visibility volumes for interactive path optimization

Visual Comput (2008) 24: 635–647DOI 10.1007/s00371-008-0244-x O R I G I N A L A R T I C L E

Manohar B. SrikanthP.C. MathiasVijay NatarajanPrakash NaiduTimothy Poston

Visibility volumes for interactive pathoptimization∗

Published online: 17 May 2008© Springer-Verlag 2008

V. Natarajan (�)Department of Computer Science andAutomation, Supercomputer Educationand Research Centre, Indian Instituteof Science, Bangalore 560012, [email protected]

M.B. SrikanthDepartment of Mechanical Engineering,Massachusetts Institute of Technology,Cambridge, [email protected]

P.C. MathiasNMR Research Centre, SupercomputerEducation and Research Centre, IndianInstitute of Science, Bangalore 560012,[email protected]

P. NaiduUniversity of Toronto, Toronto, [email protected]

T. PostonNational Institute for Advanced Studies,Indian Institute of Science Campus,Bangalore 560012, India

Abstract We describe a real-timesystem that supports design ofoptimal flight paths over terrains.These paths either maximize viewcoverage or minimize vehicle expo-sure to ground. A volume-rendereddisplay of multi-viewpoint visibilityand a haptic interface assists theuser in selecting, assessing, andrefining the computed flight path.We design a three-dimensional scalarfield representing the visibility ofa point above the terrain, describe anefficient algorithm to compute thevisibility field, and develop visualand haptic schemes to interact withthe visibility field. Given the originand destination, the desired flightpath is computed using an efficientsimulation of an articulated ropeunder the influence of the visibilitygradient. The simulation frameworkalso accepts user input, via thehaptic interface, thereby allowingmanual refinement of the flightpath.

Keywords Visibility analysis ·Flight paths · Force-directed ropes ·Haptics · Multimodal interaction

1 Introduction

Visibility is an important criterion for mission planningover terrains, both for civilian and military applications.It is vital to present a user with clear and complete in-

∗This work was conducted when the first author was with the Indian In-stitute of Science and the fourth author was with the Centre for ArtificialIntelligence and Robotics, Bangalore, India.

formation about the critical covert/exposed regions withina terrain for effective design and organization of a mission.A simple terrain visualization application that measuresvisibility of individual points on the terrain from an in-dividual camera location makes it hard to consider theentire terrain and the whole mission. For example, the vis-ibility of air-borne vehicles to potential observers on theland is an important criterion and particularly crucial forsurveillance and stealth navigation. Several applications

636 M.B. Srikanth et al.

including large areal illumination, rope-way design, cablelayout design, antenna layout design, and swarm roboticshave a similar requirement. We present a visualization andinteraction system that enables mission planners to assessvisibility over terrains and hence design an optimal flightpath over the terrain.

Significant work has been carried out within the GISand computational geometry communities on the devel-opment of sophisticated techniques for visibility analysis.Traditionally, this has meant the computation of visibleportions of a terrain from a given point, or determiningif an object is visible from a given view point. The an-swer to the former is a collection of regions, whereas thelatter is a query resulting in a yes/no answer. We are in-terested in queries of the first kind. One challenge is toefficiently store the result of the query and visualize it ina useful manner. We address this challenge and developa system that allows a user to design optimal flight paths.Several issues need to be addressed in the design of sucha system:

– Visibility measure. What is a good measure of visibil-ity? How do we compute the visibility measure effi-ciently?

– Visualization. What effective visualization techniquescan help a user comprehend visibility information, andaid him or her in planning the mission?

Fig. 1. Visualization pipeline. Given a terrain height field T , we compute the visibility field Fp once. We use standard volume renderingtechniques for real-time visualization of Fp. The rope simulation component is a vital component of the systems, where the path is opti-mized based on the gradient of Fp, called the visibility force field. A haptic device is used to interact with the rope and thus allows theuser to explore the multiple equilibria. To avoid penetration of the rope with the terrain, we compute a terrain force field, which repelspaths away from the terrain, and augments it with the visibility force field

– Interaction. What modalities of interaction best en-hance visualization and make data exploration easy forthe user?

– Level-of-detail. How do the various methods scalewith resolution of the terrain and visibility range?

