Error Propagation

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Sven Utcke

Error Propagation

in Geometry-Based Grouping

Dissertation zur Erlangung des Doktorgrades der Fakult ¨at f ¨ur Angewandte Wissenschaften

der Albert-Ludwigs-Universit ¨at Freiburg im Breisgau

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2

::::::::

Prof. Dr.

_:::::

Jan

_::::::_:::::

G.

:::::::::::::

Korvink

:::::::::::::::

(Dekan)

::::::::

Prof. Dr.

_:::::

Thomas

:::::::::::::::::::::::::::::

Ottmann

:::::::::::::::

(Vorsitz)

::::::::

Prof. Dr.

_:::::

Wolfram

::::::::::::::::::::::::::::

Burgard

:::::::::::::::

(Beisitz)

::::::::

Prof. Dr.

_:::::

Hans

_::::::::::::::::::::::::::

Burkhardt

::::::::::::::::::::

(Gutachter)

::::::::

Prof. Dr.

_:::::

Bernd

_:::::::::::::::::::::::::::

Neumann

:::::::::::::::::::::::::::::

(Zweitgutachter)

:::::

25.

_::::::::::

April

_::::::::::

2006

_:

Name der Dekanin oder des Dekansdie

Namen der Referentinnen oder Referentendas Datum

der Promotion

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3

Acknowledgements

::::The_:::::::work::::::::::::described_:::in_:::::this_::::::::thesis,_::::::and_::::its:::::::::::::description_:::::::itself,_::::::::spans_::::::over_::::::::nearly_::a

::::::::

decade._:::::::::During_:::::this_::::::time_:::::::many_:::::::::people:::::::::::::::accompanied_::::me_:::::and_::::my_:::::::work:::::::::::::::supervisors

::::and:::::::::::::colleagues,_:::::::::friends_:::::and:::::::::::relatives,:::::::::::::::::acquaintances_:::::and::::::::::::strangers_::::::have_:::::left_::::::their

::::::::

impact,_::::be_:::it:::::::::::::stimulating_:::or::::::::::::::restricting,_::::::::::inspiring_:::or:::::::::::::correcting,_::::::::aiding_:::or:::::::::::abetting,

:::::::::::::

encouraging_::::or:::::::::::::forbearing,_:::::::loving_:::or_:::::::::caring.:::::::::::Everyone_:::of_:::::::them_::::my_::::::::::heartfelt_:::::::::thanks!

:::::

Prof.::::::::::::Burkhardt_:::::and_::::::Prof. :::::::::::Neumann_:I_::::::have_:::to_::::::::thank_:::for_::::::::giving_::::me_::::the::::::::::::::possibilities

::::and_::::::wide_:::::::::support_::::for_::::the_::::::::::research_:::::::which_:::::has_::::led_:::to_:::::this_:::::::thesis._::I_::::::have_::::::::always_:::::::::enjoyed

:::::::::

working_::::for_::::::Prof.:::::::::::::Burkhardt,_:::::who::::::::::::invariably::::::::::::supported_:::::my::::::::::::::::independence,_:::::and_:::::I’m

::::::::::

indebted_:::to_::::::Prof.::::::::::::Neumann,_:::::who_:::::::::offered_::::me_::a_::::::::::position_::::::::among_::::his_::::::staff_:::::::when_::::::Prof.

::::::::::::

Burkhardt_::::left:::::::::::Hamburg_:::to_:::::::follow_::a_:::::call_::::::from_::::the::::::::::::University_:::of:::::::::::Freiburg._{: :}I_::::also_::::::wish

::to_:::::::thank_:::::the:::::::::::numerous::::::::::::colleagues_:::::and_:::::::fellow:::::::::::::researchers_:::::who_::::::::helped_:::::::along_::::my_:::::::thesis

:::::::::

through_:::::::many_:::::::::fruitful:::::::::::::discussions_:::::and_::::::who,_:::in_:::::::many_:::::::cases,_::::::took_::::on:::::::::::::themselves_::::the

::::::::::::::

considerable_:::::::::burden_:::of:::::::::::::::proofreading_:::::::::various_:::::::parts_:::of_:::::this_:::::::thesis_::::at_:::::::::various_::::::::stages.

::::My::::::::::::particular_::::::::thanks_::::go_:::to_::::::::::Andrew:::::::::::::Zisserman,_::::::who_:::in_:::::::many_::::::::::respects_:::::::::shaped_::::my

:::::::::

research_::::::::::interests_::::::and:::::::::::::::methodology_:::::and_::::::::whose::::::::::::inimitable_::::::::people_:::::::skills_:::::will_::::::::always

::::::

serve

:::as

:::an

::::::::::::::unreachable

::::::::::example;

::::::::others

:::::who

::::::went

::::far

::::::::beyond

:::::the

::::call

:::of

::::::duty

::::::::include

:::::::

