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1 INTRODUCTION

Since the early 1990s, there has been a growing interest in

the emerging technology of four-dimensional modelling of in-

dustrial projects (GSA BIM 2009, Heesom & Mahdjoubi 2004).

In comparison with traditional CAD and project management

systems, 4D modelling provides for more comprehensive vis-

ual analysis of the projects. Importantly, it assumes not only

aesthetic visualization of the project phases, but also multifac-

tor planning and scheduling of the projects. Its major benefits

are as follows:

— improved communications among stakeholders due to

the simulation of project activities ‗in progress‘ and

their immediate visualization as elements of dynamic

scenes;

— trustworthiness of project schedules carefully evalu-

ated and verified against potential spatio-temporal

clashes;

— optimum use of critical resources taking into account

not only temporal, but also a spatial factor. In par-

ticular, it assumes improved logistics at the project

site.

In final respect, 4D modelling enables better planning and

helps to identify and remove errors at the earlier phases of

design and planning, reducing potential waste during project

realization. Providing a commonly shared vision of design in-

tent, project plan and its current status, the technology elimi-

nates misunderstanding among stakeholders and promotes to

the project success (Sriprasert & Dawood 2002, Retik 1993).

As free and commercial applications like Synchro, Eu-

roSTEP 4D Linker, VirtualSTEP 4D Planner, Bentley Schedule

Simulator, Intergraph Schedule Review, Autodesk Navis-

Works, Common Point Project 4D, Visual Engineering VPS,

VTT 4D CAD, CSA PMvision, Domos 4D-suit are becoming

more accessible, numerous attempts to employ the technol-

ogy in different industrial domains and programs are under-

taken (Seliga 2007).

Nevertheless, most applications are faced with the funda-

mental problem of the performance degradation for large pro-

ject data consolidating huge 3D models and detailed schedul-

ing information. The structural hierarchical decomposition of a

whole project into subprojects, work breakdown structures

and activities can be adopted as an effective approach to

scheduling theory problems. However, it is not well suited for

such computationally intensive problems as ray tracing, colli-

sion detection, motion planning usually carried out during 4D

modelling sessions.

Octrees, k-d trees, BSP-trees, BRep-indices, tetrahedral

meshes, and regular grids are all examples of the spatial de-

composition that assumes preliminary subdivision of the space

occupied by the scene objects (Jimenez et al. 2001, Klosowski

et al. 1998). As a result, spatial localisation of objects and

search of nearby objects in the scene can be significantly im-

proved allowing various time consuming applications like view

volume clipping or occlusion culling. Unfortunately, spatial de-

composition cannot be directly applied for dynamic scenes ap-

pearing in 4D modelling and planning applications. Because of

temporal changes, statically deployed spatial structures have

to be permanently updated while the objects are replaced,

appear or are removed in the scene space.

In this paper we discuss an approach to 4D modelling

through spatio-temporal decomposition allowing effective ac-

cess to large project data relevant to a given focus time and

analysed space domains. Three partitioning schemas exploit-

ing peculiarities of spatial and temporal coherence of pseudo-

dynamic scenes are presented and estimated against typical

requests and performance requirements.

Previously there were attempts undertaken to partition hi-

erarchically both the time and space domains. Data structures

like Finkelstein‘s tree (Finkelstein et al. 1996), space-

partitioning time tree (Zhiyan & Chiang 2009), Shen‘s tree

(Shen et al. 1999) have been proposed and investigated

mainly in the context of fast volume rendering and generating

4D MODELLING OF LARGE INDUSTRIAL PROJECTS

USING SPATIO-TEMPORAL DECOMPOSITION

V.A. Semenov Institute for System Programming RAS, Moscow, Russia

Tom Dengenis, CEO Synchro Ltd., Coventry, UK

(c) Copyright August 1, 2010

ABSTRACT: There has been a growing interest in the emerging technology of four-dimensional modelling and planning of indus-

trial projects. In comparison with traditional 3D CAD and project management systems, 4D modelling provides for more compre-

