By chance is not enough: Preserving relative density through non uniform sampling

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by  Enrico Bertini

By chance is not enough: preserving relative density through non uniform sampling
EnricoBertini,GiuseppeSantucci
DipartimentodiInformaticaeSistemistica-Universita`diRoma“LaSapienza”
ViaSalaria,113-00198Roma,Italy-{bertini,santucci}@dis.uniroma1.it
Abstract malmodelwediscussinthepapergivespreciseindications
7 Dealingwithvisualizationscontaininglargedatasetis ontherightamountofdatasamplingneededtoproducea
1 achallengingissueand,inthefieldofInformationVisual- representation preserving the most important image char-
0
ization,almosteveryvisualtechniquerevealsitsdrawback acteristics, i.e., relative densities that are one of the main
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whenvisualizinglargenumberofitems. Todealwiththis cluestheusercangraspfrom2Dscatter-plots.

n problemweintroduceaformalenvironment,modelingina Thecontributionofthispaperistwofold: (1)itpresents
a
virtual space the image features we are interested in (e.g, aformalmodelthatallowsfordefiningandmeasuringdata
J
absoluteandrelativedensity,clusters,etc.) andwedefine density both in terms of a virtual space and of a physical
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some metrics able to characterize the image decay. Such space(e.g.,adisplay)and(2)itdefinesanovelautomatic
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metrics drive our automatic techniques (i.e., not uniform non uniform sampling technique driven by some metrics

] sampling) rescuing the image features and making them definedabovethepreviousfigures.
R
visible to the user. In this paper we focus on 2D scatter- The paper is structured as follows: Section 2 analyzes
G
plots,devisinganovelnonuniformdatasamplingstrategy related works, Section 3 describes the model we use to
s. abletopreserveinaneffectivewayrelativedensities. characterize clutter and density, formalizing the problem
c andintroducingthemetricsweareinterestedin,Section4
[ Keywords—visualclutter,metrics,non-uniformsampling.
describes our non uniform sampling technique, Section 5

1 discussestheresultsobtainedapplyingourtechniquestoa
1 Introduction
v realdataset,and,finally,Section6presentssomeconclu-
0 Visualizinglargedatasetsresults,veryoften,inaclut-
sions,openproblemsandfuturework.
1 tered image in which a lot of graphical elements overlap
1 andmanypixelsbecomeoverplotted,hidingfromtheuser 2 RelatedWork
7
themainimagevisualfeatures. ThispaperdealswithissuesconcerningmetricsforIn-
0
We deal with this problem providing a formal frame- formationVisualizationandtechniquestoaddresstheprob-
.
1 work to measure the amount of decay resulting from a lem of overlapping pixels and visual clutter in computer
0
given visualization, then we build, upon these measures, displays. In the following we illustrate the research pro-
7
1 an automatic non uniform sampling strategy that aims at posals closer to our approach and their relationship with
: reducingsuchadegradation. Wefocusonaverycommon ourwork.
v
visualtechnique,2Dscatter-plots,analyzingthelossofin-
i 2.1 MetricsforInformationVisualization
X formationderivedbyoverlappingpixels.
As expressed in [6], Information Visualization needs
r Inthispaperweimproveandextendsomepreliminary
a metricsabletogivepreciseindicationsonhoweffectively
resultspresentedin[1],definingaformalmodelthatesti-
avisualizationpresentsdataandtomeasureitsgoodness.
matestheamountofoverlappingelementsinagivenarea
Some preliminary ideas have been proposed considering
andtheremainingfreespace. Thesepiecesofinformation
bothformalmeasurementsandguidelinestofollow.
give an objective indication of what is eventually visual-
Tufte proposes in [9] some measures to estimate the
ized on the physical device; exploiting such measures we
qualityof2Drepresentationsofstaticdata. Measureslike
canestimatethequalityofthedisplayedgraphicdevising
the lie factor, that is the ratio of the size of an effect as
techniquesabletorecoverthedecayedvisualization.