This paper’s contributions address all these issues.While visualization generally refers only to image syn-thesis, augmenting it with haptic cues (motion captureand force feedback) significantly enhances the informa-tion flow between human and the computer. We imple-ment and demonstrate the benefit of multimodal interac-tion in our system.

Contributions. We develop a real-time system for visi-bility-guided interactive planning of optimal paths overterrains. Given a terrain, we generate a 3D scalar fielddefined on the volume above the terrain that measuresvisibility from various locations on the terrain. The largesize of the terrain necessitates an efficient algorithm forcomputing this visibility field. We develop methods foroptimal path computation over the terrain, path assess-ment, and haptic interaction. Our system also allows user-specified constraints and user-controlled path refinement.We compute a visibility force field from the visibility field,to guide the path optimization process. Paths are modeledas mechanical ropes subject to visibility forces. The user

Visibility volumes for interactive path optimization 637

provides expert input through a haptic device and interac-tively plans a flight path. Figure 1 shows our visualizationand interaction framework.

Section 2 presents prior work on visibility computa-tion, haptics, and path optimization. Section 3 describesthe visibility measure and its computation. Section 4 de-scribes the use of visualization techniques to present thevisibility information, optimal path computation, path as-sessment, and haptic interaction. Results of our experi-ments on terrains are presented in Sect. 5. Section 6 con-cludes the paper.

2 Related work

Visualization of visibility information has been little ad-dressed in the literature, except for trivial plotting andgraphing techniques. However, several results are avail-able on point-wise visibility computation, visibility analy-sis, and their application to GIS, robotic, and air-vehiclepath planning applications. The surveys by Bittner andWonka [3] and Cohen-Or et al. [5] provide a taxonomyof visibility problems and cover several applications. DeFloriani and others have done significant work in thefield of visibility analysis, with specific focus on efficientalgorithms for terrain visibility [11, 12, 17]. Stewart [21]describes an efficient algorithm for computing the ap-proximate horizon on terrains by dividing the terrain intosectors around the given point and considering sampleswithin each sector. This algorithm is primarily used forshadow generation and illumination computation. Hence,visibility computation is restricted to viewpoints on theterrain. Our application, on the other hand, requires thecomputation of a visibility field over the entire volumeabove the terrain. Visibility algorithms depend on the na-ture of terrain encoding, popular ones being regular squaregrid (RSG) and triangulated irregular network (TIN). Inthe case of TINs, visibility computation involves findingthe horizon and building a horizon tree [6] to solve vis-ibility queries for a point in logarithmic time. Buildingthe horizon tree takes O(Nα(N) log N) time, where N isthe number of sample points on the terrain. In the caseof RSG representations, determining visibility is computa-tionally expensive. Franklin [13] determines the visibilityof a point by interpolating the slopes of nearby pointsin O(N) time. Teng and Davis [22] discuss the same ap-proach for parallel computation. A similar incrementalwave-like approach for visibility calculation is adopted byBen-Moshe et al. [2] for TINs, however they consider onlya sample set of points. Teng et al. [23, 24] compute singlepoint visibility on a hypercube machine with N processorsin O(log N) time. Mills and Fox [16] also present solu-tions to intervisibility problems on fast parallel machines.Our approach is unique in that we construct an intermedi-ate data structure once and use it for two purposes: interac-tive visualization of visibility and dynamic path planning.

Covert path planning generates paths that are hidden,to the extent possible, from observers on the ground. Mar-zouqi and Jarvis [15] propose an approach for generat-ing paths that are sensitive to visibility from observers inan indoor environment with vertical walls and obstacles.Their approach is based on a distance transform, a mapthat supplies each point of the image with a distance to thenearest obstacle point. It uses a cost function to determinethe path. However, they do not address the benefits of in-troducing visual presentations to aid a human in the loopfor mission planning. Moreover, their technique is primar-ily useful for motion on a planar surface.