Simon_::::::::Julier,_::::Nic_:::::::::Pillow,_:::::Jeff::::::::::::Uhlmann,_:::::and_:::::::::Michael_:::::::N¨olle.:::::::::::Andreas_::::::::::Bieniek,_::::::Marc

::::::::

Schael,_::::::Sven::::::::::::Siggelkow,_:::::and_::::::::Gerald:::::::::::Schreiber_:::::::made_::::my_::::::work_:::at_:::::Ti-I_::::::and,_:::in_::::the_:::::case

::of

:::::the

:::::first

:::::::three,

::::::::LMB,

::::::most

:::::::::::pleasing,

:::::and

::::::::Ullrich

::::::::K¨othe

:::::and

:::::::Hans

:::::::Meine

:::::did

::::the

::::::

same_:::for_:::::me_:::at_:::::::::KOGS;_::::the_:::::last_:::::two_::::::often_:::::also_:::::had_:::to_:::::::serve_:::as_::a:::::::::::sounding_:::::::board_::::for

::::new_:::::::ideas_:::—_:::as_:::::::::Ullrich_::::::often_::::::used_::::me_:::to_:::::::sound_:::::out_::::his_:::::::ideas._::::::::Thank_:::::you_:::::very_:::::::much

:::for_::::::your:::::::::::::::comradeship!_::

::::But_::::::most_:::of_:::all_::::my_::::::::thanks_::::::::belong_:::to_::::my_::::::::family:_::::to_::::my_:::::::::mother,_::::::::Christl::::::::::::::::::Utcke-Hamann,

:::::::

whose::::::::::::::unshakeable_::::::trust_:::in_:::me_:::::and_::::my_::::::::::abilities,_:::as_:::::well_:::as_::::her_::::::::::constant_:::::and::::::::::::devotional

:::::::::

support_:::::::made_::::me_:::::into_::::the_:::::::::person_::I_::::am_::::::::today._:::::::::Vielen,_:::::::vielen_:::::::Dank_:::::::::Mama,_::::Du_:::::bist

:::die_::::::::beste!_:::::And_:::to_::::my_::::::wife,_::::::Gabi:::::::::::Beutner,_:::for_:::::the:::::::::::countless_::::::hours_::I_:::::was_:::::::::excused_::::::from

:::::::::::

household_:::::and_:::::::other_:::::::chores_:::in_:::::::order_:::to_::::::work_::::on_::::my_:::::::thesis_::::::(and::::::::::::sometimes_::::::did),_:::::and

:::for_:::::the::::::::::::::longanimity_:::::and_::::::::::patience_::::::with_:::::::which_:::::she_::::::bore_::::my:::::::::::sulkiness_:::::and::::::::::::fretfulness

:::::::

during_:::::the_::::::::bleaker:::::::::::moments_:::of_::::my_:::::::work;_:::::and_:::::::finally_:::to_::::my_::::::son,_::::::::Moritz_::::::::Utcke,_:::::::whose

::::::

birth_:::::was_::::the_::::::final_::::::::::impetus_::::::::which::::::::::::ultimately_:::::had_:::::me_:::::::finish_:::::this_::::::::thesis.:: :::::::::Without

:::::

you,:::::this

:::::::thesis

:::::::would

:::::not

::::be.

:

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4

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5

Zusammenfassung

::In

::::::::dieser

::::::::Arbeit

::::::::::::beschreibe

::::ich

:::::::::meinen

:::::::::Ansatz

::::zur

::::::::::::::::Kombination

:::::von

::::::::::::Methoden

::::der

::::::::::::::::::::::

Fehlerfortpflanzung_:::::mit:::::::::::mehreren::::::::::::::::Algorithmen,_:::::die_:::::das::::::::::::::::::::::Geometrie-basierte_:grouping von_::::::::::::::::Strukturen:::::::::::erlauben._:::::Von_:::::der::::::::::::bekannten:::::::::::Literatur:::::::::::::::unterscheidet_:::::sich_:::::::meine

:::::::

Arbeit

:::::vor

::::::allem

:::::::durch

::::die

::::::::::::::::::::::::Schwerpunktsetzung

::::auf

::::::::::::::::::Anwendbarkeit:

::::die

:::::::tats¨achliche_:::::::

:::::::::::

praktische::::::::::::::Anwendung_::::::zeigt:::::::::::deutlich,_::::::::welche_::::::zus¨atzlichen_::::::::_:::::M¨oglichkeiten:::::::::::_::::::man_:::::::durch

::::::::::::::::::::::

Fehlerfortpflanzung_::::::::::gewinnt;::::::::::::::andererseits_::::::habe_::::ich,_::::::statt_::::::starr_:::an_::::der_::::::::::exakten_:::L¨o_:::::sung

:::::::::::::

festzuhalten

:::::::(die,

::::wo

:::::m¨o_::::::glich,

:::::::nat¨u rlich::::

:::::::::::gegeben

:::::::wird)

:::::::auch

::::::::::::::untersucht,

::::::::welche

::::::::::::::::

