La estructura de la propiedad y la morfogénesis urbana

In this section, a decision threshold Dt, is estimated so that (6.10) can be used to

accept or reject the candidate match, Gcand. Dt can be a fixed threshold learnt by

presenting the SRS with a series of training scenes of the environment before actual scene recognition or it can be estimated from Πs. As this work is concerned with

natural outdoor scenes with a large dynamic range, the latter method is chosen as it is adaptive to various environments and does not require any training images from the environment which may be totally unknown or outdated. Furthermore, some positive scenes may have only a few matches due to large distortions or dim illumination that degrade Gc so much so that using a fixed Dtbecomes impractical,

6.2 Determining scene equivalence from a database 138

as is shown in Fig. 6.1.

The first step in estimating Dt is to construct the Decision matrix, ∆s from

the best few matches in Πs. These matches are determined based on two criteria,

in terms of their Gc and also in terms of N%test:

1. From the first column vector of Πs that contains N%test, the Nref matches

are sorted based on N%test. The matches with N%test > t%, where t% is a

fixed percentage threshold, are retained.

2. Next, using the last column vector of Πs, the elements in Gc are ranked so

that only the Ntop Gcs are retained. Ntop is a fixed number that determines

how many best few matches are retained.

3. Finally, ∆sis obtained by combining the results in the first two steps so that

only matches that are significant (the intersections of the first two steps) in both N%test and Gc are retained for the estimation of Dt.

In this work, the values for the fixed parameters (t%, Ntop) are set at (10%, 5)

respectively. The number of rows that remain is denoted as Nbest and this forms

the Nbest× 8 ∆s where the structure is detailed below:

Definition 6.12. Structure of ∆s The Decision matrix, ∆s is a Nbest × 8 ma-

trix with the following structure: ∆s =N%∆ Sxρ∆ S y

ρ∆ Szρ∆ Kxτ ∆ K y

τ ∆ Kzτ ∆ Gc∆



6.2 Determining scene equivalence from a database 139

between ∆s and Πs.

A Threshold vector, Ξs, containing 7 elements is constructed from ∆s that has

the same structure as a row in Πs (without the Gc):

Ξs =N%Ξ SρΞx S y ρΞ S z ρΞ K x τ Ξ K y τ Ξ K z τ Ξ  _(6.11)

The elements in Ξs, once determined, are used to compute directly the estimate

of the Dt defined as:

Dt=

N%Ξ

200 × ( ˜SρΞ+ ˜Kτ Ξ)

(6.12)

where ( ˜SρΞ, ˜Kτ Ξ) are derived from the means of the rank correlations in Ξs:

           ˜ SρΞ = 1₃P i S_ρΞi , i ∈ {x, y, z} ˜ Kτ Ξ = 1₃ P i Ki τ Ξ, i ∈ {x, y, z} (6.13)

The rest of the section describes how the elements in Ξs are derived from ∆s.

The first element of Ξs, N%Ξ, is given the value of the candidate match, N%cand

6.2 Determining scene equivalence from a database 140

not, the largest N%∆ in N%∆ is used:

N%Ξ =          N%cand if Gcand ∈ ∆s max (N%∆) otherwise (6.14)

The reason for this assignment rule is very simple - if Gcand ∈ ∆/ s, it is likely to

be unreliable and should be rejected. Using the largest N%∆ will give us a Dt that

is likely to be larger than Gcand since N%Ξ determines partially the value of Dt

(6.12). Invoking (6.10) allows the proposed SRS to effectively reject the unrelible Gcand.

The rest of the elements in Ξs are determined in a three step process:

1. Collect the rank correlations over the three spatial directions together to form a composite rank correlation matrix, denoted as (Σρ∆, Λτ ∆):

         Σρ∆ =Sxρ∆ S y ρ∆ S z ρ∆  Λτ ∆ = [Kxτ ∆ K y τ ∆ K z τ ∆] (6.15)

6.2 Determining scene equivalence from a database 141

med(Σρ∆), med(Λτ ∆) and take the minimum among the two values to deter-

mine a threshold for significant rank correlations, trank:

trank = min (med(Σρ∆), med(Λτ ∆)) (6.16)

The value of trank is limited to a maximum value so that a sufficient number

of rank correlations can be used to estimate Dt from Ξs (see the next step).

A trank that is too large yields too few rank correlations for the subsequent

computations to be reliable. In this work, trank is limited to 0.6.

3. Using trank, the statistics of the rank correlation elements in Ξs are deter-

mined by computing once again the median of these rank correlation entries that are larger than trank in ∆s.

         Si ρΞ= medSiρ∆|Siρ∆. trank , i ∈ {x, y, z} Ki τ Ξ = med {Kiτ ∆|Kiτ ∆. trank} , i ∈ {x, y, z} (6.17)

where the . operator represents a ‘>’ comparison between the elements of a vector on the LHS with a scalar on the RHS. Using trank ensures that only

the most significant rank correlations that contribute to the best matches in ∆s are used in the computation of Dt in (6.12).

6.2 Determining scene equivalence from a database 142

accept or reject the input test scene by comparing Dt and the candidate match

score Gcand (6.10). The next section explains briefly how Dt actually works in

providing a reasonable adaptive threshold for the scene decision module.