Local Coherence based Fast Speckle Reducing Anisotropic Diffusion

Bo Wang, Chaowei Tan, Dong C. Liu

Introduction

Due to the nature of ultrasound imaging, speckle as a dominant noise decrease the image contrast resolution of ultrasound image. Most of the traditional anisotropic diffusion methods for speckle reduction have good effect in SR, but are very slow in speed because they use an explicit scheme to discretize the anisotropic diffusion. In the Speckle Reducing Anisotropic Diffusion (SRAD) and the Detail Preserved Anisotropic Diffusion (DPAD), the authors use the statistics of speckle to control the diffusivity at each point according to the similarity of that area with the fully developed speckle area. Although the methods are slow, they are very good in reducing speckle.

In this paper, we propose a new fast local coherence based anisotropic diffusion method for speckle reduction referred to the scheme in the Real-Time Speckle Reduction (RTAD) method. At each pixel, we use the local coherence which can be obtained by eigenvalue decomposition of structure tensor to estimate whether this pixel is in speckle region, according to the value of local coherence, the diffusivity function of our method will smooth the pixel. Moreover, we use semi-implicit Additive Operator Splitting (AOS) scheme to do discretization. So the time step size (TSS) which controls the extent of blurring can be assigned values much larger than in the explicit Scheme.

Overview of Methods

Slow Anisotropic Diffusion based Speckle Reduction Methods

• Anisotropic Diffusion of Perona and Malik

The anisotropic diffusion method proposed by Perona and Malik on continuous domain is as follows,

where ▽ is the gradient operator, div is the divergence operator, || denotes the magnitude, c(.) is the diffusivity function, and I_n is the initial image. The dicretization formation used in Perona and Malik is a explicit scheme, it is as follows,

where I^t_s denotes the gray value of a pixel which position s is in a 2D image. λis the TSS which value is from 0 to 0.25. This method needs to iterate hundreds times to get desirable smoothing results.

• Speckle Reducing Anisotropic Diffusion

Yu and Acton derived SRAD from Lee filter, and then they modified the anisotropic diffusion formulation as follows,

where I_n is the initial image, c(q) is the diffusivity function which is a function of local statistics in the image, and the equation is,

where q(.) and q₀(.) are all the ratio of gray intensity variance and mean in windows. The ratio of q₀(.) is calculated in the fully developed speckle area, and the window of q(.) is current moving window. In the fully developed speckle area, q --> q₀ and c(q) --> 1, the SRAD will process like a isotropic diffusion, and smooth the speckle noise. If current window is at edges or borders, q >> q₀ and c(q) --> 0, then the filter could have enhancing effect to the contours. SRAD uses explicit discretization scheme, and it is still slow.

Fast Speckle Reducing Anisotropic Diffusion Model

• Semi-implicit AOS Scheme for Discretizing Anisotropic Diffusion Equation

Based on the CLMC filter, Weickert used the AOS scheme to propose a fast anisotropic diffusion and integrated the following matrix-vector notation as,

where τ is the TSS, I^t+1 and I^t are the image gray value vectors of the current and next time steps respectively, obtained by stacking each column vector of the K by L image on top of another into one long column vector of length KL. A(I^t) is a tridiagonal matix, with A(I^t) = [a_ij(I^k)] and,

g(.) is the diffusivity function and N(.) is the set of the two neighbours of pixel (boundary pixels have only one neighbor). This equation could be written as the iteration explicit scheme, the 2D explicit scheme is,

where U is a KL by KL identity matrix. In order to reduce time consumption, Weickert proposed the semi-implicit AOS scheme, the 2D formulation is,

Inspired by this idea, we could develop our method.

• Local Coherence based Fast Speckle Reducing Anisotropic Diffusion

By simplifying the discretization equation of RTAD as follows,

In our method, we only use local coherence to control the diffusion, and utilize a scalar value instead of above 2×2 matrix to control diffusion. Our diffusion equation is as,

where l(.) is the local coherence function which is the squared difference of eigenvalues of multiscale structure tensor, it is as,

Our discrete equation is as,

And could convert to a matrix-vector notation and adopting AOS scheme, and comes down to the following iteration scheme,

where U is a KL by KL identity matrix,

Our method has the same formation and idea as CLMC filter. And we could use very large TSS to do anisotropic diffusion, which means that less calculation is needed compared with explicit scheme ones.

Experiment and Results

We have tested lots of ultrasound phantom and in vivo images which contains diverse shapes and sizes of structural details, the results are in the following figures.

Conclusion

• 1. To get the same result of speckle reduction, the number of iteration of our method is much less than explicit SRAD method.

  2. Anisotropic diffusion is an edge-preserving smoothing algorithm that was successfully adapted to ultrasonic speckle reduction by using local statistics to determine the extent of blurring. And the semi-implicit RTAD scheme is introduced to substantially increase the calculation speed and used a matrix to control the diffusion.

  3. We propose a scalar based diffusivity using the eigenvalues of the structure matrix that not only matches RTAD in quality but further improves speed.