Image denoising using bilateral filter in subsampled pyramid and nonsubsampled directional filter bank domain

Abstract

A vital task in image denoising is to preserve edges and image features while removing noise. This paper presents an efficient algorithm for noise removal by incorporating an adaptive bilateral filter in the subsampled pyramid and nonsubsampled directional filter bank (SPNSDFB). This filter bank decomposes the noisy image into subbands of different frequency and orientation. Owing to its multiscale, multidirectional and lack of shift variance capability, it provides an efficient representation of intrinsic geometric structures of an image. By the fusion of the bilateral filter in SPNSDFB domain and optimum selection of parameters of the adaptive bilateral filter, the proposed algorithm minimizes mean square error (MSE) between the original image and the denoised image even at high noise densities. Experimental results show that the algorithm is found to be competitive in denoising performance due to its better edge preservation and improves peak signal-to-noise-ratio and image visual impression.

Keywords

Directional filter bank (DFB)subsampled pyramid (SP)nonsubsampled directional filter bank (NSDFB)fan filter

1 Introduction

Digital images are naturally embedded with essential geometric structures of lines, contours and edges. They are perturbed by noise during image acquisition due to the high sensitivity of recent cameras and transmission by an ineluctable ambient condition. In fact, image noise reduces the effectiveness of visual interpretation and data extraction tasks like segmentation, feature extraction and classification, pattern recognition and more. This leads to the notion of an image denoising algorithm to retrieve the original image from an incomplete, indirect and noisy image. In particular, edge-preserving image denoising approaches have received increasing attention in daily life applications such as satellite television, medical images like Computed Tomography (CT), Magnetic Resonance Imaging (MRI), mammography, diagnostic ultrasound imaging, nuclear medical imaging with Single Photon Emission Computed Tomography (SPECT) and Positron Emission Tomography (PET) as well as in the areas of research and technology such as geographical information system. It is well known that there are several efficient denoising approaches by classically preserving the essential geometric details and edges. Still it is not an easy task to separate the noise significantly from images with distinguishable characteristics in nature.

Over the past decades, several image denoising algorithms were developed based on spatial domain technique. Spatial filters such as mean filter, Gaussian filter, and Wiener filter [1] can effectively remove noise from the image. However, they fail to preserve meaningful edges significantly. In an attempt to remedy this problem, many researchers have developed several partial differential equation (PDE)-based spatial filters [2, 3]. They suffer from the staircase-like appearance in the denoised image. In eliminating staircase artifacts, total variation-based filters [4, 5], anisotropic filters [6], and fuzzy-based filters [7 –9] have shown the remarkable success. They can also detect and maintain edges while smoothing the image. But anisotropic diffusion blurs the images and increases the loss of texture details. An alternative way to preserve edges is bilateral filtering [10 –17] in pixel domain which combines domain and range filtering. Although seemingly efficient for eliminating staircase artifacts, it does not perform well for high-resolution images with thin lines or textures. Another important issue is that variance of pixel intensities is large in noise corrupted image. As a result, bilateral filter introduces false edges in the denoised image.

In recent years, researchers proposed several techniques based on multiresolution wavelet transform [18 –21] to separate the noise components from the image. Due to its ability in extracting directional information from horizontal, vertical, and diagonal orientation, it provides an optimal representation of point discontinuities, but it is not effective in capturing high dimensional line discontinuities like curves and contours. It has been recognized that shift invariance is the most important property in image denoising application. Lack of shift invariance in wavelet representation produces low frequency noise and ringing artifacts at the vicinity of edges in the denoised image. As a consequence, it affects the visual interpretation of textures and smooth regions. To overcome this, several multiscale and multiple orientation transforms like steerable pyramid [22], complex wavelet [23], curvelet [24], brushlet [25], and two- dimensional (2-D) directional wavelets [26] were proposed. They can capture linear structures, curves, and contours of the image efficiently with good energy compaction. But they are redundant at the expense of increased computational cost in image denoising application.

Recently, Do et al. pioneered a multiscale and multidirectional contourlet transform [27] based on Laplacian pyramid and directional filter bank (DFB) to achieve good directional selectivity. By approximating an image from coarse to fine resolution, it provides the orthogonal basis of the image. Its elongated support with different aspect ratio can effectively represent piecewise smooth images and smooth contours at a cost of small redundancy of about 33%. But its instability with respect to the shift of the input signal creates edge ringing artifacts. Moreover, overlapping spectra of the sampling process causes scratch phenomena. As a consequence, the denoised image has artifacts, but fewer than wavelets. However, it is not the best method in terms of PSNR. An advantage of increasing redundancy is the improvement in denoising performance by reducing visual artifacts. Accordingly, redundant transforms like nonsubsampled contourlet transform (NSCT) [28] and translation invariant contourlet transform [29] were proposed based on geometrical structures and shift invariance. They produce better denoising due to directionality and nonlinear approximation property. But they have the weakness of high redundancy at the increased computational cost, thereby making these schemes less intractable in denoising of high dimensional images.

