A Laplacian based image filtering using switching noise detector
- Ali Ranjbaran^{1}Email author,
- Anwar Hasni Abu Hassan^{1},
- Mahboobe Jafarpour^{1} and
- Bahar Ranjbaran^{1}
https://doi.org/10.1186/s40064-015-0846-5
© Ranjbaran et al.; licensee Springer. 2015
Received: 23 December 2014
Accepted: 22 January 2015
Published: 8 March 2015
Abstract
This paper presents a Laplacian-based image filtering method. Using a local noise estimator function in an energy functional minimizing scheme we show that Laplacian that has been known as an edge detection function can be used for noise removal applications. The algorithm can be implemented on a 3x3 window and easily tuned by number of iterations. Image denoising is simplified to the reduction of the pixels value with their related Laplacian value weighted by local noise estimator. The only parameter which controls smoothness is the number of iterations. Noise reduction quality of the introduced method is evaluated and compared with some classic algorithms like Wiener and Total Variation based filters for Gaussian noise. And also the method compared with the state-of-the-art method BM3D for some images. The algorithm appears to be easy, fast and comparable with many classic denoising algorithms for Gaussian noise.
Keywords
Introduction
Denoising is one of the most important issues in image processing. The most popular noise removal methods are Adaptive Median Filtering (AMF), Total Variation (TV) based algorithms, Kernel based methods, Bilateral and Guided filtering and recently BM3D state-of-the-art in natural image denoising. In this work, we introduce a noise removal approach using a local noise estimator. We use the noise estimator in a minimization energy functional scheme. We obtain an iterative image denoising process using Laplacian. Denoising can be seen as adding values of pixels with their relative Laplacian weighted by local noise estimator.
Section 2 is related work. Section 3 represents our idea for denoising. We show how we can define a noise estimator using the sign of change in intensity of pixels in a 3x3 window. Using local noise estimator modified by Gaussian weight, we define an energy functional, drive the final equation and use it in an iterative denoising process. We show that although Laplacian is known as edge detector, it can be used for noise removal purposes. The algorithm is implemented in section 4. In section 5, results are shown and described. Moreover, figures of the denoising method are shown in comparison with Wiener, ROF and BM3D filters based on TV value and visual performance.
Related work
A Total Variation based noise removal method (ROF) (Rudin et al. 1992) defines an energy functional that preserves edges of the image and smoothes Gaussian noisy area, based on the total variation norm minimizer. The TV regularization technique is a suitable method that can be extended to different noisy conditions such as Laplace and Poisson (Chan and Esedo Lu 2005; Li et al. 1994). The Split Bregman method (Goldstein and Osher 2008) is fast, reliable and extendable to different models of noise distribution. Split Bregman is a basic and effective tool in solving many functional-based problems such as Compressed Sensing (CS) (Candès et al. 2006). In recent years, the usage of kernel-based techniques in image denoising has developed the quality of noise removal results. The image used in kernel functioning is called the guidance image. One of the most popular approaches using the guidance image is bilateral filter (Petschnigg et al. 2004). Other important kernel-based methods are Data-adaptive kernel regression (Takeda et al. 2007), Non-Local Means (Buades et al. 2005) and Optimal Spatial Adaptation (Kervrann et al. 2006). Another novel state of the art method was recently introduced as guided filter (He et al. 2010). The US patent 6229578 “Edge Detection Based Noise Removal Algorithm” (Acharya et al. 2001) is a denoising method based on using edge detector. This method removes noise by distinguishing between edge and non-edge pixels and applying a first noise removal technique to pixels classified as non-edge and a second noise removal technique to pixels classified as edge pixels. BM3D is a well-engineered algorithm which represents the current state-of-the-art method for denoising images corrupted by Additive White Gaussian Noise (AWGN) (Dabov et al. 2007; Dabov et al. 2006; Dabov et al. 2008; Dabov et al. 2009; Chen and Wu 2010). In another strategy, denoised image is considered as a linear combination of the original image and its average when the coefficients are determined by an edge detector (Ranjbaran et al. 2013).
