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Visualizing quantitative time-dependent changes in the topography requires relying on a series of discrete given multi-temporal topographic datasets that were acquired on a given time-line. The reality of physical phenomenon occurring during the acquisition times is complex when trying to mutually model the datasets; thus, different levels of spatial inter-relations and geometric inconsistencies among the datasets exist. Any straight forward simulation will result in a truncated, ill-correct and un-smooth visualization. A desired quantitative and qualitative modelling is presumed to describe morphologic changes that occurred, so it can be utilized to carry out more precise and true-to-nature visualization tasks, while trying to best describe the reality transition as it occurred. This research paper suggests adopting a fully automatic hierarchical modelling mechanism, hence implementing several levels of spatial correspondence between the topographic datasets. This quantification is then utilized for the datasets morphing and blending tasks required for intermediate scene visualization. The establishment of a digital model that stores the local spatial transformation parameterization correspondences between the topographic datasets is realized. Along with designated interpolation concepts, this complete process ensures that the visualized transition from one topographic dataset to the other via the quantified correspondences is smooth and continuous, while maintaining morphological and topological relations.

Time-dependent visualization of changes in physical entities, such as topographic datasets that present morphologic alterations, requires mutually modelling a series of discrete given datasets. The presumption is that these topographic datasets, such as Digital Terrain Models (DTMs), were acquired on a given time-line and that they represent approximately the same coverage area. An infinite set of topographic datasets, which present a time interval of close to zero (

Two topographic datasets representing the same morphologic entity with different positions (

Understanding and modelling the complex inter-relations that exists between topographic datasets is therefore essential. This paper introduces a novel hierarchical modelling algorithm that establishes a quantitative mutual modelling via sets of spatial correspondences that best describes the morphologic changes occurred; thus, it enables us to carry out more precise, true-to-nature and realistic animation visualization and simulation tasks. These are essential for a reliable 3D Geographic Information Systems (GIS) workflow, while trying to best describe and simulate the reality transition as it occurred.

When considering geospatial GIS work environments, the basic requirement is that it enables support of different types of data-models,

Modelling complexity—the topographic reality that is based on 3D physical objects is normally very complex: geometric and semantic information—to name a few;

Multi-resolution and multi-representation—when dealing with multi-epoch and multi-source terrain representation the physical objects representing terrain are rarely of the same nature;

Presentation and appearance—usually when representing 3D terrain data the viewer will demand the representation to be as true-to-nature as possible and not derived by data symbols and annotation;

Animation—terrain animation has to preserve the quality and accuracy of given data, as well as be realistic and continuous throughout the simulation;

Simulations and multi-temporal representations—chosen algorithm needs to handle the general problem of physical objects (or skeletons) derived deformation instead of treating each area of anatomy as a special case;

Topologies and morphologies—are to be preserved or otherwise created if derived by the deformation process, while maintaining the realistic nature of representation.

Techniques and methods, which involve the transformation of one object to another object (e.g., topographic entities), while creating a continuous transition of shape sequences that changes gradually, are being developed in recent decades on many aspects in the field of computer graphics (e.g., [

Methods of substitution between objects,

Animated topography via linear “flip-page” transition showing artificial morphologic entities: cliff (

Procedures, such as shape blending or shape morphing, utilize a process in which one object is combined with another object, while trying to stretch, rotate and move it in order to create a maximum matching between the two objects. Two major problems or tasks have to be addressed for this assignment [

These problems are relatively straightforward to solve in the case of spatial rigid body models, such as a face, which mainly differ in their perspective point of view (e.g., [

Existing concepts and techniques aim at morphing one shape into another shape by concentrating on the shape itself,

Hierarchical modelling designed for integration of two (or more) homogenous DTMs was proposed in the work of [

Hierarchical modelling schema, depicting three-working levels: global (registration, level 1), local (matching, level 2) and precise position-derived data-integration and interpolation (level 3).

Workflow of the hierarchical modelling mechanism and its main stages. DTM, Digital Terrain Models.

