Improving signal-to-noise ratio performance of compressive imaging based on spatial correlation

Abstract

In this paper, compressive imaging based on spatial correlation (CISC), which uses second-order correlation with the measurement matrix, is introduced to improve the signal-to-noise ratio performance of compressive imaging (CI). Numerical simulations and experiments are performed as well. Referred to the results, it can be seen that CISC performs much better than CI in three common noise environments. This provides the great opportunity to pave the way for real applications.

Appendix 1

In this second-order correlation function, the product of sensing matrix $\varPhi$ shown in Eq. (4) and its transpose $\varPhi^{T}$ is expressed as:

$$\begin{aligned} \varOmega = \varPhi \varPhi^{T} = \left[ {\begin{array}{*{20}c} {\text{cov} \left( {a\left( 1 \right),a\left( 1 \right)} \right)} & {\text{cov} \left( {a\left( 1 \right),a\left( 2 \right)} \right)} & \cdots & {\text{cov} \left( {a\left( 1 \right),a\left( N \right)} \right)} \\ {\text{cov} \left( {a\left( 2 \right),a\left( 1 \right)} \right)} & {\text{cov} \left( {a\left( 2 \right),a\left( 2 \right)} \right)} & \cdots & {\text{cov} \left( {a\left( 2 \right),a\left( N \right)} \right)} \\ \cdots & \cdots & \cdots & \cdots \\ {\text{cov} \left( {a\left( N \right),a\left( 1 \right)} \right)} & {\text{cov} \left( {a\left( N \right),a\left( 2 \right)} \right)} & \cdots & {\text{cov} \left( {a\left( N \right),a\left( N \right)} \right)} \\ \end{array} } \right] \times \left[ {\begin{array}{*{20}c} {\text{cov} \left( {a\left( 1 \right),a\left( 1 \right)} \right)} & {\text{cov} \left( {a\left( 2 \right),a\left( 1 \right)} \right)} & \cdots & {\text{cov} \left( {a\left( N \right),a\left( 1 \right)} \right)} \\ {\text{cov} \left( {a\left( 1 \right),a\left( 2 \right)} \right)} & {\text{cov} \left( {a\left( 2 \right),a\left( 2 \right)} \right)} & \cdots & {\text{cov} \left( {a\left( N \right),a\left( 2 \right)} \right)} \\ \cdots & \cdots & \cdots & \cdots \\ {\text{cov} \left( {a\left( 1 \right),a\left( N \right)} \right)} & {\text{cov} \left( {a\left( 2 \right),a\left( N \right)} \right)} & \cdots & {\text{cov} \left( {a\left( N \right),a\left( N \right)} \right)} \\ \end{array} } \right] \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\; = \left[ {\begin{array}{*{20}c} {\sum\nolimits_{i = 1}^{N} {\text{cov} \left( {a\left( 1 \right),a\left( i \right)} \right)\text{cov} \left( {a\left( 1 \right),a\left( i \right)} \right)} } & {\sum\nolimits_{i = 1}^{N} {\text{cov} \left( {a\left( 1 \right),a\left( i \right)} \right)\text{cov} \left( {a\left( 2 \right),a\left( i \right)} \right)} } & \cdots & {\sum\nolimits_{i = 1}^{N} {\text{cov} \left( {a\left( 1 \right),a\left( i \right)} \right)\text{cov} \left( {a\left( N \right),a\left( i \right)} \right)} } \\ {\sum\nolimits_{i = 1}^{N} {\text{cov} \left( {a\left( 2 \right),a\left( i \right)} \right)\text{cov} \left( {a\left( 1 \right),a\left( i \right)} \right)} } & {\sum\nolimits_{i = 1}^{N} {\text{cov} \left( {a\left( 2 \right),a\left( i \right)} \right)\text{cov} \left( {a\left( 2 \right),a\left( i \right)} \right)} } & \cdots & {\sum\nolimits_{i = 1}^{N} {\text{cov} \left( {a\left( 2 \right),a\left( i \right)} \right)\text{cov} \left( {a\left( N \right),a\left( i \right)} \right)} } \\ \cdots & \cdots & \cdots & \cdots \\ {\sum\nolimits_{i = 1}^{N} {\text{cov} \left( {a\left( N \right),a\left( i \right)} \right)\text{cov} \left( {a\left( 1 \right),a\left( i \right)} \right)} } & {\sum\nolimits_{i = 1}^{N} {\text{cov} \left( {a\left( N \right),a\left( i \right)} \right)\text{cov} \left( {a\left( 2 \right),a\left( i \right)} \right)} } & \cdots & {\sum\nolimits_{i = 1}^{N} {\text{cov} \left( {a\left( N \right),a\left( i \right)} \right)\text{cov} \left( {a\left( N \right),a\left( i \right)} \right)} } \\ \end{array} } \right] \hfill \\ \end{aligned}$$

(9)

In the design of random patterns, every element in the distribution is generated by a random toolbox, so that the columns of measurement matrix A are independent and satisfy the same distribution. Consequently, we have

$$\sum\limits_{i = 1}^{N} {\left( {r\left( {a\left( t \right),a\left( i \right)} \right)} \right)}^{2} = \beta$$

(10)

where $t = 1,2, \cdots ,N$,$\beta$ is a constant and $r\left( {a\left( t \right),a\left( i \right)} \right)$ is the correlation coefficient between the $t$th column and $i$th column in the measurement matrix A. Then, the elements in diagonal line of the product $\varOmega$ are expressed as:

$$\begin{aligned} \sum\limits_{i = 1}^{N} {\left( {\text{cov} \left( {a\left( t \right),a\left( i \right)} \right)} \right)}^{2} = & \sum\limits_{i = 1}^{N} {\sigma^{4} \left( {r\left( {a\left( t \right),a\left( i \right)} \right)} \right)}^{2} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; =&\, \sigma^{4} \beta \hfill \\ \end{aligned}$$

(11)

where $\sigma^{2}$ is the variance of the selected distribution.

Because of the independence of the columns in measurement matrix A, the product of $r\left( {a\left( t \right),a\left( i \right)} \right)$ and $r\left( {a\left( {t^{\prime}} \right),a\left( i \right)} \right)$ with $t \ne t^{\prime}$ satisfies uniform distribution in the open interval $\left( { - 1, + 1} \right)$, so that we can obtain

$$\sum\limits_{i = 1}^{N} {r\left( {a\left( t \right),a\left( i \right)} \right)r\left( {a\left( {t^{\prime}} \right),a\left( i \right)} \right)} \approx 0$$

(12)

Finally, the elements of the sensing matrix $\varOmega \left( {t,t^{\prime}} \right)$, $t \ne t^{\prime}$ are equal to zero and we can obtain $\varOmega \approx \sigma^{4} \beta E$ which proves that the sensing matrix $\varPhi$ is orthogonal.