In recent years, many authors have proposed image denoising algorithm based on thresholding methods [30 –32] like global thresholding, normal shrink, SURE shrink, visushrink etc. on transform coefficients. And also, noise estimation methods like Bayesian least-squares estimate based on Gaussian scale mixture model (BLS-GSM) [21], and a bivariate shrinkage function using maximum a posteriori estimate (MAP) [33] are applied to obtain noise-free transform coefficients but they produce low frequency noise and edge ringing artifacts in the denoised image. Instead of thresholding strategies, spatial filters have recently shown increasing interests to apply on the noisy image subbands in the transform domain to filter out the noise. Accordingly, multiresolution bilateral filtering [34, 35] were proposed by filtering the approximation coefficients with the bilateral filter and thresholding the detail subbands using either hard threshold or BLS-GSM. In these methods, the bilateral filter is applied to remove low frequency noise only.

To achieve more sparse representation of images, a shift invariant, non-redundant, multiscale and direction sensitive transform [36] was proposed by combining subsampled pyramid and nonsubsampled directional filter bank (SPNSDFB) [36]. It is a variant of contourlet transform and NSCT. Its denoising performance is good by applying threshold on transform coefficients. To enhance its performance further, this paper presents a novel denoising algorithm based on the adaptive bilateral filter in SPNSDFB transform domain. SPNSDFB transform decomposes the noisy image into approximation subband and detail subbands of different frequency and orientation at each scale. It decorrelates prominent features like contours, edges, and smooth regions in the image into low frequency and high frequency directional subbands. Then the nonlinear adaptive bilateral filter which does spatial averaging without smoothing edges is applied on detail subbands of SPNSDFB to remove noise. An important characteristic is that our denoising scheme does not take into account the local statistical or structural characteristics of the two noisy pixels directly and variance of SPNSDFB transform coefficient is less than the variance of pixel intensities. As a result, it eliminates the occurrence of false edges in the denoised image and enhances the edge preserving ability of the adaptive bilateral filter. It can denoise high resolution images like medical and satellite images with better edge preservation. The performance of the proposed algorithm is evaluated and compared with other techniques in terms of peak signal-to-noise-ratio (PSNR) and visual interpretation for the test images like Barbara, Lena, circuit, and pepper. It outperforms the other techniques at low and high noise densities.

This paper is organized as follows. Section 2 gives the theory of the proposed method elaborating the structure of SP and NSDFB and bilateral filter. The experimental results are discussed in Section 3. Finally, Section 4 concludes the paper.

2 Proposed method

The proposed structure is shown in Fig. 1. It incorporates the bilateral filter in the SPNSDFB domain to enhance the denoising performance of SPNSDFB filter bank. The SPNSDFB transform approximates the original image by a linear combination of orthogonal basis elements. This multiscale transform provides good sparsity for spatially localized details like edges, corners etc. Also, this transform leads to a significant application in image denoising by extracting high frequency edges, corners, and contours embedded in smooth regions from the image corrupted by high frequency noise.

2.1 Subsampled pyramid and nonsubsampled directional filter bank

The subsampled pyramid and nonsubsampled DFB (SPNSDFB) consists of two structures namely, subsampled pyramid (SP) and nonsubsampled directional filter bank (NSDFB) as shown in Fig. 1. The NSDFB structure is designed using DFB without interpolators and decimators as in conventional DFB. The NSDFB captures the high frequency information, but it is not able to capture the low frequency content. Hence, it is combined with the SP that provides multiscale decomposition to remove the low frequencies of the image before applying the NSDFB. The combined approach is computationally efficient with good angular resolution

2.1.1 Subsampled pyramid filter bank

A subsampled pyramid (SP) is a 2-channel filter bank that decomposes the image into a coarse image and a bandpass image at each scale. In SP, the combined operation of low-pass filter and decimator gives a coarse approximation of the original image.