Auswirkungen_::::die::::::::::::::Verwendung_:::::von_:::N¨a:::::::::::herungsl¨o_::::::::sungen_:::::::haben_::::::kann_:::—_:::::und_:::in_::::::::::welchen,

::in_::::der:::::::::::Literatur_::::::::::teilweise_::::::recht_:::hä_::::ufig::::::::::::::::::anzutreffenden,_::::Fällen_::::_:::::::solche_::::Näherungslö_::::::::::_::::::::sungen

:::::::::::::

verheerende

:::::::::::::::::Auswirkungen

::::auf

::::die

:::::::::::::Korrektheit

:::::::(oder

::::::sogar

::::::::::::Existenz)

::::des

:::::::::::::Ergebnisses

:::::::

haben_:::k¨o_::::::nnen._:

::::::::

Warum_:::::::::glaube_::::ich,_::::::dass_:::::::solch_:::::eine_::::::::Arbeit_::::n¨otig_:::_::::::oder_::::::auch_:::::nur_:::n¨u_::::::tzlich_::::::sein_:::::::kann?

:::::::

Zumal_::::::doch_::::die::::::::::::::Grundlagen_::::der::::::::::::::::::::::Fehlerfortpflanzung_:::::::(wenn_::::::auch_::::::nicht_:::in_::::der:::::::::::::projektiven

:::::::::::::

Geometrie)_::::seit_:::::::vielen:::::::::::::::Jahrzehnten_:::::::::bekannt_::::::sind_:::::und_::::oft_:::::::genug_::::::::bereits_:::in_::::der_::::::::Schule

:::::::::::::

unterrichtet_::::::::::werden?_::::::Einer_:::::der_:::::Gr¨unde_::::_::f¨u_:r_::::die_:::::::::geringe::::::::::::::Verbreitung_::::der::::::::::::::::::::::Fehlerfortpflanzung

::::::

unter

:::::::::::::::::::Bildverarbeitern

::::::liegt

:::::::::meiner

:::::::::::Meinung

:::::::nach

:::in

:::::der

:::::::::::::::vorhandenen

::::::::::::Literatur,

::::::

deren_::::::::::Interesse_::::::stets_::::der_:korrekten_:::L¨osung_:::::_:::::gilt,_::::::ohne_::::::Blick_::::auf_::::die_:praktische::::::::::::::::::Anwendbarkeit.

:::Im:::::::::::::Gegensatz_:::::::::hierzu_::::ist_::::die::::::::::::::vorliegende_:::::::::Arbeit_:::::aus_:::::der_::::::::Praxis_::::f¨u_:r_:::::die_::::::::Praxis

:::::::::::::

entstanden:

::::ich

::::::zeige

:::::::::anhand

::::von

:::::::::::::Beispielen,

:::::dass

:::::sich

::::::viele

:::::::::::Probleme

::::::tats¨a_::::::chlich

:ein- facher l¨o_::sen_:::_::::::::lassen,_:::::::wenn_:::::man::::::::::::::Grundlagen_::::der:::::::::::::::::::::::Fehlerfortpflanzung_:::::ber¨u:::::::::::cksichtigt_:::—

:::::

oder_::::::sogar_:::::nur_:::::::dann;_::::ich_:::::::denke_::::die::::::::::::::Anwendung_::::auf:::::::::::::::Zebrastreifen_:::in_:::::::::Kapitel_::5_::::::::meiner

::::::::::::::

Dissertation

:::ist

:::so

::::ein

:::::::::::Beispiel.

:::::::Dabei

:::::::::behalte

::::ich

::::::::jedoch

:::::::stets

::::die

::::::::::::::algebraische

:::::und

::::::::::::::::

algorithmische::::::::::::::Komplexit¨a_:t_:::::der::::::::::::::verwendeten::::::::::::Verfahren_:::::::sowie_::::die_:Notwendigkeit zu_::

:::::

ihrer:::::::::::::::Verwendung_:::::::(oder,_::::::auch_::::das_:::::::kann:::::::::::passieren,_::::die:::::::::::::mangelnde::::::::::::::::::Notwendigkeit)_:::im

::::::

Auge.

:::::Aus

:::::::::diesem

:::::::Grund

:::::::::::::beschreibe

:::ich

:::::::nicht

::::nur

::::die

:::::::::::::::Kombination

::::von

:::::::::::::::::::::::Fehlerfortpflanzung

::::und:::::::::::::projektiver::::::::::::Geometrie_:::::(die_:::f¨u_:r_::::den::::::::::::::::::uneingeweihten_:::::::einige::::::::::::::::::Schwierigkeiten_::::::::::bereith¨alt)_::

:::::::::

sondern:::::::::::::::demonstriere_::::die:::::::::::::Anwendung_:::::::dieser::::::::::::Prinzipien_:::::::::anhand_:::::von_::3_:::::sehr::::::::::::::::verschiedenen

::::::::::::