Virtual-reality interfaces offer several advantages forscientific visualization [4, 7]. Multidimensional scientificdata are visualized using flexible virtual objects that pro-vide an abstract representation of the locations of equilib-ria. Direct manipulation of the virtual objects allows intu-itive exploration of the data and facilitates the discoveryof features, which would be otherwise difficult to find viaconventional visualization methods. In order to implementan effective multimodal scientific visualization applica-tion in the virtual environment, issues of responsivenessand fast updates must be addressed. Haptic feedback isa promising interaction modality for a variety of appli-cations [20]. The primary benefit of haptics is increasedrealism of simulation and improved operator performance.When presented with a proper combination of visual andhaptic information, the visual cues are reinforced by theforce feedback thereby making the interaction more real-istic [10]. Some of the challenges for making interactioncomfortable, fast, and effective are described in [8]. A ma-jority of the previous methods for haptic displays of vol-ume data properties are based on a fundamental relation-ship between the haptic probe position/velocity and thedata at that point [1, 14, 27]. Algorithms for constrainedpoint-based 3-doF haptic rendering have been developedfor scalar density data [9, 18]. Haptic stability is an issuein constrained haptic interaction where the haptic probemakes a rigid contact with the surfaces [25].

Our system uses rope simulation for interactive pathoptimization. The mechanics of rope and threads is hardto simulate at real-time rates, particularly for force feed-back, which must be much faster than visual. Our ropesimulation is based on multi-DoF articulated rigid bodydynamics, which gives frame rates close to haptic require-ments and restricts the rope to be inextensible.

3 Visibility measure and its computation

The input to our system is a terrain height field, definedas a bivariate function T (x, y) over domain D in R2.A point P, with coordinates (Px, Py, Pz), is called a can-didate point if Pz ≥ T (x, y). Two candidate points P, Qare said to be mutually visible if for all points R in theinterior of the line segment PQ, Rz > T (Rx, Ry). Define


a binary function as

V(P, Q) ={

1 if P visible to Q0 otherwise.

Efficient computation of V(P, Q) requires an interme-diate data structure that enables fast queries. For a givencandidate point P, we define a binary bivariate function

SP(x, y) = V(P, [x, y, T (x, y)])that identifies regions visible from P. The discrete ver-sion of SP is called a shadowgram. A radius of influ-ence limits the range of visibility to a specified distancefrom P. The visibility of P is measured as the total terrainarea visible from P assuming equal likelihood of pres-ence of observers anywhere on the terrain. We introducea real-valued weight function W(x, y) to incorporate thehigher/lower likelihood of observers at certain locations,and hence we define our visibility field Fp at a point P as

Fp(P) =∫∫

(x,y)∈D SP(x, y) AT (x, y) W(x, y) dxdy∫∫(x,y)∈D AT (x, y) dxdy

AT (x, y) =√

1+(

∂T

∂x

)2

+(

∂T

∂y

)2

where AT (x, y) is an area element at point (x, y) in thedomain D.

The trivariate scalar field Fp measures the visibility ofcandidate points from the terrain. It is important to notethat it monotonically increases along the positive z-axis.We use this property to design an efficient algorithm forcomputing it.

We define the measure of visibility Fc for a parametricspace curve φ as the integral of Fp along φ

Fc(φ) =1∫

0

Fp(φ(t)) ω(t) dt

where ω is an importance function defined on the curvethat captures user-specified information along the curve.Given two curves, Fc allows us to compare their relativevisibility and hence to identify the better path.

While designing a path φ between points, we aim tominimize or maximize the functional Fc with respect to φ.For example, minimization is appropriate for stealth navi-gation and maximization is appropriate for surveillanceapplications. Other optimization constraints can be incor-porated including total length of curve, curvature, heightfrom ground, clearance from terrain surface, etc.

Discrete visibility measures. We consider a regular squaregrid (RSG) defined as a regular grid inducing a partition ofthe domain D into equally sized rectangles. Let T be thediscrete analog of T , i.e., T(i, j) denotes the terrain height

for a grid point. Let

Fp[P] =∑

i∑

j SP[i, j] AT [i, j] W[i, j]∑i∑

j AT [i, j]where SP, AT , and W are discrete analogs of SP, AT ,and W , respectively. Fp can be computed and stored ina regular voxel grid. The discrete analog of Fc is

Fc[φ] =∑

i

Fp[φ[i]] ω(i).

Visibility computation. Computing Fp exactly could betime consuming: a naïve computation for a 256 ×256 ×256 voxel grid using standard ray-tracers takes approxi-mately 100 days on a dual Xeon workstation! We willdescribe an algorithm for fast computation of an approx-imation of Fp.

