ieee transactions on visualization and computer … · resizing by manually specifying regions of...

IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 1

ViSizer: A Visualization Resizing FrameworkYingcai Wu, Xiaotong Liu, Shixia Liu, Member, IEEE , and Kwan-Liu Ma, Fellow, IEEE

Abstract —Visualization resizing is useful for many applications where users may use different display devices. General resizingtechniques (e.g., uniform scaling) and image resizing techniques suffer from several drawbacks, as they do not consider thecontent of the visualizations. This work introduces ViSizer, a perception-based framework for automatically resizing a visualizationto fit any display. We formulate an energy function based on a perception model (feature congestion), which aims to determine theoptimal deformation for every local region. We subsequently transform the problem into an optimization problem by the energyfunction. An efficient algorithm is introduced to iteratively solve the problem, allowing for automatic visualization resizing.

Index Terms —Resizing, Visualization Framework, Perception, Focus+Context, Nonlinear Least Squares Optimization

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1 INTRODUCTION

R ESEARCH into visualization resizing is becomingparticularly important with the advance of col-

laborative visual analysis, in which users might usedifferent display devices. Although modern visualiza-tion methods can regenerate a new visualization if thedisplay changes, the common regeneration methodsdo not always work. Some methods such as tag cloudsand force-directed graph algorithms may regenerate atotally different layout to fit the new display, whichis unacceptable in a collaborative application. As aresult, a generic resizing framework is needed toefficiently produce consistent visualizations, such thatembedded useful patterns in the resized visualiza-tions can still be revealed as effectively as the original.Additionally, such a framework can relieve the burdenof designing a visualization, as the developers nolonger need to consider the re-scaling problem.

There are several possible solutions to resizing avisualization. One simple approach is uniform scaling.Unfortunately, this would not work if the visualiza-tion is resized to a different aspect ratio. To tackle thisproblem, the visualization can simply be cropped toensure that the uniform scaling can coincide with thenew aspect ratio. However, this method may discardimportant or useful context information. Some otherstraightforward solutions also often fail to producedesired results. For instance, an alternative approachfor graph layout resizing is to scale node coordinateshomogeneously, maintain their visual size, and use afast overlap removal mechanism [11]. Unfortunately,this approach does not work when the target displayis too small to hold all graph nodes without shrinking

• Yingcai Wu and Kwan-Liu Ma are with the Department of ComputerScience, University of California, Davis, 2121 Kemper Hall, OneShields Avenue, Davis, CA 95616. E-mail: {ycwu,ma}@cs.ucdavis.edu

• Xiaotong Liu and Shixia Liu are with the Microsoft Re-search Asia, Beijing, China. E-mail: [email protected],[email protected].

some of them. Moreover, the links between the graphnodes are likely to be occluded by the dense graphnodes that are relatively large in the target display.

This work presents a perception-based framework,ViSizer, for effectively resizing a visualization for anydisplay using an image warping approach [35]. Themajority of image resizing methods such as seamcarving [1] keep important regions unchanged, lead-ing to failure when the region sizes are larger than thetarget image sizes. In contrast, the optimized scale-and-stretch method [35] can address this problem byscaling important regions uniformly and deforminghomogeneous context. ViSizer employs a similar de-formation scheme, but it is much more flexible. Itcan be viewed as a multi-focus+context visualizationtechnique by allowing users to explicitly specify theexpected scaling factors for the regions of interest inthe target visualization.

Importantly, ViSizer employs a new perception-based significance measure designed for visualiza-tion. The measure can estimate the visual cluttermagnitude and guide the resizing process to avoidcompressing visually cluttered items. A new energyfunction is defined based on the measure to transformthe resizing problem into a nonlinear least squaresoptimization problem. The optimization problem canthen be solved by an efficient iterative algorithm.

The major contributions of this work are as follows:• Study a new problem of how to effectively resize a

visualization for any display.• Transform the visualization resizing problem into

an optimization problem with a novel perception-based energy function.

• Design and develop a generic framework for auto-matic visualization resizing.

2 RELATED WORK

Image resizing methods can be generally classified asdiscrete or continuous methods [31]. Discrete meth-


ods, i.e., seam carving [1], resize an image by ju-diciously inserting or removing left-to-right or top-to-bottom seams. Continuous methods [35], [36] as-sociate an image with a grid and resize the imageby deforming the grid non-homogeneously. Thesetechniques are not optimal for visualization resiz-ing. First, visualizations have special layouts withinteractive visual elements rather than static pixels.Acceptable deformation in images may be viewed asa serious distortion to the layouts. For example, non-homogeneous deformation of words in a word cloudmay decrease their readability. Second, visualizationresizing is more constrained by visual clutter - thestate in which excess and disorganized items degradevisual task performance. This performance degrada-tion is due to the difficulty in recognizing or searchingfor an item interfering with other surrounding items,especially when the item spacing is small [33].

Automatic resizing methods based on constraints[17] are commonly used for resizing objects in Graphi-cal User Interfaces (GUIs) due to their expressiveness.However, they could become difficult to use whenthere are many GUI objects or when the constraintsare too complex to specify. Manually-authored meth-ods such as Artistic Resizing [10] are also widelyemployed to resize GUI objects based on providedexamples, thus allowing a designer to customize theresizing behaviors. They are primarily employed forresizing a limited number of relatively simple graph-ics objects in GUIs. In contrast, ViSizer mainly aims atresizing a visualization with a large number of visualitems that are usually distributed irregularly.

Our method is an automatic resizing techniquebased on nonlinear least squares optimization. Com-pared with the automatic GUI resizing techniques,our method automatically formulates the constraintsby the perception-based clutter measure and thus itdoes not require manually-specified constraints. Ourmethod can also be regarded as a manually-authoredtechnique because users are allowed to customize theresizing by manually specifying regions of interestand assigning expected scaling factors for the regions.Therefore, it takes advantage of both automatic andmanually-authored resizing techniques.

Visual Clutter is an important factor for designingan effective visualization and user interface. Baldassiet al. [3] showed that visual clutter misleads users toproblematic judgments and to more confidence in er-roneous decisions. Researchers traditionally measuredvisual clutter based on information density or thenumber of elements [32]. Some researchers arguedthat this traditional method was not a good mea-sure of clutter, because the number of elements canbe ill-defined [27]. Our framework uses the featurecongestion method based on local feature variance[27] because it is effective for predicting clutter and ismuch more efficient than other quantitative methods.

