Fuzzy inference rule based image despeckling using adaptive maximum likelihood estimation

Abstract

Despite multitude of research ponders on despeckling ultrasound images, contriving an efficient despeckling method still exist as an open challenge. The presence of speckle noise in ultrasound images complicates the accuracy of disease diagnosis. The classical Rayleigh Maximum Likelihood Estimation (RMLE) based despeckling filter was exclusively proposed to despeckle ultrasound images. It provides proper tradeoff between speckle suppression and edge preservation by discriminating the image region as edge and background correspondingly. Many despeckling filters utilize various statistical measures which remains uncertain in order to distinguish between image edge region and background region. This proposed filter harness the Fuzzy inference rules based image connectivity measure to avoid ambiguity in image region discrimination and adapts an appropriate tuning parameter to modify classical RMLE despeckling. The proposed method involves three steps in which the first step uses Fuzzy inference rules to categorize the type of image area as edge and background. Second step involves recursive optimal selection of appropriate filter tuning parameter by using adaptive technique. Third step involves estimation of noise-free pixels by using RMLE formulation. Quantitative evaluation was made by considering various performance metrics to compare the proposed filter and existing filters. The obtained result demonstrates that the proposed filter predominantly preserves the edges structures and clinical features.

Keywords

Fuzzy connectedness despeckling maximum likelihood estimator Rayleigh distribution spatial tuning

1 Introduction

Ultrasonography has been widely used in clinical radiology for observing the health status of all age groups. The diverse merits of ultrasound modalities are non-invasive imaging procedure with the use of safe non-ionizing imaging radiations, cost effectiveness, feature of providing pathologically significant diagnostic information and much more. The major drawback is the presence of inherent speckle noise. Speckle noise pattern randomly intrudes along with the ultrasound image through phase sensitivity of the transducer in the form of interference. Interference basically occurs owing to the coherent summation of back-scattered echoes emerging from biological tissues [1]. It greatly obscures the robust application of computerized image processing algorithms in ultra-sound images. Hence, there exists a demand for an efficient pre-processing step to unambiguously make clinical diagnosis from ultrasound images. The prime objective of an efficient despeckling filter is to have proper trade-off between speckle suppression and edge preservation of clinically significant structures.

Literature strongly supports many number of despeckling algorithms applicable in both spatial and frequency domain. Mainly spatial domain filters were categorized into two types, namely filters based on statistical methods and filters based on speckle noise modeling. The Lee filter is a statistical linear model built to remove multiplicative speckle noise from a digital image [2]. It utilizes a significant local statistical feature named Equivalent Number of Looks (ENL) to distinguish between heterogeneous edge regions from smooth regions and also to appropriately control the smoothing process. Frost [3] proposed a statistically adaptive weighted mean filter with exponentially shaped kernel. The primary impression about this filter is that it achieves balance between mean filtering and high pass filtering while progressing from heterogeneous region to homogenous area. Bayesian Non-Local means speckle filter proposed in [4], intends to replace local comparison pixels with non-local contrast patches. This method is formulated with a new adaptive distance measure to remove speckle noise based on Bayesian framework and this filter strives to preserve the edge features. The simple and most widely used Median filters [5] and generalizations made to the robust median filters [6] are used to remove impulsive noise efficiently. Topological median filters [6] use the notion of Fuzzy connectedness and Binary connectedness based relations to compute the topological median of light and dark connected pixel intensities. Fuzzy Logic and Fuzzy inference based despeckling filters have been adapted mainly to remove impulsive noise [7, 8] from pictures. Fuzzy logic based Anisotropic Diffusion filter used fuzzy reasoning methodology to modify the classical diffusion algorithm by dynamically adapting the coefficient of variation[9, 10].

Maximum Likelihood Estimation technique is robust with diverse merits and has been widely used [11 –14]. Tuncer and Kenneth [15] proposed a Rayleigh Maximum likelihood despeckling filter exclusively for speckle suppression in ultrasound images. Generally despeckling filters identify the difference between the edge and background region of the image by calculating variance measure of neighborhood pixel intensity, which is a confusing parameter and remains uncertain in case of noise and edgy region. This property of the filter to discriminate edge and background region is essential to manipulate the balance in imparting feature enhancement and speckle reduction. Our previous work [16] describes the modification made to enhance the classical RMLE despeckling filter. This paper describes the extension of [16] using Fuzzy connectedness based Inference rules and recursive adaptation of tuning parameter to modify the classic RMLE filter.

