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Joint low-rank prior and difference of Gaussian filter for magnetic resonance image denoising

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Abstract

The low-rank matrix approximation (LRMA) is an efficient image denoising method to reduce additive Gaussian noise. However, the existing low-rank matrix approximation does not perform well in terms of Rician noise removal for magnetic resonance imaging (MRI). To this end, we propose a novel MR image denoising approach based on the extended difference of Gaussian (DoG) filter and nonlocal low-rank regularization. In the proposed method, a novel nonlocal self-similarity evaluation with the tight frame is exploited to improve the patch matching. To remove the Rician noise and preserve the edge details, the extended DoG filter is exploited to the nonlocal low-rank regularization model. The experimental results demonstrate that the proposed method can preserve more edge and fine structures while removing noise in MR image as compared with some of the existing methods.

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Authors and Affiliations

  1. The School of Electronic and Information Engineering, South China University of Technology, Guangzhou, 510640, People’s Republic of China

    Zhen Chen, Zhiheng Zhou & Saifullah Adnan

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  1. Zhen Chen
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  2. Zhiheng Zhou
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Correspondence to Zhen Chen.

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Appendix:

Appendix:

By the sparse representation of tight frame \(\textit {\textbf {D}}_{l}^{H}\textit {\textbf {D}}_{l}=\boldsymbol {I}\), let \(\boldsymbol {L}_{\textbf {Z}_{l}}\) and \(\boldsymbol {L}_{\textbf {S}_{l}}\) be the transform domain representation of Zl and Sl, respectively, i.e., \(\boldsymbol {L}_{\textbf {Z}_{l}}=\textit {\textbf {D}}_{l}\textbf {Z}_{l}\), \(\boldsymbol {L}_{\textbf {S}_{l}}=\textit {\textbf {D}}_{l}\textbf {S}_{l}\). Then, the minimization problem of Eq. 12 can be given by:

$$ \begin{array}{ll} &\underset{\textbf{S}_{l}}{\min}\frac{1}{2}\|\textbf{Z}_{l}-\textbf{S}_{l}\|_{F}^{2}+\lambda\sum\limits_{i=1}^{r} w_{l_{i}}\sigma_{i}(\textbf{S}_{l})\\ =&\underset{\textbf{S}_{l}}{\min}\frac{1}{2}\|\textbf{\textit{D}}_{l}\textbf{Z}_{l}-\textbf{\textit{D}}_{l}\textbf{S}_{l}\|_{F}^{2}+\lambda\sum\limits_{i=1}^{r} w_{l_{i}}\sigma_{i}(\boldsymbol{L}_{\textbf{S}_{l}})\\ \overset{(a)}{=}&\underset{\boldsymbol{L}_{\textbf{S}_{l}}}{\min}\frac{1}{2}\|\boldsymbol{L}_{\textbf{Z}_{l}}-\boldsymbol{L}_{\textbf{S}_{l}}\|_{F}^{2}+\lambda\sum\limits_{i=1}^{r} w_{l_{i}}\sigma_{i}(\boldsymbol{L}_{\textbf{S}_{l}}) \end{array} $$
(33)

where (a) is from the tight frame and

$$\begin{array}{ll} \left\|\textbf{Z}_{l}-\textbf{S}_{l}\right\|_{F}^{2} &=\textit{tr}\left( (\textbf{Z}_{l}-\textbf{S}_{l})\textbf{\textit{D}}_{l}^{H}\textbf{\textit{D}}_{l}(\textbf{Z}_{l}-\textbf{S}_{l})\right)\\ &=\left\|\boldsymbol{L}_{\textbf{Z}_{l}}-\boldsymbol{L}_{\boldsymbol{L}_{\textbf{S}_{l}}}\right\|_{F}^{2}. \end{array}$$

Let the SVD of \(\boldsymbol {L}_{\textbf {Z}_{l}}\) be \(\boldsymbol {L}_{\textbf {Z}_{l}}=\textbf {U}_{l}\boldsymbol {\varSigma }_{\textbf {Z}_{l}}\textbf {V}_{l}^{H}\) and define the matrix \(\boldsymbol {\varPhi }_{l}\triangleq \textbf {U}_{l}^{H}\boldsymbol {L}_{\textbf {S}_{l}}\textbf {V}_{l}\), then the objective function in Eq. 33 can be rewritten as:

