Computational and Mathematical Methods in Medicine

Volume 2017 (2017), Article ID 1769834, 13 pages

https://doi.org/10.1155/2017/1769834

## Retinal Image Denoising via Bilateral Filter with a Spatial Kernel of Optimally Oriented Line Spread Function

^{1}School of Information Science and Engineering, Shandong Normal University, Jinan 250014, China^{2}Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA 19104, USA^{3}Institute of Life Sciences, Shandong Normal University, Jinan 250014, China^{4}Key Laboratory of Intelligent Information Processing, Shandong Normal University, Jinan 250014, China^{5}School of Psychology, Shandong Normal University, Jinan 250014, China

Correspondence should be addressed to Yuanjie Zheng

Received 25 August 2016; Revised 30 November 2016; Accepted 13 December 2016; Published 5 February 2017

Academic Editor: Marc Thilo Figge

Copyright © 2017 Yunlong He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Filtering belongs to the most fundamental operations of retinal image processing and for which the value of the filtered image at a given location is a function of the values in a local window centered at this location. However, preserving thin retinal vessels during the filtering process is challenging due to vessels’ small area and weak contrast compared to background, caused by the limited resolution of imaging and less blood flow in the vessel. In this paper, we present a novel retinal image denoising approach which is able to preserve the details of retinal vessels while effectively eliminating image noise. Specifically, our approach is carried out by determining an optimal spatial kernel for the bilateral filter, which is represented by a line spread function with an orientation and scale adjusted adaptively to the local vessel structure. Moreover, this approach can also be served as a preprocessing tool for improving the accuracy of the vessel detection technique. Experimental results show the superiority of our approach over state-of-the-art image denoising techniques such as the bilateral filter.

#### 1. Introduction

Ocular diseases may evoke changes of retinal vascular structures. To diagnose them, analyzing and interpreting the characteristics of blood vessels in retinal images play an important role [1, 2]. To obtain an automatic system for assessing retinal vessels, automated vessel detection from retinal images is a fundamental step and has long been an important research topic in fields of computer vision and medical image analysis.

Given a retinal image, the performance of a vessel detection technique depends on how it deals with two fundamental challenges: image denoising where the underlying goal is to suppress noise in image and vascular detection which aims to differentiate vessels from other components of the retina. Regarding image denoising, a great variety of techniques have been proposed, including the classical Gaussian-filtering based approaches [3–6] and the edge-preserving methods [7–10]. The latter are more widely used in denoising retinal images due to their potentials in preserving the vascular structure of the retinal vessels. As a representative edge-preserving technique, bilateral filter (BLF) [7] belongs to the most popularly used techniques due to its simplicity and effectiveness. It is accomplished by combining a spatial kernel and a range kernel which measure the spatial distance and intensity difference between pixels, respectively. Regarding vascular structure detection, various techniques have been investigated, including matched filter (MF) which employs the Gaussian-shape of the cross-section of the vessel [11–13], ridge detection which relies on the ridge shape of the vessel centerlines [1, 14, 15], classification which is based on a variety of image features and machine learning algorithms [16, 17], and vesselness measures that recognize tubular vascular structures using the eigenvalues of the Hessian matrix [18–20]. Among these techniques, MF [11] and Frangi’s filter (FR) [19] are two of the most widely used vessel detection methods. MF is a simple yet effective method to detect the retinal vessels by using a Gaussian-like kernel. In essence, the intensity distribution of this kernel can be mathematically expressed as an oriented line spread function (LSF) that spreads as a Gaussian function in the theory of imaging systems [21–23]. FR performs well in detecting the tube-like structure from other components by measuring the eigenvalues of Hessian matrix computed at each pixel of a retinal image.

Disregarding the good results reported in the related works, accurate detection of retinal vessels remains an open challenge. The main reason comes from the fact that retinal vessels can be ruined during image denoising process due to the introduced negative effects of blur or vessel-structure corruption. As widely known, blur occurs when classical Gaussian based filtering is performed on the image. The edge-preserving image denoising methods (e.g., the BLF) are capable of keeping crisp edges. However, most of them may fail in maintaining retinal vessels because they ignore the special tube-like structure of the retinal vessels. This can happen especially for thin vessels due to their weak contrast and small area relative to the background and the difference in their spatial distribution compared with a common crisp edge.