shown graphically to its size in the data, are examples of
Toeliminatethesenseofclutter,weemployalowgrain
first attempts to systematically provide indications about
nonuniformsamplingtechniquedealingwiththechalleng-
thequalityoftheimagedisplayed. Tufte’sproposalhow-
ing issue of devising the right amount of sampling in or-
everappliestopaperbased2Dvisualizationsanddoesnot
dertopreservethevisualcharacteristicsoftheunderlying
directlyapplytointeractivecomputer-basedimages. Brath
data. Itisquiteevident,infact,thatatoostrongsampling
in[7],startingfromTufte’sproposal,definesnewmetrics
is useless and destroys the less dense areas, while a too
for static digital 3D images. He proposes metrics such as
lightsamplingdoesnotreducetheimageclutter. Thefor-
1
datadensity(numberofdatapoints/numberofpixels)that
resembleTufte’sapproachtogetherwithnewones,aiming
at measuring the visual image complexity. The occlusion
percentage, for example, has connections with our work.
It provides a measure of occluded elements in the visual
spacesuggestingtoreducesuchavalueasmuchaspossi-
ble.Thesemetricsareinterestingandaremoreappropriate
for describing digital representations. However, as stated
bytheauthor,theyarestillimmatureandneedrefinements.
While the above metrics aim at measuring a general
goodnessoratcomparingdifferentvisualsystems,ouraim
istomeasuretheaccuracyofthevisualization,thatis,how
well it represents the characteristics hidden inside data.
They present some similarities with past metrics but op-
erate at a lower level dealing with pixels and data points,
Figure1: Plottingmailparcels
providingmeasuresthatcandirectlybeexploitedtodrive
correctiveactions. Itisworthtonotethat,onthecontrary
of the above proposals, we will show how the suggested sity”. Even if interesting, this idea is not free of draw-
metrics can be exploited in practice to take quantitative backs. In particular, when the data present particular dis-
decisionsaboutcorrectiveactionsandenhancethecurrent tributions,i.e.,thedatasethasbothveryhighandverylow
visualization. density areas, choosing the right amount of sampling is a
challenging task. Depending on the amount of sampling
2.2 Dealingwithoverlappingpixelsandclutter
two problems can arise: 1) If the sampling is too strong
The problem of eliminating visual clutter and overlap-
the areas in which the density is under a certain level be-
ping pixels to produce intelligible graphics has been ad-
comecompletelyempty;2)Ifthesamplingistooweakthe
dressedbymanyproposals.
areaswithhigherdensitieswillstilllookallthesame(i.e.,
Jitteringasstatedin[8],isawidelyadoptedtechnique
completelysaturated)andconsequentlythedensitydiffer-
thatpermitstomakeapparentpixelsthatnaturallymapinto
encesamongthemwillbenotperceivedbytheuser.Afirst
the same position into the screen. The idea is to slightly
proposalinthisdirectionisin[1],whereanautomaticuni-
changethepositionofoverlappingpointsinordertorender
formsamplingtechniqueispresented,abletocomputethe
them all visible. Similarly, space-filling pixel-based tech-
optimalsamplingratiow.r.t. somequalitymetrics.
niques[5]distributedatapointsalongpredefinedcurvesto
Ourapproachdiffersfromtheaboveproposalsforthree
avoidoverlappingpixels,shiftingthemtopositionsthatare
mainaspects:
ascloseaspossibletotheoriginalone.