Let x (n₁, n₂) be the image to be decomposed and h₁ (n₁, n₂) be the impulse response of the filter. Then the low-pass filtered image is given by

$\begin{matrix} y_{LP} (n_{1}, n_{2}) \\ = 1 / MN \times \sum_{m^{'} = 0}^{M - 1} \\ \sum_{n^{'} = 0}^{N - 1} x (m^{'}, n^{'}) h_{1} (n_{1} - m^{'}, n_{2} - n^{'}) \end{matrix}$ (1) where n₁ = 0, 1, 2 … … M - 1 and n₂ = 0, 1, 2 … … N - 1.

The discrete Fourier transform of filtered and downsampled image is,

$\begin{matrix} Y (ω) & = & 1 / | M |^{\times} \\ \sum_{k \in N (M^{T})} Y_{LP} (M^{- T} ω - 2 π M^{- T} k) \end{matrix}$ (2) where N (M^T) is the set of all integer lattice points and ω = [ω₁, ω₂] ^T.

Based on this coarse version, the original image is predicted by interpolating and filtering the coarse image. Then the difference between the original image and the predicted image is the bandpass image or detail image.

2.1.2 Nonsubsampled directional filter bank

The NSDFB is a three stage tree-structured filter bank that splits the bandpass image into directional subbands by partitioning the frequency plane in wedge-shaped region. An l-level NSDFB structure decomposes the image into 2^l subbands. The first and second level of tree structure consists of two-band filter bank structure with complementary fan-shaped filters. The third level of NSDFB structure is constructed by combining two band fan filter bank structure with pre/post resampling operations $\begin{matrix} R_{0} & = & (\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}) R_{1} = (\begin{matrix} 1 & - 1 \\ 0 & 1 \end{matrix}) \\ R_{2} & = & (\begin{matrix} 1 & 0 \\ - 1 & 1 \end{matrix}) R_{3} = (\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}) . \end{matrix}$

It extracts 2-D directional information of an image with reduced complexity.

An effective and simple way to design two- dimensional (2-D) analysis/synthesis fan-shaped filters is by McClellan mapping approach. In this approach, one-dimensional (1-D) analysis/synthesis filters are constructed by using one-dimensional (1-D) finite impulse response (FIR) filters with Kaiser window [37] which has better frequency selectivity with high stopband attenuation. Then, by McClellan mapping approach [38, 39], the 1-D FIR filter is transformed into 2-D filter satisfying Bezout identity. The fan filters employed in this filter bank have wedge-shaped passband region as shown in Fig. 2.

2.2 Image Denoising using bilateral filter

Natural images are inhomogeneous. It contains high frequency edges, corners, and contours embedded in smooth regions. Since noise corrupting the image is of high frequency, removal of noise results in loss of high frequency details in the denoised image. By applying spatially adaptive nonlinear operation having good edge retaining capability in the transform domain, image denoising performance can be improved. Accordingly, the bilateral filter is applied in the SPNSDFB transform domain.

Bilateral filter is a nonlinear and locally adaptive filter that combines domain and range filtering. It is a noniterative filter where the weights are computed based on the spatial and photometric distance between center pixels and the neighboring pixels. The spatial and intensity weighting functions are Gaussian. Bilateral filter averages neighbors with similar intensities. The weights computed with respect to the relative intensity between center and its neighbor intensity values vary abruptly with the change in intensity value due to noise thereby creating false edges. This effect is minimized by applying the bilateral filter on the decomposed subbands of the noisy image with the SPNSDFB transform. This approach includes smoothing within similar regions, edge preservation, separation of large structure and fine detail and elimination of outliers in image denoising application.

Suppose an image f is corrupted by an additive white Gaussian noise n, then the noisy image is given by $y = f + n$ (3)

SPNSDFB subbands are obtained by decomposing the noisy image y. Bilateral filter is applied on SPNSDFB noisy subbands to obtain better denoising performance. The coefficient value at each location of the subbands is replaced by a weighted average of coefficient values from nearby coefficient value. It is given by

$\begin{matrix} I (x) & = & 1 / c \sum_{y \in N (x)} exp (- | y - x |^{2} / - 2 σ_{d}^{2}) \\ \times exp (- | I (y) - I (x) |^{2} / - 2 σ_{r}^{2}) I (y) \end{matrix}$ (4)

$\begin{matrix} c & = & \sum_{y \in N (x)} exp (- {| y - x |}^{2} / - 2 σ_{d}^{2}) \\ \times exp (- {| I (y) - I (x) |}^{2} / - 2 σ_{r}^{2}) \end{matrix}$ (5) where σ_d and σ_r are the two parameters controlling the fall-off weights in spatial and transform coefficient domains respectively, N(x) is a spatial neighborhood of coefficient I(x) and c is the normalization constant. Coefficients which are very different in value from the central value, are weighted less even though they are in close proximity to the central value. Both σ_r and σ_d determine the level of smoothness. Increasing the value of σ_d results in over smoothening of the image thereby decreasing the Mean-square-error (MSE). Setting σ_r to zero reduces the bilateral filter to a simple Gaussian smoothing filter. There is no explicit rule for selecting the parameters σ_r and σ_d. By the proper selection of σ_r and σ_d, denoising performance can be improved.