hensive visual analysis of the projects. As free and commercial applications are becoming more accessible, numerous attempts to

employ it in different industrial domains and programs are undertaken. Nevertheless, most applications are faced with the funda-

mental problem of the performance degradation at large-scale projects. In this paper we discuss an approach to 4D modelling us-

ing spatio-temporal decomposition and providing special facilities to effectively access project data and to execute typical re-

quests relevant to given time and space domains. Three partitioning schemas are presented and analysed to satisfy performance

requirements. The schemas exploit peculiarities of spatial and temporal coherence of pseudo-dynamic scenes appearing in 4D

modelling and planning applications. Theoretical complexity estimates are given to prove the suitability of the schemas for effi-

cient modelling of large industrial projects.

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multi-resolution videos. Adaptive and irregular octrees provide

interesting examples of dynamic partitioning strategies allow-

ing for real-time modifications of a scene while retaining many

of the benefits of static spatial schemas (Shagam & Pfeiffer

2003, Sudarsky & Gotsman 1999, Whang et al. 1995).

The mentioned ideas and principles of spatio-temporal de-

composition have been widely utilised in computer graphics

methods intended, in particular, for collision detection in dy-

namic scenes. Our contribution is an adoption of these princi-

ples to problems of 4D modelling and planning of large pro-

jects as well as comparative complexity analysis of three

partitioning schemas looking promising for the discussed pur-

poses.

The rest of the paper is organized as follows. Section 2 de-

scribes peculiarities of pseudo-dynamic scenes appearing in

the considered applications for 4D project modelling and plan-

ning. In section 3 we present three schemas for spatio-

temporal decomposition of project data and point out the dif-

ferences from known results obtained mainly in computer

graphics. In Section 4 theoretical estimates of complexity for

all the schemas are derived in conformity to typical visualiza-

tion scenarios and requests. Section 5 provides a comparative

analysis and recommendations on practical use of the pre-

sented schemas. In conclusions the final results obtained are

summarized.

2 PROBLEM STATEMENT

Performance is a crucial factor for 4D modelling and planning

applications targeted at the scheduling of large industrial pro-

jects. Visualization of complex dynamic scenes as well as iden-

tification and resolution of spatio-temporal clashes may re-

quire significant CPU resources. These problems are common

for many geometric reasoning applications like CAD/CAM, ro-

botics and automation, computer graphics, and virtual reality.

But the considered applications have their own peculiarities

having a strong impact on performance optimization through

spatio-temporal decomposition. We consider that the scenes

originating from 4D modelling and planning applications have

the following characteristics:

High complexity: the scenes may consist of thousands

and millions of objects with their own 3D model representa-

tions and dynamic behaviours. The objects can be both rela-

tively simple shapes and assemblies with sub-assemblies, in

the result of which the complexity of individual objects and

scenes can be essentially varied.

Mixed geometry: the objects may be canonical geometry

primitives, algebraic implicit and parametric surfaces like

quadrics, NURBS and Bezier patches, convex and non-convex

polyhedrons, solid bodies given by constructive solid geometry

(CSG) or boundary representation (BREP). No matter which

geometry models are used for the scene specification.

Pseudo-dynamic motion: All the object motions are dis-

crete in time and known in advance (in contrast to real-time

simulation in virtual reality environments or motion planning

applications). Due to high complexity, detailed specification

and analysis of all continuous motions in real industrial pro-

jects look redundant and unrealistic. In the scope of the

adopted model both discrete displacement, and instantaneous

appearance, and disappearance of objects are allowed.

Let us discuss how spatio-temporary decomposition

methods can be applied for 4D modelling of large projects

with scenes having the mentioned characteristics. First we de-

fine underlying data structures utilized by the presented parti-

tioning schemas and then introduce the basic concepts of the

scene model applied for deriving theoretical estimates of com-

plexity for typical model requests.