2 Mathematical and theoretical background

According to statistical analysis of speckle noise [17], Rayleigh probability distribution function is optimally used to model the amplitude of fully developed speckle noise in ultrasound images. Independently and identically distributed Rayleigh probability density function is given by equation (1), $p_{η} (x, σ_{η}) = \frac{x}{σ_{η}^{2}} \exp (\frac{- x^{2}}{2 σ_{η}^{2}})$ (1)

Let x represents quadrature components of ultrasound waves randomly back-scattered from ultrasound imaging modality. $σ_{η}^{2}$ represents the shape parameter of Rayleigh distribution. The generalized mathematical model of multiplicative speckle noise in ultrasound image is expressed by, $N (u, v) = O (u, v) * η (u, v) \forall u, v$ (2)

For u = 1, 2, 3 . . . X and v = 1, 2, 3 . . . Y. In equation (2), N (u, v) represents the acquired noisy ultrasound image, O (u, v) represents the original image that is free of noise component, η (u, v) represents the multiplicative speckled component that corrupts the original image.

2.1 Rayleigh maximum likelihood estimation

The proposed method utilizes the potent RMLE approach [15] to estimate the noise free image intensity $\hat{O} (u, v)$ from the speckled noisy image N (u, v). Let ω be a set of pixel intensities inside a kernel moving over entire image of O (u, v), where u and v span the pixel intensities of the scene. RMLE used for estimating noise free pixel intensity from a Rayleigh distributed statistical model is given by, $\hat{O} (u, v) = \sqrt{\frac{1}{2 ∥ ω ∥ σ_{η}^{2}} \sum N^{2} (u, v)}$ (3)

In the above equation, ω represents 3x3 kernel neighborhood, ∥ .∥ denotes the cardinality, $\hat{O} (u, v)$ represents the despeckled image and $2 σ_{η}^{2}$ represents the tuning parameter which in future will be assigned as α. This RMLE despeckling approach adaptively switches between two different tuning parameters.

2.2 Concept of fuzzy connectedness

The notion of Fuzzy Connectedness describes the hanging togetherness endowed among the pixel intensities in a digital image [18]. This property can be used to discriminate object from the background unambiguously. The pixel values in a grayscale image ranges from 0 to 255. Hence the process of fuzzification converts the coordinate pixel intensities defined by (u, v) space into fuzzy spels defined by (m, n) space ranging between 0 to 1. Those fuzzy spels μ_F (m, n) is obtained by the relationship given by $μ_{F} (m, n) = N (u, v) / 255$ (4)

It is essential to consider two significant fuzzy relations namely fuzzy spel adjacency and fuzzy spel affinity. Fuzzy adjacency relation between any two selected spel increases with decrease in the distance between them [19]. We make use of generalized geometric distance measure using Murkowski distance for computing fuzzy spel adjacency. The fuzzy spel adjacency is represented by the following equation with a pixel position distance of l and a constant value C₁. $μ_{α} (m, n) = \frac{1}{1 + C_{1} \sqrt{\sum_{i = 1}^{l} (p_{i} - q_{i})^{2}}}$ (5)

Fuzzy spel affinity relation μ_k (m, n) is used to investigate the similarity and homogeneity measure among the spels. It can be computed for spels only having high score of fuzzy adjacent values with a constant value C₂. $μ_{k} (m, n) = \frac{μ_{α} (m, n)}{1 + C_{2} μ_{F} (m, n)}$ (6)

Let us consider x = p₀ and y = p₈ within 3x3 kernel ω neighborhood as shown in Fig. 1 [16]. The concept of path denoted by ξ_xy existing inside the kernel between two spels say x and y is defined as $ξ_{xy} = p_{0}, p_{1}, p_{2}, . . p_{d}$ (7)

