Reconstructability Analysis

Now, we analyze the reconstructability of the proposed method. That is, we determine under which conditions the solution of Equation (4.4) generates accurate 3D points. The direct analysis of Eq (4.4) is difficult, since it needs to determine in which situation the adjacencypwith minimum cost, out of NN−2 _{possible adjacencies (Wikipedia, 2014), corresponds to the real object class}

trajectory. However, we find that having the motion tangent constraint reduces the possibility of finding the incorrect adjacencyp. Hence, we focus on the reconstructability of the continuous method in Equation (4.7) given the adjacencyp.

Assume we already know the ground truth 3D pointX∗_i of objecti,i= 1, . . . , N. Given that X∗_i is present on the viewing rayXi, we move the camera centerCi toX∗i along the rayXi(t)in directionri. Then any point on the line that passes throughX∗_i and has ray directionri can be represented asXi(si) = Xi∗+siri, wheresiis the signed distance along the viewing ray (not the positive distance as defined by theti). Then Equation (4.7) can be reformulated as

min s

X

(i,j)∈p

kdi,j ×(Xi(si)−Xj(sj))k22+λkXi(si)−Xj(sj)k22, (4.8)

where s = [s1, . . . , sN]. Though si is signed distance and ti is positive distance, minimizing Equation (4.7) and Equation (4.8) still output the same 3D point positions, as long as the computed 3D points in Equation (4.8) are in front of the camera centers. We will see that this is normally true, since the computed 3D points are typically close to their ground truth position if the system is well-conditioned.

We denote the solution of Equation (4.8) assopt_{. The true 3D points are ideally reconstructed}

if sopt _{= 0, since X}

i(0) equals to X∗i given sopt = 0. More specifically, sopt equals the signed Euclidean distance between the 3D points produced by Equation (4.7) and the ground truthX_i∗. Therefore,ksopt_{kis the Euclidean error of the estimated 3D points by Equation 4.7. In the remainder}

of this section, we further analyze in which situationsksopt_{kis small to better understand the quality}

𝐗

_𝑖∗

𝐗

_𝑘∗

𝐫

_𝑖

𝐝

_𝑖,𝑘

𝐫

_𝑖

-(𝐫

_𝑖

∙𝐝

_𝑖,𝑘

) 𝐝

_𝑖,𝑘 (a)λ= 0

𝐗

_𝑖∗

𝐗

_𝑘∗ 𝐝_𝑖,𝑘 (𝟏 + λ )𝐫_𝑖-(𝐫_𝑖∙ 𝐝_𝑖,𝑘) 𝐝_𝑖,𝑘 (𝟏 + λ )𝐫_𝑖 (b)λ >0 Figure 4.5: Plot of Equation (4.10) withλ= 0andλ >0.

The minimum value of Equation (4.8) is achieved at the point where the first derivative relative tosequals 0. This produces a linear equation systemAsopt _{=b, where theith row andjth column}

of matrixAis Aij =               

[(ri·di,j)di,j−(1 +λ)ri]·rj ifi6=jand(i, j)∈p

0, ifi6=jand(i, j)∈/ p

P

(i,k)∈p[1 +λ−(ri·di,k)

2_{] ifi=j.}

(4.9)

Theith element of vectorbis

bi = X (i,k)∈p(X ∗ k−X ∗ i)·[(1 +λ)ri−(ri·di,k)di,k]. (4.10) Next, we explain that if the adjacencypis correctly found, the reconstructabililty of the object class trajectory mainly depends on the condition number of the linear system defined byA. With careful observation, we can see Equation (4.9) and Equation (4.10) have the following interesting properties:

X∗_i =X∗_k,bequals 0 based on Equation (4.10). (2) Careful observation reveals that ifλis set to0, in Equation (4.10) the vector(1 +λ)ri−(di,k ·ri)di,k is perpendicular to vector X∗_i −X∗_k(Figure 4.5a), hencebi = 0. However, we will show that withλ = 0, the linear systemAs=bis unstable due to the high condition number of matrixA. (3) Furthermore, whenλincreases from 0, the two vectors slowly deviate from being perpendicular, as shown in Figure 4.5b. Therefore,bi is likely to be small ifλis close to 0.

2. Since we can not control 3D positions and there are typically small measurement errors in dij,bdoes not exactly equal to 0. This can be regarded as a small disturbance ofbaround 0. For the linear systemAsopt _{= b, one can think of the condition numberκ(A)as being}

(roughly) the rate at which the solution,sopt_{, will change with respect to a change inb. κ(A)}

is available because it solely depends onri,di,j andλ, but not on the ground truth 3D points X∗. Therefore, we can estimate the reliability of the reconstructed 3D points by computing

κ(A). Moreover, we empirically found that the condition number of matrixAis inversely related to λ. The condition number shown in Figure 4.6 is computed using 100 random cameras, and averaged over 200 trials. We can seeκ(A)is large ifλis close to 0 and drops dramatically with smallλ. Then,κ(A)decreases monotonically and slowly asλincreases. In our experiments, we chooseλ = ₁₅1 as a balance of having good chance of smallbwithout decreasing the stability of the linear system.

In conclusion, given the well-conditioned system and correct motion tangentdi,j, we are able to reconstruct the 3D positions close to the ground truth.

In document Zheng_unc_0153D_15925.pdf (Page 60-62)