Data Abstraction can be used to adapt visualiza-

tions to devices of small displays (e.g., mobile de-vices). Various data abstraction techniques [12], suchas clustering [14], point/line displacement [9], anddimensional reordering [26], have been proposed toreduce information density for alleviating the visualclutter problem. Elmqvist and Fekete [13] presenteda general hierarchical aggregation model for infor-mation visualization. These techniques can simplifyvisualizations created on large-screen devices. As aresult, the visualizations may be adapted to devicesof small screens. However, they inevitably discardinformation and are likely to fail when resizing todifferent aspect ratios. Moreover, what information isto be discarded solely relies on either the user’s abilityto navigate a view with less clutter or the heuristicrules embedded in a visualization [27].

Focus+Context Visualization such as Fisheye [15]is a popular solution for visualizing data on mobiledevices [19]. Sarkar and Brown [29] explored a Fish-eye lens method for viewing and browsing graphs.A metaphor called rubber sheet stretching was alsointroduced to visualize graphs within small displayareas [30]. Carpendale et al. [5], [6] presented a newmagnification technique using a three-dimensionalpliable surface. Keahey and Robertson [20] introducedefficient techniques to combine multiple transforma-tions. Image warping techniques were used to de-form a street-level map to fit the associated schematicmap [4]. Jenny and Hurni [18] employed a deforma-tion method to visually analyze the planimetric andgeodetic accuracy of the old map. Munzner et al.[25] used Fisheye to ensure landmark visibility andconstant frame rates for scalable tree comparison.

These Focus+Context techniques are useful in visu-alization, but they may lead to target acquisition prob-lems and impaired spatial comprehension [7]. Zanellaet al. [38] suggested using grids and shading to tacklethese problems. Our method can be regarded as afocus+context technique, but we novelly apply thetechnique in resizing visualizations. It allows users tospecify the expected scaling factors of regions of inter-est. Furthermore, the important regions are uniformlyscaled, and the distortions are distributed across thewhole visualization rather than only the local regionsas handled in the existing techniques. Guided by aperception-based significance map, ViSizer can alsominimize the chance of task performance degradationcaused by visual clutter.

3 DESIGN METHODOLOGY

This section presents our methodology for designinga resizing framework. We start from investigatingthe challenges raised by several use scenarios, andthen discussing the design constraints and visual vari-ables for the framework. Our approach, the flexibledistortion control mechanism, as well as the systemoverview are subsequently presented.


3.1 Design Challenges

There are several typical scenarios where our visual-ization resizing framework is useful.

• Several users collaborate on one visual analyticsproject using computing devices with different dis-play sizes and aspect ratios.

• A user facing a large (e.g., wall-sized) display usesher hand-held device for visual item selection.

• A user may have different computing devices withdifferent displays to work at different places.

These scenarios present a few challenges for design-ing an effective resizing approach. First, the resizedvisualization must be consistent with the originalone. Visualization inconsistency may convey incorrectinformation, mislead the discussion in the collabora-tion, or even draw a wrong conclusion. This posesa challenge to some visualization techniques such astag clouds and graph layout methods that usuallycreate different layouts for different displays. Second,the technique must be efficient. In a collaborationscenario, a new user may join in the collaborationat any time and the visualization under discussionshould be resized to fit his display in real time, suchthat he can start the collaboration immediately. There-fore, the algorithms that require excessive time toregenerate layouts is inappropriate for the application.Third, the method should naturally support multi-focus+context visualization as well as necessary visualcues for users to comprehend the geometric distortion.Usually, a user using a hand-held device does nothave an appropriate display to show the original vi-sualization and a deformed version is needed. Finally,it should avoid introducing additional clutter when avisualization is resized to a smaller display.

3.2 Design Constraints

Different visualizations usually have different con-straints and requirements on their use of space. Itis difficult or even impossible to design a genericresizing framework that can suit all different visu-alizations. This work mainly focuses on non-space-filling visualizations such as word clouds, graphs,and scatterplots. Space-filling visualizations such astreemaps have more strict spatial and geometric con-straints than non-space-filling visualizations. For in-stance, radial space-filling visualizations have strictcircular layouts, thus limiting the flexibility of usinggeometric deformation to fit a certain aspect ratio.This prevents us from utilizing empty or unimportantregions for preserving significant regions. Further-more, the spatial and geometric constraints are quitedifferent, which presents a big obstacle to creating ageneral framework. Therefore, our framework primar-ily aims at non-space-filling visualizations.

3.3 Visual Variables

Visual items such as points and lines are the basicelements for creating a visualization. Each visual itemowns a set of visual variables such as color andposition to encode multidimensional information ofa data item. Visual variable encodings are consideredas a basis for visualization. The effectiveness of theresizing framework highly depends on which visualvariable encodings are modified in the resizing pro-cess and whether or not these encodings are preservedafter resizing. This requires that we should carefullydetermine which spatial visual variables are to bechanged or to be preserved in the resizing process.

Spatial visual variables, such as position, area,length, angle, slope, density, and shape [23], havespatial properties, which can be changed more or lessby geometric deformation. In contrast, non-spatial vi-sual attributes such as color and texture are invariantto deformation. As the framework utilizes geometricdeformation to resize a visualization, spatial visualvariables of the visual items are modified. This canresult in undesired distortion to the original visual-ization and may mislead a user to draw a wrongconclusion in quantitative visualizations.

Our framework focuses on visualization taskswhere users merely need to discern data patterns suchas distributions in scatterplots rather than interpretdata values quantitatively and accurately. Addition-ally, it provides a method to facilitate user collabora-tion and interactions in visualizations shown in dif-ferent displays. We argue that in these tasks changingspatial visual variables to a certain extent is allowedwhen data patterns are preserved in significant re-gions. To simplify the discussion, in this work wemainly change visual variables: position, area, or bothto resize a visualization. All other visual variablesremain the same during the resizing process.

A scatterplot, for example, uses x- and y- positions(or coordinates) to encode two-dimension informa-tion. In many qualitative visualization tasks, the infor-mation is transformed from higher dimensional spaceby, for example, multidimensional scaling techniques(see Fig. 8). The scatterplot is used to show onlythe overall pattern and trend of the pattern. It isunnecessary to accurately and quantitatively interpretthe positional visual variable. Therefore, changing thepositional visual variable is allowed to maintain theoverall pattern. This observation can also apply toother visual variables such as area in some scenarios,where there is no need for accurate and quantitativeinterpretation of the visual variables. For example,when a user facing a large (e.g., wall-sized) displayuses her hand-held device to select graph nodes, itwould be reasonable to enlarge the area where theimportant nodes locate to facilitate selection.


