A new family of Gaussian filters with adaptive lobe location and smoothing strength for efficient image restoration
 Hassene Seddik^{1}Email author
https://doi.org/10.1186/16876180201425
© Seddik; licensee Springer. 2014
Received: 2 October 2013
Accepted: 18 February 2014
Published: 1 March 2014
Abstract
Noise can occur during image capture, transmission, or processing phases. Image denoising is a very important step in image processing, and many approaches are developed in order to achieve this goal such as the Gaussian filter which is efficient in noise removal. Its smoothing efficiency depends on the value of its standard deviation. The mask representing the filter presents generally static weights with invariant lobe. In this paper, an adaptive denoising approach is proposed. The proposed approach uses a Gaussian kernel with variable width and direction called adaptive Gaussian kernel (AGK). In each processed window of the image, the smoothing strength changes according to the image content, noise kind, and intensity. In addition, the location of its lobe changes in eight different directions over the processed window. This directional variability avoids averaging details by the highest mask weights in order to preserve the edges and the borders. The recovered data is denoised efficiently without introducing blur or losing details. A comparative study with the static Gaussian filter and other recent techniques is presented to prove the efficiency of the proposed approach.
Keywords
1. Introduction
The image denoising remains an important goal in image or video preprocessing as a preliminary task for data transmitting, pattern recognition, etc. In the case of high distortions, efficient noiseremoving techniques may introduce artifacts or blur the image. Image denoising techniques using the Gaussian filter has been widely used in many fields for its ability to efficiently restore degraded data. In [1], the authors combined the following three techniques: wavelet transform, curvelet transform, and the Gaussian filter to recover the distorted image. The authors in [2] exploited the relationship between linear diffusion and Gaussian scale space to estimate optimal variances and window size of the Gaussian. An efficient technique based on the Gaussian filter with dynamic structure that targets noise is introduced in [3, 4]. Selecting the optimal value of the standard deviation in a Gaussian filter depending on few properties of the signal knowledge is proposed in [5]. In [6], an adaptive Gaussian filtering algorithm, in which the filter variance is adapted to both noise characteristics and the local variance of the signal, is studied.
The basic premise of the Gaussian technique is that different parts of an image have varying degrees of noisiness and types of edges. Therefore, each part of the image needs to be smoothed differently. For this reason, we propose to create an adaptive filter having a Gaussian core with a variable structure for each processed window. The location of the Gaussian lobe and its smoothing strength are optimized iteratively according to the noise intensity and image characteristics. These features are optimized to efficiently clean noise and preserve the image content. The paper is arranged as follows: a brief description of the Gaussian kernel in Section 2, the conception of the proposed filter in The proposed adaptive Gaussian filter and Experimental results, comparative study in Comparative study, and a summary and conclusion in Conclusions.
2. Recall of the conventional Gaussian filter properties
The main properties of the Gaussian filter are described as follows:

Gaussian smoothing is very effective for removing Gaussian noise.

The weights give higher signification to pixels near the edge (reduce edge blurring).

It is a static and linear lowpass filter.

Separability into two onedimensional (1D) filters.

Rotationally symmetric (performs the same in all directions).

The degree of smoothing is controlled by the standard deviation σ (larger σ for more intensive smoothing).Figure 2 shows the result of applying Gaussian filters with different values of σ on the Lena image. It is clear that when we increase the width parameter (σ), the borders and the details are removed.
In order to overcome this problem, we study a smart Gaussian filter with dynamic structure. In this new filter, the variation of the standard deviation is done according to the nature and characteristic of the image areas and zones. This variation is supervised by a neural network, whereas changing the couple of means (μ_{1} and μ_{2}) will vary the position of the filter lobe in order to preserve edges and borders.
3. The proposed adaptive Gaussian filter
3.1. Estimation of the adaptive smoothing strength
The optimal value of the standard deviation is manually adjusted around the computed value in the range of [2Δσ, 2Δσ]. In fact, σ_{opt} is located between the transitory and stable zone of the PSNR_{ σ } function that are separated by the computed tangent. To validate our selection, we compute the normalized crosscorrelation between the filtered and original image called C. A segment noted D 1 is drawn as a tangent on this curve with parallel direction to D. The index representing the value of the standard deviation found confirms the computed σ_{opt}. The range of σ_{opt} is constrained by a minimum threshold imposed to the PSNR called PSNR_{min} that must be maintained over 32 dB.The selected distorted patterns of the image are introduced to a multilayer perceptron (MLP) neural network which is composed of three layers ‘input, hidden layer, and output layer’. In the test phase, the neural network generates different values of standard deviations for all the introduced distorted windows. The network generates the appropriate outputs according to the noise density and kind (Figure 5).
3.2. The adaptive kernel location
The (μ_{1}, μ_{2}) represent the positions of the Gaussian core (location of the peak). In this work, we apply the Gaussian filter only on noise to avoid blurring details and borders. The steps of the kernel location variability are presented as follows:
First step: Edge detection using the canny highboost filter operator presented by the following equation:
Second step: Filter the noisy image based on a decision computed from 8 to 25 neighborhoods’ comparison:

If I(x, y) ‘the processed pixel’ belongs to an edge, compute the difference between this pixel and its eight neighbors called P(x, y) (Figure 6 and Equation 6); in this case, the values of the mean (μ_{1} and μ_{2}) are determined according to the maximum variation (gradient).

