Growing Process

This subsection explains the iterative growing process of a planar surface starting from a seed patch. Firstly, some notations that will be used need to be introduced. The indexjwill be used as superscript to indicate the iteration index of the iterative growing process, so for example the planar surface i at stage j of the growing process will be represented by S_ij or S_ij(f_ij, δ_ij). In the latter form the function f_ij is the estimated equation of the plane at the end of thej-th stage of the growing process, and it is given by ˆnj_i anddj_i. The mean square fitting error of the surface’s points with respect to the estimated fitting surface isδ_ij, and it could be evaluated using (5.2). The ground-truth surface of thei-th surface will be represent by ¯Si.

Different to the RANSAC method whose seeds are selected randomly, in the pro- posed growing stage, the previously obtained seed patch candidates will be initially arranged in ascending order of their mean square fitting error in a growing seed list

Ψ1, thus Ψ1 = {∀ψn, ψm∈Ψ1∶δn≤δm;n<m}. The first seed patch appearing in the

list will be used to initiate the first surface. Once this surface reaches its maximum extent then its growing process will stop and the growing seed list will be updated to Ψ2 by eliminating all the seed patches that are enclosed within the first detected planar

surface. This updating process will be carried out at the end of the growing process for each planar surface,S_ij, to generate a new seed growing list Ψi+1={∀ψm∈Ψi∶ψm∉Si}.

This ensures that only non-incorporated seed patches will be used in the subsequent growing of other planar surfaces.

At this point the growing process of the planeiwill be described. This plane at its initial stage is merely defined by its seed patch, thusS_i0(f_i0, δ0_i)=ψm(fm, δm), with ψm

being the first seed patch in Ψi. At thej-th iteration stage of the growing process the

neighbors ¯N_ij of the surfaceS_ij will be firstly identified. Then points belonging to other planar surfaces will be excluded from ¯N_ij to obtain a new set N_ij ={p∶_p∉Sm, m<i}.

their fitness to this plane. So the points with fitting error larger than the threshold,

Tj, will be regarded as outliers to this planar surface. Otherwise, they will be enclosed within the current plane to formS_ij+1. This could be summarized by:

S_ij+1/S_ij ={∀p ∈N_ij ∶e(p)≤Tj} (5.3) Fig.5.2 shows an example of the first two growing steps (i.e.,j=1 andj=2) of the plane

iand itsN_i1 and N_i2 neighboring sets. After each growing stage, the plane equationf_ij

will be refined to f_ij+1 by using the linear least square plane fitting approach over the whole set of pixels of the newly updated surfaceS_ij+1.

The growing process for the surfaceiwill be iteratively repeated until one of the fol- lowing two halt conditions is met: a) the setN_ij is empty, b) no point in the neighboring setN_ij fits well into the current planar surface. These two conditions indicate that the

i-th surface has grown to its maximum extent. Once the growing process stops then the surfaceiwill be finalized and it will be represented hereinafter bySi=Si(f_iki, δk_ii),

whereki represents the index of the last growing stage of this planar surface. fiki is the

final estimated equation of the surface and it is given by ˆnki

i and d ki

i .

As previously described at the end of the growing process of thei-th planar surface, a new growing seed list, Ψi+1, will be generated and a new growing process will be

initiated by using the first-ranked seed patch in the list. This growing process will be repeated until the updated seed list is empty.

D D D S0 i Finding neighboring pixels

Neighboring pixels do not belong to the current surface Growing process ……. N1 i N2 i

Plane pixels Neighboring pixels belong to the current surface

S1

i _S2

i

Figure 5.2: An example of the growing process of a planar surface;D is depth map,S_ij

is the current surface andN_ij is current neighboring pixels.

In addition, it is worth noticing that the threshold value, Tj, which is used to determine the fitness of a pixel for the current growing surface, will be dynamically updated after each iteration of the growing process by increasing its value. That is to say, at beginning the requirement for enclosing neighboring pixels is very strict since these points will become the base of the following growing stages. With the growing of

the plane, more points need to be checked, so if the threshold is still small, then with high probability this will lead to over-segment the whole depth map into many small planar surfaces. In this work, the following equation forTj has been adopted

Tj=τ(1−e−j/λ) (5.4)

whereτ is the maximum allowed “roughness” of the planar surface andλis the changing speed of the threshold.