Beispielen._::::Im::::::::::::Folgenden:::::::::::::beschreibe_::::ich_::::den_::::::::::Aufbau_::::::::meiner_:::::::::Arbeit._:

::::::

Nach:::::::::::::Einleitung_:::::und_:::::::einf¨u_:::::::::hrenden_::::::Erl¨a:::::::::::uterungen_::::zu::::::::::::::projektiver:::::::::::::Geometrie_:::::und

::::::::::::::::::::::

Fehlerfortpflanzung_:::in_:::::den:::::::::::Kapiteln_:::::1–3_::::::::::beginnt_:::::der::::::::::::Hauptteil_::::::::meiner_:::::::::Arbeit_:::in

::::::::

Kapitel_::::4,_:::in_::::::dem_:::::die::::::::::::::Verbindung:::::::::::zwischen:::::::::::::::::::::::Fehlerfortpflanzung_::::::und:::::::::::::projektiver

::::::::::::

Geometrie:::::::::::::::::::herausgearbeitet_::::::wird._:::::Die::::::::::::::::::::zugrundeliegende_::::::Idee_:::ist_:::::::nicht_::::neu_::::::und_:::::geht

:::auf:::::::::::::Kanatanis::::::::::::::N-Vektoren_::::::zur¨uck;_:::_::::::dar¨u_::::ber::::::::::::::::hinausgehend::::::::::::beschreibe_::::ich_::::::aber_::::::auch

:::die::::::::::::::Anwendung_::::der_::::::::::gleichen_::::::::::Grunds¨a_::::tze_::::auf_::::::::andere:::::::::::::::::::::Parametrierungen_:::::und_:::::leite_:::::eine

::::::

Reihe_:::::::neuer:::::::::::::Ergebnisse_:::::her,_::::wie_:::::zum_::::::::::Beispiel_:::::eine::::::::::::::::hervorragende::::::::::::::::::Approximation_::::der

:::::::::::

Kovarianz_:::::::eines_:::an_::::::::einige_:::::::Edgel::::::::::::::angepassten::::::::::::Linienst¨ucks,_::::_:::::eine::::::::::::::::::::::Abbruchbedingung

::f¨u

:r

::::::::::::::::inkrementelle

::::::::::line-fits

::::::und

:::::::einen

::::::::neuen

:::::::::::::::Algorithmus

:::f¨u

:r

:::::die

::::::::::::::Berechnung

:::::des

::::::::::::::

Doppelverh¨a_::::::::ltnisses_:::::von_::4_::::::::Linien,_:::::::::welcher:::::::::::Aufgrund_::::der::::::::::::::Verwendung_:::::von::::::::::::::::::::::Fehlerfortpflanzung

::::::

tats¨achlich_::::::_:::::::sogar_:schneller ist_::_::::als::::::::::::bisherige::::::::::::Verfahren.:::::::::::::::Desweiteren_::::::gebe_:::::ich_:::::eine

::::::

Erkl¨a_::::::rung,

:::::::::warum

::::die

:::::von

::::::::vielen

:::::::::::Autoren

:::::::::::::verwendete

:::::::sph¨arische_::::::

::::::::::::::::::Normalisierung

::::von:::::::::::::::Koordinaten_:::::::tats¨achlich_::::::_::::::einer::::::::::::::::Euklidischen::::::::::::::::::Normalisierung_::¨u_:::::::::berlegen_:::::ist;_:::::und

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6

:::::::::::

schließlich_:::::::gebe_::::ich_::::::eine_:::U¨bersicht_:::::::::_::::::darü_::::ber,_:::::wie_:::::::viele_::::der_::::hä_:::::::::ufigsten_::::::::::Messgrö_::::ßen

:::am::::::::::::::sinnvollsten:::::::::::::verglichen_:::::::::werden_:::k¨o_::::::nnen_:::—_:::::::allein_::::::::dieser_:::::::letzte::::::::::::Abschnitt_:::k¨o_:::::nnte

:::::::

bereits

::::::viele

:::::der

:::in

::::der

::::::::::::::::::::Bildverarbeitung

:::so

:::h¨a_:::::ufig

::::::::::::::::::anzutreffenden,

:::::fein

::::::::::::::eingestellten

::::::::::::

Parameter_::¨uberfl¨_::::::u_::::ssig_::::::::::machen. _::In_:::::den_:::::::daran:::::::::::::::::anschließenden_:::::drei_::::::::::Kapiteln::::::::::::beschreibe

:::ich:::::::::::::::verschiedene:::::::::::::::Anwendungs:::::::::::szenarien._:::::Die_:::::::erste::::::::::::::Anwendung_:::in_::::::::::Kapitel_::5_::::ist_::::die

::::::::::::

Erkennung

:::::von

:::::::::::::::Zebrastreifen

::::::(und

:::::::::anderer

::::::::::::::periodischer

:::::::::::::::Strukturen).