To facilitate the computation of fuzzy connectedness based parameters among spels, we define three distinct paths ξ_ω in a kernel ω namely Path1-ξ₁, Path2-ξ₂ and Path 3-ξ₃ connecting any two spels say x and y. The number of arrows denoted in Fig. 1 corresponds to distinct fuzzy adjacency and affinity relations to be computed for each path. In order to determine the degree of fuzzy connectedness between any two generic spels say x and y, it is necessary to identify all the possible paths existing between x and y in the kernel neighborhood ω. The connected path ξ_xy inside kernel ω is considered as a chain of spels 〈p₀, p₁, ..., p_d〉. Where d represents the number of pixels in path. The path strength denoted by S_ξ (ω) is an important measure and is computed by assigning lowest degree of membership value among spels in the specified path [21] and can be defined by $S_{ξ} (ω) = \min {μ_{F} (m, n)}$ (8)

Since we considered three possible paths it is necessary to measure three distinct path strength for each path namely S_ξ1, S_ξ2 and S_ξ3. The degree of fuzzy connectedness represented by μ_χ (ω) is calculated by assigning the greatest path strength among the three possible ones. $μ_{χ} (ω) = \max {S_{ξ 1} (ω), S_{ξ 2} (ω), S_{ξ 3} (ω)}$ (9)

3 Methodology

The ultrasound data set used for this experimental study includes both clinical ultrasound images and synthetic images. This experimental study was conducted with the consideration of ethical standards mentioned by Helsinki declaration. The flowchart of proposed methodology shown in Fig. 2 comprises of three prominent design steps. Step 1 involves framing fuzzy inference rules to distinguish type of image region, Step 2 involves iterative adaptive scheme to substitute filter tuning parameter and Step 3 involves noise-free pixel intensity estimation.

3.1 STEP I: Fuzzy inference

The main objective of this step is to frame fuzzy inference rules in order to categorize the image region under the process within the neighborhood as edge or background. Binaee et al. [21] proposed an image region detection scheme that uses the fuzzy similarity based local gradients methodology to determine the type of image region. The fuzzified fuzzy spels are grouped into two categories using trapezoidal membership functions (MFs). Those two groups were represented by two linguistic variables namely Weak and Strong using trapezoidal MFs illustrated in Fig. 3. The shape of MFs depends on the nature of noise inherent in medical images [21].

Table 1 lists various fuzzy relations such as Fuzzy adjacency, Fuzzy affinity, path strength and Degree of fuzzy connectedness within 3x3 kernels. All those fuzzy values are computed using equations (5) to (8). All these fuzzy relation values are essentially computed in order to apply the fuzzy IF-THEN rules summarized in Table 2 and 3. The Fuzzy IF-THEN rules are framed in two ways namely intermediate rules and final rules. Table 2 lists the intermediate fuzzy rules computed for path strength calculations and degree of connectivity calculations. Table 3 shows the final fuzzy rule framed to determine the presence of edge and background region. To understand the fuzzy rules, we consider an example with a 3x3 test kernel consisting of pixels in p₀ and p₈ positions. Both are said to be connected only if either the cumulative strength of S_ξ1 OR S_ξ2 OR S_ξ3 should be strong or the cumulative strength of S_ξ1 AND S_ξ2 AND S_ξ3 should be strong. The process of classifying edge and background region immensely depends on path strength S_ξ and degree of connectivity μ_χ (ω). This step helps to discriminate whether the test region lies between edge and smooth region in order to proceed in choosing appropriate tuning parameter.

3.2 STEP II: Adaptive filter tuning

Spatial dynamic tuning of the filter parameter is highly useful for the proposed filtering method to work with adaptive scheme. The MLE mentioned in equation (3) contains the very useful tuning parameter as $2 α σ_{η}^{2}$ . The basic notion of tuning the filter parameter is to invariably enhance the structures of the ultrasound image. The motivation behind using this adaptive tuning approach in the proposed filter is ensured upon ad-miring at the desirable detail enhancement and noise attenuation characteristics achieved during this experimental study. Figure 4 illustrates the effect of despeckling process accompanied with various tuning parameters. It shows that the mean intensity values of original image and four despeckled images using tuning parameters T1 to T4. Where T1 : α_E = 4/Π, α_S = 4/Π ; T2 : α_E = 4/Π, α_S = 1.6 ; T3 : α_E = 4/Π, α_S = 2.5 ; T4 : α_E = 4/Π, α_S = 5. This approach is similar to an optimization technique and backtracking algorithm.