The utilized time tree is a self-balancing binary search

tree. It provides an effective data structure for mutable or-

dered sequence of time points. The root of the time tree cor-

responds to the entire time domain over which a dynamic

scene is analyzed. The time domain is recursively partitioned

into intervals for the subtrees by time points until the intervals

become single points in the tree leaves. Without regard to

particular AVL, red-black or AA structures the binary search

trees allow the lookup, insertion, and deletion operations in

logarithmic time and tree traversal — in linear time on the

number of points.

The applied regular octree is a spatial data structure that

successively partitions a space domain into 8 equally sized

cells. Starting from a root, cells are successively subdivided

into smaller cells under certain conditions, namely, when a cell

contains more than a specified number of objects and it con-

tains at least one object occupying the relative volume less

than the given threshold. Compared to methods that do not

partition space, octrees can improve execution times for que-

rying and processing data. Typical basic operations over oc-

trees are point location, region location, and neighbour

searches that have logarithmic complexity on the number of

cells.

Both time tree and octree structures are well suited for

solving various computational problems like collision detection

and view volume clipping. In different ways they exploit the

same evident principle: one needs to analyze only those

events and related objects that are in feasible domains of the

space and time partitions.

Although 4D modelling and planning applications provide

for specific functionalities, most of them employ common re-

quests connected with retrieval of data relevant to given time

and space domains. Such requests are especially important

for advanced 4D systems targeted on large-scale projects and

avoiding allocation of whole project data in computer RAM. In-

stead, the data are dynamically loaded as they become feasi-

ble for the performed functions and are released as soon as

they are unnecessary.

In this paper four types of requests are considered:

deployment of auxiliary structures for fast retrieving and

accessing project data (Q1);

reconstruction of the whole scene (or feasible fragments)

by given focus time (Q2);

volume clipping of the scene by given view camera frustum

(Q3);

animation of the project progress using a moving camera

(Q4).

It is suggested that the first request Q1 is executed before

4D modelling session. Once being deployed, auxiliary struc-

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tures can be reused to execute subsequent repeated requests

in a more effective way.

The scene reconstruction and volume clipping requests as-

sume incremental updating of the scene or its feasible frag-

ments. It is suggested that the scene representation has been

already determined for current time and fixed camera position

and it is required to update it taking into account the changed

focus time and the moved camera position.

The scene reconstruction request Q2 corresponds to the

first option and can be interpreted by looking at the presented

screenshot of Synchro system (see the figure 1). The graphi-

cal user interface of the Synchro combines and coordinates

both Gantt chart traditional for most project management ap-

plications and 3D views typical for CAD systems. By shifting

the focus time line at the Gantt chart, the user can observe

the project progress in 3D views. View camera positions are

preliminary chosen by the user so that most interesting and

critical issues of the project plan can be thoroughly investi-

gated. There is no need to update and to visualise the whole

scene representation for this purpose. Only objects located in

view camera frustum should be requested, involved in the

scene and then be visualised.

The volume clipping request Q3 corresponds to the second

option and can be illustrated by the same figure. Focus time

line remains fixed at the Gantt chart as camera position in one

of the 3D views is changed. This type request is usually in-

voked when the user tries to investigate the model snapshot

by applying zooming, translating and rotating operations over

the scene. It can be done explicitly or implicitly in the scope of

well-known navigation paradigms like ‗walk‘, ‗examine‘, ‗bird‘s

eye‘.

Figure 1. The screenshot of the Synchro system

The animation request Q4 reproduces another meaningful

use case. For example, the Synchro system provides for ad-

vanced tools for preparing animations of 4D project plans.

During the animation the camera can be smoothly moved so

that the observer can see the project plan issues in time se-

quence from most convenient perspectives.

Thus, the request Q2 implies the retrieval of project data

relevant to varied focus time and fixed space domain, the re-

quest Q3 — to the varied space domain and fixed focus time

and, finally, the request Q4 — to simultaneously varied focus

time and space domain. The request Q1 is related to the pre-

liminary deployment of auxiliary data structures needed for

further effective retrieval and access.