3 Results and discussion

To assess the denoising performance of proposed algorithm, two important factors are to be considered: the amount of noise diminution and the preservation of prominent image features and edges. We utilize PSNR as an objective measure of performance and visual quality as a subjective measure to detect artifacts. The traditional objective measure known as PSNR is defined as 10 ${log}_{10} \frac{255^{2}}{MSE}$ where MSE is the mean square error between the original image and denoised image.

We have performed several experiments quantitatively and qualitatively by adding white Gaussian noise with standard deviations ranging from 5 to 25 to some features enriched 8-bit gray scale images like Lena (512×512), Barbara (512×512), peppers (512×512), circuit (256×256) etc. The images ‘Lena’ and ‘Peppers’ are mainly composed of smooth regions and edges and the image ‘Barbara” with abundant inhomogeneous textures. SPNSDFB is used for image disintegration because of its perfect reconstruction, shift invariance property and well-balanced frequency response with the usage of high frequency selectivity analysis/synthesis filters designed from FIR filters with Kaiser window. At first, a three level SPNSDFB disintegrates the noise corrupted image into a coarse image and eight detail subbands. The decomposed subbands are orthogonal basis decorrelating the prominent features and detecting edge related information of the noisy image. Then the edge preserving bilateral filter applied on the detail subbands removes noise. SPNSDFB reconstructs the denoised detail subbands to obtain the denoised image. We have evaluated the performance of the denoised image by visual inspection and the objective measure, PSNR. For more accurate PSNR values, we repeated each experiments ten times and found the average PSNR values.

3.1 Selection of parameters σ_d and σ_r

We investigated the potential usage of the bilateral filter in SPNSDFB domain for image denoising. This denoising scheme performs well by the proper selection of the parameters σ_d and σ_r of the bilateral filter. They are chosen by experimental observation on trial and error basis. At first, σ_d is varied from 0 to 25 with various constant values of σ_r and noise standard deviation σ = 15 etc. When σ_d varies from 0 to 20, PSNR value increases gradually and reaches an optimum PSNR value at σ_d = 20 as shown in Fig. 3. Then σ_r is varied from 1 to 6 with constant values of σ_d = 20 and noise standard deviation σ = 15, 20, 25 etc., PSNR reaches a maximum value when σ_r = 2 as shown in Fig. 4. We would like to point out that the parameters σ_d and σ_r are selected to be 20 and 2 respectively to obtain better denoising performance. In addition to these parameters, our denoising scheme computes the denoised coefficients based on the spatial and photometric distance between the center coefficients and neighboring coefficients of the window of size 5×5 for each highpass directional subbands of the SPNSDFB transform.

3.2 Comparison with different denoising Schemes

We upgrade our approach by the way of finding a suitable shrinkage method in SPNSDFB domain. At first, the global threshold is chosen to be T = 3σ, where σ is the standard deviation of the noisy image and SPNSDFB coefficients less than the threshold value are discarded while retaining others without any modification. We notice that the global thresholding scheme yields abrupt artifacts with decreased global contrast in the reconstructed image, especially at high noise density. Then we consider subbands adaptive soft thresholding, normal shrink, to be applied on the highpass directional subbands of SPNSDFB. It is clear that normal shrink produces significantly noticeable artifacts such as Gibbs like ringing artifacts in the vicinity of discontinuities and specks in smooth regions, but it gives more visually pleasant images as compared to global thresholding. Further, the Wiener filter is used in the SPNSDFB domain. It results in blurring with image smoothing. Bilateral filter in spatial domain results in the denoised image with the good visual quality for low resolution images. It introduces false edges in the denoised image. It is observed that the incorporation of the bilateral filter in SPNSDFB domain eliminates the false edges with better edge preservation. More precisely, the proposed denoising scheme restores the image features and edges significantly thereby enhancing the visual quality of the denoised image. Figure 5 shows denoising results in terms of visual quality for the selected region of Barbara image to view clearly the image features and structures.