Given a terrain point P, define the visibility hull (VHP)as a surface that separates space into two regions such thatall points above VHP are visible from P and points beloware not. It is constructed using Algorithm 1, an incremen-tal procedure that computes the height of the visibility hullwithin rings of increasing size centered at P. The visibil-ity hull for the first ring is identically equal to the terrain.Within each ring, an interpolation stencil specifies the ruleto extend the existing hull. The extended hull is checkedfor intersection with the terrain height field and the slopesat the ring are updated accordingly. The value of Fp atvoxels that intersect the visibility hull are incremented bya value equal to the area factor at P. Figure 2 illustratesthe visibility hull computation procedure. This incremen-tal construction of the visibility hull leads to an efficientcomputation of the visibility field.

Algorithm 1. Visibility hull computation

1. Input: Terrain T , ring buffer RB to store line slopes withrespect to the XY plane, area factor A, point P

2. Output: Updated Fp for voxels intersecting the visibility hullVHP

3. Initialize RB to zero4. Set slope of all cells in first ring equal to slope of line joining

P with mid-point of the cell5. for each ring do6. for each cell in ring do7. Determine slope of cell using interpolation stencil8. L ← line with slope equal to RB[cellx, celly] and

whose orthogonal projection onto the XY plane is theline joining [Px, Py, 0] and [cellx, celly, 0]

9. Determine height H of L by computing voxel[cellx, celly, H] that intersects L

10. if H > T [cellx, celly] then11. Increment Fp[cellx, celly, H] by A12. else13. Q ← [cellx, celly, T [cellx, celly]]14. RB[cellx, celly] ← slope(PQ)15. Increment Fp[Q] by A16. end if17. end for18. end for


Fig. 2a,b. Visibility hulls. a The visibility hull of P = h0,0 is computed at cells within rings of increasing size. An interpolation stencilspecifies the rule to extend the hull to subsequent rings. For example, the height of the hull cell (2, 3) is computed as 2

3 m1,2 + 13 m2,2.

b Various views of the visibility hull VHP . All points above VHP are visible from P

To compute and store the visibility field, we createa voxel grid representing the volume above the domain D.A voxel stores a scalar value, initialized to zero, repre-senting the visibility Fp at that point. The contribution ofa terrain point P = T [i, j] to the visibility field is com-puted as follows: Compute the visibility hull VHP andaccumulate area factor A[i, j] to scalar value stored atvoxels that intersect VHP . The value of Fp at a voxel isnow given by the sum of scalar values stored at all voxelsbelow it. So, Fp can be computed now in a single sweepover voxels from bottom to top.

We introduce a notion of limited visibility by defininga cube of influence. For a point P on the terrain, definean axis aligned cube C with sides 2r such that P is themidpoint of the bottom face of C.

Visibility is now defined as

V C(P, Q) ={

V(P, Q) if Q is inside C0 otherwise.

The size of C determines the extent of visibility of a pointand could vary spatially over the terrain. For the purpose

of optimizing paths and for effective visualization, we re-quire a level-of-detail representation for Fp obtained usingdifferent resolutions of the terrain and sizes for C. A largerC captures global visibility smearing out the local effects,while a smaller C takes local visibility into account. Weconsider the terrain at coarser resolution when C is largeand at higher resolution for smaller C. A small cube couldadversely affect the path optimizer because of possiblysmall variations in visibility within the extent of C. There-fore, it is important to find a good balance between cubesize and terrain resolution.

4 Visibility-aware paths

Visualization. The visibility field is visualized using exist-ing volume rendering techniques where the user designsa transfer function to highlight specific regions of inter-est. Appropriate choice of a transfer function is crucialfor effective visualization. We also implement a dynamictransfer function that changes smoothly over time therebyproviding temporal cues to the variation of visibility in


Fig. 3a–l. Volume rendered visibility field (images were rendered using VolView from Kitware). a Grayscale color-mapped image ofa synthetically generated terrain. b Surface rendering of the terrain. c, d, e A vertical slice of the visibility field and volume rendered im-ages of the visibility field. f, g, h Vertical slice and volume rendered images of visibility field computed using a smaller cube of influence.i Grayscale color-mapped image of a terrain with two extended peaks and three sharp peaks. j Area factors of the terrain. k, l Volumerendered images of the visibility field

the region of interest. Figure 3 shows a volume renderedimage for a synthetically generated terrain.

