Bayesian Sparse Reconstruction Method of Compressed Sensing in the Presence of Impulsive Noise

Abstract

The majority of existing recovery algorithms in the framework of compressed sensing are not robust to the impulsive noise. However, the impulsive noise is always present in the actual communication and signal processing system. In this paper, we propose a method named ‘Bayesian sparse reconstruction’ to recover the sparse signal from the measurement vector which is corrupted by the impulsive noise. The Bayesian sparse reconstruction method is composed of five parts, which are the preliminary detection of the location set of impulses, the impulsive noise fast relevance vector machine algorithm, the step of pruning, Bayesian impulse detection algorithm and the maximum a posteriori estimate of the sparse vector. The Bayesian sparse reconstruction method can achieve effective signal recovery in the presence of impulsive noise, depending on the mutual influence of the impulsive noise fast relevance vector machine algorithm, the step of pruning and the Bayesian impulse detection algorithm. Experimental results show that the Bayesian sparse reconstruction method is robust to the impulsive noise and effective in the additive white Gaussian noise environment.

Appendices

Appendix A

(1) Derivation of Eq. (73):

When the index i∈{1,2,…,M} is added to the preliminary candidate set F to constitute the new candidate set $\tilde{F}$, the matrix $\tilde{\boldsymbol{\vartheta}}$ can be represented as

$$ \tilde{\boldsymbol{\vartheta}} = \bar{\boldsymbol{\varPhi}} _{\tilde{F}}\bar{\boldsymbol{A}}^{ - 1}\bar{\boldsymbol{\varPhi}} _{\tilde{F}}^{\mathrm{T}} + \sigma _{o}^{2} \boldsymbol{I}. $$

(95)

The relationship between the matrix $\tilde{\boldsymbol{\vartheta}}$ and the matrix ϑ is

$$ \tilde{\boldsymbol{\vartheta}} = \left [ \arraycolsep=5pt \begin{array}{@{}cc@{}} \boldsymbol{\vartheta}& \bar{\boldsymbol{\varPhi}} _{F}\bar{\boldsymbol{A}}^{ - 1}\boldsymbol{\varphi} _{i}^{\mathrm{T}} \\[3pt] \boldsymbol{\varphi}_{i}\bar{\boldsymbol{A}}^{ - 1}\bar{\boldsymbol{\varPhi}} _{F}^{\mathrm{T}} & \boldsymbol{\varphi} _{i}\bar{\boldsymbol{A}}^{ - 1}\boldsymbol{\varphi} _{i}^{\mathrm{T}} + \sigma_{o}^{2} \end{array} \right ]. $$

(96)

Then we can apply property of the matrix inversion in a block form to the matrix $\tilde{\boldsymbol{\vartheta}}$ in Eq. (96) and the inverse of the matrix $\tilde{\vartheta}$ can also be represented in block form as

$$\tilde{\boldsymbol{\vartheta}} ^{ - 1} = \left [ \arraycolsep=5pt \begin{array}{@{}cc@{}} \boldsymbol{X}_{1} & \boldsymbol{X}_{2} \\ \boldsymbol{X}_{3} & \boldsymbol{X}_{4} \end{array} \right ] $$

where

and

$$\boldsymbol{X}_{3} = - \boldsymbol{X}_{4}\boldsymbol{ \varphi}_{i}\bar{\boldsymbol{A}}^{ - 1}\bar{\boldsymbol{ \varPhi}} _{F}^{\mathrm{T}}\boldsymbol{\vartheta} ^{ - 1}. $$

Combining Eqs. (71) and (72), we have Eq. (73) established.

(2) Derivation of Eq. (76):

The update matrix of $\tilde{\boldsymbol{\varTheta}}$ can be written as

$$ \tilde{\boldsymbol{\varTheta}} \stackrel{\Delta}{=} \boldsymbol{I} - \bar{\boldsymbol{\varPhi}} _{\tilde {F}}^{\mathrm{T}}\tilde{\boldsymbol{ \vartheta}} ^{ - 1}\bar{\boldsymbol{\varPhi}} _{\tilde{F}}\bar{ \boldsymbol{A}}^{ - 1}. $$

(97)

We apply the matrix $\tilde{\boldsymbol{\vartheta}} ^{ - 1}$in Eq. (73) to Eq. (97) and obtain

(98)

Combining Eqs. (74) and (75), we have Eq. (76) established.