3 PARTITION SCHEMAS

Pseudo-dynamic, event-driven model has been adopted by the

most of 4D applications. It assumes that all the scene changes

occur in fixed points of time and space domains and are

caused by predetermined events or activities.

Such model is quite natural for project management meth-

odology. Activities and work breakdown structures (WBS) can

be directly associated with some elements of the 3D scene

and profiles like ‗installation‘, ‗removing‘, ‗displacement‘,

‗maintenance‘. Then the project plan can be visually inter-

preted and modelled by showing, removing or moving corre-

sponding elements of the scene in fixed time and space

points. Each activity or WBS is assumed to give rise to a pair

of events occurred in its start and finish time points.

Figure 2. The representation of scene events via Gantt chart

The figure 2 illustrates how a project plan represented at

the Gantt chart can be interpreted within the discussed

pseudo-dynamic, event-driven model. A project activity ‗Activ-

ity2‘ starts and finishes at ‗Time1‘ and ‗Time2‘ points marked

by ticks at the calendar axis. The 3D element ‗Element‘ has

been assigned to the activity with the defined profile ‗Mainte-

nance‘. This profile realizes the behaviour similar to temporary

installation of equipment or to temporary displacement of ma-

terials at the project site. The profile is emulated by a pair of

events corresponding to appearance of the 3D element at the

activity start point and its disappearance at the finish point.

The element is shown in the scene position ‗Loc‘ while the ac-

tivity is performed.

In the consideration below the events are represented by

seven-tuple structure { ID, TYPE, ELM, TIME, LOC, NODE,

CELL }, where ID — event identifier, TYPE — event type (that

takes one of the values ‗Appearance‘, ‗Disappearance‘, or

‗Displacement‘), ELM — reference on the associated 3D ele-

ment, TIME — event time, and LOC — spatial location of the

element in the scene (usually defined by transformation ma-

trices).

If time tree and octree have been computed and deployed,

then the event structure can be extended by corresponding

Calendar axis Time1 Time2

Gantt chart

Activity1

Activity2 (Element, Temporary)

Other Activities…

Event sequence

Event( ID1, Appearence, Element, Time1, Loc, Node1, Cell )

Event( ID2, Disappearance, Element, Time2, Loc, Node2, Cell )

Other Events…

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references NODE and CELL identifying the time tree node and

the octree cell associated with the event. Thereby, spatio-

temporal localisation of events can be immediately carried out

using the auxiliary tree structures and then can be applied for

efficient request processing. Nevertheless, there may be dif-

ferent partitioning schemas utilising similar tree structures.

Three dynamic partitioning schemas are discussed below.

3.1 Dynamic spatio-temporal partitioning schema

The first presented DST schema combines a time tree de-

ployed for whole project and an isolated dynamic octree

whose representation is permanently updated to correspond

to a given focus time of the project.

The time tree is computed once using all the time points of

events appeared during the modelled period. Time tree nodes

are interrelated with corresponding event tuples by unidirec-

tional associations. Thereby, events active in a given time in-

terval can be effectively retrieved using the time tree lookup.

The octree cells contain lists of 3D elements which are

present in the scene for a given focus time and are placed in

the corresponding tree cells. Spatial localisation of 3D ele-

ments can be done through sequential search of the octants of

minimal size surrounding the elements or their axis aligned

bounding boxes (AABB). In this paper we assume that the

elements are placed in the octants having approximately the

same size. But if a localised element or its AABB intersects

underlying planes of high level octants, then the element is

replicated in adjacent octants and the process is continued at

lower levels until a cell contains more than a specified number

of elements and at least one element takes the relative vol-

ume less than the given threshold.

Figure 3. The DST partitioning schema

The dynamic octree is updated or recomputed every time

when a project focus time is changed. Therefore, its structure

and content may undergo to significant changes during 4D

modelling session. Nevertheless, the element lists contained

by the octree cells correspond to complete representation of

the whole scene for any fixed focus time. The figure 3 illus-

trates how octree cells, time tree nodes, 3D elements and

event tuples are interrelated each other.