Path planning refers to generating paths, flight paths inour case, that obey kinematic constraints of the mobile ob-ject and also the environmental constraints such as obsta-cle avoidance, etc. Abundant literature is available for pathgeneration between two given points in space that avoidobstacles while satisfying kinematic constraints. We con-sider the problem of computing paths between a given pairof points that optimizes visibility over the path. Our ap-proach is based on the simulation of a novel force-directedrope (FDR) for interactive path optimization. We definevisibility force as the gradient of Fp that acts upon a FDR.In its equilibrium state, the FDR optimizes Fc. Since Fpis discontinuous on the terrain surface, we compute thegradient ∇Fp for points above the terrain surface using

backward differences.

∇Fp ={(

∂Fp∂x ,

∂Fp∂y ,

∂Fp∂z

)z > T(x, y)

(0, 0, 0) otherwise.

Interaction is key in mission planning. A user-assistedinteractive approach significantly enhances the temporalvisualization necessary for effective path planning. Therope dynamics gives the user a visual cue to visibilityalong the path. Force feedback provides the user with hap-tic cues of visibility on the whole path, specifically at thegrasped point.

Force-directed rope. A force-directed rope is a mechanic-ally simulated rope that responds to user interaction simi-lar to a normal rope, but is subjected to various constraints


Fig. 4a–d. Force directed rope. a A simple spring-mass system: a chain of particles, each particle demonstrating 3-dof motion. Diagonalsprings resist bending. b, c Articulated rope consisting of rigid capsules connected by spring loaded ball-socket joints. d Sensable Phan-tom Premium haptic device used in our system. A grasping holder was developed and attached to the Phantom device resulting in anintuitive interface for rope manipulation

besides its internal forces. We now describe two imple-mentations of the FDR. The first technique models theFDR as a spring-mass system (Fig. 4) where the rope isdecimated to mass points and springs between them thatimpose constraints on the motion of the mass elements.The compliant nature of this model allows the rope to ex-tend to minimize or maximize visibility. The spring’s stiff-ness parameter controls the extent to which the rope can beextended. The second implementation uses rigid capsulesconnected by ball-socket joints to model an inextensiblerope. The choice of a particular type of rope model is de-cided by the mission type (surveillance or stealth) and thetype of constraints imposed.

Spring-mass system. Let Xi(n) and Vi(n) be the positionand velocity of particle i with mass m at instance n. Theforce Fi applied on i would move it to a new positionXi(n +1) and acquire a new velocity. We compute theseusing a modified Verlet integration [26]

Xi(n +1) = (2−d)Xi(n)−(1−d)Xi(n −1)+ Fi(n)∆t2

m,

Vi(n +1) = Vi(n)+ Fi(n)+ Fi(n +1)

2

∆t2

m,

where 0 ≤ d ≤ 1 is the damping factor and ∆t (= 1ms)denotes the time step. Fi(n) is the sum of the internalspring forces and the external force field described inEq. 1. Spring Si applies a force

FSi,i+1 = KS(‖Xi − Xi+1‖− Li,i+1)(Xi − Xi+1)

‖Xi − Xi+1‖+ KV (Vi − Vi+1),

on mass points mi and mi+1, where Li,i+1 is the restlength, KS is spring stiffness, and KV is the damping con-stant of the spring.

The spring model is simple and fast, but it may intro-duce aliasing artifacts such as jagged edges because forcesare sampled only at mass points and not on the piecewise-linear path. Therefore, a large number of mass points arerequired for numerical convergence.