(3) Derivation of Eq. (78):

In order to derive Eq. (78), we first introduce an intermediate variable which can be denoted as

$$ \boldsymbol{W}_{i} = \boldsymbol{I} + \beta_{2} \bar{\boldsymbol{\varPhi}} _{\bar{F}}^{\mathrm{T}} \bar{\boldsymbol{ \varPhi}} _{\bar{F}}\bar{\boldsymbol{A}}^{ - 1}\tilde{\boldsymbol{ \varTheta}}. $$

(99)

Applying Eq. (76) to Eq. (99) we have

(100)

In line with the matrix inversion lemma, we have Eq. (77) established.

Then the matrix $\tilde{\boldsymbol{w}}$ can be represented as

$$ \tilde{\boldsymbol{w}} = \boldsymbol{W}_{i} - \beta_{2}\boldsymbol{\varphi}_{i}^{\mathrm{T}} \boldsymbol{ \varphi}_{i}\bar{\boldsymbol{A}}^{ - 1}\tilde{\boldsymbol{ \varTheta}}. $$

(101)

Applying the matrix inversion lemma to Eq. (101) we have Eq. (78) established.

(4) Derivation of Eq. (79):

The determinant of the covariance matrix C ₁ satisfies

$$ \det(\boldsymbol{C}_{1}) = \sigma_{g}^{2L_{2}} \det(\boldsymbol{\vartheta})\det(\boldsymbol{w}). $$

(102)

In line with Eq. (96), we have

$$\det(\tilde{\boldsymbol{\vartheta}} ) = \det(\boldsymbol{\vartheta}) \bigl( \boldsymbol{\varphi}_{i}\bar{\boldsymbol{A}}^{ - 1}\boldsymbol{ \varphi}_{i}^{\mathrm{T}} + \sigma _{o}^{2} - \boldsymbol{\varphi}_{i}\bar{\boldsymbol{A}}^{ - 1}\bar{ \boldsymbol{\varPhi}} _{F}^{\mathrm{T}} \boldsymbol{ \vartheta}^{ - 1}\bar{\boldsymbol{\varPhi}} _{F}\bar{ \boldsymbol{A}}^{ - 1}\boldsymbol{\varphi}_{i}^{\mathrm{T}} \bigr) = b_{i}^{ - 1}\det(\boldsymbol{\vartheta}). $$

In line with Eq. (101), we have

$$ \det(\tilde{\boldsymbol{w}}) = \det(\boldsymbol{W}_{i}) \bigl(1 - \beta_{2}\boldsymbol{\varphi}_{i}\bar{ \boldsymbol{A}}^{ - 1}\tilde{\boldsymbol{\varTheta}} \boldsymbol{W}_{i}^{ - 1} \boldsymbol{\varphi}_{i}^{\mathrm{T}}\bigr). $$

(103)

In line with Eq. (100), we have

(104)

Combining Eq. (103) with Eq. (104), we have

Therefore,

Appendix B

(1) Derivation of Eq. (86):

When the index i∈{1,2,…,M} is added to the preliminary candidate set F to constitute the new candidate set $\tilde{F}$, the matrix $\tilde{\boldsymbol{\varXi}}$ can be represented as

$$ \tilde{\boldsymbol{\varXi}} = \bar{\boldsymbol{\varPhi}} _{\bar{\tilde{F}}}^{\mathrm{T}}\tilde{\boldsymbol{Z}}\bar{\boldsymbol{\varPhi}} _{\bar{\tilde{F}}} = \beta _{2}\bar{\boldsymbol{\varPhi}} _{\bar{\tilde{F}}}^{\mathrm{T}}\bar{\boldsymbol{\varPhi}} _{\bar{\tilde{F}}} - \beta_{2}^{2}\bar{\boldsymbol{\varPhi}} _{\bar{\tilde{F}}}^{\mathrm{T}} \bar{\boldsymbol{\varPhi}} _{\bar{\tilde{F}}}\bar{\boldsymbol{A}}^{ - 1}\tilde {\boldsymbol{\varTheta}} \tilde{\boldsymbol{w}}^{ - 1}\bar{\boldsymbol{ \varPhi}} _{\bar{\tilde{F}}}^{\mathrm{T}}\bar{\boldsymbol{\varPhi}} _{\bar{\tilde{F}}}. $$