3.2 Extended spatio-temporal partitioning schema

The entended spatio-temporal (EST) partitioning schema can

be considered as an extension of the DST schema. It looks

very similar to DST due to both the time tree and octree are

deployed as primary structures (see the figure 4). But it has

principal distinctions.

Figure 4. The EST partitioning schema

The first distinction is that octree cells contain reference

lists for events appeared in the corresponding octants (more

exactly, multiple associations with event tuples). These addi-

tional references can be computed in the beginning of the

modelling session and then be reused for localisation of scene

elements. As both the time tree and the octree are deployed

once, the EST schema should be considered as static. Con-

trary to the DST schema assuming permanent changes of the

octree structure and content during modelling sessions, the

octree structure and event lists are computed once. Neverthe-

less, more detailed analysis shows that dynamic lists of 3D

elements should be remained in each cell to be capable to in-

crementally reconstruct the scene for any given focus time. In

this respect both DST and EST schemas share common or-

ganisation principles.

The second distinction is connected with the way how the

element lists are formed and updated. Contrary to the DST

schema assuming that the cell lists relate to a common focus

time point, element lists in the EST schema may relate to dif-

ferent time points. Thereby, even being consolidated, the

element lists may not reproduce the scene representation cor-

rectly.

Indeed, there is no need to reconstruct the whole scene

for a fixed focus time and to update it every time when the

focus is changed. Instead, only the visible cells should be

processed and the element lists are updated. Therefore, each

octree cell stores own time point for which its element list has

been formed. If the cell space becomes visible and its content

is to be investigated, then the element list is updated taking

into account all the events appeared between last and recent

focus times.

3D scene

Cell

Event sequence ID1 ID2 ID3 ID…

Node1 Node2

ell

Element

Time tree Dynamic octree

3D scene

Cell

Event sequence ID1 ID2 ID3 ID…

Node1 Node2

ell

Element

Time tree Static octree

dynamic lists

5

3.3 Advanced spatio-temporal partitioning schema

Finally, we present advanced spatio-temporal (AST) partition-

ing schema that can be considered as a further evolution of

the DST and EST schemas. Its main distinction is that the AST

schema first partitions the space domain by an octree and

then for each octree cell partitions the time domain by a time

tree (see the figure 5). By the construction it is very similar to

Shen‘s tree. The key difference of the proposed schema and

its principal advantage is that nodes of deployed time trees

are associated with scene events rather than scene objects.

Because of the events identify changes in the scene, the AST

partitioning schema avoids the replication of identical ele-

ments peculiar to Shen‘s tree.

As opposite to the DST and EST schemas, the AST as-

sumes that the octree is primary and time trees are secon-

dary. Therefore, only time tree nodes in the AST are associ-

ated with events. The AST schema remains lists of 3D

elements similar to the EST schema. It is deployed once in the

beginning of the modelling session, but the element lists are

formed and updated dynamically depending on parameters of

the last spatio-temporal request. It is very suitable for the re-

quests combining parameters of both time and space do-

mains, but useless for time requests because of the necessity

to lookup whole spatial structures to collect the events ap-

peared in a given time interval. This capability has been lost in

comparison with the EST schema whose time tree nodes con-

tain direct references on the associated event tuples.

Figure 5. The AST partitioning schema

4 PERFORMANCE ANALYSIS

To derive practically meaningful estimates of performance, let

a dynamic scene is represented by identical elements and by

similar appearance events so that each event adds a new

element in the scene and the final number of elements equals

to N. It is suggested that the elements are uniformly distrib-

uted over the scene space and the events are equiprobable

during the whole modelled time period.

Volume factor ν = (Δx∙Δy∙Δz)/(x∙y∙z) expresses relative

volume occupied by every element (more exactly, AABB sur-

rounding the element) with respect to overall scene dimen-

sions x, y, z. Linear factor δ = max(Δx/x, Δy/y, Δz/z) ex-

presses maximum relative size of the bounding box. Under

the additional assumption about the scene and elements pro-

portions Δx/x = Δy/y= Δz/z = δ, the volume factor takes the

value ν = δ3.

