size function
DESCRIPTION
Size Function. Jianwei Hu 2007-05-23. Author. Patrizio Frosini Ricercatore presso la Facolt à di Ingegneria dell'Universit à di Bologna Dipartimento di Matematica, Piazza di Porta San Donato, 5, BOLOGNA . http://www.dm.unibo.it/~frosini/. References. - PowerPoint PPT PresentationTRANSCRIPT
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Size FunctionSize FunctionJianwei HuJianwei Hu2007-05-232007-05-23
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AuthorAuthor
Patrizio Frosini Patrizio Frosini
• Ricercatore presso la Facoltà di Ingegneria dell'Università di BolognaRicercatore presso la Facoltà di Ingegneria dell'Università di Bologna• Dipartimento di Matematica, Piazza di Porta San Donato, 5, BOLOGNA Dipartimento di Matematica, Piazza di Porta San Donato, 5, BOLOGNA
http://www.dm.unibo.it/~frosini/http://www.dm.unibo.it/~frosini/
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ReferencesReferences1.1. Frosini, P.Frosini, P., A distance for similarity classes of submanifolds of a , A distance for similarity classes of submanifolds of a
Euclidean space, Bull. Austral. Math. Soc. 42, 3 (1990), 407-416. Euclidean space, Bull. Austral. Math. Soc. 42, 3 (1990), 407-416. 2.2. Verri, A., Uras, C., Verri, A., Uras, C., Frosini, P.Frosini, P., Ferri, M., On the use of size functions , Ferri, M., On the use of size functions
for shape analysis, Biol. Cybern. 70, (1993), 99-107. for shape analysis, Biol. Cybern. 70, (1993), 99-107. 3.3. Frosini, P.Frosini, P., Landi, C., Size Theory as a Topological Tool for Computer , Landi, C., Size Theory as a Topological Tool for Computer
Vision, Pattern Recognition and Image Analysis, Vol. 9, No. 4, 596-Vision, Pattern Recognition and Image Analysis, Vol. 9, No. 4, 596-603, 1999.603, 1999.
4.4. Frosini, P.Frosini, P., Pittore, M., New methods for reducing size graphs, , Pittore, M., New methods for reducing size graphs, Intern. J. Computer Math. 70, 505-517, 1999. Intern. J. Computer Math. 70, 505-517, 1999.
5.5. Frosini, P.Frosini, P., Landi, C., Size functions and formal series, Applicable , Landi, C., Size functions and formal series, Applicable Algebra in Engin. Communic. Comput., 12(4) (2001), 327-349. Algebra in Engin. Communic. Comput., 12(4) (2001), 327-349.
6.6. Cerri, A., Ferri, M., Giorgi, D., Cerri, A., Ferri, M., Giorgi, D., Retrieval of trademark images by means Retrieval of trademark images by means of size functions, Graph. Models, 68 (2006), 451-471. of size functions, Graph. Models, 68 (2006), 451-471.
7.7. d'Amico, M., d'Amico, M., Frosini, P.Frosini, P., and Landi, C., Using matching distance in , and Landi, C., Using matching distance in Size Theory: a survey, International Journal of Imaging Systems and Size Theory: a survey, International Journal of Imaging Systems and Technology, Vol. 16 (2006) , No. 5, 154–161. Technology, Vol. 16 (2006) , No. 5, 154–161.
8.8. Donatini, P., Donatini, P., Frosini, P.Frosini, P., Natural pseudodistances between closed , Natural pseudodistances between closed surfaces, Journal of the European Mathematical Society, Vol. 9 surfaces, Journal of the European Mathematical Society, Vol. 9 (2007), No. 2, 231–253. (2007), No. 2, 231–253.
9.9. d'Amico, M., d'Amico, M., Frosini, P.Frosini, P., and Landi, C., Natural pseudo-distance and , and Landi, C., Natural pseudo-distance and optimal matching between reduced size functions (submitted). optimal matching between reduced size functions (submitted).