A short review of this mechanism and its main stages is given and depicted in the workflow in

Global rough registration, whereas choosing a common framework for both DTMs (and, thus, solving the datum and coordinate system ambiguities existing between DTMs before performing transformation, as outlined in

Matching—since global registration is achieved and datum ambiguities are resolved, local matching is carried out while implementing the Iterative Closest Point (ICP) algorithm for rigid surfaces matching. This stage is essential for achieving precise reciprocal modelling framework between the two datasets, thus establishing localized transformation quantification;

Integration schema, which consists of the modelling inter-relations quantification evaluated in the local matching. Since transformation is extracted via ICP on a local level, this quantification is needed to be densified for all points existing in that level,

This stage encapsulates the processes associated with the hierarchical modelling of level 1 and level 2, which are designed to extract varied-scale data-relations and correlations in an automatic manner, ^{2} (approximately 150,000 points for interest point identification); level 2—1 km^{2} (approximately 2,000 points in the ICP process); and level 3—resolution (point) level.

Each area (_{xj}_{yj}_{zj}_{j}_{j},ω_{j}

Digital Transformation Parameters Model (DTPM) transformation matrix (middle turquoise); and source and target DTMs (front and background). Each matrix cell (

Hypothetically, when transforming a DTM grid-point from the source topographic dataset, while using the stored transformation parameters, should coincide with the terrain presented by the target topographic dataset. The (vertex) path from source to target is actually a hypothetical transformation between the two given physical objects that can be regarded as a time-dependent one, and any intermediate scene on that path is basically unique temporal terrain visualization.

Still, the resolution of the DTPM matrix is significantly lower than the given resolution(s) of the modeled topographic datasets. Usually, the topographic datasets used store grid-points in a resolution of a few meters up to several dozen. The DTPM resolution, however, is derived from the frame size used in the ICP matching process; thus, storing the parameters in a resolution of several hundred meters (e.g., 1 km^{2}). The fact that position-derived unique transformation values are required for each point that exists in one topographic dataset (source) for transformation to the other topographic dataset (target),

Two interpolation procedures can be pointed out:

Consequently, after the implementation of the “among” interpolation, an “in-between” interpolation is carried out on the values to appreciate spatially the position of the intermediate terrain within that space. The “in-between” (or path) interpolation is designed for the calculation of weighted six parameter transformation values from source topography (

The next section presents the algorithms and the computational concepts designed for multi-temporal visualization that were implemented and integrated.

The entire modelled area is divided into frames. Each congruent frame—one of each terrain representation—is matched via the ICP algorithm independently and separately. This localized implementation is more effective in monitoring and modelling local random incongruities, and trends exists. Thus, the estimation of the rigid body transformation(s) that aligns both models best is attained. This quantification is later used for the transformation—and deformation—of one object to the other.

ICP matching is accomplished via the minimization of a goal function based on least squares matching (LSM). This process measures the squares sum of the spatial Euclidean distances or errors, existing between geometries presented by congruent frames:

In order to perform least squares estimation, _{x}_{y}_{z}_{x}_{y}_{z}^{T} and

The least squares solution produces as the generalized Gauss-Markov model the unbiased minimum variance estimation for the parameters, depicted in Equation (2):
_{0}^{2} denotes the variance factor,

Due to the fact that both DTMs are non-rigid physical bodies, several aspects are considered:

Wide-coverage DTMs represent different data-characteristics—namely, level-of-detail and resolution—implying that existing ICP homologous points is not at all explicit;

DTMs acquired on different times (epochs) will surely represent different terrain topography and morphology (either natural or artificial activities);

Data and measurement errors can reflect on the points’ positional certainty on a relatively large scale.