:::Es

:::::::::handelt

:::::sich

::::hier_:::::um_:::::eine::::::::::::::Anwendung_:::::von_:::::der_::::ich_:::::::::glaube,_::::::dass_::::sie_:::so_::::::ohne:::::::::::::::::::::::Fehlerfortpflanzung

::::::

nicht_::::m¨o_:::::glich_::::::::::gewesen_:::::w¨are;_:::::::::::::::besonders:::::::::::::interessant_::::an_:::::::dieser:::::::::::::::Anwendung_::::ist,_:::::wie

::::::

einige_::::::::wenige:::::::::::::::::::Konfidenz-Tests_:::::eine_::::::::::Vielzahl_:::::::::manuell_:::zu_::::w¨ahlender_::::::::::::::::::::Parameter_::::::::::ersetzen

:::k¨onnen,_:::::::::::::::::wodurch_::::ein_:::::::::extrem_::::::::::stabiles_:::::::::System::::::::::::::entstanden_:::::ist._{: ::::}Die::::::::::::::::Algorithmen,

:::die_:::in_::::::::::Kapitel_::6::::::::::::::beschrieben_::::::::::werden,_::::::::besch¨a_::::::ftigen_:::::sich_:::::mit_:::::der:::::::::::::::::Segmentierung_:::::von

:::H¨a:::::::::::::userfronten

::::::::::::::::(orthogonalen

::::::und

::::::::::::parallelen

::::::::::::::Strukturen)

:::in

::::::::::::::::Einzelbildern.

::::Es

::::::wird

:::::

kein_:::::::::fertiger::::::::::::::Algorithmus_:::::pr¨asentiert,_::::::::::::::::::::::stattdessen_::::::wird_:::::::dieses:::::::::::Szenario_::::::::::genutzt,_::::um

::::eine_:::::::::Anzahl:::::::::::::::::::unterschiedlicher_:::::und_::::auf::::::::::::::::::::unterschiedlichen_::::::::Skalen::::::::::::::operierender::::::::::::Techniken

:::zu

:::::::::::::vergleichen.

:::::Der

:::::::::::::::Schwerpunkt

:::::liegt

::::auf

::::der

:::::::::::::::Bestimmung

::::::::::::kollinearer

::::::::::::::::::Liniensegmente

::::und_:::::von::::::::::::::::::Fluchtpunkten. Das_::::_:::::::letzte:::::::::::::::::::::::Anwendungskapitel,_::::::::Kapitel_:::7,::::::::::::beschreibt::::::::::::schließlich

:::::

Teile_::::der:::::::::::::::::::::::::::::Segmentierungsroutinen,_:::die_:::::::::meinen_::¨altesten_:::::::::::::::::::::::Publikationen_::u¨_::::ber_::::die:::::::::::::Erkennung

::::::::::::::::::::::::::

rotationssymmetrischer

::::::::::Objekte

::::::::::::::::::zugrundeliegen.

::::Ein

::::::::::::::wesentliches

:::::::::::Merkmal

:::ist

:::::::dabei

::::das_:::::Bild_:::::der:::::::::::::::::::Rotationsachse._::::::::Dieses_:::l¨asst_:::_:::::sich:::::::::::::theoretisch_::::als_::::::eine_::::::Linie_::::::::durch_::::die

::::::::::::::::

Schnittpunkte_::::von:::::::::::::::Bitangenten::::::::::::berechnen._::::Da_::::::diese_::::::::jedoch_::::::::::erheblich_:::in_::::::ihrer::::::::::::::Genauigkeit

::::::::::

variieren

:::k¨onnen,_::::::

:::::::haben

::::wir

:::::hier

::::ein

:::::::::::::exzellentes

:::::::::Beispiel,

:::::um

::::::::::::::verschiedene

:::::::::::::::Algorithmen

:::zu:::::::::::::vergleichen;_::::ich_:::::::zeige,_:::::wie_:::::::selbst_::::ein::::::::::::bekannter_:::::und_:::h¨a_::::ufig::::::::::::genutzter::::::::::::::Algorithmus

::::wie_::::die_::::::::::kleinste_:::::::::Summe_::::der::::::::::::::::::Fehlerquadrate_::::zu::::::::::::::::::unbrauchbaren::::::::::::::Ergebnissen_:::f¨u_:::::hren

::::::

kann,

::::::wenn

::::die

::::::::::::::::::::zugrundeliegende

:::::::::::Annahme

:::::::::unabh¨a_::::::::ngiger,

::::::::::isotroper

:::::und

:::::::::::::::::gleichverteilter

:::::::

Fehler_::::::nicht_::::::::::zutrifft,_:::::und_::::::stelle_:::::::::bessere:::::::::::::::Alternativen_:::::vor._:

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Symbols

x, X : scalars.

x, X : vectors. In a transformation, capital letters usually indicate the source of a transformation, small letters indicate the target.

P : matrix.

Σ : covariance matrix.

J_yx : Jacobian; matrix of first derivatives of y with respect to x. This is a matrix proper ifxandyare both vectors, a vector (either row or column) if one of the two is a scalar variable, and a scalar if bothxandyare scalar variables.