$$ \begin{array}{ll} &\frac{1}{2}\left\|\boldsymbol{L}_{\textbf{Z}_{l}}-\boldsymbol{L}_{\textbf{S}_{l}}\right\|_{F}^{2}+\lambda\sum\limits_{i=1}^{r} w_{l_{i}}\sigma_{i}(\boldsymbol{\varPhi}_{l})\\ =& \frac{1}{2}\left\|\textbf{U}_{l}(\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{\varPhi}_{l})\textbf{V}_{l}^{H}\right\|_{F}^{2}+\lambda\sum\limits_{i=1}^{r} w_{l_{i}}\sigma_{i}(\boldsymbol{\varPhi}_{l})\\ =& \frac{1}{2}\left\|\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{\varPhi}_{l}\right\|_{F}^{2}+\lambda\sum\limits_{i=1}^{r} w_{l_{i}}\sigma_{i}(\boldsymbol{\varPhi}_{l})\\ \overset{(a)}{=}& \frac{1}{2}\left( \text{tr}(\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}^{2})+\text{tr}(\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}^{2})\right)+\text{tr}(\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}\textbf{U}_{\boldsymbol{\varPhi}_{l}}\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\textbf{V}_{\boldsymbol{\varPhi}_{l}}^{T})\\ &+\lambda\sum\limits_{i=1}^{r} w_{l_{i}}\sigma_{i}(\boldsymbol{\varPhi}_{l})\\ \overset{(b)}{\leq}&\frac{1}{2}\left( \text{tr}(\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}^{2})+\text{tr}(\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}^{2})\right)+\sum\limits_{i=1}^{r}\sigma_{i}(\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}})\sigma_{i}(\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}})\\ &+\lambda\sum\limits_{i=1}^{r} w_{l_{i}}\sigma_{i}(\boldsymbol{\varPhi}_{l}) \end{array} $$
(34)

where (a) is from the SVD of the Φl that \(\boldsymbol {\varPhi }_{l}=\textbf {U}_{\boldsymbol {\varPhi }_{l}}\boldsymbol {\varSigma }_{\boldsymbol {\varPhi }_{l}}\textbf {V}_{\boldsymbol {\varPhi }_{l}}^{T}\) and (b) follows from the fact that

$$ \begin{array}{ll} \text{tr}(\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}\textbf{U}_{\boldsymbol{\varPhi}_{l}}\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\textbf{V}_{\boldsymbol{\varPhi}_{l}}^{T})&\leq\sum\limits_{i=1}^{r}\sigma_{i}(\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}})\sigma_{i}(\textbf{U}_{\boldsymbol{\varPhi}_{l}}\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\textbf{V}_{\boldsymbol{\varPhi}_{l}}^{T})\\ &\leq\sum\limits_{i=1}^{r}\sigma_{i}(\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}})\sigma_{i}(\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}) \end{array} $$
(35)

and σi(⋅) denotes the i-th largest singular value of a matrix.

Equation 35 holds if and only if \(\boldsymbol {\varPhi }_{l}=\boldsymbol {\varSigma }_{\boldsymbol {\varPhi }_{l}}\). Therefore, the rank minimization problem (33) is equivalent to the following minimization:

$$ \begin{array}{ll} \underset{\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}}{\min}\frac{1}{2}\left\|\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\right\|_{F}^{2}+\lambda\left\|\boldsymbol{W}_{l}\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\right\|_{*} \end{array} $$
(36)

where \(\boldsymbol {W}_{l}=\text {diag}\{w_{l_{1}},w_{l_{2}},...,w_{l_{r}}\}\) is a diagonal matrix.

Next, we will prove the minimization \(\frac {1}{2}\|\boldsymbol {\varSigma }_{_{\textbf {Z}_{l}}}-\boldsymbol {\varSigma }_{\boldsymbol {\varPhi }_{l}}\|_{F}^{2}+\|\boldsymbol {W}_{l}\boldsymbol {\varSigma }_{\boldsymbol {\varPhi }_{l}}\|_{*}\) in Eq. 36 is equivalent to

$$ \begin{array}{ll} \underset{\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}}{\min}\frac{1}{2}\left\|\boldsymbol{W}_{l}\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{W}_{l}\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\right\|_{F}^{2}+\lambda\|\boldsymbol{W}_{l}\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\|_{*} \end{array} $$
(37)

By the property of matrix norm, we have:

$$ \begin{array}{ll} &\frac{1}{\left\|\boldsymbol{W}_{l}^{-1}\right\|_{F}^{2}}\left( \left\|\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\right\|_{F}^{2}+\lambda\left\|\boldsymbol{W}_{l}\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\right\|_{*}\right)\\ \leq& \frac{1}{\left\|\boldsymbol{W}_{l}^{-1}\right\|_{F}^{2}}\left( \left\|\boldsymbol{W}_{l}^{-1}\right\|_{F}^{2}\left\|\boldsymbol{W}_{l}(\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}})\right\|_{F}^{2}+\lambda\|\boldsymbol{W}_{l}\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\|_{*}\right)\\ =& \left\|\boldsymbol{W}_{l}(\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}})\right\|_{F}^{2}+\frac{\lambda}{\|\boldsymbol{W}_{l}^{-1}\|_{F}^{2}}\|\boldsymbol{W}_{l}\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\|_{*} \end{array} $$
(38)