(105)

Applying Eq. (77) to Eq. (78) we have

In line with Eq. (84), we have

$$ \tilde{\boldsymbol{w}}^{ - 1} = \boldsymbol{w}^{ - 1} + \boldsymbol{f}_{i}. $$

(106)

Moreover,

$$ \boldsymbol{\varPhi}_{\bar{\tilde{F}}}^{\mathrm{T}}\boldsymbol{ \varPhi}_{\bar{\tilde{F}}} = \boldsymbol{\varPhi}_{\bar {F}}^{\mathrm{T}} \boldsymbol{\varPhi}_{\bar{F}} - \boldsymbol{\varphi} _{i}^{\mathrm{T}} \boldsymbol{\varphi}_{i}. $$

(107)

Applying Eqs. (76), (106) and (107) to Eq. (105) we have Eq. (86) established.

(2) Derivation of Eq. (88):

The matrix $\tilde{\boldsymbol{G}}$ can be represented as

$$ \tilde{\boldsymbol{G}} = \boldsymbol{y}_{\tilde{F}}^{\mathrm{T}} \tilde{\boldsymbol{\vartheta}} ^{ - 1}\bar{\boldsymbol{\varPhi}} _{\tilde{F}}. $$

(108)

Applying Eq. (73) to Eq. (108) we have

$$\tilde{\boldsymbol{G}} = \left [ \begin{array}{c} \boldsymbol{y}_{F} \\ y_{i} \end{array} \right ]^{\mathrm{T}}\left [ \arraycolsep=5pt \begin{array}{@{}cc@{}} \boldsymbol{\vartheta}^{ - 1} + b_{i}\boldsymbol{d}_{i}\boldsymbol{d}_{i}^{\mathrm{T}} & - b_{i}\boldsymbol{d}_{i} \\ -b_{i}\boldsymbol{d}_{i}^{\mathrm{T}} & b_{i} \end{array} \right ] \left [ \begin{array}{c} \bar{\boldsymbol{\varPhi}} _{F} \\ \boldsymbol{\varphi}_{i} \end{array} \right ] = \boldsymbol{G} + \boldsymbol{g}_{i}. $$

Thus, we have Eq. (88) established.

(3) Derivation of Eq. (90):

The matrix $\tilde{\boldsymbol{R}}$ can be represented as

$$ \tilde{\boldsymbol{R}} = \boldsymbol{y}_{\bar{\tilde{F}}}^{\mathrm{T}} \tilde{\boldsymbol{Z}}\bar{\boldsymbol{\varPhi}} _{\bar{\tilde {F}}} = \beta _{2}\boldsymbol{y}_{\bar{\tilde{F}}}^{\mathrm{T}}\bar{\boldsymbol{ \varPhi}} _{\bar{\tilde{F}}} - \beta _{2}^{2} \boldsymbol{y}_{\bar{\tilde{F}}}^{\mathrm{T}}\bar{\boldsymbol{\varPhi}} _{\bar{\tilde{F}}}\bar{\boldsymbol{A}}^{ - 1}\tilde{\boldsymbol{\varTheta}} \tilde{\boldsymbol{w}}^{ - 1}\bar{\boldsymbol{\varPhi}} _{\bar{\tilde{F}}}^{\mathrm{T}}\bar{\boldsymbol{\varPhi}} _{\bar{\tilde{F}}}. $$

(109)

Moreover, we have

$$ \boldsymbol{y}_{\bar{\tilde{F}}}^{\mathrm{T}}\boldsymbol{ \varPhi}_{\bar{\tilde{F}}} = \boldsymbol{y}_{\bar{F}}^{\mathrm{T}}\boldsymbol{ \varPhi} _{\bar{F}} - y_{i}\boldsymbol{\varphi} _{i}. $$

(110)

Applying Eq. (76), (106), (107) and (110) to Eq. (109), we have Eq. (90) established.