Transparency is also an interesting technique to over- • itprovidesasoundmodelfordefininginbothavir-
comeocclusionandreduceclutter,bothinin3D[12]and tualandphysicalspaceseveralmetricsintendedspecif-
2D[3]visualizations. However, whendealingwithpixel- icallyfordigitalimages;
basedvisualizationsitisnotpossibletoconveytransparency
atthelevelofsinglepixels,thoughitisuseless. • it provides, on the basis of the above figures, some
Constantdensityvisualization[10][11]isaninteresting quantitativeinformationabouttheimagedecay;
techniquetodealwithclutter. Exploitingtheideaofgen-
• it exploits such numerical results for automatically
eralizedfisheyeviews[4],itconsistsingivingmoredetails
computing where, how, and how much to sample
tolessdenseareasandlessdetailstodenserareas,allowing
preserving,asmuchaspossible,acertainvisualchar-
thescreenspacetobeoptimallyutilizedandtoreduceclut-
acteristic.
ter.Theproblemswiththisapproacharethatitrequiresthe
usertointeractwiththesystem,theoveralltrendofdatais 3 ModelingVisualDensityandClutter
generallylost,andsomedistortionsareintroduced.
In this section we present the formal framework that
Sampling is used in [2] to reduce the density of visual
aimsatmodelingtheclutterproducedbyover-plottingdata.
representation. As the authors state, if the sampling is
Some preliminary issues about the matter are in [1]; here
madeinrandomway,thedistributionispreservedandthough
weshowarefinementofthatresults.
it is still possible to grasp some useful information about
We consider a 2D space in which we plot elements by
data correlation and distributions, permitting “to see the
associatingapixeltoeachdataelementmappingtwodata
overall trends in the visualization but at a reduced den-
2
attributes on the spatial coordinates. As an example, Fig- i.e.,selectingnelementsfromasetofpelementsallowing
ure1showsabout160,000mailparcelsplottedontheX-Y repetitions(dispositionswithrepetitions: pn).
plane according to their weight (X axis) and volume (Y Calculatingthe#configwithexactlykcollisionsisper-
axis). Itisworthnotingthat,evenifthenumberofplotted formed in three steps. First we calculate all the possible
itemsislittle, theareaclosetotheoriginisverycrowded ways of selecting n−k non colliding points from p pix-
(usually parcels are very light and little), so a great num- els (combinations without repetitions: (cid:0) p (cid:1)). After that,
n−k
berofcollisionsispresentinthatarea: themostcrowded for each of such combinations, we calculate all the possi-
areacontainsmorethat50,000(about30%)ofthewhole blewaysofhittingk timesoneormoreofthen−k non
datasetcompressedinlessthan1%ofthewholescreen. collidingpointsinordertoobtainexactlykcollisions,that
Exploiting well known results coming from the calcu- corresponds to selecting k elements form a set of n − k
lus of probability, we derive a function that estimates the elements with repetitions (combinations with repetitions:
amountofcollidingpointsand,asaconsequence,theamount (cid:0)n−k+k−1(cid:1)). Finally, because of we are interested in all
k
of free available space. More formally, two points are the possible dispositions, we need to count the permuta-
in collision when their projection is on the same physical tions (PERM) of these combinations. Unfortunately, be-
pixel. In order to derive such a function, we imagine to causeofthevariablenumberofduplicates(e.g.,itispossi-
tossndatapointsinarandomwayonafixedareaofppix- bletohavekcollisionshittingk+1timesthesamepixelp ,
i
els. Thisassumptionisquitereasonableifweconductour orktimesp andtwotimespixelp , ork-1timesp , two
i j i
analysisonsmallareas. timespixelp ,andtwotimespixelp andsoon)wewere
j k
Toconstructsuchfunctionsweuseaprobabilisticmodel noabletoexpresssuchpermutationsbyacloseformula.