∝ : proportional to.

∞ : infinity.

IR : set of real numbers.

( )⁻ :

:::::::::::::::pseudoinversepseudo-inverse.

( )⁻_n :

:::::::::::::::pseudoinverse_:pseudo-inverse computed by setting all eigenvalues except the first n to zero.

| | : determinant.

| |n×n : determinant of the upper left n×n matrix.

k k : norm.

( )^T : transpose.

( )^−T : inverse of the transpose (or, of course, transpose of the inverse).

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12 Symbols

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Chapter 1 Introduction

The last thing we decide in writing a book is what to put first.

Blaise Pascal, 1623–1662

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14 Grouping and Error Propagation

1.1 Grouping and Error Propagation

This thesis describes the approach used for, and the improvements possible by, the use of error_::::::::::::::::::::propagation_:error-propagation in conjunction with several algorithms for the grouping of structures based on geometric entities. But rather than rigidly favouring the exact solution each and every time¹ I have put particular weight on practicability, demonstrating the relative gain for many approaches and giving shortcuts where the results are not marred by their use; but also demonstrating how common shortcuts used by many authors can lead to disaster if the underlying assumptions are violated.

1.1.1 Why Error Propagation?

Why do I believe that such a thesis is necessary and indeed valuable? The principles of linear error propagation, which I will use in this thesis, have been known for a long time, often enough they are even taught in school; they are the staple of photogrammetrists, geodesists, physicists, as well as many other scientists. But — they are rarely enough used in computer vision. True, a number of publications exist, starting with Kanatani’s work [70, 75] more than 13 years ago, and with F¨orstner’s contribution to the_::“Handbook of Computational Geometry for Pattern Recognition, Computer Vision, Neurocomputing and Robotics_:” [49] as the latest, very nice, example²; but by and large error_:::::::::::::::::::propagation_:error-propagationhas been all but ignored by the computer vision community.

I believe that the reason for this disregard is twofold: for one thingerror_:::::::::::::::::::propagation error-propagation is simply unknown in computer vision circles, and if Kanatani didn’t manage to change this then surely this thesis won’t be able to either. But I also believe that _:::::error:::::::::::::::propagation_:error-propagation is seen as an unnecessary complication: “Let me solve this really complicated and important problem first, and then I can worry about details like error propagation” seems to be the attitude of many a researcher, or even “Sorry, but error propagation is much too slow for any real(-time) application”. And such a mind-set is unfortunately fostered by authors like Kanatani, who are more interested incorrect than inpracticable solutions. And it is here that I hope this thesis could have a small impact: demonstrating that many problems are indeed much easier solved using _:::::error:::::::::::::::propagationerror-propagation, or indeed only solvable using error_:::::::::::::::::::propagation_:error-propagation— I believe that the application described in Section 5 is such an example — but all the time with a firm eye on computational complexity as well as the necessity for error_:::::::::::::::::::propagation error-propagation (or, as it sometimes happenshappen, the lack of it). It is to this_:::::::::

end that I not only describe the combination of error propagation with projective geometry, which for the unwary keeps a number of stumbling blocks at hand, but

1Exact in its derivation, that is.

2Chapter 4.1 lists more literature on the subject

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The Outline of this Thesis 15 also demonstrate 3 very different application domains. In the following I’ll describe the outline of this thesis in more detail.

1.2 The Outline of this Thesis

::::The_::::::flow_:::of_:::::this_:::::::thesis_::::::goes_::::::from_:::::the::::::::::::theoretical::::::::::::::foundations:::::::::::::(projective::::::::::::geometry,

:::::

error:::::::::::::::propagation,_:::::and_:::::::their:::::::::::::::combination)_:::to:::::::::::practical::::::::::::::applications:::::::::::::showcasing_:::::one

::or

:::::::more

:::of

:::::the

::::::::::::previously

::::::::::::described

:::::::::::::theoretical

:::::::::::::principles;

::::::::within

:::::the

:::::::::::::application

:::::::::

chapters_::I_:::go_:::::::from_::::the_::::2D_:::::case_:::of_::a_:::::::single_::::::::planar:::::::::::::::homography_:::to_:::::the_:::::case_:::of_::::::::several

:::::::::::::::

homographies_::::all_:::::::::within_:::::one_::::::::image_:::::and_:::::::from_:::::::there_::::to_::::the_::::::case_::::of_::::an_::::::even_:::::less

:::::::::::

restricted

::::::class

:::of

:::::::::objects,

::::::::::surfaces

:::of

::::::::::::revolution.