(4) Derivation of Eqs. (91)–(93):

We define

(111)

(112)

and

$$ \tau_{3}(F) = - \boldsymbol{y}_{\bar{F}}^{\mathrm{T}} \boldsymbol{Zy}_{\bar{F}} = - \beta_{2}\boldsymbol{y}_{\bar {F}}^{\mathrm{T}} \boldsymbol{y}_{\bar{F}} + \beta _{2}^{2} \boldsymbol{y}_{\bar{F}}^{\mathrm{T}}\bar{\boldsymbol{\varPhi}} _{\bar{F}}\bar{\boldsymbol{A}}^{ - 1} \boldsymbol{\varTheta} \boldsymbol{w}^{ - 1}\boldsymbol{\varPhi} _{\bar{F}}^{\mathrm{T}}\boldsymbol{y}_{\bar{F}}. $$

(113)

Moreover, we have

$$\rho_{2}(F) = \tau_{1}(F) + \tau_{2}(F) + \tau_{3}(F). $$

In addition, we define

$$ pv_{1} = \tau_{1}(\tilde{F}) - \tau_{1}(F). $$

(114)

The quantity $\tau_{1}(\tilde{F})$ can be denoted as

$$\tau_{1}(\tilde{F}) = - \bigl(\boldsymbol{y}_{\tilde{F}}^{\mathrm{T}} \tilde{\boldsymbol{\vartheta}} ^{ - 1}\boldsymbol{y}_{\tilde{F}} + \tilde{\boldsymbol{G}}\bar{\boldsymbol{A}}^{ - 1}\tilde{\boldsymbol{ \varXi}} \bar{\boldsymbol{A}}^{ - 1}\tilde{\boldsymbol{G}}^{\mathrm{T}} \bigr). $$

The quantity $\boldsymbol{y}_{ \tilde{F}}^{\mathrm{T}}\tilde{\boldsymbol{\vartheta}} ^{ - 1}\boldsymbol{y}_{\tilde{F}}$ can be represented as

(115)

Applying Eqs. (86), (88) and (115) to Eq. (114) we have Eq. (91) established.

We define

$$pv_{2} = \tau_{2}(\tilde{F}) - \tau_{2}(F). $$

Thus, we have

$$pv_{2} = 2\tilde{\boldsymbol{R}}\bar{\boldsymbol{A}}^{ - 1} \tilde{\boldsymbol{G}}^{\mathrm{T}} - 2\boldsymbol{R}\bar{\boldsymbol{A}}^{ - 1} \boldsymbol{G}^{\mathrm{T}} = 2\bigl(\boldsymbol{r}_{i}\bar{ \boldsymbol{A}}^{ - 1}\tilde{\boldsymbol{G}}^{\mathrm{T}} + \boldsymbol{R} \bar{\boldsymbol{A}}^{ - 1}\boldsymbol{g}_{i}^{\mathrm{T}} \bigr). $$

We define

$$ pv_{3} = \tau_{3}(\tilde{F}) - \tau_{3}(F). $$

(116)

The quantity $\tau_{3}(\tilde{F})$ can be denoted as

$$\tau_{3}(\tilde{F}) = - \beta_{2}\boldsymbol{y}_{\bar{\tilde{F}}}^{\mathrm{T}} \boldsymbol{y}_{\bar{\tilde{F}}} + \beta _{2}^{2} \boldsymbol{y}_{\bar{\tilde{F}}}^{\mathrm{T}} \bar{\boldsymbol{\varPhi}}_{\bar{\tilde{F}}}\,\bar{\boldsymbol{A}}^{ -1}\tilde{\boldsymbol{\varTheta}} \tilde{\boldsymbol{w}}^{- 1}\bar{\boldsymbol{\varPhi}} _{\bar{\tilde{F}}}^{\mathrm{T}}\boldsymbol{y}_{\bar{\tilde{F}}}. $$

Moreover, we have

$$ \boldsymbol{y}_{\bar{\tilde{F}}}^{\mathrm{T}}\boldsymbol{y}_{\bar{\tilde{F}}} = \boldsymbol{y}_{\bar{F}}^{\mathrm{T}}\boldsymbol{y}_{\bar{F}} - y_{i}^{2}. $$

(117)

Applying Eqs. (76), (106), (110) and (117) to Eq. (116) we have Eq. (93) established.