Users plan optimization for participatory urban texture documentation

Abstract

We envision participatory texture documentation (PTD) as a process in which a group of users (dedicated individuals and/or general public) with camera-equipped mobile phones participate in collaborative collection of urban texture information. PTD enables inexpensive, scalable and high quality urban texture documentation. We propose to implement PTD in two steps. At the first step, termed viewpoint selection, a minimum number of viewpoints in the urban environment are selected from which the texture of the entire urban environment (the part visible to cameras) with a desirable quality can be collected/captured. At the second step, called viewpoint assignment, the selected viewpoints are assigned to the participating users such that given a limited number of users with various constraints (e.g., restricted available time) users can collectively capture the maximum amount of texture information within a limited time interval. In this paper, we define each of these steps and prove that both are NP-hard problems. Accordingly, we propose efficient algorithms to implement the viewpoint selection and assignment problems. We study, profile and verify our proposed solutions comparatively by both rigorous analysis and extensive experiments.

Appendix: Calculating \(A_{p,S_c}\)

The image of each cell c consists of a set of pixels P _c . With a pinhole model, the region represented by each pixel p ∈ P _c is in the form of a four sided polygon which area depends on a set of parameters S _c . S _c includes camera focal length f, camera view angle θ (the angle between camera image plane and c) and camera distance to c represented by d . Finally, we denote the area represented by p ∈ P _c on c, according to the setting in S _c , by $A_{p,S_c}.$ Here, we elaborate on how to calculate $A_{p,S_c}.$First, assume the camera image plane is parallel to c, i.e., view angle is zero and θ = 0 (we later consider the general case). Consider a pixel p with the height of p _v and the width of p _h . In this case, the polygon represented by p is a rectangle with the side lengths of l _v (vertical side) and l _h (horizontal side). l _v is coplanar with c and has the length of

$$ l_v=\frac{p_v\times d_c}{d_p}, $$

(5)

where d _c is the distance between camera lens and point o on c where o, camera lens and center of p _v are on the same line (Fig. 20). d _p is the distance between midpoint of p _v and the pinhole (both d _p and d _c can be calculated based on the parameters in S _c ). Equation 5 is derived based on the similarity of triangles formed by l _v , p _v and the pinhole. In a similar way, we can find the length of l _h which is coplanar with c. Therefore, $A_{p,S_c}$ is derived as follows:

$$ A_{p,S_c}=l_h\times l_v=\frac{p_h\times p_v\times d_c^2}{d_p^2}. $$

(6)

$A_{p,S_c}$ is not exactly the same for all the pixels in P _c and depends on d _c and d _p . Note that p _h and p _v are the intrinsic camera properties. Equation 6 holds for a set of cells whose images are completely captured on the camera image plane. To find this set, we calculate the field of view of camera [15]. The field of view of a camera is the portion of scene space that actually projects onto the camera image plane. For a camera with the image plane diameter of I _d , the field of view φ is $2\arctan\frac{I_d}{2f}$ [15]. The image of a cell within the field of view of is completely captured on the camera plane and Eq. 6 holds for it.

Fig. 20

A pixel p with the side length p _v where p _v represents the length of l _v on cell c

We now consider the case in which the view angle is non-zero (− 90 ≤ θ < 0 or 0 < θ ≤ 90). In this case, the area represented by p is not necessarily a rectangle but still a four sided polygon where each side of the polygon is represented by a side of p. Assume the bottom side of p is denoted by p _b . We explain how to calculate l _b (the side of the polygon represented by p _b ) and the approach can be extended to calculate the other sides. Consider a plane which contains l _b and p _b . We call this plane $P_\bot.$ The projection of θ on $P_\bot$ is denoted by $\theta_\bot$ (Fig. 21). Based on the figure, l _b  = x ₁ + x ₂ where x ₁ and x ₂ can be calculated according to the law of sines [36], i.e.,

$$ x_1=\frac{d_{c_\bot}\sin\big(\frac{\phi_p}{2}\big)}{\cos\big(\frac{\phi_p}{2}-\theta_\bot\big)}~~~\text{and}~~~ x_2=\frac{d_{c_\bot}\sin\big(\frac{\phi_p}{2}\big)}{\cos\big(\frac{\phi_p}{2}+\theta_\bot\big)}. $$

ϕ _p is the field of view for p (the field of view of a camera which only consists of pixel p) and can be derived similar to field of view of camera. $d_{c_\bot}$ is the distance between camera lens and point o, where o is at the intersection of l _b and the line passing through camera lens image on $P_\bot$ and the center of p _b . Similarly, we can calculate the other sides of the polygon represented by p. Additionally, we can find the angle between the sides based on the equation of planes form from the image sides and pixel sides. Consequently, we can derive $A_{p,S_c}.$

Fig. 21

The top-view of a camera pixel p and a cell c