Table 1 compares the denoising performance of the proposed method with other denoising schemes like the bilateral filter in the spatial domain and global thresholding, normal shrink, and Wiener filtering in the SPNSDFB domain in terms of PSNR. The PSNR value of the proposed method is consistently higher than that of the other methods due to its edge retaining ability. For Barbara image with noise density σ= 20, the PSNR value is about 0.2623 dB greater than normal shrink, 0.5952 dB greater than Global thresholding, 0.0173 dB greater than Wiener filter and 1.8273 dB greater than the spatial bilateral filter. Figure 6 shows the amount by which output PSNR changes as the function of input PSNR for the proposed method and compares with other methods for Lena and Barbara images. Similarly, Fig. 7 compares the increase in output PSNR from the input PSNR as a function of the standard deviation of the noise. They clearly emphasis that the proposed method outperforms the other methods due to its to better directional representation. Visual interpretation and PSNR values of the denoised results clearly justifies the choice of the bilateral filter in the SPNSDFB domain.

3.3 Comparison with existing denoising schemes in the literature

For the sake of comparison, we provide the discrete wavelet transform (DWT), contourlet transform (CT) and NSCT denoising result using global thresholding at different noise densities. Moreover, we employed some of the state-of-the -art methods in the literature such as Multiresolution Bilateral filtering (MBF) [34] and Bilateral filter and Gaussian scale mixtures in Pyramidal dual-tree directional filter bank (BF and GSM in PDTDFB) [35]. The DWT results in substantial reduction in performance due to both the lack of shift invariance and mixed orientation in the diagonal band. It produces low frequency noise and edge ringing artifacts resulting in degradation in the visual quality of textures and smooth regions in the image. The contourlet transform based image denoising results in scratch phenomena due to frequency aliasing problem. In MBF and BF and GSM in PDTDFB domain, we notice the occurrence of Pseudo-Gibbs artifacts in the vicinity of edges. In particular, our method is shift invariant and also the detail subbands of SPNSDFB are better correlated with image features. As a result, the proposed denoising scheme always provides better visual quality when compared with CT, NSCT, MBF and BF and GSM in PDTDFB domain.

Figure 8 Compares the visual quality of the denoising results for selected region of Barbara image at noise density σ = 20. It clearly indicates the better recovery of linear structures and retention of image details with fewer artifacts by our denoising scheme compared with other techniques. As a result, the proposed denoising scheme always provides better visual quality when compared with CT, NSCT, MBF and BF and GSM in PDTDFB domain. Table 2 shows the PSNR values of denoising results at different noise densities. Clearly, SPNSDFB has uniformly better PSNR than the DWT, CT, NSCT, MBF and BF and GSM in PDTDFB domain.

4 Conclusion

We proposed an edge preserving image denoising algorithm that integrates the bilateral filter in the SPNSDFB domain. The SPNSDFB implements a shift invariant, multiscale and multidirectional transform by combining SP and NSDFB for an efficient representation of images with less redundancy. The NSDFB structure is constructed without using decimators and interpolators thereby reducing computational complexity with less redundancy. FIR filters with Kaiser window having better selectivity and high stopband attenuation are used to design fan filters in NSDFB structure. As a result, SPNSDFB structure decorrelates the image features more efficiently by disintegrating the image into eight directional subbands at three scales. And also, the adaptive bilateral filter is a better edge preserving filter with image smoothing. By the optimum selection of parameters σ_d and σ_r of bilateral filter and obtaining denoised coefficients based on spatial and photometric distance between center coefficient and the neighboring coefficient of SPNSDFB transform, our approach allows us to keep meaningful edges, texture details and image structures and eliminating the outliers thereby providing good perceptual quality and high PSNR value while eliminating most of the noise in smooth regions. Comparison with other state of-the-art denoising techniques demonstrates the efficiency of our approach which gives the best output PSNRs and good visual quality for most of the images.

There is no explicit rule for finding the bilateral filter parameter. In future, Instead of selecting the parameters of bilateral filter on trial and error basis, an adaptive least mean square algorithm can be used to optimize the parameters by finding an empirical relationship between them to achieve better denoising performance.

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Image denoising using bilateral filter in subsampled pyramid and nonsubsampled directional filter bank domain

Abstract

Keywords

1 Introduction

2 Proposed method

2.1 Subsampled pyramid and nonsubsampled directional filter bank

2.1.1 Subsampled pyramid filter bank

2.2 Image Denoising using bilateral filter

3.1 Selection of parameters σ d and σ r

3.2 Comparison with different denoising Schemes

3.3 Comparison with existing denoising schemes in the literature

4 Conclusion

References

3.1 Selection of parameters σ_d and σ_r