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Bayesian learning of a search region for pedestrian detection

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Abstract

An efficient pedestrian detection method is proposed for intelligent vehicles in this paper. The proposed method learns the region in which pedestrians are likely to be detected and narrows down the search to the likely region. The likely region is modeled as a Gaussian distribution on the y-axis and its parameters are updated by a Bayesian approach. Thus, the proposed method starts with an exhaustive full search, but gradually narrows down the search by focusing on the likely region. The learning of the likely region is formulated as a Bayesian learning problem and the likely region is analytically derived. The proposed method is combined with two popular pedestrian detection methods, Haar-like Adaboost and HOG-LSVM, and some experiments are conducted with the Caltech pedestrian dataset. The experiments show that the proposed method not only reduces computation time, but also enhances performance by rejecting false positive results.

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Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2013R1A2A2A01015624).

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Correspondence to Euntai Kim.

Appendices

Appendix 1

The posterior is updated by multiplying the likelihood (4) and prior (5) and we obtain

$$ \begin{array}{c}\hfill p\left(\mu, \phi \left|{C}_t\right.\right)\propto p\left({C}_t\left|\mu, \phi \right.\right)p\left(\mu, \phi \right)\hfill \\ {}\hfill \propto \left[{\phi}^{\frac{N}{2}} \exp \left\{-\frac{\phi }{2}{\displaystyle \sum_{n=1}^N{\left({c_{t,}}_n-\mu \right)}^2}\right\}\right]\times \left[{\phi}^{\alpha -\frac{1}{2}} \exp \left\{-\frac{\phi }{2}\left(\frac{1}{B}{\left(\mu -b\right)}^2+2\lambda \right)\right\}\right]\hfill \\ {}\hfill ={\phi}^{\alpha +\frac{N}{2}-\frac{1}{2}} \exp \left\{-\frac{\phi }{2}\left({\displaystyle \sum_{n=1}^N{\left({c_{t,}}_n-\mu \right)}^2}+\frac{1}{B}{\left(\mu -b\right)}^2+2\lambda \right)\right\}.\hfill \end{array} $$
(17)

Here, let us define

$$ f\left(\mu \right)=\frac{1}{2}\left[{\displaystyle \sum_{n=1}^N{\left({c_{t,}}_n-\mu \right)}^2}+\frac{1}{B}{\left(\mu -b\right)}^2+2\lambda \right]. $$
(18)

f(μ) is a quadratic function in μ and, after some arithmetic manipulation, we obtain

$$ f\left(\mu \right)=\frac{1}{2}\left[\frac{1}{B_t}{\left(\mu -{b}_t\right)}^2+2{\lambda}_t\right], $$
(19)

where

$$ \frac{1}{B_t}=\left(N+\frac{1}{B}\right), $$
(20)
$$ {b}_t={B}_t\left({\displaystyle \sum_{n=1}^N{c}_{t,n}^2+\frac{b}{B}}\right), $$
(21)
$$ {\lambda}_t=\frac{1}{2}\left[{\displaystyle \sum_{n=1}^N{c}_{t,n}^2+\frac{b^2}{B}-\frac{b_t^2}{B_t}}\right]+\lambda . $$
(22)

Then, the posterior becomes

$$ \begin{array}{cc}\hfill p\left(\mu, \phi \left|{C}_t\right.\right)\hfill & \hfill \propto {\phi}^{\alpha +\frac{N}{2}-1} \exp \left\{-\frac{\phi }{2}\left(\frac{1}{B_t}{\left(\mu -{b}_t\right)}^2+2{\lambda}_t\right)\right\}.\hfill \end{array} $$
(23)

By comparing the posterior with the prior (5), if we define \( {\alpha}_t=\alpha +\frac{N}{2}, \) then

$$ \mu, \phi \left|{C}_t\right.\sim NG\left({b}_t,{B}_t,{\alpha}_t,{\lambda}_t\right). $$

Appendix 2

Since ϕ ~ G(α, λ), the pdf of \( \frac{1}{\phi } \) is given by the inverse gamma distribution, which is defined by

$$ \frac{1}{\phi}\sim {p}_{IG}\left(x\left|\alpha, \lambda \right.\right)=\frac{\lambda^{\alpha }}{\varGamma \left(\alpha \right)}{x}^{-\alpha -1}{e}^{-\frac{\lambda }{x}}. $$
(24)

Then,

$$ \begin{array}{c}\hfill E\left(\frac{1}{\sqrt{\phi }}\right)={\displaystyle {\int}_{x=0}^{\infty }{x}^{\frac{1}{2}}}{p}_{IG}\left(x\left|\alpha, \lambda \right.\right)dx\hfill \\ {}\hfill ={\displaystyle {\int}_{x=0}^{\infty }{x}^{\frac{1}{2}}}\cdot \frac{\lambda^{\alpha }}{\varGamma \left(\alpha \right)}{x}^{-\alpha -1}{e}^{-\frac{\lambda }{x}}dx\hfill \\ {}\hfill =\frac{\lambda^{\alpha }}{\varGamma \left(\alpha \right)}{\displaystyle {\int}_{x=0}^{\infty }{x}^{-\alpha +\frac{3}{2}}\left(-\frac{1}{\lambda}\right){e}^{-\frac{\lambda }{x}}\left(-\frac{\lambda }{x^2}\right)dx.}\hfill \end{array} $$
(25)

By introducing a new variable \( u=\frac{\lambda }{x} \) and using \( \left(-\frac{\lambda }{x^2}\right)dx=du \), we can rewrite the above equation as

$$ \begin{array}{c}\hfill E\left(\frac{1}{\sqrt{\phi }}\right)=\frac{\lambda^{\alpha }}{\varGamma \left(\alpha \right)}{\displaystyle {\int}_{u=0}^{\infty }{\left(\frac{\lambda }{u}\right)}^{-\alpha +\frac{3}{2}}\left(\frac{1}{\lambda}\right){e}^{-u}du}\ \hfill \\ {}\hfill =\frac{\lambda^{\alpha }}{\varGamma \left(\alpha \right)}{\displaystyle {\int}_0^{\infty }{(u)}^{\alpha -\frac{1}{2}-1}{e}^{-u}du}\hfill \\ {}\hfill =\frac{\lambda^{\frac{1}{2}}}{\varGamma \left(\alpha \right)}\cdot \varGamma \left(\alpha -\frac{1}{2}\right)\hfill \end{array} $$
(26)

since \( \varGamma (z)={\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-t}{t}^{z-1}dt}. \)

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Baek, J., Hong, S., Kim, J. et al. Bayesian learning of a search region for pedestrian detection. Multimed Tools Appl 75, 863–885 (2016). https://doi.org/10.1007/s11042-014-2329-z

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  • DOI: https://doi.org/10.1007/s11042-014-2329-z

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