Run Optimization Input with an Initial Grid Create Adaptive Grid Resized Result

Smooth Scaling Factors

Optimize Node Opsition

Converge?Clutter Map DOI Map

Fig. 1. System overview: ViSizer first creates a significance map based on the degree of interest (DOI) map andthe visual clutter map, and produces a significance-aware grid; it then searches for an optimal transformation tothe grid and adjusts the visualization with the deformed grid.

3.4 Method and Distortion Control

We design an efficient and flexible resizing frameworkwith a seamless integration of a multi-focus+contextmechanism for non-filling-space visualizations. Bychanging the spatial variables, the framework enablesdifferent levels of distortion in the resized results tomeet different user requirements.• Distortion-free: all visual items except the empty

space in the non-space-filling visualization are uni-formly scaled. In other words, we modify the po-sitions of visual items, such that the empty spaceis greatly compressed while the relative positionsof the visual items are preserved. This is particu-larly useful for the tasks of visualizing overall datapatterns rather than quantitative analysis.

• Controlled distortion (multi-focus+context visual-ization): the visual items will be deformed basedon the expected scaling factors specified by users.In particular, we change both the positions as wellas the areas of the visual items to fit a new displaywhile allowing for the multi-focus+context effect.

Furthermore, the framework uses background gridsto facilitate users’ comprehension of the geometricdistortion to improve the accuracy, as suggested byother researchers [22], [38].

The primary benefit of this framework is that it isflexible and can meet different resizing requirements.Users can determine whether distortion is allowed ornot. By measuring the visual clutter in the original vi-sualization, the framework can avoid compressing thecluttered regions in the resized result. Additionally, itcan also relieve the burden of visualization designersfor handling the re-scaling problem.

3.5 System Overview

Fig. 1 shows an overview of ViSizer. ViSizer employsa grid-guided resizing optimization scheme. It par-titions a visualization with a grid, then iterativelyadjusts the grid in an judicious manner under someconstraints to achieve an optimal deformation of thegrid. Finally, the visualization can be resized accord-ing to the deformed grid by forward mapping. Wechoose to deform the grid rather than the visualizationin the optimization because of the efficiency and

flexibility of the grid-guided method. The efficiencyis achieved through the iterative optimization schemewidely used in image warping and resizing, whilethe flexibility is achieved by the energy function as-sociated with the grid-guided optimization method.Moreover, the grid can provide sufficient visual cuesfor a viewer to comprehend the deformation.

ViSizer includes two parts: pre-processing and op-timization. In the pre-processing part, a significancemap, a combination of a DOI map and a visual cluttermap, is created to encode the significance value ofevery quad in the grid. Next, a significance-aware oradaptive grid is created based on the significance mapto reduce linearization artifacts and to approximatethe nonlinear deformation better. In the optimizationpart, the resizing problem is transformed into a non-linear least squares optimization problem through anenergy function based on the significance map, quaddeformation, and edge bending. ViSizer solves the op-timization problem iteratively to find a good solution.The scaling factor for every grid quad will be adjustedat each iteration to minimize the potential distortion.The iteration repeats until a certain convergence con-dition is reached, i.e., all vertex movements are verysmall in the current iteration. Finally, the optimizationgenerates a deformed grid and it is utilized to adjustthe visualization accordingly.

4 PRE-PROCESSING

In pre-processing, the framework first associates aninput visualization with a uniform grid used forwarping the visualization. The input visualizationconsists of a bitmap image of the visualization andall visual items of the visualization. It then createsa significance map for encoding the significance ofdifferent regions in the visualization. Finally, the gridis adjusted to be significance-aware, which means thatmore important regions are covered by more quads.

4.1 Significance Measure

The significance measure is an image-based measureand is a core part of the resizing framework. It isused to create a significance map for guiding thesignificance-aware grid adjustment and to determine


Fig. 2. Left to right: illustrative examples (left) andtheir clutter maps (right) for showing the detected colorclutter, orientation clutter, and density clutter (from topto bottom).

the vertex movements in the optimization process.The significance of each local region can be estimatedby the measure based on the DOI and the magnitudeof clutter of the visual items in the region. Only quadsthat are both locally important and cluttered shouldbe protected against distortion.

4.1.1 Degree of InterestDegree of interest (DOI) was first introduced by Fur-nas [15] to indicate that visual items in visualizationhave different levels of importance. Clearly, DOI isapplication-specific and different applications mayhave different definitions. With appropriately definedDOI, important regions can be differentiated fromless important regions, which allows us to distributedistortion to less important regions. For example, aDOI map for a scatterplot is shown in Fig. 1; itassigns a higher level of importance to the top leftcluttered region. For simplicity and clarification, inscatterplots (except Fig. 1), we view all visual itemsequally important. In word clouds and graphs, theimportance of a visual item is assigned based on thesize of the item (word or graph node).

4.1.2 Clutter Estimation

The DOI map is used to preserve regions of interestin visualization resizing. However, relying only onthe DOI is insufficient for determining the shrinking

or stretching operations of a visualization. This isbecause some regions may become crowded withexcess, unorganized visual items when a user repeat-edly resizes the visualization. As a consequence, thevisualization would be cluttered and the performanceof visual tasks, such as visual searching, could bedegraded [33]. Fig. 3(f) shows an example in which vi-sual clutter becomes severe when the words in the redellipses get closer and closer. To tackle this problem,a quantitative measure of visual clutter estimation isintroduced. In this scenario, the regions with highmagnitudes of clutter should not be shrunk to avoidbeing even more cluttered.

Our framework employs an efficient method calledFeature Congestion [27] to estimate the clutter magni-tude in every local region. This method can producean image called clutter map with the same resolution ofthe visualization for revealing the clutter magnitudeat every pixel. It uses the level of feature congestionto indicate the degree of clutter in an image. The con-gestion level can be measured by a statistical saliencymodel based on the observation that unusual itemsare usually salient [27]. Whether or not an item is un-usual depends on how different the feature vector ofthe item is from the local distribution of other featurevectors. A feature vector is composed of the color, theluminance-contrast, and the orientation of the item.Thus, the statistical saliency for a feature vector X

can be evaluated by the Mahalanobis distance [24] as

∆ =√

(X − µ)TS−1(X − µ) (1)

where µ and S denote the mean matrix and covariancematrix of the local feature distribution, respectively.

