Speckle Noise Removal: A Local Structure Preserving Approach

Abstract

This paper proposes a speckle noise removal approach for clinical ultrasound images by doing outlier removal and smoothening operations alternately. During the initial investigation, it was found that the log-transformed ultrasound image follows Fisher–Tippett distribution and has fixed median absolute deviation (MAD). Hence, the noise in log-transformed ultrasound images behaves like white Gaussian noise with transients or outliers. Therefore, the de-noising problem can be considered as the removal of outliers followed by smoothening. These two processes are unified in one framework by defining a Bayesian Maximum-a-Posteriori (MAP) estimation function. This function has two terms: fidelity and regularizer. The fidelity is derived using the proposed generalized Fisher–Tippett distribution, whereas a weighted total variation is used as a regularizer. A regularizer weigh scheme is introduced to preserve edges in the images. The weights are computed using echo-texture graded local-oriented structure information present in an image. To obtain tissue-specific echo-texture, fuzzy C-means clustering is deployed for grouping similar tissue echo-textures. This grouping will help to discriminate the proper boundary of the tissue. To extract the original image, the MAP function is minimized and is performed using the generalized Bregman alternate method of multipliers. Ten different existing techniques are used to compare the performance of the proposed method on both phantom and clinical ultrasound images. The proposed approach achieved a signal-to-noise ratio in the range of 5–10 and a peak signal-to-noise ratio in the range of 67–70. Structural preservation metrics like figure of merit came out to be as high as 0.8. Moreover, using the proposed approach lower signal suppression index and higher effective number of lookup values are achieved for the restored clinical ultrasound images. The proposed algorithm can provide better piecewise smoothness and high contrast in despeckled images. Along with it, the edges are seen to be well preserved. Both qualitative and quantitative analysis support the efficacy of the approach compared to state-of-the-art methods.

Appendix: Proof of Theorem 1

Let us assume that N follows Nakagami distribution given by

$$\begin{aligned} {p_N}(\eta ) = \frac{{2{m^m}}}{{\Gamma (m){\Omega ^m}}}{\eta ^{2m - 1}}{e^{\frac{m}{\Omega }{\eta ^2}}},\quad \eta > 0. \end{aligned}$$

(37)

Then applying log transformation to noise, we have the form

$$\begin{aligned} W=D\ln N. \end{aligned}$$

This can be rewritten as

$$\begin{aligned} W=\ln N^{\frac{1}{s}}, \end{aligned}$$

where $s=\frac{1}{D}$.

According to [40], if N follows Nakagami distribution, then $H=N^{\frac{1}{s}}$ will follow generalized Nakagami distribution given by

$$\begin{aligned} {p_H}(h) = \frac{{2s{m^m}}}{{\Gamma (m){\Omega ^m}}}{h^{2sm - 1}}{e^{\frac{m}{\Omega }{h^{2s}}}},\quad h > 0, \end{aligned}$$

where m, s and $\Omega$ are shape, correction factor and scaling parameters of the distribution respectively. h is the value at the $\textbf{x}$th location of H.

Thus for $W=\ln H$, the probability distribution is given by

$$\begin{aligned} {p_W}(w) = \frac{{{P_w}({W^{ - 1}})}}{{\left| {\frac{{dw}}{{dh}}} \right| }}, \end{aligned}$$

or

$$\begin{aligned} {p_W}(w)& = {} \frac{{2s{m^m}}}{{\Gamma (m){\Omega ^m}}}{\left( {{e^w}} \right) ^{2sm}}{e^{ - \frac{m}{\Omega }{{\left( {{e^w}} \right) }^{2s}}}}\\& = {} \frac{{2s{m^m}}}{{\Gamma (m)}}{e^{\left[ {\ln \left[ {{{\left( {\frac{1}{\Omega }{e^{2sw}}} \right) }^m}{e^{ - {\hspace{1.0pt}} \frac{m}{\Omega }{e^{2sw}}}}} \right] } \right] }}\\& = {} \frac{{2s{m^m}}}{{\Gamma (m)}}{e^{\left[ {\left[ {m\left( {2sw - \ln \Omega } \right) {\hspace{1.0pt}} \, - {\hspace{1.0pt}} \frac{m}{{{e^{\ln \Omega }}}}{e^{2sw}}} \right] } \right] }}\\& = {} \frac{{2s{m^m}}}{{\Gamma (m)}}{e^{\left[ {m\left( {2sw - \ln \Omega } \right) {\hspace{1.0pt}} \, - {\hspace{1.0pt}} m\left( {{e^{\left( {2sw - \ln \Omega } \right) }}} \right) } \right] .}} \end{aligned}$$