Let us define auxiliary parameters α, β, γ characterizing

the 4D modelling session. The parameter α > 0 defines rela-

tive volume density of the scene for a final time point as α = ν

N. The volume of view frustum with respect to the whole

scene space is given by the relative parameter β [0,1]. And

the relation Δt / t of the analyzed time interval Δt to whole

modelling period t is defined by the parameter γ [0,1].

All the performance estimates are derived for the average

case under the assumptions about the scene model peculiari-

ties. Let CTime is the average cost of point classification in time

tree nodes, CBox is the average cost of spatial localisation of

the given AABB in octree cells (relatively underlying subdivi-

sion planes of each higher level octant). In similar way, CCone

is the cost of testing whether the given AABB is located inside

view frustum (admitting intersections with its planes) or ex-

actly outside it. For simplicity, costs for navigating over the

trees, costs for gathering of elements and events as separate

collections are taken negligible and eliminated from the per-

formance analysis. Costs for forming and rendering of 3D

scenes are not related to the partitioning schemas and are be-

yond of the consideration too.

4.1 Deployment request Q1

First of all, let us estimate deployment costs for the presented

partitioning schemas. Costs for octree deployment can be ob-

tained independently from the time tree costs for all the

schemas. Octree deployment costs are presented only for the

EST, AST schemas. These costs are missing for the DST

schema as the octree structure and its content are perma-

nently updated during the 4D modelling session. According to

the accepted scenario, at a start time it does contain neither

elements nor cells and, therefore, does not require any com-

putations. At a finish time, it contains all the scene elements

placed in the deployed octant cells. Such costs are related to

other request types.

Under the scene model assumptions, identical elements

are localized in the leaf octants at the level L = log(1/δ) if α

1 and L(N) = 1/3 log(N) in the opposite case α < 1. Here L is

a height of the octree. The probability of simultaneous inter-

section of localized element and three underlying planes of oc-

tant of level l [1, L] equals to δl3, where δl = 2l δ is an ef-

fective relative size of the element AABB. By the octree

construction, it causes the element replication and placement

in eight children octants of the level l + 1. In similar way, the

probability of intersection of two planes and replication of the

element in four octants is equal to 3δl2 (1 – δl)

. The probabil-

ity of intersection of one plane and replication of the element

in two octants counts 3δl (1 – δl). And finally, the probability

of the element localisation exactly in one octant is (1 – δl)3.

Thereby, the average distribution of the element number

at the level l is given by the formula:

Rl = 8 δl3 + 12 δl

2 (1– δl) + 6 δl (1– δl) + (1– δl)3

Element

3D scene

Cell

Event sequence ID1 ID2 ID3 ID…

Node1 Node2

ell

Time trees

Static octree

dynamic lists

6

The average cost of element localisation in the whole oc-

tree is given by CBox R(L), where a summarizing function R(L)

is defined as Σl=1L l∙Rl. Then, the total costs for the octree de-

ployment take the value CBox R(L)N as each appearance event

requires exactly one localisation operation for the correspond-

ing element. It can be shown that the function R(L) is ex-

pressed by the formula:

R(L) = (1/98)*(5 δ323L+4 – 35 δ323L+4L – 80 δ3 – 49

δ222L+3 + 147 δ222L+3L + 392 δ2 – 147 δ2L+2 + 147 δ2L+2L +

588 δ + 49 L2 + 49 L)

with upper limit given by R(L) ≤ 4*(L+1)L.

The time trees are static structures in all the presented

schemas and the deployment costs can be directly calculated.

The binary search trees in the DST and EST schemas contain

exactly N nodes and their computing can be done in CTime

log(N!). In the AST schema time trees are formed in each of

8L leaf octants with the average number of events equal to

N/8L. Therefore, total costs to deploy all such trees within the

AST schema count CTime 8L log((N/8L)!).

By concluding, the deployment costs for DST, EST and AST

schemas are expressed by the corresponding formulae:

C1DST = CTime log(N!)