basedontheparametersjustdescribed,thatherewesum- ¿From the above expression we derived, through a C
marizeforthesakeofclarity: program, a series of functions (see Figure 5) showing the
behavior of the observed area as the number of plotted
• nisthenumberofpointswewanttoplot; points increases. More precisely, we compute the avail-
ablefreespaced(Yaxis,aspercentagew.r.t. p),themean
• pisthenumberofavailablepixels;
ofcollidingelementsk(Yaxis,aspercentagew.r.t. n)for
• kisthenumberofcollisions; anygivennumberofplottedpointsn(Xaxis,aspercentage
w.r.t. p). Forexample,ifwehaveanareaof64pixels,the
• disthenumberoffreepixels. graphtellusthatplotting200%(128)ofppointswillpro-
duce an average of 56.7% (72.5) collisions. On the other
Theprobabilityofhavingexactlykcollisionsplottingn
hand, if we plot 128 points having 72.5 collisions we can
points on an area of p pixels, Pr(k,n,p), is given by the
computethefreepixelsd,asd=64−(128−72.5)=8.5
followingfunction:
(13.3%).
The behavior of the functions is quite intuitive: as the
PERM[( p )(n−k+k−1)]
n−k k ifn≤pandk ∈[0,n−1] numberofplottedpointsnincreasesthepercentageofcol-
pn
orn>pandk ∈[n−p,n−1] lisions increases as well while the free space decreases;
roughlyspeaking,wecansaythatoverplottingfourtimes
0 ifn>pandk ∈[0,n−p] the screen results in a totally saturated display (1.6% of
freespace).
Thefunctionisdefinedonlyfork <n,becauseitisim- Such functions can tell us how much we are saturat-
possibletohavemorecollisionsthanplottedpoints. More- ingthespaceor,asamorecomplexpossibility,thewayin
over, it is easy to understand that in some cases the prob- whichthedisplayisabletorepresentrelativedensitiesand
ability isequal to zero: if n > p, becauseof we areplot- howmuchtosamplethedatatoguaranteeaprefixedvisu-
ting more points than available pixels, we must necessar- alizationquality.Thisresultisexploitedinthenextsection
ilyhavesomecollisions. Forexample,ifwehaveanarea andweclarifyitthroughanexample. Assumethatweare
of 8×8 pixels and we plot 66 points, we must necessar- plotting n points on the area A turning on p pixels and
1 1
ily have at least 2 collisions, so Pr(0,66,64) = 0 and 2npointsontheareaA turningonp pixels. Inprinciple,
2 2
Pr(1,66,64)=0. theusershouldperceiveareaA ascontainingmore(i.e.,
2
The basic idea of the formula is to calculate, given p twice as many) points as area A . Because of collisions,
1
pixels and n plotted points, the ratio between the number p ≤ 2p and as n increases the user initially looses the
2 1
ofpossiblecasesshowingexactlykcollisionsandthetotal informationthatareaA containstwiceasmanypointsas
2
numberofpossibleconfigurations. A andforgreatervaluesofntheuserisnotabletograsp
1
Thelatteriscomputedconsideringallthepossibleways anydifferencebetweenA andA . Asanumericalexam-
1 2
inwhichitispossibletochoosenpointsamongppixels,
3
values will be really assumed by the sample areas. For
eachvaluewecancomputethenumberofsampleareasin
which that value is present and an histogram showing the
distributionofthevariousdatadensitiescanbecomputed.
Forexample,ifweplot100datapointsintoanareaof10
sample areas, we could have the following configuration:
3sampleareaswith20datapoints,2sampleareaswith15
datapoints,2sampleareaswith5datapoints.
Represented density is defined as rd = pi,j where
i,j A
p is the number of distinct active pixels that fall into
i,j
SA . The number of different values that a sample area
i,j
canassumeisheavilydependentonthesizeofsamplear-
eas. If we adopt sample areas of size 8x8 pixels, as de-
scribedbefore,thenumberofdifferentnotnullrepresented
densitiesis64. Thus,wecanrepresentatmost64different
Figure2: Collidingelementspercentage
representeddensityvalues.Itisquiteobviousthat,because
ple, ifweplot64and128pointsontwo8×8areas, the ofcollisions,rd ≤d .
i,j i,j
pixelsturnedoninthetwoareaswillbe40.55and55.43, Usingtheabovedefinitionswedevisedaneffectiveset
so the ratio of displayed pixels is only 1.36. In order to ofqualitymetricswhosecompletediscussion,however,is
preservethevisualimpressionthatareaA containstwice outofthescopeofthispaper(see[1]forapracticaluseof
2
asmanypointsasA acceptingadecayof20percentwe thesequalitymetricsforuniformsamplingstrategies).