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OutlineOutline• General Concepts of Size FunctionGeneral Concepts of Size Function
• DefinitionDefinition• Invariant PropertiesInvariant Properties
• Comparing Size FunctionComparing Size Function• Corner Points & Formal SeriesCorner Points & Formal Series
• Reducing Size GraphsReducing Size Graphs• -reduction-reduction• ⊿⊿-reduction-reduction
• Measuring FunctionsMeasuring Functions• ApplicationsApplications
• Images Retrieval Images Retrieval • 3D Shape Matching3D Shape Matching
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What are Size FunctionsWhat are Size Functions• Size Functions are a new kind of Size Functions are a new kind of
mathematical transformmathematical transform• Size Functions are a mathematical tool for Size Functions are a mathematical tool for
describing and comparing shapes of describing and comparing shapes of topological spacestopological spaces
• Shape Size graph Natural Shape Size graph Natural numbernumber
http://vis.dm.unibo.it/sizefcts/FAQ/faq.htmhttp://vis.dm.unibo.it/sizefcts/FAQ/faq.htm
measuring function
size function
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DefinitionsDefinitions• Definition 1: Size Pair Definition 1: Size Pair
• is a compact topological space.is a compact topological space.• is a continuous function from to the set (called is a continuous function from to the set (called
measuring function).measuring function).• Definition 2: homotopyDefinition 2: homotopy
For every we define a relation in by For every we define a relation in by setting setting if and if and only if either or there exists a continuous path only if either or there exists a continuous path such that such that and for every . In this second case and for every . In this second case we shall say that and are we shall say that and are
homotopic and call a homotopy from homotopic and call a homotopy from to . to .
( , )M jMj M ¡
( )yj £ -y Î ¡ yj £@ M
( , )yP Q P Q Mj £@ Î P Q=:[0,1] Mg ®
(0) , (1)P Qg g= = ( ( )) yj g t £ [0,1]t ÎP Q
( )yj £ - g ( )yj £ -P Q
The BULLETIN of the Australian Mathematical Society 1990The BULLETIN of the Australian Mathematical Society 1990
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Definitions (Contd.)Definitions (Contd.)• Remark 3:Remark 3:
For every we shall denote by the set For every we shall denote by the set ..
• Definition 4: Size FunctionDefinition 4: Size FunctionConsider the function defined by Consider the function defined by setting equal to the (finite or infinite) setting equal to the (finite or infinite) number of equivalence classes in which is number of equivalence classes in which is divided by the equivalence relation . Such a divided by the equivalence relation . Such a function will be called the size function associated function will be called the size function associated with the size pair .with the size pair .
x Î ¡ M xj £{ }: ( )P M P xjÎ £
( , )( , )Ml x yj
( , )( , )Ml x yj
( , ) : { }Ml j ´ ® È ¥¡ ¡ ¥
M xj £yj £@
( , )M j
The BULLETIN of the Australian Mathematical Society 1990The BULLETIN of the Australian Mathematical Society 1990
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ExampleExample
http://vis.dm.unibo.it/sizefcts/FAQ/faq.htmhttp://vis.dm.unibo.it/sizefcts/FAQ/faq.htm
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Invariant PropertiesInvariant Properties• Euclidean InvarianceEuclidean Invariance
Biological Cybernetics 1993Biological Cybernetics 1993
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Invariant PropertiesInvariant Properties• ““Ad hoc” InvarianceAd hoc” Invariance
Biological Cybernetics 1993Biological Cybernetics 1993
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Resistant to NoiseResistant to Noise
Biological Cybernetics 1993Biological Cybernetics 1993
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Resistant to Occlusions Resistant to Occlusions
Biological Cybernetics 1993Biological Cybernetics 1993
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Concepts for ComparisonConcepts for Comparison• Cornerpoint Cornerpoint
• • •
• Formal SeriesFormal Series• 3A+B+4C+5D+E3A+B+4C+5D+E
( , )x y( , ) ( , ) ( , ) ( , )( , ) ( , ) ( , ) ( , ) 0M M M Ml x y l x y l x y l x yj j j ja b a b a b a bé ù é ù+ - - - - - + + - - + >ê ú ê úë û ë û( , ) min ( , ) min( , ) ( , ) 0M Ml x y l x yj jb b- - + >( , ) max ( , ) max( , ) ( , ) 0M Ml x y l x yj ja a+ - - >
Applicable Algebra in Engineering, Communication and Computing 2001Applicable Algebra in Engineering, Communication and Computing 2001
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How to CompareHow to CompareCompare formal series and Compare formal series and
