Download - Interactive Visual Classiﬁcation with Euler Diagrams

Interactive Visual Classification with Euler Diagrams

Gennaro Cordasco,Rosario De Chiara

Università degli Studi di Salerno, Italy

Andrew Fish*

University of Brighton, UK

*funded by UK EPSRC grant EP/E011160: Visualization with Euler Diagrams.

Andrew Fish, University of Brighton, UK

Overview

• Classification Problem– related Euler Diagram applications

• Diagram Abstraction Problem• Concepts needed– e.g. weakly reducible, marked points, …

• On-line algorithms – complexity analysis

• EulerVC application demo

2

Andrew Fish, University of Brighton, UK 3

Classification Problem

• Resource classification is often challenging for users, and commonly hierarchical classifications are not sufficient for their needs.

• Free-form tagging approaches provide a flat space, utilising different tagging and visualisation mechanisms.

• What about using non-hierarchical classification structures, and what about the use of (Euler) diagrams…?


Related Euler Diagram Applications

• LHS [Inria]: Visualizing the numbers of documents matching a query from a library database, facilitating query modifications.

• RHS [Salerno]: File organisation with VennFS allows the user to draw Euler Diagrams in order to organize files within categories.

4


The main problem

• To compute the set of zones associated to a given collection of curves

• Zones are not so easy to describe, they can be non-convex, non-simply connected,…

• How to update the zone set when we add curves?

5

6

Euler Diagram ExampleLHS: A concrete diagram with 4 contours (simple closed

curves in plane) and 6 zones (regions inside a set of contours and outside the rest).

RHS: Depicts the abstract diagram/set system, which is the abstraction of the LHS: d= {A,B,C,D}, {∅,{B},{C},{D},{A,B},{B,C}}

ABC

D{A,B}

{B} {B,C}

{C} {D}

∅


Euler Diagram Problems: static

Concrete EDAbstract ED/ Set System

Abstraction

Generation


Euler Diagram Problems: dynamic

Concrete EDAbstract ED

Abstraction

Generation

TransformationsTransformations


Euler Diagram Problems: dynamic

Concrete EDAbstract ED

Abstraction

Generation

TransformationsTransformations

Wellformedness conditions


ED wellformedness conditions • contours are simple closed curves,

each with a single, unique label; • contours meet transversely (so no

tangential meetings or concurrency) and at most two contours meet at a single point;

• zones cannot be disconnected– i.e. each minimal region is a zone.

• Or, if you prefer… wellformed Euler diagrams can be thought of as “simple Euler diagrams”; i.e. simple Venn diagrams where some regions of intersection can be empty/missing”.


Fast Diagram InterpretationGiven a wellformed concrete Euler diagram d:1. compute the abstract Euler diagram for d

A

B

4 Zones:{A}{A,B}{B}∅


Fast Diagram InterpretationGiven a wellformed concrete Euler diagram d:1. compute the abstract Euler diagram for d 2. evaluate if the addition of a new contour or removal of an

existing contour yields a wellformed diagram (and update the abstract diagram accordingly).

A

B

A

B

C

C


• On-line approach– We consider the diagram construction using the

natural operations of contour addition/removal.

Fast Diagram Interpretation

A

B

A

B

A

C

B

C

Timeline


• On-line approach– We consider the diagram construction using the

natural operations of contour addition/removal.– What class of diagrams is this?

Fast Diagram Interpretation

A

B

A

B

A

C

B

C

Timeline


Weakly reducible Euler Diagrams• Reducible Venn diagrams [e.g. see Ruskey 97]

– the removal of one of its curves yields a Venn diagram

• Reducible Euler diagrams– the removal of one of its curves yields a WF Euler diagram

• Completely reducible Euler diagrams – There is a sequence of curve removals that yields the empty

ED through WF diagrams (or a sequence of curve additions from the empty ED).

• Weakly reducible Euler diagrams – There is a sequence of curve removals and additions that

yields the empty ED through WF diagrams.


Weakly reducible Euler Diagrams

ReducibleWeakly Reducible

Completely Reducible

WF Euler diagrams

?

?





WF Euler diagrams

?

?





WF Euler diagrams

?

?

Link to Proof

20

Wellformedness online verification• Theorem 1: Let d = C, Z, be a weakly reducible Euler

diagram, then |Z| = ip(d) + cc(d) + 1 where – cc(d) denotes the number of components– ip(d) denotes number of intersection points

1 component6 intersection points 1+6+1=8 zones






• Proof: by induction for single component case– Use that the addition of a new contour generates x > 1 intersection points splitting the new contour into x segments, each of which splits a different zone (otherwise the component was not connected) and so we have x new zones.

