size function jianwei hu 2007-05-23. author patrizio frosini ricercatore presso la facolt à di...

Size FunctionSize Function

Jianwei HuJianwei Hu2007-05-232007-05-23

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/

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).

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

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

measuring function

size functionsize function

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: homotopyFor 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

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

ExampleExample

http://vis.dm.unibo.it/sizefcts/FAQ/faq.htmhttp://vis.dm.unibo.it/sizefcts/FAQ/faq.htm

Invariant PropertiesInvariant Properties• Euclidean InvarianceEuclidean Invariance

Biological Cybernetics 1993Biological Cybernetics 1993

Invariant PropertiesInvariant Properties• ““Ad hoc” InvarianceAd hoc” Invariance


Resistant to NoiseResistant to Noise


Resistant to Occlusions Resistant to Occlusions


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

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= ××× ××× ××× ×××P

1 21 2 1 2 1 21 1 1 2 2 2{ , , , , , , , , , , , , }kn n n

k k kQ Q Q Q Q Q Q Q Q= ××× ××× ××× ×××Q

F 'P 'Q

f Î F' '

(( , ),( ', ')) min{max{ ' , '},max{ , }}2 2

y 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

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

-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

(*)iBiv

( ) ( )iv wj j-

(*)L

L(*)


-reduction-reduction


⊿⊿-reduction-reduction• Three simple Three simple ⊿-moves⊿-moves


⊿⊿-reduction-reduction


⊿⊿-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


-reduction-reduction vs ⊿-reduction vs ⊿-reduction

KO KO ⊿⊿ ⊿⊿KO KO


-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


Measuring FunctionsMeasuring Functions• Distance from pointsDistance from points• ProjectionsProjections• JumpsJumps

Graphical Models 2006Graphical Models 2006

Images RetrievalImages Retrieval

Graphical Models 2006Graphical Models 2006

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

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.

size function jianwei hu 2007-05-23. author patrizio frosini ricercatore presso la facolt à di...

Documents