2.2.1. The Proposed Synthetic Algorithm

The synthetic algorithm uses image processing technology and the illumination intensity-based assumption. Equations (1)–(9) and Figure 2 depict the illumination intensitybased assumptions and the corresponding process based on the geometrics and mathematics. Figure 2a abstracts a typical navigation scenario of the planetary rovers using a sketch. The light source (*I*) can be approximated as the sun in the scenario. The angles between the rays *i*1, *i*2, and *i*<sup>3</sup> of the light and the horizontal ground (*g*) are *θ*1, *θ*2, and *θ*3, respectively. When *I* is significantly far away from the ground *g*, this research considers that all light rays are parallel to each other, so the angles *θ*1, *θ*2, and *θ*<sup>3</sup> of *i*1, *i*2, and *i*<sup>3</sup> and the horizontal ground are equal (see Equation (1)).

$$\begin{cases} \quad \dot{i}\_1 \parallel \dot{i}\_2 \parallel \dot{i}\_{3\prime} \\ \quad \theta\_1 = \theta\_2 = \theta\_{3\prime} \end{cases} \quad \text{if } I \to +\infty. \tag{1}$$

**Figure 2.** The sketches of the planetary rover navigations. (**a**) refers to the typical scenario of the planetary rover. (**b**) refers to the abstracted scenario through applying Equation (1) to (**a**). (**c**) refers to the abstracted scenario with a small and closer rock landmark compared to (**b**). *G*1, *G*2, *G*3, *P*, *p*1, and *p*<sup>2</sup> in the (**b**) scenario correspond to *G*1, *G*2, *G*3, *P*, *p*1, and *p*<sup>2</sup> in the (**c**) scenario, respectively.

Figure 2b shows the abstracted sketch of Figure 2a through applying Equation (1). The angles between all rays (*i*) and *g* all equal to *θ*. This research defines *ρ* to refer to the density of rays, which also refers to the illumination intensity (*L*) in the unit area on the ground. Therefore, the *L* on a specific ground area equals the multiplication between the area of the region (*S*) and *ρ* (see Equation (2)).

$$L = \rho \ast \ S \tag{2}$$

The solid blue lines (*p*<sup>1</sup> and *p*2) in Figure 2b refer to the rock area captured by the navigation camera. The dashed line (*lb*) refers to the normal line perpendicular to the phase plane. The solid black line segment (*PG*3) refers to the corresponding rock on the image. Although the rock occupies the same image region as the ground *G*1*G*3, the *L* of the rock is different than the ground without rock because of the difference between *G*2*G*<sup>3</sup> and *G*1*G*3. In Equation (3), *LG*1*G*<sup>3</sup> refers to the *L* in the *G*1*G*<sup>3</sup> area, and *PG*<sup>3</sup> refers to the *PG*<sup>3</sup> area.

$$
\rho = \frac{L\_{G\_1 G\_3}}{PG\_3} \tag{3}
$$

Notably, all images involved in this section refer to the grayscale images. Thus, *PG*<sup>3</sup> is a grayscale image. This research assumes the image grayscale value of the *PG*<sup>3</sup> area relates to two parameters, corresponding density (*ρ*) and the surface optical properties (*Popt*) of the object (*cT*).

i. The above discussion uses Equation (2) to achieve the desired illumination intensity, while *ρ* is difficult to obtain from a grayscale image. However, the known information is the corresponding image grayscale value (*G*1*G*3) and the area of *PG*3. It is noteworthy that *G*1*G*<sup>3</sup> and *G*2*G*<sup>3</sup> appear in the same image region. This research assumes that the ratio (*ρ*) between the sum grayscale in *G*1*G*<sup>3</sup> and the area of *PG*<sup>3</sup> can approximate the value of *ρ* (see Equation (4)).

