Condensed anisotropic diffusion for speckle reducton and enhancement in ultrasonography
 Kalaivani Shanmugam^{1}Email author and
 Wahidabanu RSD^{2}
https://doi.org/10.1186/16875281201212
© Shanmugam and RSD; licensee Springer. 2012
Received: 20 July 2011
Accepted: 18 June 2012
Published: 18 July 2012
Abstract
This article proposes a technique for speckle reduction in medical ultrasound (US) imaging which preserves the point and linear features with the added advantage of energy condensation regulator. Whatever be the post processing task on US image, the image should undergo a preprocessing step called despeckling. Nowadays, though the US machines are available with builtin speckle reduction facility, they are suffered by many practical limitations such as limited dynamic range of the display, limited number of unique directions that an US beam scan follow to average an image and limited size of transducer, etc. The proposed diffusion model can be used as a visual enhancement tool for interpretation as well as a preprocessing task for further diagnosis. This method incorporates two terms: diffusion and regulator. The anisotropic diffusion preserves and enhances edges and local details. The regularization enables the correction of feature broadening distortion which is the common problem in secondorder diffusionbased methods. In this scheme, the diffusion matrix is designed using local coordinate transformation and the feature broadening correction term is derived from energy function. Performance of the proposed method has been illustrated using synthetic and real US data. Experiments indicate better speckle reduction and effective preservation of edges and local details.
Keywords
Introduction
For more than two decades, ultrasonography has been considered as one of the most powerful techniques for imaging organs and soft tissue structures in the human body. Today, it is being used at an everincreasing rate in the field of medical diagnostic technology. Ultrasonography is often preferred over other medical imaging modalities because it is noninvasive, portable, versatile, does not use ionizing radiations, and also relatively of lowcost. The images produced by commercial ultrasound (US) systems are usually optimized for visual interpretation because of its realtime usage. However, the usefulness of medical ultrasonography is degraded by signaldependent noise called ‘speckle’ which is multiplicative in nature.
Many filters have been developed to cope up with speckle, with differences lying in the assumptions about the speckle model [1]. The methods described by Lee [2], Frost et al. [3], and Kuan et al. [4, 5] are based on multiplicative model and simple logarithmic operation converts the speckle into additive noise. Filtering based on anisotropic diffusion (AD) was introduced by Perona and Malik [6] who had constituted a powerful tool for signal and image enhancement. When AD is introduced for first time an undesirable effect called “pin hole effect” may result and this is addressed by Monteil and Beghdadi [7] through optical flow technique. Later, Yu and Acton [8] have proposed a novel filtering scheme based on the filters first described by Lee and Frost. The authors find a relation between the former and the AD equation and give rise to a speckle removal filter, which they call speckle reducing anisotropic diffusion (SRAD). This filter has shown very good performance with different levels of speckle. However, SRAD tends to broaden thin linear features and point features. These features carry useful information for diagnosis and the problem need to be corrected.
To overcome the feature broadening problem, a method has been proposed by Acton [9]. This method combines the strength of SRAD and deconvolution restoration. This technique assumed that feature distortion is caused by the convolution of the point spread function of the imaging system with the underlying feature. Hence, deconvolution sharpens features, while SRAD removes the speckle. This method showed promising results on synthesized US data, although no results were reported for real data. A generic framework to find the matrixvalued counterparts of the Perona–Malik PDE with various diffusivity functions is proposed by Burgeth et al. [10].
Fourthorder partial differential equation (PDE)based despeckling method has been proposed in [11]. This can reduces the speckle and also able to keep the image edge better, but this method requires more number of iterations to converge.
Regularization methods have been used in realvalued image restoration [12, 13], as well as image reconstruction problems such as medical tomography [14, 15] to obtain improved image estimates in the face of data degradation. The simplest and the most common approach is to use quadratic functions of the unknown quantities. These methods lead to computationally straightforward optimization problems, but they suppress useful features in the resulting imagery, such as edges. Recently, considerable effort has been spent in designing alternative, nonquadratic constraints which preserve such features. Methods based on these nonquadratic constraints have successfully been used in edgepreserving regularization in image restoration [12] and computerassisted tomography [13–16].
In this article, a new method has been proposed to reduce speckle in US images by incorporating a nonquadratic regularization into nonlinear coherent diffusion to preserve and enhance edges, local details, and to correct the feature broadening distortion. The proposed model carries two terms: the first is coherent diffusion term that reduces the speckle by nonlinear coherent diffusion, which utilizes the diffusion tensor derived from coordinate transformation. The second term is called regulator, which enhance the performance of coherent diffusion as well as it enables the correction of feature broadening distortion. Therefore, our model performs simultaneous speckle reduction, structure enhancement, and feature broadening correction with minimum computational cost.
Background of diffusion
The basic idea in the use of PDEs in image processing is to deform an image, a curve, or a surface in a PDE framework and to approach the expected result as a solution to this equation.
Let I:Ω → ℜ be a scalarvalued image (gray level image) with Ω ⊂ ℜ^{ p }. Gradient of the image characterizes the difference in gray value. In Biomedical imaging, besides noise also edges result in a large gradient at fine scales. The direct approach to reduce variations in the image I would be to reduce the gradient of the image globally.
where ∂Ω is boundary of Ω and n is normal vector of boundary. Euler–Lagrange equation makes the link between PDEs evolution and gradient descent for continual minimization.
