:SINGHUA��NIVERSIT� �IF��������CBG�FI���CB���G��� CB�-C���������IGG��CFAI�� 3�B��� ,I��4ICE��B� ��������B��IB����B�3�B� 1G�B��I��2B�J�FG��L 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 1
.I���B� � ��������3CF�G � �AUSS��URFACE��ECONSTRUCTION � �ESULTS�AND��OMPARISONS � �ONCLUSION 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 2
�� ��������3CF�G n �URFACE�RECONSTRUCTION�FROM�POINT�CLOUDS �ELL�STUDIED • n )PPLICATIONS 4ANUFACTURE • )NIMATION • �ISUALI�ATION • -TC� • 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 3
�� ��������3CF�G n �OMBINATORIAL�METHODS �TILI�E��PART�OF �THE�INPUT�SAMPLE�POINTS�AS�VERTICES� • �SUALL��SENSITIVE�TO�NOISE�AND�MA��PRODUCE��AGGED�SURFACES� • [Amenta et al. 2002] [Kolluri et al. 2004] [Xiong et al. 2014] etc… 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 4
�� ��������3CF�G n 1MPLICIT�METHODS -STIMATE�IMPLICIT�FUNCTIONS�FROM�THE�INPUT�SAMPLES�AND�EXTRACT� • ISO�SURFACES�TO�GENERATE�TRIANGLE�MESHES� F ( q ) =0 F ( q )<0 0 F ( q )>0 <0 Samples F ( q ) 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 5
�� ��������3CF�G n 1MPLICIT�METHODS 0OW�TO�CHOOSE�IMPLICIT�FUNCTION�AND�HOW�TO�SOLVE�IT( • F ( q ) =0 F ( q )<0 0 F ( q )>0 <0 Samples F ( q ) 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 6
�� ��������3CF�G [Hoppe et al . 1992] [Curless and Levoy 1996] [Carr et al . 2001] [Kazhdan et al . 2013] [Calakli and Taubin 2011] [Kazhdan et al . 2006] … … 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 7
�� �B�����CF��IB���CB % ∈ ' ( \ * 0 Σ n 1NDICATOR�FUNCTION� ! = # % ∈ /* 1/2 Σ 1 % ∈ Σ 0! = 1 Samples Indicator function Indicator function gradients 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 8
�� �B�����CF��IB���CB����CF���AG n �OISSON�SURFACE�RECONSTRUCTION�?2A�HDAN ET�AL� ����� �ONVOLUTION�OF�THE�INDICATOR�FUNCTION • 6VER�SMOOTHED • n �CREENED��OISSON�SURFACE�RECONSTRUCTION?2A�HDAN ET� AL� ����� )�SCALAR�FUNCTION�FITTING�TERM • 6VER�FITTING • 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 9
���� ���B�����G��B����IB���CB n �IGNED�DISTANCE�FUNCTION�METHODS )�SIGNED�DISTANCE�FUNCTION�IS�SMOOTH�NEAR�THE�SURFACE • -AS��TO�INTERPOLATE • �ENSITIVE�TO�NOISE • 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 10
�� ���L�F���GC�I��CB n 6BSERVATION 1NDICATOR�FUNCTION�ROBUST�TO�NOISE • �IGNED�DISTANCE�FUNCTION�EAS��TO�INTERPOLATE�NEAR�THE�SURFACE • n 6B�ECTIVE )WA��FROM�THE�SURFACE�INDICATOR�FUNCTION • �EAR�THE�SURFACE�SIGNED�DISTANCE�FUNCTION • �ALANCE�BETWEEN�DATA�FIDELIT��AND�RESILIENC��AGAINST�NOISE • 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 11
�� ���L�F���GC�I��CB n 6UR�METHOD �ASED�ON��AUSS�3EMMA�IN�THE�POTENTIAL�THEOR�� • -STIMATED�DIRECTL��FROM�AN�EXPLICIT�INTEGRAL�FORMULA�WITHOUT� • SOLVING�AN��LINEAR�S�STEM� -AS��TO�PARALLELI�E�WITH�SMALL�OVERHEAD� • 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 12
.I���B� � �ELATED��ORKS � ��IGG��IF��������CBG�FI���CB � �ESULTS�AND��OMPARISONS � �ONCLUSION 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 13
�� ��IGG�,�AA� n 3ET Σ BE�AN�OPEN�REGION�IN� ℝ ( � * Σ DENOTES�THE�CLOSURE�OF� Σ �� �ONSIDER�THE�FOLLOWING�DOUBLE�LA�ER�POTENTIAL��.OR�AN�� % ∈ ℝ ( , 4 ∈ /Σ [ /8 ! % = 5 %, 4 9: 4 . /1 4 67 n 8 IS�THE�FUNDAMENTAL�SOLUTION�OF�THE�3APLACE�E�UATION��WHICH� CAN�BE�STATED�EXPLICITL��AS 1 8 %, 4 = − . 4> % − 4 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 14