Articulated rope. The high resolution requirement inspring-mass systems can be overcome if the rope ismodeled as an articulated rigid body [19]. N rigid cap-sules {Ci}N−1

i=0 are connected end-to-end by sphericaljoints {Φi}N−2

i=0 to impose constraints on their motion (seeFig. 4). Joint Φi connects Ci with Ci+1. Each joint Φihas three springs {Si, j}2

j=0 that model resistance to bend-ing and axial twisting. The motion of the capsules obeyssix degree-of-freedom linear and rotational dynamics. Thehaptic proxy point is coupled to the grasped point on the

Fig. 5. Terrain force field TFF is computed around the terrain up toa constant offset from the surface in both directions. TFF repels therope away from the surface of the terrain


rope via a soft-joint, which allows for compliance andkeeps the rope from getting stretched beyond its limits. Allsimulation events are coordinated using a timer. The pos-ition Xi and orientation qi of Ci are updated using Euler

Fig. 6a–c. Haptic interaction. a In the first interaction scheme, the rope is fixed at the end points, the user grasps the rope and displaces it.b In the second scheme, the rope is spooled at one end and the user drags the other end toward the destination along a desired path.c The haptic device in action

time integration.

Xi[n +1] = Xi[n]+∆t · Vi[n]qi[n +1] = q̂i(∆t ·ωi[n]) ·qi[n]


where q̂(ω) = [cos(|ω|/2), sin(|ω|/2) ·ω/|ω|] is a unitquaternion. The velocity and angular momentum are up-dated using forward Euler time integration.

Force field. The capsules exhibit linear and rotational mo-tion under the influence of a force field

G = k1∇Fp + k2TFF+ k3U1 + k4U2 + . . . , (1)

where ∇Fp is the gradient of the visibility field, TFF isthe terrain force field that keeps the FDR away from thesurface of the terrain, and U1, U2, . . . are additional userforces. The weights k1, k2, . . . control the influence ofeach force on the rope and their signs control the directionof motion of the elements of the FDR. The linear compon-ent of the force acting on Ci is computed as the averageover the length of the capsule, and the rotational force(torque) is computed about the center of gravity of Ci .Both components are used to compute the new positionand orientation of the capsules. The spherical joint main-tains the connectivity between capsules and the springsresist bending thereby producing smoother ropes. Inexten-sibility of this rope model is useful when there is a con-straint on the total length of the path. This model is supe-rior to the spring-mass model in that it does not producealiasing artifacts.

Terrain force field. The primary force acting on the rope isdue to the visibility field. To prevent the rope from pen-etrating the terrain during free reconfiguration (withouthaptic intervention) we need to detect collisions betweenthe FDR and the terrain surface. Instead of detecting thecollisions explicitly, we design a terrain force field that re-pels the FDR away from the terrain.

To compute the repulsion force at a space point, welocate the nearest terrain point. The line joining a pointin space with its closest terrain neighbor aligns withthe normal to the terrain at the terrain neighbor. There-fore, in order to identify nearest terrain neighbors forall space points, we compute the intersections betweenoffsets of the triangulated terrain surface and voxels inthe volume above. All voxels that intersect a triangleof the terrain surface store the closest point on that tri-angle in the first step. Next, the triangle mesh is off-set in the direction of the vertex normals by a smalldistance followed by the intersection test. If the inter-section test reveals a closer point, the voxel is updatedappropriately. We repeat the above step until all vox-els in the neighborhood of the terrain are traversed.Figure 5 shows the TFF computed using our method.Though computing TFF is time consuming, it is doneonly once for a terrain. This is particularly advanta-geous for real-time rope dynamics. TFF computationcan be replaced by a terrain collision detection and re-sponse system, which will be too slow for real-time ropedynamics.

Haptics. A kinesthetic 3-dof haptic device is used tointeract with the FDR. The user modifies the FDR con-figuration by grasping a point on the FDR and movingit to the desired location. The forces felt in the hapticdevice indicate visibility over the path. A strong forceindicates motion away from a desired location (localminima/maxima). The haptic device has two modes of in-teraction with the rope. In the first mode, the rope is fixedat the two end points, the user grabs and displaces therope thereby physically exploring multiple equilibria. Inthe second mode of interaction, the rope is spooled at oneend and the other end is dragged in the desired path, to-wards the destination using the haptic device. During thisprocess, the rope reconfigures to optimize for visibility(see Fig. 6).