The chief significance of using this tuning parameter is that, it acts as a decisive element to operate the filter in two different modes namely maximum filter mode and minimum filter mode respectively. If the fuzzy inference system identifies image region in a kernel as heterogeneous area, then the fuzzy rule is applied accordingly to operate the filter as maximum filter with edge kernel weight α_E to guarantee the feature enhancement. And if the kernel region is identified as homogeneous area, smooth kernel weight α_S is used to operate the filter as minimum filter [15].

In order to ensure easy discrimination between edge and smooth region, the proposed filter assigns threshold as $σ_{η}^{2} = \sqrt{2 / Π}$ , with this assumption authors of [15] in classical RMLE despeckling have truncated the tuning parameter as a constant value of α = 4/Π.

Initially during the estimation process of noise-free pixel intensity in the proposed method, the filter does not stick with a fixed tuning parameter. Figure 5 illustrates the comparative plot of mean intensity values of images processed with above mentioned four tuning parameters. This figure apparently highlights the use of the tuning parameter because, the original image and image processed with T2 tuning parameter has same mean intensity values. This evaluation justifies the necessity of adaptive scheme for the purpose of choosing appropriate tuning parameter. With reference to the same figure, if the optimization of filter tuning parameters selects appropriate tuning, it may also help in improving the Signal to Noise Ratio (SNR) of the filtered image. The inference generated by the first step is used to assist the second step to select the filter tuning parameter appropriate weights. The proposed filter intends to minimize the Error of despeckling process with the use of iterative adaptive tuning.

The proposed algorithm efficiently adapts the kernel weights of neighborhood to switch from one value to another and ultimately sticks with the kernel weight which yields smallest Error. The kernel weight denoted by $2 σ_{η}^{2}$ used in RMLE approach is optimized until the error between the original image and the despeckled image reaches desired stopping criteria λ.In many of the optimization problems, we consider a fitness function in order to achieve the desired optimization. Similarly in this experimental study, we formulate the error function defined by equation (10) as a factor of convergence criterion to stop adaptation of tuning parameter within the kernel neighborhood. $Err = 10 \log_{10} \frac{Var [O (u, v)]}{\frac{1}{ω} \sum [\hat{O} (u, v) - O (u, v)]^{2}} \leq λ$ (10)

3.3 STEP III: Maximum likelihood filtering

This step solely depends on filtering out the speckle noisy patterns from the corrupted image. Before performing this step, the algorithm simply switches back to manipulate the pixel intensity values instead of fuzzy spels. Hence the concept of defuzzication is not actually required in this proposed filtering approach because fuzzified spel values were used for the purpose of making inference to distinguish the type of image region. This step involves estimation of noise free pixel intensity using the MLE formulation defined by Equation (3) in Section 2.1. The efficiency of this MLE despeckling technique relies on adaptive scheme for optimal selection appropriate tuning parameter denoted by $2 σ_{η}^{2}$ in Equation (3). This process of adaptive tuning based parameter selection is recursively accomplished by combining process of step 2 and step 3.

4 Results and discussion

This experimental study involves experimentation of various despeckling techniques in both synthetic images and real time clinical images. The clinical ultrasound data set used for this experimental study was obtained from GE Logic 400 imager equipment with the specifications of curvilinear transducer producing 2 ultrasound beaming frequency around 3 to 5 MHz. To validate the performance of the proposed filter, it was compared with various filters such as Bayesian Non Local Means (BNLM) filter, Frost filter, Median filter, Lee filter and classical RMLE filter. In order to intensively report the performance measure, we denoised the series of images corrupted with different densities of speckle noise ranging from 0.01 to 0.08 levels of noise variance. Various performance metrics established to validate the competent performance of the proposed filter includes Structure similarity index (SSIM), Feature similarity index (FSIM) and Universal Quality Index (UQI).