1
havetosamplethedata(64and128points)asmuchas50 The above metrics, together with the statistical results
per cent resulting in 32 and 64 points that, once plotted, giveusthemeanstodeviseanautomaticnonuniformsam-
turn on 25.32 and 40.55 pixels, i.e., a ratio of 1.6 (20 per plingtechniquedescribedinthenextsection.
centofdecay).
4 Nonuniformsampling
3.1 Datadensitiesandrepresenteddensity In[1]auniformsamplingstrategyhasbeenpresented,
Thepreviousresultsgiveusawaytocontrolandmea- showingitsabilityinimprovinganimagereadability. Ap-
surethenumberofcollidingelements. Beforeintroducing plyingthesameamountofsamplingtothewholeimageis
our optimization strategy, we need to clarify our scenario quite straightforward but presents several drawbacks. As
andtointroducenewfiguresanddefinitions. anexample,itisquiteobviousthatsamplingareaspresent-
Weassumetheimageisdisplayedonarectangulararea ingverylowdatadensityisuselessandpotentiallydanger-
(measured in inches) and that small squares of area A di- ous. Moreover it is quite evident that the most important
videthespaceinm×nsampleareas(SA)wheredensity cluesausercangraspfrom2Dscatter-plotsaredifferences
is measured. Given a particular monitor, resolution and indensitiesandouropinionisthatanonuniformsampling
size affect the values used in calculations. In the follow- canpreserveinamoreefficientwaysuchdifferences.
ingweassumethatweareusingamonitorof1280x1024 Theproblemofrepresentingrelativedensitiesistheone
pixelsandsizeof13”x10.5”. Usingthesefigureswehave ofcreatinganoptimalmappingbetweenthesetoftheac-
1,310,720 pixels and if we choose SA of side l = 0,08 tualdatadensitiesandthesetofavailablerepresentedden-
inch, the area is covered by 20.480 (128x160) sample ar- sities. Each data density must be associated to one of the
easwhosedimensioninpixelis8×8. Weconsidersmall 64(underthehypothesisof8×8sampleareas)available
areasbecauseofitmakestheuniformdistributionassump- represented densities. Any given visualization is one par-
tionquiterealistic. ticular mapping. Consider the case in which a visualiza-
For each SA , where 1 ≤ i ≤ m and 1 ≤ j ≤ n, tion is obtained by displaying a large data set. It likely
i,j
wecalculatetwodifferentdensities: realdatadensity(or, correspondstoamappinginwhichhigherdensitiesareall
shorter,datadensity)andrepresenteddensity. mappedontofewsinglerepresenteddensities, theonesin
Datadensityisdefinedasd = ni,j wheren isthe whichquiteallpixelsareactive(panesaturation). Thisis
i,j A i,j
number ofdata points thatfall into samplearea A . For why in that areas relative densities can not be perceived:
i,j
agivenvisualization,thesetofdatadensitiesisfiniteand alargenumberofhighdatadensitiesismappedontovery
discrete. In fact, if we plot a number n of data elements close values. Our idea is to investigate how these map-
into the display, each SA assumes a value d that is pingscouldbechangedinordertopresenttotheusermore
i,j i,j
withinthefiniteanddiscretesetofvalues:0, 1, 2,..., n. informationaboutrelativedensitiesaccepting,toacertain
A A A
In general, for any given visualization, a subset of these extent,somedistortion.