A




• Corollary of Proof: The addition of a simple closed curve A to a weakly reducible diagram d which only meets d in transverse intersections at points which are not intersection points of d yields a WF Euler diagram iff d+A satisfies the “zones condition” above.

A




• Corollary of Proof: The addition of a simple closed curve A to a weakly reducible diagram d which only meets d in transverse intersections at points which are not intersection points of d yields a WF Euler diagram iff d+A satisfies the “zones condition” above.

• The point(s): – We have enough points to “mark” the zone set– Can check wellformedness after curve addition by counting

A


Euler Diagram Abstraction

For our interface, we want both:– Quick construction method– Quick interpretation method

So, we consider the use of ellipses.Given two ellipses A and B, one can quickly find:

– their intersection points (in particular, since in a wellformed diagram tangential points are not allowed, we will have 0, 2 or 4 intersection points);

– their relationship (that is, if they overlap, if one is contained in the other or if they are disjoint)


Marked Points• We can represent each zone of d using a single marked

point; that is, we associate to each zone z Z(d) a single point such that where cl(z) is the

closure of z.

A

B

C

D E

2( )m x m R ( ) ( )m x cl zm


Marked Points

A

B

C

D E

• We can represent each zone of d using a single marked point; that is, we associate to each zone z Z(d) a single point such that where cl(z) is the

closure of z.

2( )m x m R ( ) ( )m x cl zm


Marked Points

A

B

C

D E


closure of z.

2( )m x m R ( ) ( )m x cl zm


Marked Points

A

B

C

D E

• We can represent each zone of d using a single marked point; that is, we associate to each zone z Z(d) a single point such that where cl(z) is the closure of z. And we can update this set of marked points appropriately upon the addition or removal of curves.

2( )m x m R ( ) ( )m x cl zm


Ellipse Addition

A

B

C

D

d

Draw a new

contour E Compute

E's relationship with

d

Is d +E wellformed

?

Add E to d

Compute Split zones

Generate new

Zones

Update Marked Points

Update covered Zones

Cont(E)Over(E

)Inter(E)

Reject A

noyes

d is a wellformed diagram

E

EE

E

E

d +E is a wellformed diagram


Ellipse Addition

A

B

C

D

d

Draw a new

contour E Compute


d

Is d +E wellformed

?

Add E to d

Compute Split zones

Generate new

Zones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


E

EE

E

E


Zd={, {A}, {B}, {C}, {A,B}, {A,C}, {B,C}, {C,D}, {A,B,C}}


Ellipse Additiond is a wellformed diagram

A

B

C

D

d

Zd={, {A}, {B}, {C}, {A,B}, {A,C}, {B,C}, {C,D}, {A,B,C}}

Draw a new

contour E Compute


d

Is d +E wellformed

?

Add E to d

Compute Split zones

Generate new

Zones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


E

EE

E

E



Draw a new

contour E Compute


d

Is d +E wellformed

?

Add E to d

Compute Split zones

Generate new

Zones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


E

EE

E

E


Ellipse Addition

A

B

C

D E

x1

x2

x3

x4x5

x6

d

Compute E's relationship with d

Zd={,{A},{B},{C},{A,B},{A,C},{B,C},{C,D},{A,B,C}}Cont(E)=Over(E)={A,B,C}Inter(E)={x1,x2,x3,x4,x5,x6}


Draw a new

contour E Compute


d

Is d +E wellformed

?

Add E to d

Compute Split zones

Generate new

Zones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


E

EE

E

E


Ellipse Addition

A

B

C

D E

x1

x2

x3

x4x5

x6

d

Is d +E wellformed ?

By using Thm 1



Draw a new

contour E Compute


d

Is d +E wellformed

?

Add E to d

Compute Split zones

Generate new

Zones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


E

EE

E

E


Ellipse AdditionCompute Split zones


Each arc splits exactly one zone

A

B

C

D E

dx2

x3

x4x5

x6

x1


Compute Split zones

Split Zones={, {C}, {B,C},{B},{A,B},{A}}


Draw a new

contour E Compute


d

Is d +E wellformed

?

Add E to d

Compute Split zones

Generate new

Zones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


E

EE

E

E


Ellipse Addition

A

B

C

D E

dx2

x3

x4x5

x6

x1



Generate new zones

Split Zones={, {C}, {B,C},{B},{A,B},{A}}New Zones={{E}, {C,E}, {B,C,E}, {B,E}, {A,B,E}, {A,E}}

Draw a new

contour E

Compute E's

relationship with

d

Is d +E wellformed

?