Elsewhere, we process the selected window using a filter with support size equal to (6 × σ_{opt} + 1) × (6 × σ_{opt} + 1) and the appropriate smoothing strength.$\mathit{f}\left(\mathit{x}\right)=\mathit{a}exp\left(\mathit{\alpha x}\right)\cdot sin\left(\mathit{wx}\right)$(5)
For 1 to 8 (number of neighbors for each processed pixel in a window of size ‘3 × 3’, the maximum variation max[P(x,y)] is computed to determine automatically the values of the means (μ_{1} and μ_{2}) and define the location of the Gaussian lobe.
Sharp changes in an image can be associated to edges, or noise and such changes correspond to higher gradients. To consider that a pixel I(x, y) belongs to an edge and not as noise, we must satisfy two conditions:

High gradient variation between this pixel and its 8 or 25 neighbors for (3 × 3) or (5 × 5) window sizeConnection continuity between different pixels considered as edges as presented in Figure 7
 1.
${\overrightarrow{\mathit{\tau}}}_{\mathit{x},\mathit{y}}$ is always orthogonal to the tangent of the image edge.
 2.
${\overrightarrow{\mathit{\tau}}}_{\mathit{x},\mathit{y}}=max\u2225\overrightarrow{\mathit{P}}\left(\mathit{x},\mathit{y}\right)\u2225$: The lobe displacement follows the maximum gradient of the image windows where the central pixel of this window belongs to the detected edge.
where the couple (N, M) represents the image size.
4. Experimental results
MSE is the mean square error, d is the maximal coded image intensity, n and m are the image sizes, and f and r are the original and the filtered image.
4.1. Filtering salt and pepper noise
Comparison between the AGK and the static Gaussian filter
Noise densities  AGK PSNR (dB)  Static PSNR (dB)  ∆PSNR (dB) 

0.01  36.84  34.48  2.35 
0.02  37.13  33.56  3.54 
0.03  37.9  33.2  4.88 
0.04  37.8  32.6  5.2 
0.05  38.24  32.13  6.11 
0.06  38.84  31.78  7.05 
0.07  38.97  31.46  7.51 
0.08  39.49  31.23  8.26 
0.09  39.83  31.06  8.77 
PSNR variation
S&P densities  AGK PSNR  Static PSNR  ∆PSNR 

0.05  38.06  34.77  3.29 
0.06  38.19  34.2  3.99 
0.07  38.20  33.87  4.33 
0.08  38.28  33.32  4.96 
0.09  38.12  33.18  4.94 
0.1  38.05  33.03  5.02 
0.2  37.397  31.35  6.04 
0.3  35.6  3.58  5.01 
0.4  33.94  30.24  3.7 
0.5  32.8  30  2.79 
4.2. Speckle noise
The proposed AGK filter generates better denoising results than the conventional filter. This efficiency was illustrated by different examples of filtered image. All the content and details were preserved and well perceptible after the filtering process.
5. Comparative study
Comparison results between the proposed technique, the conventional filter, and the AGSS filtering process
Images  AGSS technique  Static Gaussian filter  Proposed technique (AGK)  ∆ P 1  ∆ P 2 

Boat  31.9  32.818  40.11  8.21  7.29 
Lena  23.1  33.0982  37.957  14.86  4.86 
House  28.7  33.2332  37.891  9.19  4.66 
Barbara  35.5  32.7839  38.331  2.83  5.55 
Peppers  20.3  33.6212  40.274  19.97  6.65 
6. Conclusions
In this paper, a new lowpass filter with Gaussian core is presented. An adaptive Gaussian kernel based on dynamic core with variable structure is shaped. This new kernel conserves all the mathematical characteristics of the static Gaussian filter. The smoothing strength and the support size are supervised in each processed window of the image by a neural network to achieve the best filtering results. At the same time, the Gaussian lobe moves continuously in eight directions with the appropriate magnitude to avoid averaging of higher filter weights to preserve borders and edges. A comparative study is conducted to prove the efficiency of the proposed approach. Different image tests are shown with zoomed zones to validate the efficiency of this filter.
Declarations
Authors’ Affiliations
References
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Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.