�� ��IGG�,�AA� n 3ET Σ BE�AN�OPEN�REGION�IN� ℝ ( � * Σ DENOTES�THE�CLOSURE�OF� Σ �� �ONSIDER�THE�FOLLOWING�DOUBLE�LA�ER�POTENTIAL��.OR�AN�� % ∈ ℝ ( , 4 ∈ /Σ [ /8 ! % = 5 %, 4 9: 4 . /1 4 67 n �OTE�THAT 6? B EFA G@ A %, 4 = − � H 6@ A CD EFA WHICH�WE�CALL�THE�KERNEL�FUNCTION��AND�DENOTE�B�� I(%, 4) � 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 15
�� ��IGG�,�AA� n :HEN[THE�INDICATOR�FUNCTION�CAN�BE�FORMALI�ED�AS�� χ % ≈ − 1 x − y G N y 4π O y. A. ( x − y P∈Q n �HERE� y. A IS�A�SMALL�REGION�NEAR�THE�SAMPLE� y [THE�SET� y. A P∈Q COVER�THE�SURFACE /Σ � % ∈ ' ( \ * 0 Σ 1NDICATOR�FUNCTION� !(%) = # % ∈ /* 1/2 Σ 1 % ∈ Σ 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 16
������2,1��� ��G�CB��BI��L ��� 1 Indicator Function Value �INGULARIT� ��� �LOBALNESS ��� 0 Distance to the Surface 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 17
���� ��G�CB��BI��L n 6RIGINAL�KERNEL�� I %, 4 = EFA G@(A) H CDG EFA V I %, 4 , % − 4 ≥ V % n W I %, 4 = −Y % − 4 G 1 4 % − 4 < V % , Y ∈ 0,1 4> G V ( (%) 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 18
���� ��G�CB��BI��L W ! % = 5 \ I %, 4 d:(4) + 5 I %, 4 d: 4 _ E,` E ∩67 67\ _ E,` E ! % \ V % + Y9 ( (%) = 1 2 + 1 2 − Y 9 % 4V ( (%) + O (| V % | ) 4 1 4V % + 9 ( % 2 + 9 % 1 Y = 1 4V ( % ! % ≈ \ 2 + 9 % 1 0 Y = 0 Distance to the surface 2V % 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 19
������2,1��� ��G�CB��BI��L ��� �INGULARIT� ��� �LOBALNESS ��� 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 20
������2,1��� B EFA G@ A surface Indicator function � d %, 4 = − EFA H 4. e ( CD % − 4 c ,ISCONTINUIT� f ��� Indicator Function Value ��B�I��F��L ��� Distance to 1 the Surface �LOBALNESS ��� 0 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 21
���� ��G���B���F���CB k % j i %′ 4 k %’ h iFB Sample Point 4 h i j i %′ 4 h iFB h i 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 22
������2,1��� ,ISCONTINUIT� ��� �INGULARIT� ��� ��C���B�GG ��� 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 23
���� ��C���B�GG Complexity � m(n o ) 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 24
���� ��G��-I���DC���-���C� n ,ATA��TRUCTURE�)DAPTIVE�6CTREE n 6B�ECTIVE�-STIMATE�FUNCTION�VALUE�FOR�EACH�CELL 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 25
���� ��G��-I���DC���-���C� n pqrs > uℎhwrℎxy9 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 26
���� ��G��-I���DC���-���C� n pqrs < uℎhwrℎxy9 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 27
���� ��G��-I���DC���-���C� { o Complexity � m(nyzn) { B 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 28
.I���B� � �ELATED��ORKS � �AUSS��URFACE��ECONSTRUCTION � ��GI��G��B���CAD�F�GCBG � �ONCLUSION 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 29
� ���CF���AG��CF��CAD�F�GCBG n �OISSON�����2A�HDAN ET�AL ����������� n �CREENED��OISSON��� �2A�HDAN ET�AL ��������:6� n ��,���ALAKLI ET�AL ����������. n ,ICTIONAR��LEARNING���IONG ET�AL ��������:6� 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 30
� ���IF��L n ,ATA��AMPLES�ON�UNIT�SPHERE������ SAMPLES 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 31
� ���IF��L n ,ATA�)IM�HAPE 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 32
� ���IF��L n ,ATA� Berger et al. A benchmark for surface reconstruction ?:6������� 2019/8/9 Surface Reconstruction Based on Modified Gauss Formula 33
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