Add E to d

Compute Split zones

Generate new

Zones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


E

EE

E

E


Ellipse AdditionZd={,{A},{B},{C},{A,B},{A,C},{B,C},{C,D},{A,B,C}}Cont(E)=Over(E)={A,B,C}Inter(E)={x1,x2,x3,x4,x5,x6}

A

B

C

D E

dx2

x3

x4x5

x6

x1


Update marked points

Points x2 and x3 are swapped with y1 and y2, respectively. For instance, y1 previously marking {C} in d now marks {C,E}, whilst {C} is marked by x2.

Draw a new

contour E

Compute E's

relationship with

d

Is d +E wellformed

?

Add E to d

Compute Split zones

Generate new

Zones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


E

EE

E

E


Ellipse Addition

A

B

C

D E

x1

x2

x3

x4x5

x6

d

y1

y2



Update covered zones

Zones {A,C} and {A,B,C} are not split and their marking points, y3 and y4, are in interior(E), so these zones need to be updated and become {A,C,E} and {A,B,C,E}.

Draw a new

contour E

Compute E's

relationship with

d

Is d +E wellformed

?

Add E to d

Compute Split zones

Generate new

Zones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


E

EE

E

E


Ellipse Addition

A

B

C

D E

x1

x2

x3

x4x5

x6

d

y1

y2y3

y4



Ellipse Additiond+E is a wellformed diagram

Zd+E={, {A}, {B}, {C}, {E}, {A,B}, {A,E}, {B,C}, {B,E}, {C,D}, {C,E}, {A,B,E}, {A,C,E}, {B,C,E}, {A,B,C,E}}

Draw a new

contour E

Compute E's

relationship with

d

Is d +E wellformed

?

Add E to d

Compute Split zones

Generate new

Zones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


E

EE

E

E


A

B

C

D E


Ellipse Additiond+E is a wellformed diagram

Zd+E={, {A}, {B}, {C}, {E}, {A,B}, {A,E}, {B,C}, {B,E}, {C,D}, {C,E}, {A,B,E}, {A,C,E}, {B,C,E}, {A,B,C,E}}

Draw a new

contour E

Compute E's

relationship with

d

Is d +E wellformed

?

Add E to d

Compute Split zones

Generate new

Zones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


E

EE

E

E


A

B

C

D E

dx2

x3

x4x5

x6

x1

Skip Ellipse Deletion


Ellipse Removald=d+E

Zd={, {A}, {B}, {C}, {E}, {A,B}, {A,E}, {B,C}, {B,E}, {C,D}, {C,E}, {A,B,E}, {A,C,E}, {B,C,E}, {A,B,C,E}}

Remove a

contour C

Compute C's

relationship with d-C

Is d - C wellformed

?

Remove C from d Compute

Split zones

MergeZones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


C

C

d -C is a wellformed diagram

A

B

C

D E

C

C

C

C

C


Ellipse Removald=d+E


A

B

C

D E

d

Remove a

contour C

Compute C's


Is d - C wellformed

?


Split zones

MergeZones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


C

C


C

C

C

C

Cx1

x2

x3

x4x5

x6


Ellipse Removald is a wellformed diagram


Remove a

contour C

Compute C's


Is d - C wellformed

?


Split zones

MergeZones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


C

C


A

B

C

D E

C

C

C

C

C


Ellipse RemovalCompute C’s relationship with d-C

Zd={, {A}, {B}, {C}, {E}, {A,B}, {A,E}, {B,C}, {B,E}, {C,D}, {C,E}, {A,B,E}, {A,C,E}, {B,C,E}, {A,B,C,E}}Cont(C)=∅Over(C)={A,B,E}Inter(C)={z1,z2,z3,z4,z5,z6}

Remove a

contour C

Compute C's


Is d - C wellformed

?


Split zones

MergeZones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


C

C


A

B

C

D E

C

C

C

C

Cz2

z3

z4

z5

z6

z1


Ellipse RemovalIs d-C wellformed?


Remove a

contour C

Compute C's


Is d - C wellformed

?


Split zones

MergeZones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


C

C


A

B

C

D E

C

C

C

C

Cz2

z3

z4

z5

z6

z1

Only disconnected zones need to be checked.


Compute Split zones

Split Zones={ ,{B}, {E}, {A,E}, {B,E}, {A,B,E}}

Ellipse Removal

Remove a

contour C

Compute C's


Is d - C wellformed

?


Split zones

MergeZones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


C

C


A

B

C

D E

C

C

C

C

Cz2

z3

z4

z5

z6

z1




Merge zones

Ellipse Removal

Remove a

contour C

Compute C's


Is d - C wellformed

?


Split zones

MergeZones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


C

C


A

B

C

D E

C

C

C

C

Cz2

z3

z4

z5

z6

z1




Ellipse RemovalRemove marked points that belong to C


Remove a

contour C

Compute C's


Is d - C wellformed

?