This model uses a set of covariance ellipsoids, de-termined by the covariance matrix S, in the featurespace to represent the local feature distribution. Withthe model, the difficulty of adding the new importantitem to a local area can be simply measured by thesize of the local covariance ellipsoid represented by S.Given a type of feature space with a limited volume,such as color gamut, the larger size of the localellipsoid indicates less space for adding a new salientitem. This is because the item has to be outside of theellipsoid to ensure that it appears to be unusual tothe existing items inside the ellipsoid. In other words,there is little feature space excluding the large localellipsoid for choosing an appropriate feature vectorfor a new salient item. Thus, the feature space is likelyto be congested with a large covariance local ellipsoid.

We follow the procedure in [28] for quantitativelyestimating the degree of visual clutter across an im-age. Interested readers can refer to [28] for moredetails about the procedure. With the method, thesystem can successfully identify cluttered regions dueto the color clutter, the orientation clutter, the densityclutter clutter (see Fig. 2), or their combination (seethe clutter map in Fig. 1).


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Fig. 3. (a) Word cloud where the tiny words in grey are filtered out before resizing. (b) Visual clutter map. (c) DOImap. (d) Significance map. (e) Uniformly scaled word cloud. (f) and (g) Results resized using the DOI map andthe significance map, respectively.

4.1.3 Significance MapThe significance map is used to guide the later op-timization process (see Section 5) for visualizationresizing. The goal of the optimization is to shrink orstretch a visualization to fit any display, whereas theclutter magnitude in the visualization should not beincreased and regions with high degrees of interestshould be preserved. Therefore, the significance mapW can be set up by combining the DOI map DOI

and the clutter map C as W = DOI ∗C. The averageof pixel significance within quad q is computed asthe significance wq for the quad. Furthermore, wq isnormalized such that 0 ≤ wq ≤ 1 (a larger valueindicates higher significance).

4.2 Significance-Aware Grid and Adaptive Grid

The initial uniform grid can simplify the implemen-tation and support faster performance. However, theuniform grid places an equivalent number of quadsin every local region in spite of the significance ofthe regions, leading to linearization artifacts in theoptimization process (see Fig. 4(b), where the words inthe right side are shrunk too much). To reduce the arti-facts and better approximate the optimal deformation,we adjust the initial grid to ensure that significant re-gions are covered by more quads than less importantregions. The resulting grid is called a significance-aware grid. Two types of significance-aware gridsare derived, such that the proposed framework isapplicable to most visualizations. The first type is todirectly deform the initial grid to attract the quads ofless important regions to those of significant regions.It is adapted from the method in [35] by optimizingthe following energy function:

∑

{i,j}∈E

√

1 + wij · (vi − vj)2 (2)

where E is a set of edges in the grid, vi and vjare the positions of nodes i and j, and wij is theaverage weight of the quads that share the edge {i, j}.

In an optimal scenario in which the energy is theminimum, the nodes in the interior of the significantregions become closer, thus attracting the surroundingnodes to the regions (see Fig. 4 (c) and (f)). Comparedwith the energy function in [35], our energy functionuses

√

1 + wij rather than 1 + wij to prevent a quadattracting too many neighboring nodes (see Fig. 4(d) and (g)). We also tested with other choices suchas wij and w2

ij , and found that√

1 + wij producedbetter results in general. This optimization problem isa nonlinear least squares optimization problem andcan be solved iteratively to approximate the optimalnode positions of the grid. This approach is simplefor implementation and does not require changing thegrid topology. In addition, because this is done onlyonce in the preprocessing step, the resizing perfor-mance remains the same.

However, this method may introduce linearizationartifacts (the important words inside the red ellipsesin Fig. 4(g) are too small). To tackle this problem, anadaptive grid is used as a second type of significance-aware grid. The basic idea is simply to use a quadtree to partition the visualization and ensure that sig-nificant regions are covered by more quads. Fig. 4(h)shows the result by the adaptive grid (Fig. 4(e)). Theimplementation becomes more complicated comparedwith the first grid type, but the results look better.

5 OPTIMIZED RESIZING

This section describes the resizing algorithm of Vi-Sizer. It is adapted from a continuous image warpingmethod [35]. Our method is different from the imagewarping method in three aspects. First, a completelydifferent significance map is derived to guide theoptimization process. Second, the energy function istailored for visualizations (Section 5.2). Third, besidesautomatically finding an optimal scaling factor for afocus region, our method also allows users to specifyan expected one for the region (Section 5.3).


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Fig. 4. Results obtained by resizing Fig. 3(a) horizontally. (a) Uniformly resized result. (b) Bad result producedby the initial grid where the words on the right are overly compressed. (c) and (d) Significance-aware grids ofFig. 3(a) created using the energy functions defined in [35] and in Equation (2), respectively. (e) Adaptive grid ofFig. 3(a). (f)-(h): Results produced by the grids shown in (c), (d), and (e), respectively.

Given a visualization, we place a grid on it anddenote the grid by G = {V,E, F}, where V , E, and F

represent the nodes, edges, and quads of G, respec-tively. Let nv, ne, and nf be the numbers of nodes,edges, and quads. We have V = [vT0 , v

T1 , · · · , vTnv

]T ,and vi ∈ ℜ2 represents the node position of nodei. The optimization is to transform G judiciously forcreating a new grid G′ = {V ′, E′, F ′}, so as to ensurethe uniform scaling of salient quads and the distortiondistribution to other quads without increasing visualclutter. A new visualization can then be created byspace interpolation according to the new grid G′.

5.1 Non-Linear Least Squares Optimization

An energy function is used to transform the resizingproblem to an optimization problem. We follow [35]and define a quad deformation term for quantifyingthe non-uniform quad distortion and an edge bendingterm for evaluating the edge bending. In the follow-ing, we briefly describe the energy function and moredetails can be found in [35].