In this paper, we propose a novel image filtering approach specialized for denoising retinal images while preserving the vascular structures. Our approach is realized by determining a set of spatial kernels for the BLF algorithm, which are represented by an LSF with an orientation and scale adjusted adaptively to the local vessel structures. Our LSF function is similar to the functions used in MF. However, we incorporate it in BLF for image denoising, unlike the vessel detection purpose in MF. By combining the benefits of BLF and MF, our approach is not only noniterative, simple, and easy to implement, but also able to outperform the state-of-the-art image filtering methods in maintaining details of retinal vessels and removing image noise. The advantages of our approach are validated by the improved vessel detection accuracies as shown in our experimental results.

#### 2. Related Work

This section provides an overview of the current state-of-the-art techniques for both image denoising and vessel detection.

##### 2.1. Image Denoising

In recent years, a large variety of approaches have been presented for image denoising. These approaches can be basically classified into two categories: classical Gaussian-filtering based techniques which may lose the edge information recognized as the primary feature of an image [3–6] and edge-preserving based methods which attempt to preserve prominent edges while denoising [7–10]. Gaussian-filtering based techniques are commonly used in medical image analysis. However, as the weights of a Gaussian filter purity depend on the spatial distance, these approaches may lose prominent image edges and introduce blurring effect, which can cause troubles to vessel detection in retinal images. Edge-preserving methods were proposed to overcome the loss of prominent edges. For example, the anisotropic diffusion filter [8] and the weighted least squares filter [9] attempt to smooth images while preserving edges based on measuring the image gradient. The nonlocal means filter [10] computes filtered result relying on the similarity of intensity and the order of pixel in their neighborhoods. BLF is distinguished for its edge-preserving ability, for which a spatial kernel and a range kernel are combined and the output at each pixel relies both on the spatial distance and intensity differences [7].

Among those edge-preserving image denoising techniques, BLF is perhaps the most widely used one due to several of its strengths. First, BLF employs a computation of weighted average, which is easy to implement. Second, BLF is a noniterative and local approach, requiring less computational cost in contrast to other iterative [8, 9] and global [10] edge-preserving methods. Third, it has been validated that BLF can keep the crisp edges while removing noise in the image. BLF has been applied to a large variety of tasks including image enhancement [24], artistic rendering [25], image editing [26], optical flow estimation [27, 28], feature recognition [29], medical image denoising [30, 31], and 3D Optical Coherence Tomography retinal layer segmentation [32].

Despite the strengths of BLF, when the task refers to retinal image denoising, BLF might be degraded because it does not take the special tube-like structure of the retinal vessels into consideration. In most image denoising scenarios, BLF tends to preserve crisp edges when two special characteristics exist: prominent contrast observed at the vicinity of edge and larger area occupied by the edge structure compared with isolated noise. In contrast, thin vessels in a retinal image are different from a common crisp edge, due to their weak contrast compared to background and tiny area in the image. BLF cannot handle these special properties of vessels and therefore details of vessels would probably be missed in the filtered image. In this paper, we present a novel BLF algorithm specialized for retinal image denoising based on the properties of the vascular structures. It outperforms the original BLF method in preserving thin vessels.

##### 2.2. Vessel Detection

Recent years have witnessed several classes of approaches for retinal vessel detection and most of them depend on the special characteristics of the vascular structures. Among these approaches, matched filter approaches employ an observation that the cross-section of a retinal vessel can be approximated by a Gaussian function [11–13], ridge detection methods rely on an extraction of image ridges which can be treated as vessel centerlines [1, 14, 15], classification methods utilize a variety of image features to classify each pixel by using machine learning algorithms (e.g., a Bayesian classifier) [16, 17], and vesselness measures analyze the eigenvalues of Hessian matrix at each pixel in order to recognize tubular vascular structures [18–20].