This equation appears in many physical transport processes. In the context of heat transfer, it is called heat equation. In image processing, we may identify the concentration with the grey value at a certain location. If the diffusion tensor is constant over the whole image domain, one speaks of homogeneous diffusion whereas a spacedependent filtering is called inhomogeneous. Often the diffusion tensor is a function of the differential structure of the evolving image itself. Such a feedback leads to nonlinear diffusion filters. Diffusion which does not depend on the evolving image is called linear. Sometimes the computer vision literature deviates from the preceding notations, i.e., the homogeneous filtering is named isotropic and inhomogeneous blurring is called anisotropic, even if it uses a scalarvalued diffusivity instead of a diffusion tensor.
Anisotropic diffusion
With initial condition: $I(x,y,0)={I}_{0},I(x,y,0);{\Re}^{2}={\Re}^{+}$ is an image in the continuous domain, where (x, y) specifies the spatial position; t is an artificial time parameter; с is the diffusion constant, and ∇I is the image gradient. The с value is suggested to provide backward diffusion around intensity transitions and forward diffusion in smooth areas in favor of edge sharpening and noise removal. Edges and local details are the most interesting parts in diagnostic imaging for clinicians. Therefore, enhancement and preservation of edges and local details on denoising are very important. In Equation (9), с is a scalar function and ∇I serves only as an edge detector rather than providing smoothing.
Nonlinear coherent diffusion
where the eigen vectors ω_{1} and ω_{2} represent the directions of maximum and minimum variations and the eigen values μ_{1} and μ_{2} correspond to the strength of these variations, respectively. However, the diffusion tensor used in the nonlinear coherent diffusion model was actually depending on local statistics which are isotropic in nature, and also on the tensor provided by Gaussian smoothed image which may not effectively suppress the spatially correlated speckle noise.
Proposed technique
Condensed anisotropic diffusion
For many years, image regularization with discontinuities (edges) preservation has been studied in the computer vision community. Image regularization with PDE is again based on a measure of local parameter variations. In Equation (14), the first term is regularization term α coupled with data attachment or fidelity term (I_{i 0} − I_{ i }).
$U=({u}_{1}\text{,}\phantom{\rule{0.12em}{0ex}}{u}_{2}\text{,}\dots ,{u}_{n})$ is the n × n orthogonal matrix of the unit eigen vector columns u_{ k }, forming an orthonormal vector basis. $A=\text{diag}({\lambda}_{1}\text{,}\phantom{\rule{0.12em}{0ex}}{\lambda}_{2}\text{,}\dots ,{\lambda}_{n})$ is the corresponding diagonal matrix of the positive eigen vectors. The spectral decomposition separates the orientation features and diffusivity features A of tensor D.
In the constant regions, $\phantom{\rule{0.25em}{0ex}}{\mu}_{1}\approx {\mu}_{2}\approx 0\text{and}{\lambda}_{1}\approx {\lambda}_{2}\approx \alpha $, which gives D ≈ αI_{ d } where, I_{ d } is identity matrix. Thus, in the constant (homogeneous) region, there is no preferred diffusion direction and the diffusion tensor is isotropic. For image contours, ${\mu}_{1}>>{\mu}_{2}>>0\text{and}{\lambda}_{2}>{\lambda}_{1}>0$, diffusion tensor is anisotropic and mainly directed by the tangent vector of the image.
The proposed CAD model is composed of two components: the nonlinear coherent diffusion component and the energy condensation component. The former accounts for speckle removal and the latter reduces the broadening distortion of point and linear features. According to the number of scatterers per resolution cell, the nature of speckle pattern is classified into three categories: Fully formed speckle (FFS) pattern [20], nonrandomly distributed with long range order [21, 22], and nonrandomly distributed with short range order [23]. The region corresponds to FFS carries less tissue information, i.e., small gradient variations and the diffusion must become isotropic along all directions, i.e., λ_{1} ≈ λ_{2}. This condition can be accomplished by setting the local coherence measured by μ_{1} − μ_{2} close to zero. On the other hand, the areas of edges and local details corresponding to structured tissue carries rich information about the imaged texture, i.e., big gradient variations. Therefore, the AD is needed in domains of edges and local details, which diffuse along the tangent direction of edges and not across the edges. In Equation (21), λ_{1} is related to big gradient variation through Tukey’s biweight robust estimator [18] and $({\lambda}_{1}{\lambda}_{2})>{s}^{2}$ is related to fully structured region and diffusion occurs only in contour direction that is along t. The stopping level s^{2} can be set manually.
To emphasize thin linear and point features in US image, which bear useful information for diagnosis, an energy condensation component is included in the proposed model. In Equation (16), I_{ c } is threshold value, which is set as mean of the image function I(x,y). The weight factor β is positive and it determines the amount of speckle smoothing, point and linear feature preservation. With γ < < 1, the proposed condenser performs the following operation: First, the bright regions correspond to I ≥ I_{ c } gets fat during the diffusion process and increases the total energy rapidly. Second, the majorities of darker regions corresponding to I < I_{ c }, undergo NCD as ${\left(\frac{I}{{I}_{c}}\right)}^{\gamma}\to 0$. Thus, the condenser prevents the fattening of bright and linear structures without affecting the diffusion performed by the first term in Equation (16).
In the implementation of CAD model $\alpha =1\text{,}\phantom{\rule{0.12em}{0ex}}s=70\text{,}\phantom{\rule{0.12em}{0ex}}\beta =0.05\text{,}\phantom{\rule{0.12em}{0ex}}\gamma =0.75\text{,}\phantom{\rule{0.12em}{0ex}}{I}_{c}=\u3008I\u3009$ are chosen. After the iteration, the energy of the updated I is rescaled by a factor of $\frac{\u3008{I}_{0}\u3009}{\u3008{I}_{0}\u3009}$ , where 〈〉 is mean value. Thus, the processed image has same energy as the input I_{0}.
Discretization scheme
When β = 0, the image region undergoes nonlinear coherent diffusion. The β value should be chosen such that it prevents the fattening of bright structures without affecting normal non linear coherent diffusion in dominant image regions. The technique in [20] can be used to find the value of β.
where M, N are number of columns and rows in the processed image. By setting a threshold for MAE value, the diffusion process can be stopped. This threshold value can be adjusted by clinicians according to the purpose of speckle reduction. When the despeckling method is used as a visual aid to improve the interpretation, a small diffusion time is enough to remove the speckle. On the other hand, if the method is applied as a preprocessing step, a longer diffusion time can be adopted.
Proposed algorithm