The quad deformation energy is defined for a quadto ensure that it is uniformly scaled. Given a quadfk, its uniformly deformed version is f ′

k = skfkwhere sk is a 2 × 2 uniform scaling matrix. Wemathematically formulate the energy for all quadsF in a least-squares system as ||WFF

′ − WFSF ||2where F = [f0, f1, · · · , fnf−1

]T , and WF and S are thematrices whose element at the uth row and the vth

column is defined as

Suv =

{

sk if u = v

0 otherwiseWF,uv =

{√ωfk if u = v = k

0 otherwise

The quad can be represented as fTk = qkV where qk

is a 4× nv matrix and its element at the uth row andthe vth column is defined as

qk,uv =

1 if v = i

−1 if v = j

0 otherwise

where i and j are the node indices of the uth edgeof fk. We let Q = [qT0 , q

T1 , · · · , qTn3

f−1

]T , the quad set F

can be derived by F = QV . Therefore, the total quaddeformation energy is

||WFQV ′ −WFSQV ||2 (3)

The edge bending energy is used to preserve thegrid edge orientation. Given an edge ek ∈ E, itsuniformly scaled version is e′k = lkek where lk is a2 × 2 uniform scaling matrix. The energy for all theedges E can be defined in a least-squares system as||WEE

′ − WELE||2 where E = [eT0 , eT1 , · · · , eTne−1]

T ,and L and WE are matrices whose element at the uth

row and the vth column is defined as

Luv =

{

lk if u = v

0 otherwiseWE,uv =

{√ωek if u = v = k

0 otherwise

Let ek = vi − vj = hkV , where hk is a 1 × nv vectorand its vth element hk,v can be defined as

hk,v =

1 if v = i

−1 if v = j

0 otherwise

Let H = [hT0 , h

T1 , · · · , hT

ne−1]T , we can obtain E = HV .

Therefore, the total edge bending energy is

||WEHV ′ −WELHV ||2 (4)

The optimal nodes positions V ′ of the grid can beapproximated by minimizing the total energy:

argminV ′,S,L

||WF (QV ′ − SQV )||2 + ||WE(HV ′ − LHV )||2

It can be viewed as an over-determined system AV ′ =b(V ), where A = [QTWT

F , HTWTE ]T and b(V ) =

[V TQTSTWTF , V THTLTWT

E ]T . Hence, we minimize

argminV ′

||AV ′ − b(V )||2 (5)

It is a non-linear least squares optimization problemthat can be approximated by iteratively updating thenode positions. In our experiments, the initial guess


(a) (b) (c)(a) (b) (c)

Fig. 5. (a) Result without any smoothing. (b) and(c) Results in which the scaling factors are equalizedusing the original and our adapted energy functions,respectively. (c) is the best result because the relativesizes of the nodes (indicated by the red ellipses in (a))are preserved better than those of (a) and (b).

V 0 is obtained by homogeneously resizing the originalvisualization to the target display. See [21] for a moredetailed description for solving the problem.

5.2 Quad Transformation Smoothing

The scaling factors in each local region must besmoothed out to prevent the distortion of an impor-tant object caused by the different scaling factors ofthe surrounding regions. Fig. 5(a) shows a resizedvisualization without quad transformation smooth-ing. The salient nodes marked by the red ellipsesare clearly distorted. We formulate the smoothingproblem as an optimization problem.∑

k∈F

∑

q∈N(k)wq(s

′k−s′q)

2+∑

k∈Fwk(s

′k−sk) = 0

where N(k) denotes the quads surrounding the quadk, wq is the average significance of all nodes in thegrid, and wk indicates the significance of the quadk. Compared to the smoothing function defined in[35], our function uses wq rather than 0.5(wq +wf ) toweight (s′k − s′q)

2. Fig. 5 shows the different resizingresults using the original function (Fig. 5(b)) and ours(Fig. 5(c)). Generally, our tailored function can pro-duce better results with less distortion to the salientobjects in the red ellipses. The smooth scaling s′k canbe estimated by minimizing the function. This processis repeatedly performed after we have obtained sk atevery iteration of the optimization.

5.3 Multi-Focus+Context Visualization

The scaling factors S in b(V ) are automatically deter-mined during the optimization. This allows for creat-ing distortion-free results, as the all visual items areuniformly scaled. The distortion is largely absorbedby the empty space. Fig. 3(g) and Fig. 4(h) presenttwo examples of distortion-free results in which the

(a)

(c)

(b)

(d)

Fig. 6. Multi-focus+context visualization. (a) Resultcreated by uniform resizing. (b)-(d) Results createdwith the specified scaling factors: sk = 1, sk = 2, andsk = 4 for important graph nodes, respectively.

words are uniformly scaled based on the change ofthe display size. The spatial relations between thewords are mostly preserved and thus the results areconsistent with the original ones.

Many applications usually prefer the distortion-free results. However, focus+context techniques arestill needed in some scenarios, especially when thedisplay size is limited [19]. Techniques such as dataabstraction might have more or less limitations aswe discussed in Section 2. Our framework naturallysupports multi-focus+context visualization during theresizing process to address this issue. It allows auser to specify a desired uniform scaling factor δ

for any object in the resized visualization. Thus, thequad scaling factors of the quads covering the objectsare fixed to the constant value δ specified by theuser during the optimization. This can produce aresult similar to multi-focus+context visualization (seeFig. 6). Therefore, it can be regarded as a combinationof significance-aware focus+context visualization andvisualization resizing techniques. ViSizer transforms avisualization through a grid and thus can seamlesslyprovide the background grid to support the user’scomprehension of geometric distortion. It has beenreported that the background grid can help improvethe accuracy of visualization performance [22], [38].

6 RESULTS AND DISCUSSION

In this section, we demonstrate the effectiveness ofour framework and show how we can apply it todifferent visualizations. The techniques described inthis work were implemented by Java and Prefuse. Allresults were generated on an Apple Macbook Pro withan Intel Core i7 2.66GHz CPU and 4 GB Ram.


6.1 Experiments

6.1.1 Word Clouds

In the first experiment, we tested ViSizer for showingits usefulness for resizing word clouds. The resizingtechnique is important for word cloud visualizations.It is usually difficult to resize a word cloud to fit anew display. Uniformly scaling a word cloud to asmaller display may create a word cloud in whichmany words are too tiny to be easily recognized. Re-generating a new word cloud might be another op-tion. However, the word cloud, e.g., re-created by [34]or [8], can be totally different from the original one.As they are based on either a random algorithm [34]or a force-directed algorithm [8], they usually fail tocreate stable word clouds with a specified aspect ratio.Additionally, context-preserving word clouds [8] usethe relative positions of the words in the clouds toencode important semantic information in the originaltext. This requires that the relative positions betweenwords in the original word cloud should be preserved.