Then, it easily follows that

$$ \begin{array}{ll} &\left\|\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\right\|_{F}^{2}+\lambda\left\|\boldsymbol{W}_{l}\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\right\|_{*}\\ \leq&\left\|\boldsymbol{W}_{l}^{-1}\right\|_{F}^{2}\left\|\boldsymbol{W}_{l}(\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}})\right\|_{F}^{2}+\lambda\left\|\boldsymbol{W}_{l}\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\right\|_{*} \end{array} $$
(39)

On the other hand,

$$ \begin{array}{ll} &\frac{1}{\left\|\boldsymbol{W}_{l}\right\|_{F}^{2}}\left( \left\|\boldsymbol{W}_{l}(\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}})\right\|_{F}^{2}+\lambda\left\|\boldsymbol{W}_{l}\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\right\|_{*}\right)\\ \leq& \frac{1}{\left\|\boldsymbol{W}_{l}\right\|_{F}^{2}}\left( \left\|\boldsymbol{W}_{l}\right\|_{F}^{2}\left\|\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\right\|_{F}^{2}+\lambda\|\boldsymbol{W}_{l}\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\|_{*}\right)\\ =& \left\|\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\right\|_{F}^{2}+\frac{\lambda}{\|\boldsymbol{W}_{l}\|_{F}^{2}}\|\boldsymbol{W}_{l}\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\|_{*} \end{array} $$
(40)

It follows that

$$ \begin{array}{ll} &\left\|\boldsymbol{W}_{l}(\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}})\right\|_{F}^{2}+\lambda\left\|\boldsymbol{W}_{l}\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\right\|_{*}\\ \leq&\|\boldsymbol{W}_{l}\|_{F}^{2}\left\|\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\right\|_{F}^{2}+\lambda\|\boldsymbol{W}_{l}\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\|_{*} \end{array} $$
(41)

Suppose Ω1 and Ω2 are the solutions of the optimization problems (36) and (37), respectively. By Eq. 39, \(\forall {\boldsymbol {\varSigma }^{\prime }}_{\boldsymbol {\varPhi }_{l}}\in {\varOmega }_{1}\), for any \(\boldsymbol {\varSigma }_{\boldsymbol {\varPhi }_{l}}\in {\varOmega }_{2}\), the following relationship can be easily obtained:

$$ \begin{array}{ll} &\left\|\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\right\|_{F}^{2}+\lambda\left\|\boldsymbol{W}_{l}\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\right\|_{*}\\ \leq&\left\|\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{\varSigma}^{\prime}_{\boldsymbol{\varPhi}_{l}}\right\|_{F}^{2}+\lambda\left\|\boldsymbol{W}_{l}\boldsymbol{\varSigma}^{\prime}_{\boldsymbol{\varPhi}_{l}}\right\|_{*}\\ \leq&\left\|\boldsymbol{W}_{l}^{-1}\right\|_{F}^{2}\left\|\boldsymbol{W}_{l}(\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{\varSigma}^{\prime}_{\boldsymbol{\varPhi}_{l}})\right\|_{F}^{2}+\lambda\left\|\boldsymbol{W}_{l}\boldsymbol{\varSigma}^{\prime}_{\boldsymbol{\varPhi}_{l}}\right\|_{*} \end{array} $$
(42)

which implies that \({\varOmega }_{2}\subseteq {\varOmega }_{1}\).

According to Eq. 41, \(\forall \boldsymbol {\varSigma }_{\boldsymbol {\varPhi }_{l}}\in {\varOmega }_{2}\), for any \({\boldsymbol {\varSigma }^{\prime }}_{\boldsymbol {\varPhi }_{l}}\in {\varOmega }_{1}\), we can derive:

$$ \begin{array}{ll} &\left\|\boldsymbol{W}_{l}(\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{\varSigma}^{\prime}_{\boldsymbol{\varPhi}_{l}})\right\|_{F}^{2}+\lambda\left\|\boldsymbol{W}_{l}\boldsymbol{\varSigma}^{\prime}_{\boldsymbol{\varPhi}_{l}}\right\|_{*}\\ \leq&\left\|\boldsymbol{W}_{l}(\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}})\right\|_{F}^{2}+\lambda\left\|\boldsymbol{W}_{l}\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\right\|_{*}\\ \leq&\|\boldsymbol{W}_{l}\|_{F}^{2}\left\|\boldsymbol{\varSigma}_{_{\textbf{Z}_{l}}}-\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\right\|_{F}^{2}+\lambda\|\boldsymbol{W}_{l}\boldsymbol{\varSigma}_{\boldsymbol{\varPhi}_{l}}\|_{*} \end{array} $$
(43)

From the above result, we have \({\varOmega }_{1}\subseteq {\varOmega }_{2}\). Furthermore, we easily obtain Ω2 = Ω1 which implies that the optimization problem (33) is equivalent to the minimization (37).

It completes the proof.

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Chen, Z., Zhou, Z. & Adnan, S. Joint low-rank prior and difference of Gaussian filter for magnetic resonance image denoising. Med Biol Eng Comput 59, 607–620 (2021). https://doi.org/10.1007/s11517-020-02312-8

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