C1EST = CTime log(N!) + CBox R(L)N

C1AST = CTime 8 L log( (N/8L)! ) + CBox R(L)N

4.2 Scene reconstruction request Q2

The scene reconstruction request Q2 is executed under the

suggestion that the scene is known for the previous focus time

and it is required to update the scene for the recent focus

time. All the events appeared during this interval must be col-

lected and the corresponding updates must be done in the

scene.

The request results for the EST schema can be obtained in

two ways. Starting with the time tree, feasible events ap-

peared in the analysed time interval can be determined in CT

γN. The relative factor γ defines here the time duration over

which the scene is reconstructed. The value γ = 0 corresponds

to the confluent case when no events appear and there is no

need to update the scene. The value γ = 1 corresponds to re-

construction of the scene over the whole modelling period.

Additional checks whether the detected events appear in the

view frustum take CConeγN.

The second way assumes the primary use of the octree.

Navigation through visible octants and checks against appear-

ance of the associated events in the given time interval re-

quire CTimeβR(L)N. The factor R(L) takes into account the ex-

pected number of elements at the octree level L that differs

from the original number due to the element replication.

Applying the optimal strategy combining both methods,

the cost function takes the form:

C2EST = min( (CTime + CCone)γN, CTimeβR(L)N )

The performance estimate for the AST schema can be de-

rived in a similar way, but the lookup process should be ex-

panded on partial time trees deployed in the octree leaves.

Each such tree provides for fast binary search of those events

which appear during the given time interval in the parent oc-

tant. Gathering actual events through all visible octants and

feasible nodes, the resulting collection can be obtained and

processed to update the scene. Taking into account the uni-

form distribution of elements over the scene space and events

over the modelled time period, the expected number of events

appeared in the view frustum during the time interval is given

by the value βγR(L)N. Therefore the total costs are expressed

by the corresponding formula:

C2AST = CTime βγR(L)N

Finally, the request Q2 can be satisfied using the DST

schema. By iterating through the time tree nodes, actual

events can be identified in CTime γN. The octree is dynamically

recomputed if the focus time has been changed. For each de-

tected appearance event the associated element should be

placed into the corresponding octant according to the ac-

cepted modelling scenario. To update the octree γN elements

must be placed. As explained above, it can be done in CBox

R(L) γN. Therefore, the final performance estimate for the

DST schema takes a form:

C2DST = CTime γN + CBox R(L) γN

Here the height of the dynamic octree 1 ≤ L ≤ L depends on

the number of already placed elements as L = log(1/δ) if α γ

1, and L(N) = 1/3 log(γ N) in the opposite case α γ < 1.

4.3 Volume clipping request Q3

The volume clipping request can be executed in a way com-

mon for all the presented schemas. Again, iterating through

the octree the visible octants located in the view frustum are

determined and associated events (or elements) are collected.

Let a camera is moved so that each new view frustum takes

additional scene space volume with the relative factor β[0,1]

introduced above. To determine which octants are visible, it is

necessary to iterate approximately over β Σl=1L 8l cells from

the root down to leaves and to check whether they belong to

the view frustum or not. By expressing the sum of the geo-

metrical series the computational cost of the volume clipping

operation takes a form CCone β(8L+1-1)/7. The level parameter

corresponds to the octree height fixed for the EST and AST

schemas and varied for the DST schema depending on the

factual number of processed events and placed elements. The

determination of visible octants is a key performance issue for

the DST schema, so the following formula takes place:

C3DST = CCone β(8L+1-1)/7

The EST and AST schemas require additional updates of

the dynamic element lists. Remind that although these sche-

mas assume static deployment, the element lists have to be

modified to conform to particular time points. It avoids unnec-

essary updates of the element lists belonging to the invisible

octants and simplifies the scene reconstruction for the octants

which become visible. Therefore, to execute the volume clip-

ping request the reconstruction operation must be invocated

for those cells which become visible and whose lists do not

conform to the recent focus time. The cost estimates for the

reconstruction operation have been obtained in the previous

section. Thereby, the total costs for the volume clipping re-

quest are expressed by the following similar formulae:

C3EST = CCone β(8L+1-1)/7 + CTime βR(L)N

C3AST = CCone β(8L+1-1)/7 + CTime βγR(L)N

7

The formulae have been derived under the suggestion that the

scene reconstruction request is executed first time for the oc-

tants belonging to the changed view frustum.

