Evaluating and Optimizing Feature Combinations for Visual Loop Closure Detection

Abstract

Loop closure detection (LCD) is a key step in visual simultaneous localization and mapping systems to correct the map and relocalize the vehicle. However, it may fail when illumination variations and shifting dynamics present in the scenes. One way to improve the precision is to effectively combine multiple image features that supply complementary information. In this paper, a general method to quantitatively measure the efficacy of individual image features as well as feature combinations for LCD is proposed by calculating the statistical distance considering the distributions of feature vectors. Based on different statistical distances including Kullback-Leibler divergence, Bhattacharyya divergence and Wasserstein metric, various numerical indices capable of evaluating feature combinations are obtained and compared. An unsupervised algorithm is further proposed to optimize feature combinations by maximizing any of the indices. Experiments show that the proposed indices can measure the efficacies of image features and the resulting feature combinations maximizing the indices can improve the precision of LCD.

Appendix:

1.1 A.1 Proof of the Assertions in Section ??

Suppose that there are two features h = [h₁,h₂]^T with the statistics

$$ \boldsymbol s = \left[ \begin{array}{c} s_{1} \\ s_{2} \end{array}\right], \boldsymbol {\varSigma}= \left[ \begin{array}{cc} {\sigma_{1}^{2}} & \rho\sigma_{1}\sigma_{2}\\ \rho\sigma_{1}\sigma_{2} & {\sigma_{2}^{2}} \end{array} \right] $$

(1)

where s₁ and s₂ are positive and the correlation coefficient ρ < 1.

• 1. Suppose that D[h₂] > D[h₁], then

  $$ D[h_{2}]=\frac{{s_{2}^{2}}}{{\sigma_{2}^{2}}}>\frac{{s_{1}^{2}}}{{\sigma_{1}^{2}}}=D[h_{1}] $$

  (2)

  and equivalently, s₂/s₁ > σ₂/σ₁. Therefore

  $$ \left( 1+\frac{s_{2}}{s_{1}}\right)^{2}>\left( 1+\frac{\sigma_{2}}{\sigma_{1}}\right)^{2}>1+\left( \frac{\sigma_{2}}{\sigma_{1}}\right)^{2}+2\rho\frac{\sigma_{2}}{\sigma_{1}} $$

  (3)

  and equivalently,

  $$ D[h_{1}+h_{2}]=\frac{(s_{1}+s_{2})^{2}}{{\sigma_{1}^{2}}+{\sigma_{2}^{2}}+2\rho\sigma_{1}\sigma_{2}}>\frac{{s_{1}^{2}}}{{\sigma_{1}^{2}}}=D[h_{1}]. $$

  (4)

• 2. The optimal weight vector maximizing (??) is [9]

  $$ \boldsymbol w^{*}=\left[ \begin{array}{c} w_{1}^{*} \\ w_{2}^{*} \end{array}\right] \propto \boldsymbol{\varSigma}^{-1}\boldsymbol s\propto \left[ \begin{array}{c} d_{1}-\rho d_{2} \\ \lambda d_{2} -\rho\lambda d_{1} \end{array}\right] $$

  (5)

  where $d_{1}= \sqrt {D[h_{1}]}$, $d_{2}=\sqrt {D[h_{2}]}$ and λ = σ₁/σ₂. We define the ratio

  $$ k = \frac{w_{1}^{*}}{w_{1}^{*}+w_{2}^{*}}=\frac{d_{1}-\rho d_{2}}{d_{1}+rd_{2}-\rho(rd_{1}+d_{2})}. $$

  (6)

  A direct calculation shows that the sign of ∂k/∂D[h₁] and ∂k/∂ρ are the same as 1 − ρ² and D[h₁] − D[h₂], respectively. Therefore ∂k/∂D[h₁] > 0 always holds and ∂k/∂ρ > 0 if and only if D[h₁] > D[h₂].

1.2 A.2 Jacobian of FEIs

The Jacobian of D(w) in Eq. ?? is

$$ \frac{d D}{d\boldsymbol w}= \frac{2\boldsymbol w^{\text{T}}\boldsymbol s}{(\boldsymbol w^{\text{T}} \boldsymbol {\varSigma} \boldsymbol w)^{2}}[(\boldsymbol w^{\text{T}} \boldsymbol {\varSigma} \boldsymbol w) \boldsymbol s - (\boldsymbol w^{\text{T}}\boldsymbol s) \boldsymbol {\varSigma} \boldsymbol w]. $$

(7)

This type of FEIs includes D_FS, D_LR, the first part g₁ of D_α according to Eq. ?? and the first part of D_WS according to Eq. ??.

The Jacobian of the second part g₂ of D_α(w) can be deduced from Eq. ??,

$$ \frac{d g_{2}}{d\boldsymbol w}=\frac{d g_{2}}{d r_{\sigma}^{2}}\frac{d r_{\sigma}^{2}}{d \boldsymbol w}=\frac{r_{\sigma}^{2}-1}{r_{\sigma}^{2}(\upbeta r_{\sigma}^{2}+\alpha)}\frac{2({\sigma_{F}^{2}}\boldsymbol {\varSigma}_{T}-{\sigma_{T}^{2}}\boldsymbol {\varSigma}_{F})\boldsymbol w}{{\sigma_{F}^{4}}}. $$

(8)

For ∥w∥ = 1, the Jacobian of the second part of D_WS(w) can be deduced from Eq. ??,

$$ \frac{d}{d\boldsymbol w}\frac{\Delta \sigma^{2}}{\Vert\boldsymbol w \Vert^{2}}=2(\sigma_{F}-\sigma_{T})\left( \frac{\boldsymbol {\varSigma}_{F}}{\sigma_{F}}-\frac{\boldsymbol {\varSigma}_{T}}{\sigma_{T}}\right)\boldsymbol w-2{\Delta} \sigma^{2}\boldsymbol w. $$

(9)