Step 1: For each point (x, y) belongs to 2D space of all real numbers $(x\text{,}\phantom{\rule{0.12em}{0ex}}y)?{R}^{2}$, calculate the gradient in x and y directions and estimate the absolute gradient magnitude $\left?I\right=\sqrt{{I}_{x}^{2}+{I}_{y}^{2}}$ for local window of size w × w.

Step 2: Evaluate the diffusivity from Equation (20) and the principal components from Equations (21) and (11).

Step 3: Calculate the median of I for each coordinate.

Step 4: Solve the diffusion equation in (27) to update ${I}_{i,j}^{n+1}$ from ${I}_{i,j}^{n}$ and the calculated matrices at step n using the semi implicit scheme.

Step 5: Complete all the pixels in the image and check for stopping criteria as in (28) as a function of n. Loop until the stopping a criterion is satisfied for time step t = 0.25.
Experiments and results
The performance of the proposed method is evaluated using artificial image, simulated phantom, and real US image. In each study, the performance of the proposed CAD is compared with Perona and Malik diffusion (PM), adaptive weighted median filter (AWMF) [28], SRAD [8], nonlinear coherent diffusion (NCD) [18], median boosted anisotropic diffusion (MBAD) [29], and Laplacian pyramidbased nonlinear diffusion (LPND) [27].
where N and N_{ideal} are the numbers of detected and original edge pixels, respectively; d_{ i } is the Euclidean distance between the i th is a constant typically set to 1/9. Dynamic range of detected edge pixel and the nearest original edge pixel; λ FOM is based on all edges being found, all being placed in the correct location and no false alarms. The value is between the processed image and the ideal image. We used the canny edge detector [32] to find the edge in all processed results.
where σ_{ g }^{2}, σ_{ e }^{2} are the variances of the noise free reference image, the error between the original and denoised image, respectively.
where the standard deviation ${\sigma}_{x}={\left(\frac{1}{N1}\sum _{i=1}^{N}{({x}_{i}{\mu}_{x})}^{2}\right)}^{1/2}$ and the mean intensity${\mu}_{x}=\frac{1}{N}\sum _{i=1}^{N}{x}_{i}$, covariance ${\sigma}_{\mathit{xy}}=\frac{1}{N1}\sum _{i=1}^{N}({x}_{i}{\mu}_{x})({y}_{i}{\mu}_{y})$ are calculated using local statistics within a total of N windows. Constants C_{1}, C_{2} < < 1 to ensure stability and N is chosen as 32. The SSIM has values in the 0 to 1 range, with unity representing structurally identical images. The SSIM values are calculated only for simulated images for which the original is available for comparison.
where Ω is image region, N is pixel no. in the region. For good diffusion model, the homogeneous region in the image exhibits less contrast after diffusion than compared to the original one.
Execution time of compared algorithms
Number of iterations  Execution time (s)  