Compared with other vessel detection methods, MF [11] is simple yet effective. Its advantages originate from the Gaussian-like kernels used in its detection process. Blood vessels are characterized by having a tubular structure with a small curvature and being darker/brighter compared with the background. These special properties can be well captured by MF, leading to a good performance in detecting thin vessels within a low-contrast region [33]. As a representative method for measuring vesselness, FR [19] is a practical and state-of-the-art technique. It exploits the eigendecomposition of the Hessian matrix to discriminate the tubular structure from other components.

Different from MF and FR which aim to detect blood vessels from retinal images, our proposed method is designed to denoise retinal images while preserving the vascular structure. It also can be applied to improve the accuracy of the vessel detection techniques.

#### 3. Bilateral Filter and Matched Filter

In this section, we introduce the basic concepts of bilateral filter (BLF) [7] and matched filter (MF) [11] for convenience of describing our method in Section 4.

##### 3.1. Bilateral Filter

As a nonlinear, edge-preserving image filtering method, BLF treats the intensity value at each pixel as a weighted average of its nearby pixels’ intensity values [7]. BLF is capable of fixing the problem of Gaussian blur in traditional Gaussian-convolution based image filtering methods as it combines two components: Euclidean distance and radiometric difference expressed by the following equations:where and denote the image intensities of pixel in the output image and pixel in the input image, respectively. represents a set of pixels neighboring to pixel . and are the spatial kernel and range kernel, for which the weights are computed from the Euclidean distance and the photometric difference between pixels and , respectively. The latter is usually measured by image features such as intensity or texture [34, 35]. is a normalization term computed by (2). In (1), and both take a value inverse to the corresponding input and are expressed typically as a Gaussian function. As an example, is calculated byIn (3), is a scale parameter determining the weight distribution pattern of the kernel. A large means that the range Gaussian widens and flattens.

BLF outperforms many other image filtering algorithms due to its ability to achieve good filtering behavior while preserving crisp edges [7]. It is obtained by combining the spatial kernel and the range kernel in (1). In smooth regions, BLF performs as a Gaussian low-pass filter by averaging away the small, weakly correlated differences between pixel values caused by noise, thanks to the spatial kernel. For a sharp boundary formed by a dark region and a bright one, BLF replaces the dark pixels by an average of the dark pixels in its vicinity while ignoring bright pixels and* vice versa*, thanks to the range kernel.

However, it is not trivial for BLF to distinguish between thin vessels and noise when applying it to retinal images. This is caused by several characteristics of the specific structure of thin vessel compared with a common image edge which is formed by dark and bright regions. First, the pixels of the thin vessel occupy a smaller portion of the pixels in its local window, causing the vessel pixels to be averaged away by the spatial kernel. Second, image intensities of thin vessels are likely to be close to the background due to the limited resolution of retinal imaging and less blood flow, causing vessel pixels to be averaged away by the range kernel. Third, the spatial distribution of vessel pixels is significantly different from independent, isolated image noise, but BLF lacks functions to fully capture the related specific properties.

##### 3.2. Matched Filter

The matched filter (MF) [11] is a vessel detection method based on the observation that the cross-section of vessels can be modeled as a Gaussian function. In this method, a given retinal image is convolved with a set of Gaussian-like kernels built in different orientations, then taking the maximum response over all orientations as the result at each pixel. Since blood vessels have lower reflectance compared with other retinal tissues [36, 37], the vessels appear darker than the background in the retinal image. Yet many techniques also work on the inverted retinal images [17, 38], in which the vessels may appear brighter than the background. MF is valid for both the darker and brighter cases.

Assume the blood vessels appear darker than the background. Let denote the number of orientations. The th oriented kernel matched to a vessel at an angle can be mathematically defined aswhere are the coordinates with the origin being at the center of the kernel window and are the new coordinates computed by (5) at an angle . is the variance used to control the scale of this Gaussian function, which can be set to different values to detect both thin and thick vessels in its extended methods [13]. denotes the length of the kernel window. In essence, each value in this kernel depends on the perpendicular distance between the point and an oriented straight line passing through the center of the vessel. In particular, when , the coordinate and the direction of the vessel is aligned with the -axis and can be expressed asThis equation demonstrates the means to detect the vessels distributed in the vertical direction. To deal with vessels appearing brighter than the background, the value of the above kernels should be inverted accordingly (e.g., take the absolute value of (4)).

