Evaluation of metrics and a dynamic thresholding strategy for high precision single sensor scene matching applications

Abstract

Matching images of a scene with high precision is a core component in real time applications such as image-based navigation of unmanned aerial vehicles, terminal guidance of missiles. The aim of the research discussed in this paper is to develop a scene matching algorithm of high accuracy, robust to the variations between the images compared with scope for parallelism. After a study of state-of-the-art scene matching algorithms, a local feature-based approach was chosen to meet the stated requirements. The paper discusses the techniques for modules of the algorithm chosen and the methodology implemented after a metric study with a dynamic threshold based nearest neighbour distance ratio matching strategy (DT-NNDR). The metrics and the proposed strategy were tested for the matching of planar images from Affine covariant features group, unmanned aerial vehicle videos from a national defence agency, and satellite images from Google. The performance of strategy was compared with the existing NN-DR strategy with an experimental optimal threshold. The overall precision of the proposed strategy is 88%, making it suitable for applications which require high value of true positive.

Appendix

The theoretical justification derived for the higher performance city block compared to the Chessboard and Euclidean, with the NN-DR strategy, is as follows:

Fig. 11

Comparison of metrics

Figure 11 shows the geometrical representation of the distances between two descriptors A (d1, d2) and C (d1’, d2’) in a 2D plane. From the figure with triangle ABC, we can observe line AB stands for the difference in the first dimension |d₁-d₁’|, and BC is the difference in the second dimension |d₂-d₂’|.

By properties of the triangle

$$ AB+ BC> AC>\max \left( AB, BC\right) $$

(9)

$$ =>d\left( City\ Block\ {L}_1\right)>d\left( Euclidean\ {L}_2\right)>d\left( Chessboard\ {L}_{\infty}\right) $$

(10)

The determination of the corresponding feature for a feature p from the feature set Q can be mathematically expressed as follows:

$$ Correspondence\left(p,Q\right)=\left\{\begin{array}{c}{q}_1,\frac{d_{q_1}}{d_{q_2}}< threshold\\ {} null, otherwise\end{array}\right. $$

(11)

Here q₁ and q₂ are the first and second nearest neighbours of p from Q respectively.

Using Eq. (10)

$$ {d}_{q_1}\left({L}_1\right)>{d}_{q_1}\left({L}_2\right)>{d}_{q_1}\left({L}_{\infty}\right) $$

(12)

$$ {d}_{q_2}\left({L}_1\right)>{d}_{q_2}\left({L}_2\right)>{d}_{q_2}\left({L}_{\infty}\right) $$

(13)

Negating Eq. (13),

$$ -{d}_{q_2}\left({L}_1\right)<-{d}_{q_2}\left({L}_2\right)<-{d}_{q_2}\left({L}_{\infty}\right) $$

(14)

Since opposite inequalities can be added, Eqs. (13) and (14) can be combined as,

$$ {d}_{q_1}\left({L}_1\right)-{d}_{q_2}\left({L}_1\right)>{d}_{q_1}\left({L}_2\right)-{d}_{q_2}\left({L}_2\right)>{d}_{q_1}\left({L}_{\infty}\right)-{d}_{q_2}\left({L}_{\infty}\right) $$

(15)

We can observe from Eq. (15) that the difference between the first minimum and second minimum distances is the highest for the city block followed by Euclidean and Chessboard metrics.

Higher the difference between the numerator and the denominator, lesser will be the value of the fraction. Hence from (15),

$$ \frac{d_{q_1}\left({L}_1\right)}{d_{q_2}\left({L}_1\right)}<\frac{d_{q_1}\left({L}_2\right)}{d_{q_2}\left({L}_2\right)}<\frac{d_{q_1}\left({L}_{\infty}\right)}{d_{q_2}\left({L}_{\infty}\right)} $$

(16)

From Eq. (16), it can be observed that the ratio between the first minimum and the second minimum is the least for the city block distance metric. Hence more cases will satisfy the condition in Eq. (11) for feature correspondence, resulting in more matches by city block distance metric than the widely used Euclidean and Chessboard with the NN-DR matching strategy.