Since the interpolation mechanisms discussed in

Section 4.2.1 and

Section 4.2.2 are not specifically designed for transformation parameters interpolation stored in the DTPM, this section aspires to verify that, indeed, they can be used for this purpose. Moreover, as discussed in

Section 4.2.2, quaternion multiplication is not commutative, thus, there might exist some constraints as to the SQUAD sequence implemented; this is analyzed and described in

Section 5.2.

#### 5.1. Translation Parameters

Two examples of the suggested bi-directional third degree parabolic interpolation are depicted in

Figure 9. The source data is a DTPM transformation matrix with a resolution of 700 m, which is derived from using 1 km

^{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 (

i.e., “

among”), while shifting from the 700 m matrix grid resolution to the desired 50 m DTM resolution, preserves continuity, guaranteeing smooth interpolation within the entire coverage area. The output is a more detailed and continuous translation value calculation on the entire area required for the computation of the intermediate topography.

**Figure 9.**
Bi-directional third degree parabolic interpolation for translation values t_{x} (left) and t_{z} (right). (**b**) and (**d**) are displayed in the original 700 m DTPM resolution, while (**a**) and (**c**) are displayed in the resulting 50 m resolution needed for morphing and blending.

**Figure 9.**
Bi-directional third degree parabolic interpolation for translation values t_{x} (left) and t_{z} (right). (**b**) and (**d**) are displayed in the original 700 m DTPM resolution, while (**a**) and (**c**) are displayed in the resulting 50 m resolution needed for morphing and blending.

#### 5.2. Rotation Parameters

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

Section 5.1), an erroneous resulting interpolation difference value of one decimal degree will result in a maximum position shift of six meters.

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.

Figure 10 schematically depicts this, while the four corners represent a single DTPM cell with the rotation registration values used. An 11 by 11 grid was generated covering the cell (total of 121 positions), in which for each position, two SQUAD sequence calculations were implemented: interpolation sequence

a—two SLERPs on the two horizontal couples (

q_{1}-q_{2} and

q_{4}-q_{3}), followed by a SLERP on the resulting quaternions values; and,

b—two SLERPs on the two vertical couples (

q_{1}-q_{4} and

q_{2}-q_{3}) followed by a SLERP on the resulting quaternions values. This analysis is carried out to quantify and evaluate whether choosing a specific sequence (

a or

b) has any quantitative effect on the calculated interpolation values outcome,

i.e., measuring the attributed commutative error. Thus, for all 121 positions, two sets of four unit-quaternion coefficients values are calculated (

W,

x,

y,

z); consequently, three Euler angle values, (

φ,

κ,

ω) are generated, enabling the quantitative analysis of the differences these two sets of rotation values have.

Figure 11 depicts a mesh representation of all 121 four unit-Quaternion coefficients values calculated via the two SQUAD sequences—

a and

b. Visually comparing these two sets—left and right, respectively—no significant differences are evident. When comparing the values received, a maximum difference value of 0.01 in the W coefficient exists with significant lower difference values for all remaining coefficients. Moreover, it is apparent that the values are smooth and continuous within the entire cell area with no abrupt change in value.

Figure 12 depicts all 121 three Euler rotation angles values received via the two SQUAD sequences—

a and

b (after translating them from the quaternion 4D domain). Inspecting the values received, there are no significant differences for all three coefficients: a maximum of 0.004 decimal degrees. The values are smooth and continuous within the entire cell area, with no abrupt change in value.

**Figure 10.**
DTPM single cell with two spherical and quadrangle (SQUAD) sequences given as parabola-motion form (arcs)—orientation nodes q_{i} {i ∈ [1–4]} values (φ,κ,ω) in degrees.

**Figure 10.**
DTPM single cell with two spherical and quadrangle (SQUAD) sequences given as parabola-motion form (arcs)—orientation nodes q_{i} {i ∈ [1–4]} values (φ,κ,ω) in degrees.

**Figure 11.**
Four unit-quaternion coefficient values calculated via two different SQUAD sequences—

a (

**left**) and

b (

**right**) (

a and

b are denoted as parabola motion in

Figure 9)

.
**Figure 11.**
Four unit-quaternion coefficient values calculated via two different SQUAD sequences—

a (

**left**) and

b (

**right**) (

a and

b are denoted as parabola motion in

Figure 9)

.

**Figure 12.**
Three Euler angles values calculated via two different SQUAD sequences—

a (

**left**) and

b (

**right**) (

a and

b are denoted as parabola motion in

Figure 9)

.
**Figure 12.**
Three Euler angles values calculated via two different SQUAD sequences—

a (

**left**) and

b (

**right**) (

a and

b are denoted as parabola motion in

Figure 9)

.
Figure 13 depicts the value differences between the three Euler rotation angles received via the two SQUAD sequences. Though slight value changes do exist—average angular difference of approximately 4 × 10

^{−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 “

among” interpolation difference value will have an effect of less than 2 cm on the shifting values (1:35,000 in scale). Carried out on large coverage DTMs with resolutions of dozens to hundreds of meters, the resulting outcome is still highly reliable.

It is visible in

Figure 13 that on the border of the cell, the differences are zero, suggesting the process is correct (the SQUAD on the borders is actually translated into SLERP implementation). The Difference values increase toward the center of the cell, which is a result of the SQUAD and SLERP mathematical notions.

Table 1 depicts the mean and standard deviation (SD) values computed for these angular differences.

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.

**Figure 13.**
Difference values between the three Euler angles calculated via two different SQUAD sequences—values in decimal degrees (z-axis).

**Figure 13.**
Difference values between the three Euler angles calculated via two different SQUAD sequences—values in decimal degrees (z-axis).

**Table 1.**
Mean and SD angular difference values for the two different SQUAD sequences.

**Table 1.**
Mean and SD angular difference values for the two different SQUAD sequences.
Parameter | Difference values |
---|

Mean | SD |
---|

dφ (deg) | −0.0033 | 0.0041 |

dκ (deg) | 0.00005 | 0.0003 |

dω (deg) | −0.0042 | 0.0048 |