$$\rho \approx \overline{\rho} = \frac{L\_{\text{G}\_{1}\text{G}\_{3}}}{PG\_{3}} = \sum\_{(x,y) \in T} \left[ \frac{pixel\_{\text{img}}(x,y)}{N\_{pixel}} \right] \tag{4}$$

However, Figure 2c shows another scenario. A pronounced difference between *G*1*G*<sup>2</sup> (Figure 2b) and *G*1*G*<sup>2</sup> (Figure 2c) comes from a smaller and closer rock landmark. Therefore, the difference (Δ*ρ*) between *ρ* and *ρ* is located on *G*1*G*<sup>2</sup> (equivalent to

*G*1*G*2). It is noteworthy that *ρ* is the ratio between the sum grayscale of *G*1*G*<sup>3</sup> and *PG*3, whereas *ρ* is the ratio between the sum grayscale of *G*2*G*<sup>3</sup> to *PG*<sup>3</sup> (see Equation (5)).

$$
\Delta \rho = \overline{\rho} - \rho = \frac{L\_{G\_1 G\_3}}{PG\_3} - \frac{L\_{G\_2 G\_3}}{PG\_3} = \frac{L\_{G\_1 G\_3} - L\_{G\_2 G\_3}}{PG\_3} \tag{5}
$$

Substituting Equation (2) into Equation (5) can produce Equation (6), so Δ*ρ* is a value related to *LG*1*G*<sup>2</sup> .

$$
\Delta \rho = \overline{\rho} - \rho = \frac{\rho \ast G\_1 G\_3 - \rho \ast G\_2 G\_3}{PG\_3} = \frac{\rho \ast G\_1 G\_2}{PG\_3} = \frac{L\_{G\_1 G\_2}}{PG\_3} \tag{6}
$$

ii. The optical properties of the object surface are complex (such as surface reflectance, refracting, and absorptivity), and they do not belong to the scope of this research. Here, we use a variable *cT* to pack all factors related to optical properties. Equation (7) depicts the grayscale change caused through the optical properties.

$$P\_{opt} = f\_1(c\_T) \tag{7}$$

Recalling the objective of the synthetic algorithm, Equation (7) can only correlate the optical properties and image grayscales implicitly. Thus, this research proposes Equation (8) to approach Equation (7) artificially. Equation (8) assumes that the grayscale distribution in the target region (rock in this research) is a function of the coordinates when *ρ* is constant. This research calculates the averaged grayscale value (*imgmean*) for the corresponding image area. Then, it subtracts the grayscale values (*img*) to *imgmean* to obtain a differential grayscale "image" (*img*Δ), which is a statistical result only related to the coordinates.

$$P\_{opt} \approx \, \text{img}\_{\Delta} = \, \text{img} - \, \text{img}\_{\text{g}\,\text{mean}} \tag{8}$$

The synthetic algorithm corresponding to the rock-embedded area can be depicted using Equation (9):

*L* = *ρ* ∗ *img*<sup>Δ</sup> − *C*. (9)

The *C* refers to the constants used to correct the distance between *ρ* and *ρ*. Recalling Equation (6), Δ*ρ* positively correlates to the *LG*1*G*<sup>2</sup> . The practical area of *LG*1*G*<sup>2</sup> is a varying value that is dependent on the appearance of the target. Measuring *LG*1*G*<sup>2</sup> is challenging, but *LG*1*G*<sup>2</sup> positively correlates to *imgmean* (a brighter image causes a higher *LG*1*G*<sup>2</sup> ). Thus, this research assumes *C* is a constant that depends on *imgmean*. Table 2 depicts the values of *C*, while the detailed experiments for deciding *C* can be found in Appendix A.2 in the Appendix A. It is noteworthy that *L*, *L*, and *img*<sup>Δ</sup> all contain multiple values, which correspond to the coordinates.

**Table 2.** The constant *C* to correct *ρ* from *ρ*.


<sup>1</sup> The values correspond to the grayscale metric with 256 scales.