4
Figure3: Ascreenareamadeof100sampleareas: real(a)andrepresented(c)datadensities
In the following we use a simple numeric example to each SA , between 0 and 12. Looking at figure 3 (d) it
i,j
clarify our approach. Assume we are plotting 2264 (this iseasytodiscoverthatmorethen50%ofthevisualization
strangenumbercomesfromarandomdatageneration)points pane(54sampleareasoutof100)rangingbetween22and
onascreencomposedby400x400pixelsarrangedin100 49datadensitycollapsedonjustthreedifferentrepresented
sample areas of size 4x4 pixels. In the example we con- datadensities(10,11,12).
centrate on the number of data elements or active pixels Inordertoimprovesuchasituationwewanttoproduce
neglecting the SA area value (what we called A), that is anewmappingamongthegivendatadensitiesandthe12
justaconstant. InFigure3(a)thedatadensities(interms availablerepresenteddensities. Thiscanbedonepursuing
ofnumberofpoints)correspondingtoeachsampleareaare thegoalofpreservingthemaximumnumberofdifferences,
displayed. loosing,ontheotherend,theirextent. Inotherwords,we
Figure3(b)showstheactualvaluesofdatadensities(X want to present the user with as many difference in den-
axis) together with the associated number of sample ar- sity as possible, partially hiding the real amount of such
eas sharing each value (Y axis). As an example, we can differences.
see that the maximum data density 49 is shared by just Inordertoobtainsucharesult,startingfromfigureFig-
one sample area (SA ) and the minimum data density ure 3 (b) and considering only the 96 sample areas with
2,6
0 is shared by four sample areas (SA , SA , SA , datadensity>0wesplitthexaxisin12(i.e.,theavailable
5,6 5,8 6,4
SA ). Figure3(c),obtainedapplyingthestatisticalre- representeddensities)adjacentnonuniformintervals,each
10,3
sultsdiscussedinSection2(seeFigure5),showstheactual of them containing 96/12 = 8 sample areas. Obviously,
representeddensity(intermsofactivepixels)ranging,for because of we are working on discrete values we cannot
5
Figure4: Nonuniformsampling
guarantee that each interval contains exactly 8 sample ar- 5 Discussion
eas and we have to choose an approximation minimizing In this section we show the effectiveness of our tech-
thevariance. Afterthat,thedataelementsbelongingtothe niquecommentingtheimagesobtainedapplyingdifferent
sample areas associated with the same interval i are sam- sampling strategies. We compare the images acquired vi-
pledinawaythatproducesarepresenteddensityequalto sualizing a real dataset: the one containing 160,000 mail
i. Asanexample,thefirstintervalencompassesdataden- parcelsalreadymentionedinSection3.
sities1(sharedby6sampleareas)and2(sharedby3sam- Theimagescomefromatoolspecificallydevelopedfor
pleareas)andtheassociateddataelementsaresampledas ourpurposes. ItisaJavabasedapplicationthatpermitsto
muchasneededinordertoproducearepresenteddensity inspectseveralcharacteristicsofthedisplayedimagesuch
equalto1. Thesecondintervalencompassesdatadensity as: thedata/representeddensityofeachsamplearea,some
3, 4, and 5 (7 sample areas) and, after the sampling, the qualitymetrics,andthenumberofoverlappingpixels. Itis
resultingdatadensityis2,andsoon. alsopossibletoapplyuniformandnon-uniformsampling,
The represented densities resulting from this approach andtofiltersampleareaswithdata/representeddensityout
are depicted in Figure 4 (a); Figure 4 (b) shows the new, ofaspecificrange.