Methodology
It is predictable that in implementing the algorithm by large number of iterations λ must be a positive small value. In real condition SWN is not constant and reduced during the evolution.
Implementation
- 1.
Setting parameters : window size 3×3, λ = 0.5
- 2.
Reading image u _{0}
- 3.
Adding zero mean Gaussian Noise (imnoise code)
- 4.Computing SWN for current pixel$$ SWN=\frac{\frac{\pi }{2}+ta{n}^{-1}\left(-3{g}_x{g}_{x+\Delta x}\right)}{\pi}\kern0.5em \frac{\frac{\pi }{2}+ta{n}^{-1}\left(-3{g}_y{g}_{y+\Delta y}\right)}{\pi }\ {e}^{-{\left({g}_x+{g}_{x+\Delta x}\right)}^2}{e}^{-{\left({g}_y+{g}_{y+\Delta y}\right)}^2} $$(17)
- 5.Computing Laplacian for current pixel$$ {\nabla}^2u = \frac{u\left(x+\Delta x,y\right)+u\left(x-\Delta x,y\right)-2u\left(x,y\right)\kern0.5em }{4}+\frac{u\left(x,y+\Delta y\right)+u\left(x,y-\Delta y\right)-2u\left(x,y\right)\kern0.5em }{4} $$(18)
- 6.
Updating u = u _{0} + λ SWN ∇^{2} u _{0} for the current pixel
- 7.
Going to step 4 and continuing
- 8.
Finishing when the whole of the image is scanned for ten times.
Results and discussions
TV for 6 noisy and denoised images with noise variance 0.005
Noise variance (0.005) | TV (original mage) | TV (noisy image) | TV (ROF model) | TV (Wiener filter) | TV (our method) |
---|---|---|---|---|---|
Boat | 1.6 | 2.09 | 1.31 | 1.42 | 1.22 |
Man | 1.76 | 2.20 | 1.32 | 1.51 | 1.22 |
House | 1.3 | 1.81 | 1.21 | 1.27 | 1.14 |
Cameraman | 1.67 | 2.13 | 1.33 | 1.53 | 1.20 |
Lena | 1.26 | 1.74 | 1.16 | 1.24 | 1.12 |
Barbara | 1.57 | 2.07 | 1.15 | 1.36 | 1.10 |
TV for 6 noisy and denoised images with noise variance 0.1
Noise variance (0.1) | TV (original image) | TV(noisy image) | TV (ROF model) | TV (Wiener filter) | TV (our method) |
---|---|---|---|---|---|
Boat | 1.6 | 8.68 | 1.63 | 2.20 | 1.43 |
Man | 1.76 | 8.28 | 1.62 | 2.26 | 1.42 |
House | 1.3 | 8.72 | 1.59 | 2.12 | 1.39 |
Cameraman | 1.67 | 8.44 | 1.61 | 2.22 | 1.41 |
Lena | 1.26 | 8.11 | 1.53 | 2.12 | 1.34 |
Barbara | 1.57 | 8.50 | 1.54 | 2.14 | 1.35 |
This relation shows that in the blurred locations, identified by the blurring detector, u is restored by the forth order of differentiation in the blurred areas.
Conclusion
We presented a noise removal method using a local switching noise estimator in an energy functional minimizing process. We showed that in addition to using Laplacian in edge detection tasks, it can be used for noise removal applications. Smoothness can be easily controlled just by the number of iterations. We compared the method with some classic methods like ROF and Wiener filters based on TV value and also with the state-of-the-art BM3D. The result of denoising is totally comparable with ROF model. The computation time is equal to ROF model with 10 iterations. Based on experimental results we concluded that in relation to classic filters like ROF the method appears to be easy, fast and applicable for many noisy images. We analyzed that the technique can be applied for other image processing applications like deblurring by defining the appropriate detector and weight functions. The main disadvantage of the method is its filtering action on the area of the image including low intensity edges. The method can be improved to represent better response by defining better noise detectors.
Declarations
Authors’ Affiliations
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Copyright
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