The framework views a word as a visual item. Eachword is attached to a grid quad via an anchor point.After the gird is deformed, the anchor point positionsare adjusted by interpolating the four nodes of thequad. The size of each word is changed based on howthe size of the associated grid quad changes.

We generated a context-preserving word cloud by aforce-directed algorithm [8] using a real dataset with13, 828 news articles spanning one year (from 2008to 2009) that were related to American InternationalGroup (AIG). In the word cloud, semantically-similarwords get close to each other. As we wanted to shrinkthe word cloud for a small display, we filtered outtiny words in grey that are almost unrecognizable inthe target display (see Fig. 3 (a)). Figs. 3(b)-(d) showthe visual clutter, DOI, and significance maps of theword cloud used for guiding the resizing optimizationprocess. Fig. 3(a) (right) presents the color encodingscheme for these maps. We resized the word cloudvertically to reduce it to half its size.

Fig. 3(e) shows a uniformly resized result in whichpreviously large words become unnoticeable, not tomention the smaller ones. Fig. 3(f) is a result createdby our method guided only by the DOI map. Com-pared to Fig. 3(e), it distributed most of the distortionto the empty space, thus reserving more room forimportant keywords. However, the words inside thered ellipses were overly packed, making it challengingfor users to recognize the words quickly. We canremedy this problem with the help of the visual cluttermap (see Fig. 3(b)) that can inform the optimizationprocess of the crowding degree in every local region,preventing the words from being excessively packed.Fig. 3(g) presents the result created by using thesignificance map (Fig. 3(d)). We can clearly observethat the overcrowding problem was fixed using theperception-based clutter measure.

We further tested different types of grids. Fig. 4(a)is a uniformly deformed result in which most ofthe words become tiny. We can use a uniform gridfor resizing the word cloud in most cases. However,it cannot always produce a good result due to thelinearization artifacts in the optimization process. Fig.4(b) shows an example where only the left part of theword cloud is deformed correctly. We then used threetypes of grids (Figs. 4(c)-(e)) from the original wordcloud (Fig. 3(a)) to reduce the linearization artifacts.Fig. 4(f) shows a result based on the grid in Fig. 4(c)generated by the original energy function [35]. We cansee that the grid is distorted too much. The words inthe red ellipse in Fig. 4(f) are too separated and theirrelative sizes change a great deal.

Fig. 4(g) shows the result based on the grid inFig. 4(d) created by our adapted energy function(Eq. (2)). The grid was modestly adjusted withouttoo much distortion. However, some words originallyneighboring each other, e.g., in the red ellipse, arestill far away. To solve the problem, we used anadaptive grid (Fig. 4(e)) to resize the word cloud.Fig. 4(h) presents the result in which the words arenicely packed and their relative sizes and positionsare mostly preserved. We found that the adaptive gridgenerally worked better than other grids. However,it took more time (2.553 seconds in this experiment)than the significance-aware grids (within 1 second).

6.1.2 Graph VisualizationThe second experiment was conducted to show theusefulness of our technique in a graph visualization.Regenerating a new graph layout by, e.g., a force-directed algorithm, for a different display is usuallytime-consuming and the new layout could be totallydifferent from the original. Furthermore, most existingalgorithms do not take into account the differentdisplay aspect ratios and cannot make efficient useof the screen space. In resizing a graph, every graphnode is regarded as a visual item. Each node isattached to a grid quad via an anchor point. Afterthe grid is deformed, the anchor points are adjustedby interpolating the four nodes of the quad. The sizeof the graph node is changed based on how the sizeof the associated grid quad changes.

We tested ViSizer with two real graph datasets.One is a social network dataset from Prefuse with129 nodes and 161 edges, while the other containsmajor airline routes of Northwest Airlines in theUnited States with 235 nodes and 2, 101 edges. Weused both the size and color of a graph node toencode its degree. The graph of the social networkdata was generated by a force-directed algorithm.Fig. 6(a) presents a uniformly resized result wherenodes become too small to analyze. Figs. 6(b)-(d)demonstrate the resized results with an increasing sk(1, 2, and 4) manually specified for the importantgraph nodes. The quads covering the nodes were


Original Graph and Significance Map Resizing by

Uniform Scaling

Resizing with

Significance-aware

Grid

Resizing with

Adaptive Grid

Resizing by Uniform Scaling Resizing with Significance-aware Grid Resizing with Adaptive Grid

(a) (b) (c) (d)

(e) (f) (g)

Fig. 7. Results created by resizing a graph visualization originally shown on a 27 inch display with 1920× 1200pixels (top-left) to a 3.5 inch display with 960 × 640 pixels in different orientations using uniform scaling, ViSizerwith a significance-aware grid, and ViSizer with an adaptive grid. ViSizer makes more efficient use of the smalldisplays than uniform scaling. The adaptive and the significance-aware grids both work well in most cases formaintaining original information. However, the significance-aware grid might have a chance to produce artifacts.The node indicated by the green arrow is not well preserved in the results of the significance-aware grid.

uniformly expanded by the specified sk to produceresults similar to focus+context visualization. ViSizerdistributed the distortion to the less important nodesand empty space across the entire visualization.

Fig. 7 shows a typical use of ViSizer for resizinga graph originally shown on a 27 inch display with1920 × 1200 pixels to a 3.5 inch display with 960 ×640 pixels in horizontal and vertical orientations. Theused airline data contains spatial information for eachgraph node. Uniformly scaling the graph to the smalldisplay with a very different aspect ratio produced asqueezed visualization (see Fig. 7 (b)). In addition, itis difficult for a user to explore and interact with thegraph in the much smaller display (see Fig. 7 (b) and(e)) because the graph nodes are barely discerniblein such a display. Simply increasing the sizes of thenodes would cause the graph nodes to overlap oneanother. On the other hand, we can see that there is agreat deal of white space in the left part of the graph.Therefore, we could compress the white space to makeroom for enlarging significant regions.

