Voronoi-based Second-order Descriptor with Whitened Metric in LiDAR Place Recognition

Abstract

The pooling layer plays a vital role in aggregating local descriptors into the metrizable global descriptor in the LiDAR Place Recognition (LPR). In particular, the second-order pooling is capable of capturing higher-order interactions among local descriptors. However, its existing methods in the LPR adhere to conventional implementations and post-normalization, and incur the descriptor unsuitable for Euclidean distancing. Based on the recent interpretation that associates NetVLAD with the second-order statistics, we propose to integrate second-order pooling with the inductive bias from Voronoi cells. Our novel pooling method aggregates local descriptors to form the second-order matrix and whitens the global descriptor to implicitly measure the Mahalanobis distance while conserving the cluster property from Voronoi cells, addressing its numerical instability during learning with diverse techniques. We demonstrate its performance gains through the experiments conducted on the Oxford Robotcar and Wild-Places benchmarks and analyze the numerical effect of the proposed whitening algorithm.

Figures (6)

• Figure 1: The illustration of feature space of each Voronoi cell learned by our method. We select the query place and its top-3 closest neighbors metrized by our method from the Oxford maddern2017oxford and visualize each cell of their descriptors reduced by ICA. White dots and colored cross marks denote query and neighbors. Our whitening transforms each Voronoi cell more homogeneous and suitable for Euclidean distancing.
• Figure 2: The overall architecture. First, the input local descriptors $\mathbf{X}$ are mapped by networks $\mathcal{F}_\text{proj}$ and $\mathcal{F}_\text{score}$, multiplied into the global descriptor $\widetilde{\mathbf{X}}$. Our whitening module implemented with ZCA whitening kessy2018whitening computes the standard deviation matrix $\mathbf{S}$ from $\widetilde{\mathbf{X}}$ and normalizes each cluster's feature by multiplying $\mathbf{S}^{-1}$ over $\widetilde{\mathbf{X}}$. Finally, the global descriptor $\mathbf{Z}$ is vectorized and scaled down to $\frac{1}{\sigma}\mathbf{Z}$.
• Figure 3: Mean Average Precision at K (MAP@K) plotted up to 10 predictions in the Oxford and In-house datasets.
• Figure 4: The ROC curves by place recognition models in the Wild-Places intra-sequence evaluations. False positive rate (FPR) and true positive rate (TPR) were measured with the decision threshold in range $[0, 1]$ metrized in the descriptor space.
• Figure 5: We measured the matrix rank and effective rank roy2007effective of covariances before the whitening ($\hat{\mathbf{\Sigma}}$), after RBLW shrinkage ($\hat{\mathbf{\Sigma}}_\text{RBLW}$), and after the whitening ($\hat{\mathbf{Q}}\hat{\mathbf{\Lambda}}\hat{\mathbf{Q}^{\top}}$) using our default $(C\!=\!16,\;M\!=\!16)$ method. Red and blue lines denote the medians of matrix ranks and effective ranks each.