.2 Co-Registration: State of the Art . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Proposed Methodology: CPAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.2 Validating Found Pixels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.3 Note on Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Parameters Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3.1 Spectral Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3.2 Neighbourhood Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 CPAM Validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4.1 Illustrative Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4.2 Validation on Synthetic Data . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4.3 Triple Pass Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.5 CPAM Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.5.1 HyMap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.5.2 HySpex - DLR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.5.3 AISA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5.4 Extension to Satellite Data . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5 Filling the Gaps 77
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Available Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3 Radial Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3.2 Solving the Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3.3 Dealing with Outliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3.4 Note on Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.4 Retained Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4.1 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6 Quality Indicators for Maps 89
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.1.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.1.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.2 Basic Analysis of L3 products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.2.1 Generic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2.2 Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2.3 Specialised ’Basic’ Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2.4 Case Study: Nominal Data . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2.5 Case Study: Absolute Data . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.3 In-depth Analysis: Cohen’s Kappa Coefficient . . . . . . . . . . . . . . . . . . . . 108
6.3.1 Inter-Rater Reliability: the Kappa Coefficient . . . . . . . . . . . . . . . . 108
6.3.2 Other Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.4 A Unified Framework: Krippendorff’s α . . . . . . . . . . . . . . . . . . . . . . . 113
6.4.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.4.2 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.4.3 Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
vi
6.4.4 Note on Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.4.5 Relationships Between Krippendorff’s Alpha and Other Coefficients . . . 115
6.4.6 Conclusions on Theoretical Background . . . . . . . . . . . . . . . . . . . 115
6.5 Experimental Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7 Improving Land-Products 119
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.2 Framework & Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.2.1 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.2.2 Model’s Parameters: Origins and Analysis . . . . . . . . . . . . . . . . . . 120
7.3 Methodology: Improving Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.3.1 Which ’Empirical’ Parameters? . . . . . . . . . . . . . . . . . . . . . . . . 121
7.3.2 Optimisation Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.3.3 Over-fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.4 Optimisation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.4.1 Generic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.4.2 Other Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.4.3 Simulated Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.5.1 Classification Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.5.2 Obtaining Optimisation Parameters . . . . . . . . . . . . . . . . . . . . . 128
7.5.3 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
8 In Depth Analysis of the Variability 133
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.2 Retained Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.2.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.2.2 Existing Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.2.3 Differing Viewpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.3 Analysis of Deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.3.1 Classical Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.3.2 Which Distribution? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.4 Stable Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.4.1 Notions and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.4.2 Sum of Stable Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.4.3 Link with Other Distributions. . . . . . . . . . . . . . . . . . . . . . . . . 141
8.4.4 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.5 Retrieving Individual Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.5.1 Retained Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.5.2 Getting the Optimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.5.3 Summary: Final Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.6 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
9 Summary 149
9.1 Reminder: Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.2 Review of Proposed Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
9.3 Critical Review of the Achievements . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
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