4.4 Animation request Q4

The animation request assumes simultaneous changes of both

the focus time and the view camera position. Therefore it can

be represented by a pair of the scene reconstruction and vol-

ume clipping operations. As a result, its cost can be estimated

as a simple sum of the auxiliary request costs. It is true for

the DST schema:

C4DST = C2DST + C3DST

For the EST and AST schemas more adequate dependencies

can be identified as the volume clipping operations provide for

the partial scene reconstruction. Thus, the following cost rela-

tions can be established:

C4EST = C3EST

C4AST = C3AST

5 COMPARATIVE ANALYSIS

Let us compare the three presented partitioning schemas with

regards to typical requests and common performance re-

quirements imposed upon their execution. The comparative

analysis of the computational costs enables to formulate rec-

ommendations on the effective use of the schemas for large

project data. The complexity formulae obtained for all the in-

troduced requests and the specified schemas are summarised

in the table 1.

Table 1. The complexity costs for the partitioning schemas

Q1

C1DST = CTime log(N!)

C1EST = CTime log(N!) + CBox R(L)N

C1AST = CTime 8 L log((N/8L)!) + CBox R(L)N

Q2

C2DST = CTime γN + CBox R(L) γN

C2EST = min((CTime + CCone)γN, CTime βR(L)N) C2AST = CTime βγR(L)N

Q3

C3DST = CCone β(8L+1-1)/7

C3EST = CCone β(8L+1-1)/7 + CTime βR(L)N C3AST = CCone β(8L+1-1)/7 + CTime βγR(L)N

Q4

C4DST = C2DST + C3DST C4EST = C3EST

C4AST = C3AST

First of all, it can be concluded that the costs for deploy-

ment of auxiliary structures may be varied significantly. The

asymphtotic estimates of the complexity O(NlogN) coincide

each other for the time tree deployment, although the AST

schema looks more preferable for practical purposes. The oc-

tree deployment for the EST and AST schemas takes the same

costs O(Nlog2N) which are fully avoided by the DST schema.

The scene reconstruction can be efficiently done in

O(Nlog2N) for the requests assuming significant fragmentation

of time and space domains using the AST schema. Other

schemas have the same asymphtotic estimate with multipliers

larger than the factor 4/9CTimeβγ admitted by the AST

schema.

The DST schema is most efficient for the volume clipping

requests as it operates mainly with the partially deployed oc-

tree and avoids additional costs for the scene reconstruction in

latent octants. The AST looks more preferable than the EST as

it simplifies the checks for active events using multiple time

trees deployed in the octree leaves. If the time interval for the

reconstruction is relatively small for latent octants, then it

provides for performance advantage over the EST schema.

Finally, the AST and EST schemas are well suited for the

animation requests. The complexities are the same as those of

derived for the volume clipping operation. The DST schema

requires for total updates of the octree structure which are

unnecessary for other considered schemas. Each of the pre-

sented schemas has own benefits and drawbacks compared

with the other solutions. The criterion of uniformly high per-

formance for different type requests seems to be adopted for

the choice of optimal schema.

6 CONCLUSIONS

Thus, the spatio-temporal decomposition approach to 4D

modelling of large-scale industrial projects has been pre-

sented. The approach assumes special data structures and fa-

cilities to effectively execute typical requests relevant to given

time and space domains. Three partitioning schemas have

been presented and analysed to satisfy performance require-

ments. The schemas exploit coherence peculiarities of the

pseudo-dynamic, event-driven scenes appearing in 4D model-

ling and planning applications. The derived theoretical esti-

mates of the complexity as well as the conducted computa-

tional experiments prove benefits of the schemas and their

wide potential use.

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