It is known from Stirling’s formulae [4] that $\Gamma (m+1)={\sqrt{2\pi m} {{\left( {\frac{m}{e}} \right) }^m}}$. Thus substituting it, we get

$$\begin{aligned} {p_W}(w)& = {} \frac{{2s{m^{m + 1}}}}{{\sqrt{2\pi m} {{\left( {\frac{m}{e}} \right) }^m}}}{e^{\left[ {m\left( {2sw - \ln \Omega } \right) {\hspace{1.0pt}} \, - {\hspace{1.0pt}} m\left( {{e^{\left( {2sw - \ln \Omega } \right) }}} \right) } \right] }}\\& = {} \frac{{2sm}}{{\sqrt{2\pi m} \left( {\frac{1}{{{e^m}}}} \right) }}{e^{\left[ {m\left( {2sw - \ln \Omega } \right) {\hspace{1.0pt}} \, - {\hspace{1.0pt}} m\left( {{e^{\left( {2sw - \ln \Omega } \right) }}} \right) } \right] }}\\& = {} \frac{{2s{e^m}}}{{\sqrt{\frac{{2\pi }}{m}} }}{e^{m\left[ {\left( {2sw - \ln \Omega } \right) {\hspace{1.0pt}} \, - \left( {{e^{\left( {2sw - \ln \Omega } \right) }}} \right) } \right] }} \end{aligned}$$

which is of the form defined in Eq. (6).

On rearranging the terms in Eq. (6),we get

$$\begin{aligned} {p_W}(w) = \frac{{2s}}{{\sqrt{\frac{{2\pi }}{m}} }}{e^{m\left[ 1+{\left( {2sw - \ln \Omega } \right) - {e^{\left( {2sw - \ln \Omega } \right) }}} \right] }}. \end{aligned}$$

Replacing the second term of the exponential distribution with the first three terms of Taylor series, we have

$$\begin{aligned} {p_W}(w) \simeq \frac{{2s}}{{\sqrt{\frac{{2\pi }}{m}} }}{e^{ - \frac{m}{2}\left[ {{{\left( {2sy - \ln \Omega } \right) }^2}} \right] }}. \end{aligned}$$

Here $\Omega$ is the power of the signal. With image intensity into consideration, the power can be defined as $2\sigma ^2$ [46]. Thus,

$$\begin{aligned} {p_W}(w)& = {} \frac{{2s}}{{\sqrt{\frac{{2\pi }}{m}} }}{e^{ - \frac{m}{2}\left[ {{{\left( {2sy - \ln 2\sigma ^2 } \right) }^2}} \right] }}\\& = {} \frac{{2s}}{{\sqrt{\frac{{2\pi }}{m}} }}{e^{ - \frac{m}{2}\left[ {\frac{{{{\left( {sy - \ln \sqrt{2} \sigma } \right) }^2}}}{{0.25}}} \right] }}\\& = {} \frac{1}{{\sqrt{\frac{{2\pi 0.25 }}{{m{s^2}}}} }}{e^{ - \frac{1}{2}\left[ {\frac{{{{\left( {y - \frac{1}{s}\ln \sqrt{2} \sigma } \right) }^2}}}{{\frac{{0.25}}{{m{s^2}}}}}} \right] .}} \end{aligned}$$

This form of the pdf is the same as the pdf of Gaussian distribution with mean $\mu _w=ln\sqrt{2}\sigma$ and variance ${\sigma _w}^2=\frac{0.25}{ms^2}$. On careful analysis of the variance, it can be written as $\sigma _w=\frac{1}{\sqrt{ms^2}}MAD$ [36] where $MAD=0.5$. Thus we conclude that the transformed noise has constant median absolute deviation independent of the underlying scattering condition.