Split zones

MergeZones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


C

C


A

B

C

D E

C

C

C

C

Cz2

z3

z4

z5

z6

z1


Ellipse RemovalRemove marked points that belong to C


Remove a

contour C

Compute C's


Is d - C wellformed

?


Split zones

MergeZones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


C

C


A

B

C

D E

C

C

C

C

C


Update covered zones

Zone {C,D} is not split by C and its marked point is in interior(C), so it needs to be updated and it becomes {D}.

Ellipse RemovalZd={, {A}, {B}, {C}, {E}, {A,B}, {A,E}, {B,C}, {B,E}, {C,D}, {C,E}, {A,B,E}, {A,C,E}, {B,C,E}, {A,B,C,E}}Cont(C)=∅Over(C)={A,B,E}Inter(C)={z1,z2,z3,z4,z5,z6}

Remove a

contour C Compute

C's relationship with

d-C

Is d - C wellformed

?


Split zones

MergeZones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


C

C


A

B

C

D E

C

C

C

C

C


d-C is a wellformed diagram

Ellipse Removal

Remove a

contour C Compute

C's relationship with

d-C

Is d - C wellformed

?


Split zones

MergeZones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


C

C


C

C

C

C

C

Zd-C={, {A}, {B}, {E}, {D}, {A,B}, {A,E}, {B,E}, {A,B,E},}

A

B

D E


Complexity

• Ellipse addition and removal algorithms are similar and their complexity is

56

Complexity

Draw a new

contour E Compute


d

Is d +E wellformed

?

Add E to d

Compute Split zones

Generate new

Zones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


E

EE

E

E

d +E is a wellformed diagramO(|C|)

O(|C|)

O(|C| log |C|)O(|C|)

O(|C|)

O(|Z|)

For each other curve, compute intersection points with E; if curve C disjoint then check if marked point for C is in interior (E)

57

Complexity

Draw a new

contour E Compute


d

Is d +E wellformed

?

Add E to d

Compute Split zones

Generate new

Zones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


E

EE

E

E


O(|C|)


O(|C|)

O(|Z|)

Check no tangential/triple points created. Check WF using the number of intersection points of E with other curves (Thm 1)

58

Complexity

Draw a new

contour E Compute


d

Is d +E wellformed

?

Add E to d

Compute Split zones

Generate new

Zones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


E

EE

E

E


O(|C|)


O(|C|)

O(|Z|)

a) Each arc splits exactly 1 zone

b) # arcs = O(|C|)c) Consecutive

arcs split zones which differ by one exactly contour, so we order the points of inter(E).

59

Complexity

Draw a new

contour E Compute


d

Is d +E wellformed

?

Add E to d

Compute Split zones

Generate new

Zones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


E

EE

E

E


O(|C|)


O(|C|)

O(|Z|)

Add E to split zones

60

Complexity

Draw a new

contour E

Compute E's

relationship with

d

Is d +E wellformed

?

Add E to d

Compute Split zones

Generate new

Zones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


E

EE

E

E


O(|C|)


O(|C|)

O(|Z|)

Iterate through Inter(E) and perform set membership check

61

Complexity

Draw a new

contour E

Compute E's

relationship with

d

Is d +E wellformed

?

Add E to d

Compute Split zones

Generate new

Zones



Cont(E)Over(E

)Inter(E)

Reject A

noyes


E

EE

E

E


O(|C|)


O(|C|)

O(|Z|)

Iterate through non-split zone set and check if their marked point is in interior(E)

62

A naive approach

Draw a new

contour E

A new contour

is added

Duplicate each zone

Check Zones

d is an Euler diagram

E

E

O(1)

O(|Z|)

O(|Z|f(|C|))

O(|Z|f(|C|))

f(|C|) = time needed to check if

a given zone is present in d + E


EulerVC

• As a proof of concept we have implemented the algorithms as general purpose Java library– And used it to build a small application which

allows users to use Euler Diagrams to classify internet bookmarks on their desktop, or on Delicious…


“Delicious is a social bookmarking web service for storing, sharing, and discovering web bookmarks” (from Wikipedia)

– Users can tag each of their bookmarks with freely chosen index terms


BookmarkTags

66

EulerVC: skip slides; demo tool


EulerVC

81

EulerVC


Conclusion• Algorithms developed to solve the on-line diagram

abstraction problem.• EulerVC application constructed useful for:

– User classification of resources (requires user testing…)– Exploratory research of diagram properties like weak

reducibility (see Symmetric Venn(5) with Ellipses)

• Future plans: – Handle NWF cases by extending the use of marked pts– General resource handling (web pages, photos, files...) – Integration with EulerView idea (for larger scale)– Investigate diagram classes


Blank

Download - Interactive Visual Classiﬁcation with Euler Diagrams

Top Related