5 Experiments

We performed our experiments on both synthetically gen-erated terrains and terrains obtained from United StatesGeological Survey (USGS). Experiments on the syntheticterrain allowed us to test the algorithm in a controlled andwell understood environment. Experiments on a real ter-rain allowed us to study the effectiveness of our method.The synthetically generated terrain consisted of two par-allel prisms located along the length and near the centerof a square flat area (see Fig. 7). Given various initial con-figurations of source and destination points, the objectivewas to maximize visibility. In each case, the rope simula-tion quickly and correctly identified the desired path. Thevisibility maximizing path always passed over the valleybetween the prisms.

The terrain from USGS is available as a DEM, whichwe sampled at 512 ×512 resolution. We computed thearea factor and the visibility field over a 512×512×128voxel grid. For this experiment, we worked with two usersfrom the Institute for Robotics and Intelligent systems,Centre for Artificial Intelligence and Robotics, Bangalore:a mission planner who was the primary user and a user-interface expert who helped evaluate the usability of thehaptic interface for rope manipulation. In the following,we discuss the experiment and collective observations ofthe two users.

The experiment was performed for a terrain region inthe East Bay Hills of California near San Francisco (seeFigs. 8 and 9). We briefed the user about the system in-cluding a description of the terrain, visibility hull, visibil-ity field, and haptic interface. The task was to find a paththat minimizes/maximizes visibility while satisfying ad-ditional user-imposed criteria. The user traced initial pathsbetween two pairs of points, Q to P and Q to R, usinga 2D mouse-based entry tool (see Fig. 10). Our path trac-ing tool created a space path from the 2D path by liftingit appropriately above the terrain. Next, the user with the


Fig. 7. Rope simulation with k1 set to 1 for maximizing visibility over a synthetic terrain. The visibility field is shown as a volume ren-dering with red indicating low values and blue indicating high values. A and B are the source and destination points, respectively. (Toprow) A and B are located 10 units above ground. The rope reaches the equilibrium state in 3.5 s. (Middle row) The destination is movedto lie diagonally across the terrain. (Bottom row) A and B are placed similar to the first row but 1 unit above ground. The FDR reachesequilibrium after a longer period of time. However, in all three cases, the flight path attempts to pass above the valley in order to maximizevisibility

help of the volume visualization tool studied the distri-bution of the visibility field over the entire terrain andspecifically near the region between source and destina-tion.

Fig. 8a,b. Path planning experiments were performed on terrain re-gion in the East Bay hills of California, between Oakland and Mor-aga. Coordinates: 3746019:92′′ North, 12206016:72′′ West. Imagetaken from Google Earth (http://earth.google.com). a Oblique viewshowing valleys and peaks. b Orthographic view from top

Figure 10 shows the result of the experiment. The ropesimulation computed an optimal path given the user-tracedspace path as an initial configuration. The user controlledthe values of parameters ki to increase/decrease the rela-tive influence of the visibility field. Negative values of k1

Fig. 9. a Surface rendering and b raw height field of the locationshown in Fig. 8. The digital elevation model (DEM) is availablefrom the US Geological Survey (http://www.usgs.gov)


Fig. 10a–d. Mission planning experiment over the East Bay hills. a User supplied a set of initial paths from Q to P and R. b The 2D pathswere lifted above the terrain and the resulting space paths were fed to the rope simulation as initial configurations. c The FDR identifieda minimum visibility (green) path between Q and R. The user manipulated the rope using the haptic device to incorporate additionalconstraints, namely p1 should be hidden and p2 should be visible from the path. The rope simulation included these constraints to finda new optimal (purple) path. d Given a visibility maximization criterion, the FDR converged to the same path between Q and R from twodifferent initial configurations


helped minimize the visibility objective, Fc. The result-ing path passed through a deep valley and stayed as closeto the terrain as possible. The user then identified a pointp1 that should necessarily remain hidden from the pathand a second point p2 that should necessarily be visiblefrom the path. The haptic interface allowed the user to ma-nipulate the rope by pulling it away from p1 and towardsp2. While manipulating the rope to satisfy the two addi-tional constraints, the user tuned the rope stiffness param-eter, which essentially controls the flexibility of the rope.This flexibility allowed the user to impose additional con-straints with ease. In addition to the haptic feedback, theuser found the visibility hull to be of great help as a visualaid while specifying the additional constraints. The visi-bility field was rendered with higher transparency valuesduring the rope simulation phase in order to help the userfocus his attention on the rope. Subsequent to the rope ma-nipulation, the rope simulator computed the appropriatevisibility minimizing path, which now passed a differentvalley but remained close to the terrain. We repeated theentire experiment for maximizing visibility and obtainedthe results shown in Fig. 10d. The user found our visibilityfield-driven, haptic-assisted, user-in-loop approach to beintuitive and efficient for generating optimal paths whencompared to manual path planning.