:::::In

::::::more

::::::::detail,

::II’m starting this thesis with an overview of the state of the art in projective geometry (Chapter 2) and _::::::error::::::::::::::propagation_:error-propagation (Chapter 3) respectively. These chapters_:::::::::

chapter do not contain anything new and are for a huge part lifted straight out of [103] and a couple of other books, in spirit if not in words. If you know your way around projective geometry or_::::::error::::::::::::::propagation_:error-propagation I would recom- mend to simply skip the respective chapter, they are here for completeness,

:and as a handy reference for later work. The actual thesis starts with Chapter 4, which com- bines projective geometry and error propagation. The underlying idea is not new, and as far as the application to homogeneous coordinates is concerned can be found in [75]; however, in this chapter I also consider the application of these principles to other parameterisations than homogeneous coordinates and, starting from first principles, derive a number of new results such as an excellent approximation to the_:::

::::::::::::

covariance_::of_::a_:::::line_::::::::::segment_:::::::fitted_:::to_:::::::edgelsa fitted line-segments’ covariance, a new stopping-criterion for incremental fits based on a χ²-test, and a new algorithm for the calculation of the cross-ratio of 4 lines which due to the use of

:::::

error

::::::::::::::propagation error-propagation in fact performs faster than current algorithms. I will also give an intuitive explanation why the spherical normalisation used by many authors is indeed superior to an Euclidean normalisation; and finally I will give an overview on how to compare a number of common stochastic entities. Just this last section alone could already put away with many of the numerous, finely tuned parameters so common to computer vision algorithms.

The next three chapters describe different application scenarios. In Chapter 5 I describe the application of error-propagation principles to the grouping and recognition of

::::::

zebra

::::::::::crossings

:zebra-crossings and other repeated structure. This application was first described by me in [6], and is a nice example of an implementation which I believe would have been impossible without the use of _::::::error::::::::::::::propagation

::::due_:::to_::::the_::::::high::::::::::::variations_:::of_::a::::::::::::::::::zebra-crossing’s_:::::size_:::::and_::::::::quality_::::::even_::::::::within_::a_:::::::single

:::::::

imageerror-propagation; of particular interest here is how only a few confidence-tests can replace a host of manually chosen parameters, resulting in a uniquely stable algorithm. _::It:::::::::::describes_::::the::::::::::::::::::groundbreaking_::::::work_:::on_:::::::which_::::::later::::::::::::::publications_:::::such_:::as_:[135]

::::::

build._::In Chapter 6 I _::::::::outline_:describe an algorithm for the grouping of houses (or, indeed, any structure consisting of orthogonal and parallel elements). Over the years

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16 The Outline of this Thesis we have seen a _::::few_:number of algorithms for the reconstruction of buildings from monocular images [36, 87, 97], however, in contrast to multi-view approaches these nearly always require manual segmentation of

:::::::image

:::::::::regionsimage-regions. The algorithm outlined_:::::::::_:described in this chapter could be seen as an attempt to remedy this situation. It is, however, included in this thesis for a different reason: buildings show a number of diverse features at different scales, and I will in particular have a closer look at collinear_::::line:::::::::::segments_:line-segments of only a few pixels to several hundreds_::::::::::_:hundredth of pixels in length and distance as well as vanishing_::::::::::

:::::::

pointsvanishing-points, the image of intersection of parallel lines at infinity, which can be anywhere from literally in the image to literally at infinity. What is more, these features come with differing accuracies, and even one and the same feature can have different accuracies attached to it depending on context. This application is therefore well suited as a showcase for several differentideas_:::::_:::::and:::::::::::::approaches_:::::such_:::as_::a

::::new::::::::::::algorithm_::::for_::::the_::::::::::iterative:::::::::::::::improvement_:::of::::::::::::::::::vanishing-point_::::::::::position_:::::and_::::one_::::for

::::the

:::::::::::automatic

:::::::::::grouping

:::of

:::::::::::vanishing

::::::::points;

::a

:::::new

:::::::::::objective

:::::::::function

::::for

::::the

::::::::::(partial)

::::::::::::

calibration_:::of_::a_::::::::camera_::::::from:::::::::::::::::::vanishing-points_:::::::which_:::::::takes_::::the_::::::::::different:::::::::::::::uncertainties

::in_::::the:::::::::::positions_:::of_::::the:::::::::::vanishing_::::::::points_:::::into_:::::::::account_:::::and_:::::::::extends_:::::the_::::::usual:::::::::::Legoland

:::::::::::::

assumption

:::to

::::::more

:::::::::general

:::::::::setups;

:::an

::::::::::::extension

:::on

:::::::::::previous

::::::work

:::::::which

:::::::takes

::::the

:::::::::::::::::

vanishing-point::::::::::::::information_::::::into_:::::::::account_:::::::when_::::::::::merging:::::::::::::::::line-segments;_:::::and_::::::::finally

:a::::::::::::::comparison_:::of_::::the:::::::::::::::performance_:::of_::::::::several_::::::::::different::::::::::::::::::error-measures,_::::::both_:::::new_::::::ones

::::first

:::::::::::::introduced

:::in

:::::this

:::::::thesis

:::as

::::::well

:::as

:::::::::::::established

::::::ones

::::::from

::::the

::::::::::::literature,

::::for

::::the

:::::::::::::::

identification_:::of_::::::::::collinear_:::::line:::::::::::segmentstechniques.