4.1 Structure Similarity Index Measure (SSIM)

SSIM performance metric measures the similarity between original and despeckled images with values ranging between [1]. Some of the conventional performance metrics including Peak Signal to Noise Ratio and Mean Square Error have been proved to be conflicting with the perception of the human eye. SSIM assessment combines the visual impact of three terms namely luminance, contrast and structure of images and is defined by $SSIM (u, v) = f (lum (u, v), c (u, v), s (u, v))$ (11)

In the Equation(11) u and v represents 2 dimensional coordinates of image, lum (u, v) represents luminance comparison function, c (u, v) represents contrast comparison function and s (u, v) represents structure comparison function.

4.2 Feature Similarity Index Measure (FSIM)

FSIM quality metric calculates the similarity between the original image O (u, v) and despeckled image $\hat{O} (u, v)$ . FSIM calculation involves combining two computations. They are similarity map oriented with phase congruencies and similarity map oriented gradient map computed from image gradients. Similarity measure is calculated using Equation (12) and (13) $S_{PC} (x) = \frac{2 {PC}_{1} (x) . {PC}_{2} (x) + M_{1}}{{PC}_{1} (X)^{2} . {PC}_{2} (X)^{2} + M_{1}}$ (12) $S_{G} = \frac{2 G_{1} (x) . G_{2} (x) + M_{2}}{G_{1} (X)^{2} . G_{2} (X)^{2} + M_{2}}$ (13)

Phase congruencies of original and despeckled images are denoted by PC1 and PC2 respectively. G1 and G2 are Gradient Magnitude maps extracted from both of the images. M1 and M2 represent constant positive number. The combined similarity is given by, $S_{L} = [S_{PC} (x)]^{α} . [S_{G} (x)]^{β}$ (14) $FSIM = \frac{\sum_{x \in D} S_{L} (x) . {PC}_{m} (x)}{\sum_{x \in D} {PC}_{m} (x)}$ (15)

D represents image domain, β and α represents constant term for pooling the similarity values computed from phase congruency and gradient maps into a single value.

4.3 Universal Quality Index (UQI)

UQI quality metric quantifies the image distortion in terms of three combinations namely Loss of correlation, Luminance distortion and Contrast distortions respectively. This measure is independent of viewing conditions and individual observers. The dynamic range of this index lies between [– 1, 1] and the value nearing 1 indicates the good quality of despeckling technique carried out for comparison. Let O be the original image, $\hat{O} (u, v)$ be the despeckled image then the UQI metric is defined by $UQI = \frac{4 σ_{O \hat{O}} O \hat{O}}{(σ_{O}^{2} + σ_{\hat{O}}^{2}) [(O^{2} + {\hat{O}}^{2})]}$ (16)

The proposed filter performs effective despeckling when compared with other as it makes use of fuzzy inference scheme to differentiate edge and background region to adaptively choose optimal tuning parameter. Figure 6 shows the visual results of despeckling performed for synthetic image.The visual results of despeckling the clinical ultrasound image is shown in Fig. 7 and Fig. 8 respectively. Figure 7(c)to (h) shows the image processed with BNLM filter, Frost filter, Median filter, Lee Filter, RMLE filter and proposed filter. The Figure 8(c) to (h) shows the zoomed version of Fig. 7 to clearly visualize the edge preservation capability of various filters. Figure 9 shows the graphical comparative analysis of pro-posed filter and various filters. It illustrates the efficacy of the proposed filter in terms of performance metrics namely SSIM, FSIM and UQI respectively.

This performance analysis declares the robustness and superior filtering capability of the proposed filter in all versions of images corrupted with different levels of speckle noise variance.

5 Conclusion

The revolutionized developments achieved in ultrasound imaging modalities after the advancements of digital imaging technology necessitates the need for efficient despeckling techniques to improve the faithful diagnostic evaluation. To this end, we have proposed a method to improve classical RMLE despeckling with fuzzy connectedness based inference rules and adaptive tuning scheme by providing better despeckling. The main contributions involved in the proposed work as a modification to the classical RMLE based despeckling includes (i) utilization of the fuzzy connectivity analysis intending to reduce the vagueness in discriminating the edge and the background and (ii) use of optimal recursive tuning method to minimize the mean square error. The simulation results show that the proposed filter effectively preserves the edge structures in comparison with other state of the art despeckling methods. This despeckling facilitates physicians to improve diagnostic accuracy from the ultrasound images.

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