Stereo Matching Based on CF-EM Joint Algorithm

Abstract

Cost Filtering (CF) and Energy Minimization (EM) are two main cost aggregation methods in stereo matching. Due to global smoothness assumption, EM methods can get higher matching accuracy. However, they tend to fail in occluded areas, while locally adaptive support-weight CF method can solve it well. This paper proposed a CF-EM joint stereo matching framework on the basis of the proof that CF and EM methods can realize interconversion to each other. In this joint framework, we firstly use CF method with fully connected Markov Random Field (F-MRF) model to yield a more robust unary potential. And then, the output unary potential is used as the input to a standard EM method to compute the final disparity in Local connected MRF (L-MRF) model. Experiments results demonstrate that our method can improve the stereo matching accuracy as the achievement of energy transferring from F-MRF to L-MRF.

This work is supported by the Projects of NSFC (61371191, 61201236), and Research Project of China SARFT (2015-53).

Appendix

Proof of Lemma 2.1

Proof.

If $d = d_{p}$, $\varphi (d,d_{p} ) = 0$. We have $\left( {C_{q} (d) + w_{q,p} \varphi (d,d_{p} )} \right) = C_{q} (d_{p} )$.

If $d \ne d_{p}$, $\varphi (d,d_{p} ) = 1$, thus, $\left( {C_{q} (d) + w_{q,p} \varphi (d,d_{p} )} \right) = C_{q} (d) + w_{q,p}$.

Therefore, $m_{q,p}^{1} (d) = \mathop {\hbox{min} }\limits_{{d \in {\mathcal{L}}}} \left\{ {C_{q} (d_{p} )|d = d_{p} ,C_{q} (d) + w_{q,p} |d \ne d_{p} } \right\}$.

When $C_{q} (d_{p} ) = 1$, $C_{q} (d) = 1$, because $0 \le \omega \le 1$, thus

$$ m_{q,p}^{1} (d) = C_{q} (d_{p} ) . $$

(24)

When $C_{q} (d_{p} ) = 1$, $C_{q} (d) = 0$, thus,

$$ m_{q,p}^{1} (d) = w_{q,p} C_{q} (d) . $$

(25)

When $C_{q} (d_{p} ) = 0$, $m_{q,p}^{1} (d) = C_{q} (d_{p} )$. Because $d = d_{p}$, $w_{q,p} (d,d_{p} ) = 1$, thus,

$$ C_{q} (d_{p} ) = w_{p,q} C_{q} (d)\left| {d = d_{p} } \right. $$

(26)

Synthesize (24–26), we can get: $m_{q,p}^{1} (d) = w_{q,p} (d,d_{p} )C_{q} (d)$, Eq. (11) is proved.