Generating Fluttering Patterns with Low Autocorrelation for Coded Exposure Imaging

Abstract

The performance of coded exposure imaging critically depends on finding good binary sequences. Previous coded exposure imaging methods have mostly relied on random searching to find the binary codes, but that approach can easily fail to find good long sequences, due to the exponentially expanding search space. In this paper, we present two algorithms for generating the binary sequences, which are especially well suited for generating short and long binary sequences, respectively. We show that the concept of low autocorrelation binary sequences, which has been successfully exploited in the field of information theory, can be applied to generate shutter fluttering patterns. We also propose a new measure for good binary sequences. Based on the new measure, we introduce two new algorithms for coded exposure imaging - a modified Legendre sequence method and a memetic algorithm. Experiments using both synthetic and real data show that our new algorithms consistently generate better binary sequences for the coded exposure problem, yielding better deblurring and resolution enhancement results compared to previous methods of generating the binary codes.

Appendix

1.1 Appendix 1: Derivation of Eq. (7)

Let ${\hat{U}}$ be a fluttering shutter pattern with the elements ${\hat{u}}_{i}\in \{0, 1\}$. We denote B as $B={\hat{U}}-\mu$ with elements $b_{i}\in \{-\mu , 1-\mu \}$, where $\mu$ is the mean value of the elements in ${\hat{U}}$. Then we introduce $U=2(B+\mu -0.5)$, where $u_{i}\in \{-1, 1\}$. The difference between ${\hat{U}}$ and U is that the sequence values have changed from $\{0,1\}$ to $\{-1,1\}$.

Let ${\hat{a}}_{k}$ be the autocovariance of ${\hat{U}}$ and $t_{k}$ be the autocorrelation of B, then ${\hat{a}}_{k}=t_{k}$ (Boufounos 2007). We denote $a_{k}$ as the autocorrelation of U, which is derived as follows.

$$\begin{aligned} a_{k}= & {} \sum _{i=0}^{n-k-1}u_{i}u_{i+k}\\= & {} 4\sum _{i=0}^{n-k-1}(b_{i} + m)(b_{i+u} + m) \quad (\text {where}\quad m=\mu -0.5)\\= & {} 4\sum _{i=0}^{n-k-1}(b_{i}b_{i+k}+mb_{i}+mb_{i+k}+m^2)\\= & {} 4\sum _{i=0}^{n-k-1}b_{i}b_{i+k} + 4\sum _{i=0}^{n-k-1}(m^2+mb_{i}+mb_{i+k})\\= & {} 4\sum _{i=0}^{n-k-1}b_{i}b_{i+k} + 4\left( \sum _{i=0}^{n-k-1}m^2 +m\sum _{i=0}^{n-k-1}(b_{i}+b_{i+k})\right) \\= & {} 4t_{k}+ 4\left( (n-k)m^2 +m\sum _{i=0}^{n-k-1}(b_{i}+b_{i+k})\right) \\= & {} 4{\hat{a}}_{k}+ 4m\sum _{i=0}^{n-k-1}({\hat{u}}_{i} + {\hat{u}}_{i+k} + 3\mu - 0.5) \\&\quad \left( \text {where} \sum _{i=0}^{n-k-1}b_{i}b_{i+k} = {\hat{a}}_{k}\right) . \end{aligned}$$

As mentioned in the original paper, m becomes 0 with the assumption that the sequence is balanced with equal number of zeros and ones for optimal autocorrelation properties.

1.2 Appendix 2: Example Sequences

See Table 2

Table 2 Table of the proposed fluttering patterns