more uniform distribution of such represented densities. Figure5shows: (a)theoriginalvisualization(nosam-
We want to point out that in this new representation the pling), (b) the one obtained uniformly sampling the data
abovecollapsed54datadensitiesofFigure3(b)nowrange leaving 80% of the original dataset, (c) the one obtained
between6and12representeddensitiesallowingtheuserto uniformly sampling the data leaving 20% of the original
discovermoredensitydifferences.Ontheotherhand,asan dataset(thisvalueisthebestuniformsamplingratiocom-
example,therealdifferencebetweendatadensities29and puted by the proposal shown in [1]), (d)the one obtained
22(1.32)ispoorlymappedonrepresenteddensities7and usingnonuniformsampling. Itisquiteevidentthatatoo
6(1.16). weakuniformsampling(Fig.5(b))doesnotmakeapparent
Roughlyspeaking,wecanthinkatthewholeprocessas density differences in high density areas. Conversely, an
follows. We have at disposal p different represented den- optimized(butstilltoostrong)uniformsampling(Fig.5(c))
sitiesthatarematchedagainstk realdatadensitieswhere, makes them apparent but to the detriment of low density
usually,k >>p;thatimpliesthateachrepresenteddensity areas. In fact, the upper right area originally contained a
is in charge to represent several, different data densities, cluster that is not visible anymore. Figure 5(d) shows the
hiding differences to the user. The game is to change, by resultobtainedwhenapplyingnon-uniformsampling. The
nonuniformsampling,theoriginaldatadensities,altering featuresinthelowdensityareasarestillvisible(asinthe
theirassignmenttothepavailablerepresenteddensitiesin case of weak uniform sampling) but, at the same time, in
ordertopreservethenumberofdensitydifferences. thehighdensityareaitmakesmoreevidentdensitydiffer-
6
Figure5: Comparisonofvarioussamplingmethodsvisualizingthemailparcelsdataset
encesthatintheoriginalimagewerenotperceptible(asin resenteddensitiesassmartlyasitcan.
the case of strong uniform sampling). Figure 5(e) makes The logic behind the algorithm can be better appreci-
itclearer. Itisobtainedfilteringoutthesampleareaswith ated looking at Fig. 6 that compares the original and non
data density lower than 810 (i.e., SA with less than 810 uniformlysampledvisualizationstogetherwiththeirdensi-
points) therefore showing the most dense areas. If com- tieshistograms. Thedensitiesaremoreevenlydistributed,
pared withthe other images itis easy tonotice that while allowing the dense areas to exhibit the underlying trends.
on Figure 5(d), that pattern is perfectly clear, on images Moreover,thepeaksassociatedwiththehigherdatadensi-
(Fig. 5(a) and (b)) it is hidden in the saturated areas and ties(i.e.,62,63,64)arenotpresentanymore.
on (Fig. 5(c) it is faintly visible. Roughly speaking, we
6 Conclusionsandfuturework
can say that our technique produce at the same time the
In this paper we presented a low grain, non uniform
advantagesofbothstrongandweaksampling.
sampling sampling technique that automatically reduces
Anotherinterestingaspectworthtomention,ishowthis
visual clutter in a 2D scatter plot and preserves relative
technique can be operated. When applying uniform sam-
densities. To the best of our knowledge this approach is
pling, the choice of the amount of sampling to choose is
aquitenovelwayofsamplingvisualdata. Thetechnique
critical. Ifthesamplingfactorisselectedbyhand,theuser
exploits some statistical results and a formal model de-
hastotrymanycombinationsuntils/hefindsthevaluethat
scribing and measuring over plotting, screen occupation,
best conveys the information. To overcome this, we ap-
andbothdatadensityandrepresenteddatadensity. Sucha
pliedin[1]analgorithmtoautomaticallydevisetheamount
modelallowsforcomputingwhere,how,andhowmuchto
of sampling to apply. Exploiting the metrics presented in
samplepreservingsomeimagecharacteristics(i.e.,relative
Section 3.1 we were able to find the best sampling factor
density).
to apply, but the problem of uniform sampling still held.
Severalopenissuesrisefromthiswork:
Conversely, with non-uniform sampling there is no need
to search into a space of solutions and the algorithm runs • users must be involved. Our strategy provides pre-
autonomouslywiththeideaofassigningtheavailablerep- cise figures but we need to map them against user
7
Figure6: Nonuniformsampling
perceptions. Asanexample,stillreferringtoourap- References
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7 Acknowledgements
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