Fig. 7 (c) and (f) show the results created by ViSizerwith the significance-aware grid, while Fig. 7 (d) and(g) present the results created by ViSizer with the

adaptive grid. These results assigned more displayspace to important nodes (the larger the nodes, themore important they are) by compressing the whitespace in the graph while still preserving the overallgraph structure. Comparing these results, we canobserve that the two types of grids produced similarresults. Nevertheless, the adaptive grid (Figs. 7(d) and(g)) works slightly better than the other (Figs. 7(c)and(f)) in preserving the original information. For in-stance, the node indicated by the green arrow isdistorted in the results of the significance-aware grid.Since the spatial information and the sizes of thenodes are useful and important for analysis, it istherefore desirable to preserve the information.

6.1.3 ScatterplotsThe third experiment was conducted to demonstratethe use of our technique for scatterplots. The scatter-plots used in this experiment were created by project-ing a high dimensional data set to two dimensionalspace with multidimensional scaling. Therefore, the x

and y axes do not a concrete meaning. Fig. 8 (left)shows an original scatterplot of the data from IN-SPIRE [37] on a 21.5 inch display with 1920 × 1080


Resizing by

Uniform Scaling

Resizing by

VisSizer

Vis

InfoVis

VAST

Fig. 8. Results created by resizing a scatterplot originally shown on a 21.5 inch display with 1920×1080 pixels (left)to a 3.5 inch display with 960× 640 pixels and a 9.7 inch display with 1024× 768 pixels by uniform scaling (middle)and ViSizer (right). Through the judicious removal of white space, ViSizer assigns more space to important areasand maintains the overall pattern, thus showing the relationships among the points in clusters more clearly.

Fig. 9. Smooth transition between a visually faithful representation of a visualization and a representation meantto facilitate selection.

pixels. It reveals the topic distribution of IEEE Vis,IEEE InfoVis and, IEEE VAST proceeding papers pub-lished from 2006 to 2008. Each paper is representedby a point with a fixed size in the scatterplot. Thisscatterplot is very sparse and has a great deal of whitespace. Fig. 8 (middle) presents two uniformly scaledscatterplots that are shown on a 3.5 inch display with960×640 pixels and a 9.7 inch display with 1024×768pixels, respectively. These results (particularly thesmallest scatterplot) look squeezed. Additionally, thepoints in the scatterplots are too small and too close tobe visually distinguished from one another. For exam-ple, the points inside the cluster indicated by the greenarrow look quite cluttered in these results. It would bechallenging for a user to interact with these clutteredpoints to facilitate visual analysis. One solution mightbe to decrease the point sizes to reveal the relationsamong the points. However, this would result in verytiny points in the small scatterplots that are hardlydiscernible. Applying a simple magnifying lens doesnot work either for distortion-sensitive applications.In addition, these techniques do not take differentaspect ratios into account and cannot preserve the

overall pattern.

Fig. 8 (right) shows the results of VisSizer for thesame smaller displays. Every point in the scatterplotis regarded as a visual item and considered as equallyimportant. It is attached to a grid quad and its positionis adjusted by interpolating the four nodes of the quadafter the quad deformation. From these results we cansee that in contrast to uniform scaling, ViSizer worksbetter in preserving the overall pattern (or shape) ofthe original scatterplot. It compressed only the whitespace and allocated more room to more significantregions, thus helping to clearly reveal details of acluster (see the cluster indicated by the green arrow).This would be especially helpful when a viewer needsto interact with the points for detailed analysis ofthe data. Moreover, ViSizer employed a significance-aware grid to deform the scatterplot and thus hada strict spatial constraint for minimizing the chanceof distorting the overall pattern. The results show theeffectiveness of ViSizer for resizing a larger scatterplotto smaller ones with very different aspect ratios.

Fig. 9 presents a smooth transition between a vi-sually faithful representation of a visualization and


a representation meant to facilitate selection. Thesmooth transition of the change can provide an alter-native to background grids for enhancing the under-standing of geometric deformation introduced by theresizing process. The clutter map (see Fig. 1) showsthat all three clusters in the scatterplot are visuallycluttered. Although the two clusters (bottom-right andtop-left) exhibit a higher level of density, there ismuch more color diversity in the bottom-left clusters.According to our clutter estimation measure, bothcases suggest a certain level of feature congestion inthe feature space, thus leading to a high degree ofvisual clutter. For illustration, we manually specifiedthe DOI map and assigned a higher importance levelto the bottom-left cluster. Combining the DOI mapand the clutter map, we generated a significance mapwhich indicates that the bottom-left cluster is moresignificant than other regions.

Guided by the significance map, the scatterplot canbe prudently resized to ensure that most distortionis distributed to less significant regions, such thatsignificant regions as well as the overall layout aremaintained. From Fig. 9, we can see that the signifi-cant region and the overall layout are well preservedconsistently during the transition. In addition, ViSizermostly compress only the empty space among theclusters at the beginning. Only when no more emptyspace is available does ViSizer start to deform thetwo less significant clusters (top-left and bottom-rightclusters). This result demonstrates the effectivenessof our clutter estimation measure and shows thatViSizer can successfully generate a smooth transitionof change, which is useful for comprehension of theresizing deformation.

6.2 Time Performance

Fig. 10 shows the time of resizing different visualiza-tions using the significance-aware grid and the adap-tive grid. For each visualization, we recorded the timeneeded for resizing it to its original 95%, 90%, · · · , 5%.Table 1 presents the average time performance. Asshown in our results, the adaptive grid usually createsbetter results than the significance-aware grid (seeFig. 4 and 7). A possible reason is that the adaptivegrid has a regular shape which can better preservethe overall structure of a visualization. However, thisadvantage is gained at the cost of performance andusing the adaptive grid generally needs more timefor resizing a visualization (see Fig. 10 and Table 1).We can also observe that it took much less time toresize scatterplots than other visualizations using theadaptive grid. This is because the scatterplots used inthe experiments have more empty space than othervisualizations, thus resulting in far less grid quadsand a smaller linear system to optimize.

The implementation of the resizing framework us-ing an adaptive grid is more complicated and chal-

Fig. 10. Time (measured in milliseconds) needed forresizing different visualizations using the adaptive grid(Grid A) and the significance-aware grid (Grid B).

lenging. Based on our experimental results, we sug-gest that the adaptive grid should be used whenusers need more accurate results and do not caremuch about the performance. We recommend thesignificance-aware grid for most applications al-though it might slightly deform the items.