The visibility computation took 93 s and the hapticloop executed at a frequency of 1000 Hz. The path reachedthe equilibrium state within a span of 3–4 s, depending onthe extent to which it was perturbed. An Intel Dual Xeonworkstation running at 3.0 GHz was used for the computa-

tion and user interaction. The volume was visualized usinga second workstation.

6 Conclusions and future work

We have described an interactive system for generation,assessment, and interactive planning of flight paths thatoptimizes a given visibility criterion. Efficient precompu-tation of a visibility field and efficient simulation of anarticulated rope simulation results in interactive compu-tation of optimal paths between two end points specifiedby a user. The system allows the user to specify addi-tional constraints on the path using both visual and hapticfeedback. The seamless integration between the interac-tion mechanism and the path optimizer results in a veryeffective system. The development of a mission planningsystem that can accept several complex optimality crite-rion while providing interactive optimization capabilityremains a challenging problem.

Acknowledgement This work was done in collaboration with theInstitute for Robotics and Intelligent Systems, Centre for Arti-ficial Intelligence and Robotics (CAIR), Defense Research andDevelopment Organization (DRDO), India. The authors thank theVirtual Reality and Robotics group of CAIR for providing supportand feedback on the visibility measures. The authors also thankthe Supercomputer Education and Research Centre (SERC) of In-dian Institute of Science for providing the computational facilities.Thanks to Subrata Rakshit, Sitaram N., and V. S. Mahalingam forproviding valuable feedback on initial proposals.

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MANOHAR B. SRIKANTH is a graduate studentand research assistant at MIT. He obtained hisMS in Bangalore from the Indian Institute ofScience in 2007, after running VR and machineinteraction projects at the Defence Research andDevelopment Organisation. He currently worksin adaptive control and simulation for UAVs. Helikes multidisciplinary work that include scien-tific visualization, haptics and control. He hasseveral pending patents and published journaland conference papers.

P.C. MATHIAS is currently an associate profes-sor in the NMR Research Center, Indian Instituteof Science, Bangalore. His research and teach-ing interests include computer graphics and vi-sualization, digital signal processors and NMRinstrumentation.

VIJAY NATARAJAN is an assistant professor inthe Department of Computer Science and Au-

tomation and an associate faculty member in theSupercomputer Education and Research Centreat the Indian Institute of Science, Bangalore. Hereceived his PhD degree in computer sciencefrom Duke University in 2004. He holds a BEdegree in computer science and a MS degreein mathematics from Birla Institute of Tech-nology and Science, Pilani, India. His researchinterests include scientific visualization, compu-tational geometry, computational topology, andmeshing.

PRAKASH NAIDU received his PhD degree inengineering from the University of Toronto,Canada in 1997. He has worked in or has beenaffiliated with leading institutions, laboratories,and industry in the US, Canada, and India indifferent capacities. He has several awards andhonors including the Canadian CommonwealthFellowship and Engineering Design Awards. Heis the originator of the field of Design for Au-

tomation and his research interests include de-sign theory and methodology, design of intelli-gent systems in the fields of medical devices,biotechnology, advanced manufacturing, mecha-tronics, robotics, automation, and virtual reality.He is a member of the American Society of Me-chanical Engineers, and licensed member of theAssociation of Professional Engineers of Ontario(P.Eng.).

TIM POSTON has a mathematics PhD fromWarwick, England, and has published with co-authors from archaeologists to brain surgeons.He is an inventor on ten issued patents, fifteenpending, which range from a helical surgicalprobe to flexible ways to find, collaborate on andwork with documents. He is co-author of twotexts, on General Relativity and on CatastropheTheory, in print since the 1970s. He has workedin haptics, visualization, and the cognitive inter-action of humans and machines.

visibility volumes for interactive path optimization

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