PM  300  73.65 
AWMF  –  62.95 
SRAD  300  12.08 
NCD  300  18.06 
MBAD  300  26.82 
LPND  300  20.68 
CAD  300  10.34 
Performance measures
Method  Simulated B mode image  Field II simulated foetus image  Real US image  

SNR  MSE  FOM  SNR  MSE  FOM  SNR  FOM  
Noisy  15.867  100.56  0.0969  25.538  121.69  0.0907  23.718  0.0976 
PM  19.486  88.64  0.1092  28.937  84.85  0.1762  25.645  0.1792 
AWMF  20.097  92.63  0.2206  29.427  91.02  0.2009  27.097  0.2461 
SRAD  24.468  51.84  0.3325  32.618  58.69  0.3786  29.678  0.3357 
NCD  28.093  76.92  0.3768  34.936  78.09  0.3868  31.493  0.3668 
MBAD  32.386  58.89  0.3976  35.227  62.56  0.3902  32.386  0.4462 
LPND  36.753  45.63  0.4186  38. 538  58.63  0.4409  37.789  0.4798 
CAD  41.923  16.246  0.5032  44.419  19.56  0.5156  44.087  0.5332 
SSIM value for the compared algorithms
Method  SSIM value  

Simulated B mode image  Field II simulated foetus image  
Noisy  0.3512  0.2289 
PM  0.6584  0.5709 
AWMF  0.6987  0.5987 
SRAD  0.7529  0.7297 
NCD  0.8028  0.7832 
MBAD  0.8163  0.8652 
LPND  0.8209  0.8704 
CAD  0.9567  0.9643 
CNR value for the compared algorithms
Simulated B mode image  Field II simulated foetus image  Real US image  

Region 1  Region 2  Region 1  Region 2  Region 1  Region 2  
Original (Noisy)  0.7862  0.7382  0.6489  0.6821  0.5656  0.5365 
PM  0.5289  0.4987  0.5332  0.5903  0.4087  0.4008 
AWMF  0.6032  0.5037  0.5123  0.5952  0.4239  0.4107 
SRAD  0.3185  0.3065  0.4165  0.4360  0.3543  0.3365 
NCD  0.2568  0.2167  0.3085  0.2987  0.2987  0.2754 
MBAD  0.2367  0.1754  0.2234  0.2145  0.1967  0.1965 
LPND  0.0938  0.0838  0.1045  0.1467  0.1245  0.1376 
CAD  0.0023  0.0021  0.0043  0.0035  0.0024  0.0032 
Conclusions
In this article, we propose a new diffusion model called CAD that reduces speckle, preserves information carrying features and also avoids blocking effects, point, and linear feature broadening problems. The new CAD model carries two terms: one is coherent diffusion term for speckle reduction and for structured region, organ surface preservation. The second term is a regulator term that condenses the diffusion and emphasizes thin linear and point features. In this scheme, the diffusion matrix is designed using local coordinate transformation and the feature broadening correction term is derived from energy function. The median filter is used as a smoothing operator. In CAD, the structured tissues which carry rich of information undergo AD and the speckle pattern undergo isotropic diffusion; this flow can be controlled by setting the local coherence value close to zero. The energy condensation component is included to emphasize the information carrying point/linear features, which controls the feature fattening, effectively for bright regions. In the implementation, we have used. The MAE value is set to 0.1 for artificial and simulated data and 0.2 for real US image. The simulation takes ≈ 300 iterations to converge to a stationary solution. Thus, the proposed method can be implemented practically to enhance the visual interpretation ability of radiologist with minimum cost and this method can also be used as a pre processing tool for many image processing task such as segmentation, feature extraction, etc.
Declarations
Acknowledgement
The author would like to thank Dr. Senthilvelmurugan, Radiologist from Kauvery Medical Centre Hospital, Trichirappalli, Tamilnadu and Dr. G. Jayakumar from Sarani clinic, Pudansandhai, Tamilnadu for their support and thoughtful comments.
Authors’ Affiliations
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