• Hausdorff distanceHausdorff distance• Two sets and Two sets and •
• Matching distanceMatching distance• Two sets andTwo sets and
• is the set of all bijections from tois the set of all bijections from to• • •
1 1 2 2 h hmP mP mP+ +×××+1 1 2 2 k knQ nQ nQ+ +×××+
1 2{ , , , }hP P P××× 1 2{ , , , }kQ Q Q×××max{max min ,max min }i j i j j i i jP Q P Q- -
1 21 2 1 2 1 21 1 1 2 2 2{ , , , , , , , , , , , , }hm m m
h h hP P P P P P P P P= ××× ××× ××× ×××P1 21 2 1 2 1 2
1 1 1 2 2 2{ , , , , , , , , , , , , }kn n nk k kQ Q Q Q Q Q Q Q Q= ××× ××× ××× ×××Q
F 'P 'Qf Î F
' '(( , ),( ', ')) min{max{ ' , '},max{ , }}2 2y x y xd x y x y x x y y - -= - -
( )inf sup ( , )match i fif id d P Q=
Applicable Algebra in Engineering, Communication and Computing 2001Applicable Algebra in Engineering, Communication and Computing 2001
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Reduction of Size GraphsReduction of Size Graphs
• A global method: A global method: -reduction-reduction• A local method: ⊿-reductionA local method: ⊿-reduction
International Journal of Computer Mathematics 1999International Journal of Computer Mathematics 1999
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-reduction-reduction• is the set of one ring neighbor of is the set of one ring neighbor of • is the set for which takes the is the set for which takes the
largest valuelargest value• is the single step descent flow functionis the single step descent flow function• is the descent flow operatoris the descent flow operator• Minimum vertexMinimum vertex• Main saddle Main saddle
(*)iA(*)iB
iv( ) ( )iv wj j-
(*)LL(*)
International Journal of Computer Mathematics 1999International Journal of Computer Mathematics 1999
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-reduction-reduction
International Journal of Computer Mathematics 1999International Journal of Computer Mathematics 1999
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⊿⊿-reduction-reduction• Three simple Three simple ⊿-moves⊿-moves
International Journal of Computer Mathematics 1999International Journal of Computer Mathematics 1999
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⊿⊿-reduction-reduction
International Journal of Computer Mathematics 1999International Journal of Computer Mathematics 1999
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⊿⊿-reduction-reduction• Does a total Does a total ⊿-reduction exist?⊿-reduction exist?• Two different ways to obtain the same total Two different ways to obtain the same total
⊿-reduction ⊿-reduction
International Journal of Computer Mathematics 1999International Journal of Computer Mathematics 1999
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-reduction-reduction vs ⊿-reduction vs ⊿-reduction
KO KO ⊿⊿ ⊿⊿KO KO
International Journal of Computer Mathematics 1999International Journal of Computer Mathematics 1999
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-reduction-reduction vs ⊿-reduction vs ⊿-reduction• Sometimes Sometimes -reduction makes the size graph -reduction makes the size graph
worseworse
• The procedure of applying simple The procedure of applying simple ⊿-moves ⊿-moves cannot proceed indefinitelycannot proceed indefinitely
International Journal of Computer Mathematics 1999International Journal of Computer Mathematics 1999
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Measuring FunctionsMeasuring Functions• Distance from pointsDistance from points• ProjectionsProjections• JumpsJumps
Graphical Models 2006Graphical Models 2006
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Images RetrievalImages Retrieval
Graphical Models 2006Graphical Models 2006
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3D Shape Matching3D Shape Matching• Measuring FunctionsMeasuring Functions
• Distance from the center of mass to each vertexDistance from the center of mass to each vertex• Transformations invarianceTransformations invariance
• Distance from some fixed planesDistance from some fixed planes• Distance from the point user specifiedDistance from the point user specified
• Deformed model retrievalDeformed model retrieval• Curvature of each point (patch)Curvature of each point (patch)
• Feature sensitiveFeature sensitive
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3D Shape Matching3D Shape Matching• Size graph reductionSize graph reduction
Salient Geometric Features for Partial Shape Matching and Similarity, Salient Geometric Features for Partial Shape Matching and Similarity, Ran Gal and Daniel Cohen-or, ACM Transactions on Graphics, Vol. 25, Ran Gal and Daniel Cohen-or, ACM Transactions on Graphics, Vol. 25, No. 1, January 2006, Pages 130–150. No. 1, January 2006, Pages 130–150.
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