Chapter 7 finally describes part of the grouping algorithm underlying some of my older publications on the recognition of surfaces of revolution such as [3–5, 9], but also newer publications on their reconstruction, such as [8]. An important feature for both recognition as well as reconstruction of SORs is the object’s axis. The axis can be calculated, e. g., based on the intersections of bitangents, which can vary considerably in their accuracy; it is therefore an excellent example to_::::::::::compare_::::the

::::::::::::::

performance_:::of_::a_:::::::::number_:::of:::::::::::::established::::::::::::algorithms_::::on_::a_:::::::::number_:::of_::::::::::different_:::::::::features

::::and_:::to_:demonstrate how even a well-known and often-used algorithm like total least squares will fail if the underlying assumptions (iiid-data) are violated; much better alternatives are introduced and an extensive comparison and discussion shows the merit of

:::::

error

:::::::::::::::propagation

:error-propagation for a problem which, in similar form, one can see tackled with unsuitable tools at nearly any computer-vision conference_:,

:::::

even_:::::::today._::::::The::::::::::::::comparisons_::::are_::::::done_::::on_:::::real:::::::::::::::contour-data_::::::::derived_::::::from_:::::real_::::::::images

::::::

which

:::::::::::::previously

:::::::::::appeared

::::in

::::::::::::::publications

::::::::about

::::the

:::::::::::grouping

::::::and

:::::::::::::recognition

:::of

:::::::

SORs._:.

This thesis ends, as all theses do, with a conclusion and outlook in Chapter 8.

Due to the diverse nature of the underlying problems, ranging from projective geometry to error_::::::::::::::::::::propagationerror-propagation, from intrinsically two-dimensional problems like the recognition of repeated structure to intrinsically three-dimensional problems like the grouping of box-like_:::::and_::::::even_:::::::::(partly)_:objects on the one hand and intrinsically free-form objects (,_: surfaces of revolution_::),_:, on the other hand,

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The Outline of this Thesis 17 there is no separate chapter entitled “_::::::::::literature_:::::::::surveyliterature-survey”. Instead you can find a small overview over the then relevant literature in each chapter’s introduction, and then again whenever a direct reference can help to set the work described in context. The bibliography itself comes in two parts, starting with a list of my own relevant work on page 201 and the bibliography proper on page 203.

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18 The Outline of this Thesis

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Chapter 2 Projective Geometry

. . . experience proves that anyone who has studied geometry is in- finitely quicker to grasp difficult subjects than one who has not.

Plato, The Republic, Book 7, 375 B. C.

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20 Projective Transformations

2.1 Introduction

When working in computer vision and image understanding, one of the first things one often seeks to describe is the image formation process, i. e. how are the real world and any specific image of this world related to each other. This connection can be made elegantly by projective geometry.

Projective geometry is much older than computer vision. According to [138] the first systematic treatise on projective geometry was published 1822 by Poncelet in his Traité des propriétés projectives des figures. Prompted by Felix Klein’s Erlangen programme of 1872 [79] as well as a general interest in invariant theories, projective geometry became rather fashionable among the mathematicians of the late 19^th and early 20^th century (e. g. [39]). The book that by many in the vision community is considered the standard reference on projective geometry, Algebraic Projective Geometry by J. G. Semple and G. T. Kneebone [138], dates back to 1952. Only

::::::::::::::::

comparativelycomparablyrecent trends in computer vision require a somewhat more involved algebra; mostly tensor algebra as it is used in shape from multiple view approaches [59]. However, since this thesis concentrates on single view geometry, only standard projective geometry is used here.

This chapter describes the theory and principles of projective geometry as they apply to this thesis. Starting from 2D projective transformations, the notion of homogeneous coordinates is introduced and several subgroups of the projective group are presented (Section 2.2). This leads naturally to the discussion of different camera models in Section 2.3. Points, lines and conics are introduced (Sections 2.4 and 2.5) as well as the crossratio of four collinear points or four coincident lines respectively (Section 2.6). Finally some special transformations (canonical frames in Section 2.7 and “projective symmetry” in Section 2.8) are presented, and an alternative repre- sentation of the projective plane is introduced: the Gaussian sphere (Section 2.9), which has proven useful for error-propagation purposes or algorithms like the grouping byvanishing_::::::::::_::::::::pointsvanishing-points discussed in Section 6. This introduction is naturally a rather brief and incomplete one, the interested reader can find additional information in, e. g., [43, 69, 103, 138, 146].

2.2 Projective Transformations

Projective geometry describes a group based on central (conic) projections. Con- fining ourselves to an image’s two dimensions, each projection can be visualised as a central projection from an arbitrary plane Π⁰ onto a second plane π, compare Figure 2.1. The totality of all those projections from one plane onto another forms the projective group [138].

Since any two-dimensional plane in 3D can be transferred into any other two-

Error Propagation

Sven Utcke