TABLE 1Average time performance using different grids

Visualization Significance-Aware Gid Adaptive GridScatterplots 698ms 1557ms

Graphs 780ms 2864msWord clouds 713ms 5748ms

6.3 Discussion

ViSizer is not fully optimized for time performance,but it can be easily adapted to run in real timeby precomputing keyframes for different sizes theninterpolating these keyframes. We will optimize thecurrent system and employ a GPU-accelerated tech-nique [16] to solve the linear system involved inthe optimization. This would enable many interestinginteractive applications, including smart support forwindow resizing, for semantic zooming in ZUIs, andfor rendering legible overview insets.

As discussed in the previous experiments, ViSizercan also ease the difficulty of target selection in smalldisplays. Although researchers have developed manytarget selection techniques by reducing the distanceD or increasing the width W of the target, theyusually do not scale well to the situation in whichmultiple targets are crowed together [2], especially forsmall displays. In contrast, our framework naturallyscales well to the situation by the significance-guidedresizing method and the embedded focus+contextscheme. In particular, ViSizer can facilitate target se-lection by expanding the target and removing emptyspace between the cursor and the target. That is itcan facilitate target selection by both decreasing D

and increasing W . With the technique, one interest-ing application is that a user facing a large (suchas wall-sized) display uses her hand-held device toselect visual items, which are otherwise too tiny andindistinguishable to be selected readily. In the future,we want to extend our framework to support a more


sophisticated target selection method by consideringthe likelihood of selection.

The main advantage of ViSizer is that it can uni-formly scale important items through diverting de-formation to other regions without increasing cluttermagnitudes with the help of the perception-basedsignificance map. Moreover, by distributing deforma-tion to only the white space, ViSizer can producedistortion-free results in which every local non-emptyregion is well preserved. ViSizer deforms the wholevisualization through a gird and thus it can alsolargely retain the overall pattern of the original visu-alization. Nevertheless, the deformation does have animpact on the alignment of objects, which could resultin misinterpretation of the data and unfair compar-isons between visual items. Therefore, ViSizer may notbe appropriate for resizing visualizations requiringaccurate relative positioning of visual items. In con-trast, ViSizer can be used primarily for visualizationtasks that do not require accurate and quantitativeunderstanding of the data items. For instance, it canhelp users better discern data patterns or distributionin a smaller hand-held device.

The framework can produce multi-focus+contextresults by highlighting important regions while com-pressing others. The important regions are uniformlyscaled to the sizes specified by a user. As otherfocus+context visualizations, these distorted resultsmight lead to impaired spatial comprehension [7].ViSizer uses background grids [22], [38] to providenecessary visual cues, which helps improve compre-hension of the distortion and task accuracy. Althoughthis problem can be alleviated to some extent by thebackground grids, the inevitable distortion may havenegative impacts on visualizations that require accu-rate interpretation of visual variables such as positionand size of visual items. Distortion and misalignmentscould also prevent users from comparing data accu-rately. This is a limitation of our approach. We believethat this problem could be addressed by developingbetter user interactions, providing animated transi-tion, and/or providing better background visual cuesother than simple background grids. We plan to studythis problem and improve our framework to bettersupport comparison tasks in the future.

7 CONCLUSIONS AND FUTURE WORK

This work introduces a perception-based resizingframework for automatically resizing a visualizationto any display size without introducing additionalvisual clutter. Prominent objects can either be uni-formly scaled or fixed to a size specified by the userduring the resizing process. The deformation intro-duced by the resizing operation is distributed to lessimportant regions globally over the visualization. Ourframework targets at non-space-filling visualizationsthat usually contain some empty space among visual

items. As for space-filling visualizations, some of thevisual items (e.g., rectangles in the a treemap) canbe very large and uniform and they could thereforebe treated like empty space. We plan to extend ourframework for space-filling visualizations.

ACKNOWLEDGMENTS

This research was supported in part by the HPLabs and U.S. National Science Foundation throughgrants CCF-0808896, CNS-0716691, CCF 0811422, CCF0938114, and CCF-1025269.

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Yingcai Wu received the BEng degree incomputer science and technology from theSouth China University of Technology in2004 and the PhD degree in computer sci-ence from the Hong Kong University of Sci-ence and Technology in 2009. Since June2010, He has been a postdoctoral researcherat the Visualization and interface Design In-novation (VIDi) research group in the Univer-sity of California, Davis. Prior to his currentposition, he was a postdoctoral researcher in

the Hong Kong University of Science and Technology. His researchinterests include information visualization, visual analytics, medicalvisualization, and user interface design.

Xiaotong Liu is a Ph.D student in Depart-ment of Computer Science & Engineering ofThe Ohio State University, currently workingin Prof. Han-Wei Shen’s GRAVITY Group.His research interests lie in Information Visu-alization, Visual Analytics, Computer Graph-ics and Human Computer Interaction. Xiao-tong Liu received his Bachelor degree inSoftware Engineering at Shanghai Jiao TongUniversity in 2011, and worked at MicrosoftResearch Asia in Internet Graphics Group

with Dr. Shixia Liu and Dr. Yingcai Wu on information visualizationresearch from 2010 to 2011.

Shixia Liu is a lead researcher in the Inter-net Graphics Group at Microsoft ResearchAsia. Her research interest mainly focuses oninteractive, visual text analytics and interac-tive, visual graph analytics. She received aB.S. and M.S. in Computational Mathematicsfrom Harbin Institute of Technology, a Ph.D.in Computer Aided Design and ComputerGraphics from Tsinghua University. Beforeshe joined MSRA, she worked as a researchstaff member and research manager at IBM

China Research Lab, where she managed the departments of SmartVisual Analytics and User Experience.

Kwan-Liu Ma is a professor of computerscience and the chair of the Graduate Groupin Computer Science at the University ofCalifornia, Davis. He leads the VIDi researchgroup and directs the DOE SciDAC Institutefor Ultrascale Visualization. Professor Ma re-ceived his PhD degree in computer sciencefrom the University of Utah in 1993. He was arecipient of the PECASE award in 2000. Hisresearch interests include visualization, high-performance computing, and user interface

design. Professor Ma was a paper chair of the IEEE VisualizationConference in 2008 and 2009, and an associate editor of IEEETVCG in 2007-2011. He co-founded the IEEE Pacific VisualizationSymposium in 2008 as well as the IEEE Symposium on LargeData Analysis and Visualization (LDAV) in 2011. Professor Mapresently serves on the editorial boards of the IEEE CG&A, theJournal of Computational Science and Discoveries, and the Journalof Visualization. He is an IEEE Fellow. Contact him via email:[email protected].

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