Addressing the aforementioned yields three geometric point-to-cell constraints that are implemented in the ICP process:

The corresponding coordinates of the paired-up nearest neighbor _{i}_{0}_{2} ×_{3} ×_{0},_{1},_{2},_{3}} are calculated based on the cell’s corner heights, depicted in Equation (3) and

The line-equation, derived from the coordinates of point ^{t}_{i}_{i}^{t}_{i}_{i}

The same constraint outlined in 2 is applied, only here, the line-equation is perpendicular to the local cell-plane in frame f in the

_{i}

Schematic indexing of local DTM grid-cell corner heights: _{i}

The point transformed from ^{t}_{i}_{i}_{i}^{t}_{i}

The first constraint for point _{i}

The second and third constraints enforce a closest point by requiring the first order derivative of the plane and the line connecting the two points (Equation (5)) to be perpendicular, _{i}

Thus, the last two constraints can be written as in Equation (7):

It can be described as if these criteria of constraints shift the vector between the points, ^{t}_{i}_{i}_{i},

Schematic explanation of least squares target function minimization via three constraints: ^{t}_{i}_{i}^{t}_{i}

Summing up all (

Due to the DTPM data structure—the DTM look-like model—an interpolation that will ensure continuous parameter representation has to be chosen. Interpolating DTM heights using bi-directional third degree parabolic equations is described in [

This algorithm is designed for height interpolation; analogously, it can be altered to handle the three linear characterized translation values stored in the transformation matrix on three separate processes. Equation (8) depicts the algorithm’s equations utilized for this interpolation. Implementing this process enables us to accurately define for each grid-point of the source dataset the three translation transformation parameters and, hence, to receive a more detailed and smooth ‘translation surface’ that corresponds adequately to these translation values.

Returning to the “

_{k}_{P}_{n}_{n}

Euler angles representation are perhaps the most common parameterization of 3D orientation space. A naive approach to interpolating the rotation values might suggest independent and separate processes between each of the three Euler angles in order to produce the mediate positions needed. The outcome of this will produce a motion that is specified by a rotation that looks contracted, jerky at times, not continuous and ill-specified. This occurs mainly due to the fact that Euler angles ignore the interaction of the rolls about separate axes; hence, there is not a unique path between every two orientations across different coordinate systems; thus, there exists a dependency among the three-axis [

The notation of quaternion is therefore given to define a 3-dimensional number system by a four-dimensional one [

The naive linear interpolation between two key quaternions will result in a straight line, which ignores the natural geometry of rotation space: the interpolated rotation would result in a motion that speeds-up in the middle. This is because linear interpolation is not in motion on the hyper-sphere surface, but, instead, cuts across it. The needed motion is a motion that does not speed-up, but keeps a constant velocity (_{1} and _{2} are two key unit-quaternion orientations, _{i}—_{1} (while _{1} and _{2}.

SLERP is required for “_{0},_{1},_{2},_{3}}, three steps of the SLERP sequence can be suggested. This is equivalent to a Bezier curve with a spherical cubic interpolation, depicted in Equation (11). Still, quaternion multiplication is not a commutative one; so, the order in which this SQUAD sequence is implemented should be considered.

_{0},_{1},_{2},_{3},_{0},_{1},_{2},_{3},

Conceptual representation of spherical linear interpolation (SLERP): instead of a naive linear interpolation (

Since the interpolation mechanisms discussed in

Two examples of the suggested bi-directional third degree parabolic interpolation are depicted in ^{2} frames (with approximately 30% overlap) ICP process. The desired resolution of 50 m is derived from the used DTMs. Calculating the translation values for a specific location (

Bi-directional third degree parabolic interpolation for translation values _{x}_{z}

Since quaternion multiplication is not commutative, the order of how the SLERP interpolations are carried out within each SQUAD implementation might have influence on the resulting rotation values and, thus, affecting the reliability of the animation produced with the creation of topographic artifacts. For example, while relying on the fact that the matrix grid size used in the interpolation is 700 m wide (as in

To quantify this inadequacy, a synthetic analysis evaluation is carried out, in which large registration values of rotation angles were taken into account. Large values were deliberately used to ascertain the reliability of this interpolation mechanism used. _{1}_{2} and _{4}_{3}), followed by a SLERP on the resulting quaternions values; and, _{1}_{4} and _{2}_{3}) followed by a SLERP on the resulting quaternions values. This analysis is carried out to quantify and evaluate whether choosing a specific sequence (