Unlike the spatial kernel computed by a common Gaussian function in (3), the MF kernel uses a particular model. Indeed, in the area of imaging systems, this model can be mathematically expressed as a particular LSF [21–23] that spreads as a Gaussian function. The intensities of this particular LSF are distributed equally on one orientation with a Gaussian profile. It can model the orientation and the cross-section of the vascular structure very well. Moreover, the scale parameter can also be adjusted for different vascular widths. Therefore, even for thin vessels with low contrast, MF provides the best overall response compared with most techniques [33].

#### 4. Our Method

The common spatial kernel (as expressed in (3)) of BLF is isotropic across the whole image and we find that adjusting this kernel according to the local vessel structure can significantly improve the performance of image denoising. A blood vessel appears locally in the retinal image as a dark/bright tube with a small curvature, and the shape of its cross-section is similar to a Gaussian function. These properties can be represented by an LSF that spreads as a Gaussian function. Therefore, we propose to determine the spatial kernel of BLF by a particular LSF, which can be obtained by using the Gaussian-like kernel of MF. The proposed new BLF performs significantly better in preserving thin blood vessels in retinal images while denoising image. Moreover, with some simple experiments, we find it outperforms the original BLF in some other aspects that benefit vascular structure preservation.

##### 4.1. Bilateral Filtering with a Spatial Kernel of Optimally Oriented LSF

Given a retinal image, our filter also computes a weighted average of the local pixels based on (1). The difference lies in the definition of the spatial kernel . Mathematically, the method can be expressed aswhere is the proposed spatial kernel which is adaptive to different local orientations and scales of the blood vessels. The value of is determined by . We can obtain by computing the perpendicular distance between pixel and the straight line passing through the center of blood vessel , rather than measuring the Euclidean distance between different pixels by the spatial kernel of BLF. is also a normalization term. The core idea of our method is to design an optimal spatial kernel and combine it with the range kernel of BLF.

In order to obtain , we start with constructing several multiscale oriented Gaussian-like kernels based on MF. For each pixel at its local window in a retinal image, we use these Gaussian-like kernels to detect the vessel’s orientation and scale. The best matching one is selected as the detection result. at each pixel can be simply computed by using the detection result. Finally, combined with the range kernel results in our filter.

###### 4.1.1. Multiscale Oriented Gaussian-Like Kernels

To construct the multiscale oriented Gaussian-like kernels, the equations of MF are used. The process is similar to that in MF: compute one case aligned with the -axis using (6); then rotate it by (5) to generate oriented Gaussian-like kernels. Moreover, in order to capture vessels at a variety of widths, we apply the oriented Gaussian-like kernels at different scales; that is, we further set different scale parameters for each orientation. Then we can obtain a total of multiscale oriented Gaussian-like kernels. Suppose that the vessels appear darker than the background in the retinal image; one of these kernels is defined aswhere denotes the th oriented Gaussian-like kernel in the th scale with the parameters and . can be computed using (5) at an angle . When the vessel is brighter than the background in retinal image, the value of needs to be inverted. The calculating process is not computationally expensive since these kernels can be calculated independently of the retinal image without relying on the image intensity.

###### 4.1.2. Local Vessel Detection

Given a retinal image, the aim of local vessel detection is to detect the vessel’s orientation and the scale of each pixel at its local window. In this process, the retinal image is convolved with the multiscale oriented Gaussian-like kernels constructed in advance and then each point has responses. Then we select the one with the maximum response as the detecting result at each point. The detecting process can be expressed by the following:where is the kernel selected from different multiscale oriented Gaussian-like kernels which has the maximum response. is the intensity matrix of local window centered at pixel . and are the index numbers of orientations and scales. Symbol in (9) represents the dot product of two matrices. All of the kernels and intensity matrices need to be normalized before computing. An exampling spatial kernel specified by our approach is shown in Figure 1.