Rician Denoising Based on Correlated Local Features LMMSE Approach

Abstract

In this study we propose a novel correction scheme that filters Magnetic Resonance Images data, by using a modified Linear Minimum Mean Square Error (LMMSE) estimator which takes into account the joint information of the local features. A closed-form analytical solution for our estimator is presented and it proves to make the filtering process far simpler and faster than other estimation techniques that rely on iterative optimization scheme and require multiple data samples. An experimental validation of our correction scheme was carried out through large scale experiments using both clinical and synthetic MR images, artificially corrupted with rician noise of σ varying from 1 to 40. These noisy images were filtered using our proposed method against the classical LMMSE, the Non-Local Means filter and the Nonlocality-Reinforced Convolutional Neural Networks (NRCNN) techniques. The results show an outstanding performance of our proposed method, given the fact that from σ ≈ 12 onwards, the proposed method outperforms all other methods. Another attention-grabbing feature of our method is that its Structural Similarity does not vary sharply [0.87, 0.95] across the σ spectrum as the other three techniques, which implies that this method can work on a wider range of deteriorated images than the rest of the techniques.

Appendix: : Mathematical procedures

$$ \begin{array}{lll} B_{mse}(\hat{\theta}) & = E\Big[a^{T}(x-E(x))-(\theta-E(\theta))\Big]^{2}\\ &=E\Big[a^{T}(x-E(x))(x-E(x))^{T}a\Big]\\ &-E\Big[a^{T}(x-E(x))(\theta-E(\theta))\Big]\\ &-E\Big[(\theta-E(\theta))(x-E(x))^{T}a\Big]\\ &+ E [\Big(\theta-E(\theta)\Big)^{2}]=a^{T} C_{xx} a-a^{T} C_{x\theta} - C_{\theta x} a + C_{\theta\theta}, \end{array} $$

$$ \begin{array}{@{}rcl@{}} \begin{array}{lll} C_{AM} & = E\{(A-E\{A\})(M-E\{M\})^{T}\}\\ & = E\{(A-E\{A\})(A+n-E\{A+n\})^{T}\}\\ & = E\{(A-E\{A\})(A+n-E\{A\})^{T}\}, \text{ as } E\{n\}=0\\ & = E\{(A-\!E\{A\})(A - E\{A\})^{T}\} + E\{(A - E\{A\})n^{T}\}\\ & = C_{AA} + E\{(A-E\{A\})\}\cdot E\{n^{T}\}=C_{AA}. \end{array} \end{array} $$

$$ \begin{array}{lll} C_{MM} & = E\{(M-E\{M\})(M-E\{M\})^{T}\}\\ & = E\{(A+n-E\{A+n\})(A+n-E\{A+n\})^{T}\}\\ & = E\{(A-E\{A\}+n)(A-E\{A\}+n)^{T}\}\\ & = E\{(A-E\{A\})(A-E\{A\})^{T}+(A-E\{A\})\cdot n^{T}\\ & +n\cdot(A-E\{A\})^{T}+n\cdot n^{T}\}\\ & = C_{AA}+E\{(A-E\{A\})\}\cdot E\{n^{T}\}\\ & +E\{n\}\cdot E\{(A-E\{A\})^{T}\}+E\{n\cdot n^{T}\}\\ & = C_{AA}+{\sigma^{2}_{n}}\cdot I_{\Omega}, \end{array} $$

$$ \begin{array}{lll} C_{AA} & = E\{(A-E\{A\})(A-E\{A\})^{T}\}\\ & = E\{[X\cdot c-E\{X\cdot c\}][X\cdot c-E\{X\cdot c\}]^{T}\}\\ & = E\{[X\cdot c-X\cdot E\{c\}][X\cdot c-X\cdot E\{c\}]^{T}\}\\ & = E\{[X(c-E(c))][X(c-E(c))]^{T}\}\\ & = X\cdot E\{(c-E{c})(c-E{c})^{T}\}\cdot X^{T}\\ & = X\cdot C_{cc}\cdot X^{T}. \end{array} $$

For order 1:

$$ c_{0}=\bar u; c_{s}=\frac{\overline{s\cdot u}}{\bar{s^{2}}}; c_{t}=\frac{\overline{t\cdot u}}{\bar{t^{2}}}, $$

and then for order 2:

$$ c_{0}=\frac{(\overline{s^{2} t^{2}}^{2}-\overline{s^{4} t^{4}})\bar u+(\overline{s^{2} t^{4}}-\overline{s^{2} t^{2} t^{2}})\overline{s^{2} u}+(\overline{s^{4} t^{2}}-\overline{s^{2} t^{2} s^{2}})\overline{t^{2} u}}{(\overline {s^{4}}-\overline{s^{2} t^{2}})(2\bar{s^{2}}^{2}-\overline{s^{2} t^{2}}-\overline{s^{4}})}, $$

$$ c_{s}=\frac{\overline{s\cdot u}}{\overline{s^{2}}}; c_{t}=\frac{\overline{t\cdot u}}{\overline{t^{2}}}; c_{st}=\frac{\overline {s \cdot t\cdot u}}{\overline {s^{2} t^{2}}}; $$

$$ c_{ss}=\frac{(\overline{s^{2} t^{4}}-\overline{s^{2} t^{2} t^{2}})\bar u+(\overline{t^{2}}^{2}-\overline{t^{4}})\overline{s^{2} u}+(\overline{s^{2} t^{2}}-\overline{s^{2} t^{2}})\overline {t^{2} u}}{(\overline {s^{4}}-\overline{s^{2} t^{2}})(2\bar{s^{2}}^{2}-\overline{s^{2} t^{2}}-\overline{s^{4}})}, $$

$$ c_{tt}=\frac{(\overline{t^{2} s^{4}}-\overline{s^{2} t^{2} s^{2}})\bar u+(\overline{s^{2}}^{2}-\overline{s^{4}})\overline{t^{2} u}+(\overline{s^{2} t^{2}}-\overline {s^{2} t^{2}})\overline {s^{2} u}}{(\overline{s^{4}}-\overline{s^{2} t^{2}})(2\overline{s^{2}}^{2}-\overline{s^{2} t^{2}}-\overline{s^{4}})}. $$