DTPM single cell with two spherical and quadrangle (SQUAD) sequences given as parabola-motion form (arcs)—orientation nodes _{i}

Four unit-quaternion coefficient values calculated via two different SQUAD sequences—

Three Euler angles values calculated via two different SQUAD sequences—

^{−3} decimal degrees—these values are not significant enough to have an effect on the procedures suggested here: with DTPM cell size used in the interpolation of 700 m wide, the resulting “

It is visible in

It is worth emphasizing that the rotation values stored in the DTPM are normally smaller than the ones used in this analysis: normally, several decimal degrees with respect to the dozens used here. Thus, it can be concluded that using quaternion space and SLERP and SQUAD interpolation concepts is reliable and produces qualitative results; thus, it contributes to the mechanism and concepts implemented here.

Difference values between the three Euler angles calculated via two different SQUAD sequences—values in decimal degrees (z-axis).

Mean and SD angular difference values for the two different SQUAD sequences.

Parameter | Difference values | |
---|---|---|

Mean | SD | |

−0.0033 | 0.0041 | |

0.00005 | 0.0003 | |

−0.0042 | 0.0048 |

The entire process was programmed with a Matlab R2013a working environment and implemented on a standard PC working station (Windows 7 with i5 processor and 4G RAM). The duration of processing time is dependent on the DTM coverage area analyzed. A single process, ^{2} takes less than 60 s. Though not developed to have real-time capabilities, with the required programming language and processing unit (C++ and server workstation, for example), this scheme might present near real-time potential for the processing of a relatively small to medium area; based on previous experience and tests, the entire processing time can be reduced to a sub-second.

One can assume that two given topographic datasets represent two epochs: _{0}_{1}_{i}_{i}

Intermediate scene _{i}^{1}/_{3} (bottom row) produced, while using data stored in source (_{0} = 0.0, _{1} = 1.0,

Intermediate scene _{i}^{2}/_{3} (_{0} = 0.0, _{1} = 1.0,

^{2}. The source DTM was produced via photogrammetric means using low resolution satellite imagery, while the target DTM was produced via digitization of a 1:50,000 height contours map. Both DTMs present a resolution of 50 m and approximately the same level of accuracy. The top row in _{0} = 0.0), while the bottom row in _{1} = 1.0). The complete fully-automatic hierarchical modeling mechanism is implemented, resulting in the extraction of the complete different levels of geometric interrelations and correspondence of both DTMs stored in the DTPM (translation values extracted are: _{x}_{y}_{z}_{i }^{1}/_{3} and _{i} =^{ 2}/_{3}, is presented—bottom row in

Moreover, a series of intermediate scenes can be produced, while visualizing from _{0} = 0.0 to _{1} = 1.0 at

Standard off-the-shelf GIS and animation applications designed to model topographic datasets are mostly based merely on the datasets’ mutual coordinate reference systems and on simplified geometric set of rules. This is usually in contrast to the physical reality and alterations these datasets model and represent. The commonly used “flip-page” concept does not try to attempt solving mutual existing relations, but rather only to produce an abstract and simplified simulation and representation. Shape blending and shape morphing algorithms will usually require a manual intervention, solving the complexity required.

The proposed hierarchical modeling mechanism, on the other hand, uses a different level of correlations and interrelations that are automatically

The hierarchical modeling mechanism enables us to quantify precisely the inaccuracies and discrepancies exist between the topographic datasets via the concept of using several consecutive levels of spatial modeling. This enables us to extract the existing mutual correlations and, hence, monitoring and modeling phenomenon that are characterized only locally for a more reliable and qualitative true-to-nature multi-temporal visualization. It is worth noting that this concept is not solely restricted to visualization implementations only, as other various accurate and reliable geo-oriented GIS analysis tasks can be implemented based on this novel mechanism.

The authors declare no conflict of interest.