# Bilateral image denoising in the Laplacian subbands

- Bora Jin
^{1}, - Su Jeong You
^{2}and - Nam Ik Cho
^{1}Email author

**2015**:26

https://doi.org/10.1186/s13640-015-0082-5

© Jin et al. 2015

**Received: **23 September 2014

**Accepted: **23 July 2015

**Published: **6 August 2015

## Abstract

This paper presents an image denoising algorithm, which applies bilateral filtering (BLF) in the Laplacian subbands. It is noted that the subband images have wider area of photometric similarity than the original, and hence, they can be more benefited by the BLF than the original. Specifically, an image is Gaussian filtered to obtain a low band image, and the low band image is subtracted from the original to have the high band signal, which forms the Laplacian subbands. For the high band image denoising, we derive an adaptive kernel that is dependent on the edge intensity and photometric similarity of subband images. The high band image is convolved with this kernel and then added to the denoised low band signal, which produces the denoised image. We also propose to process the denoised high band signal by the gradient histogram preservation method, for sharpening the edges with less noise amplification. Experimental results show that the proposed denoising method provides higher PSNR than the original BLF and other multi-resolution denoising algorithms. Since the high band image is also effectively denoised in this process, the sharpened image by high band modification is also visually more pleasing when compared with the results of the conventional sharpening methods.

### Keywords

Bilateral filter Denoising Image enhancement## 1 Introduction

Image denoising is a fundamental process in image formation, transmission and display systems, and thus a huge number of methods have been developed. The overview of classical linear filtering and some of recently developed nonlinear methods can be found in [1], where the relations of different nonlinear methods are also well explained. For suppressing the noise while keeping the edges, the state-of-the-art methods use the similarities of pixels locally or globally. For example, a simple yet effective local-similarity method is the bilateral filtering (BLF) [2, 3], and the representatives of global similarity methods are nonlocal means (NLM) filtering [4] and block matching 3D (BM3D) algorithm [5].

The edge-sharpening is also an important topic in image processing, which enhances the visual quality of images [6–9]. One of the classical edge enhancement methods is to use the “unsharpening filter”, where an image is low pass filtered and subtracted from the original, which leaves the high band signal that contains the edges. The high band signal is then amplified and added to the low pass filtered image, which is the edge-sharpened result. When the Gaussian filter is used for the low pass filtering, its subtraction from the original is the Laplacian of Gaussian, and thus the subband images so formed are called the Laplacian Pyramids [6]. In this process, since the noise in the high band can also be amplified, it is necessary to denoise all the subband images in the Laplacian pyramid.

In this paper, we modify the BLF for denoising the Laplacian subband images, which is aimed as a new denoising algorithm that works better than the original BLF and also as an efficient method of suppressing high band noise when sharpening the edges in the Laplacian pyramids. The idea of applying the BLF to the Laplacian subband images is based on the observation that the BLF works better when there are more photometric similarities in the images, and the subband images have wider area of photometric similarity than the original. However, since the properties of subband images are different from the original, we need to design a new filtering kernel, which is one of the modifications proposed in this paper. Also, for the edge enhancement with noise suppression, we propose a new enhancement technique which restores the strength of edges that are smoothed by filtering and then adds the restored edges to the high band signal.

Experiments on the images corrupted by pseudo white Gaussian noise, shot noise, and mixture noise are performed, and it is shown that the proposed method improves PSNR than the original BLF and other local neighbor methods based on the subband decomposition. For evaluating the results for the real camera noises, we capture noisy images under low light conditions and compare the visual qualities. Also, considering that the addition of multiple tripod captured images as a reference, we compare the PSNR. Comparisons on real noisy images also show that the BLF in the Laplacian subbands improves the denoising performance than the original BLF. Also, modification of the edge coefficients in the high band gives sharpened images with less noise amplification than the conventional edge-sharpening method in the Laplacian pyramids. When compared with the nonlocal approaches such as NLM [4] and BM3D [5], the proposed method shows lower or similar PSNR for the white Gaussian noise like any other local adaptive filters. However, for the real noise and mixed noise, the proposed method shows comparable or sometimes higher PSNR than the nonlocal methods while requiring much less computations due to the nature of local filtering. In summary, the proposed method shows better results than the conventional BLF and other subband filtering schemes such as [10, 11], which are the representatives of local adaptive filtering methods, and shows comparable results to the nonlocal methods for the nonstationary noises while requiring less computations. Hence, the Laplacian subband BLF can be a reasonable choice for denoising and enhancing the images when fast or real-time implementation is needed.

## 2 Related works

### 2.1 Laplacian subbands

*I*, the Gaussian filter is applied iteratively with downsampling at every step. This process can be described as

*↓*

_{2}(·) denotes the downsampling by 2 and

*G*

*a*

*u*

*s*

*s*

*i*

*a*

*n*(·) is the Gaussian filtering. Then, the Laplacian subbands are defined as

where *↑*
_{2}(·) denotes the upsampling by 2 and *n* is the level of pyramid. In this paper, we use just two levels of Laplacian subband (*n*=1), where *L*
_{1} denotes the high-frequency subband and *L*
_{2}=*G*
_{1} represents the low-frequency subband.

### 2.2 Bilateral filter

*p*and

*q*denote pixel positions,

*N*

_{ p }is the neighbor of

*p*,

*I*(

*q*) is the intensity of input image at a pixel

*q*,

*W*is the normalizing factor \(W=\sum _{q \in N_{p}} w(p,q),\) and

*w*(

*p*,

*q*) is the kernel of the BLF defined as [2]

where *σ*
_{
d
} is the bandwidth for the spatial distance and *σ*
_{
r
} for the photometric distance. For successfully reducing noise variance while keeping the edges, it is important to find the balance between *σ*
_{
d
} and *σ*
_{
r
}, and also to find an appropriate size of the neighbor.

## 3 Bilateral filtering in the Laplacian subbands

### 3.1 Example of subband BLF for a 1-D signal

We first show a simple denoising example with a synthetic 1-D signal, which motivates to apply the BLF to Laplacian subbands. Note that the kernel of bilateral filter in Eq. (4) is consisted of two terms, i.e., geometric and photometric terms. From this, we can see that the photometric weights would be kept large for wider area when a pixel *p* is in the flat area where *I*(*p*) and *I*(*q*) are similar, and hence many neighboring pixels can contribute for the denoising. On the contrary, when the pixel is in the non-flat area where ∥*I*(*p*)−*I*(*q*)∥ is large, the photometric weights diminish and thus the neighboring pixels less contribute for the denoising.

Mean squared errors of the BLF results for the original and subband signals

Area | Original | Original BLF | Subband BLF |
---|---|---|---|

Overall ([0,255]) | 9.12 | 3.24 | 2.11 |

Flat ([0,39]) | 10.8 | 4.58 | 2.86 |

Edge ([80,119]) | 6.11 | 1.55 | 2.64 |

Slope ([150,189]) | 8.53 | 3.45 | 1.67 |

### 3.2 Proposed subband BLF

*L*

_{2}and high band

*L*

_{1}. For the low band image

*L*

_{2}, we apply the conventional BLF with

*σ*

_{ d }=1.8 and

*σ*

_{ r }=

*σ*as suggested in [10], where

*σ*is the noise variance. As stated above, we concentrate on the filtering scheme for the high band image

*L*

_{1}, especially at the edge area. The basic idea is to give larger weights to the pixels that have similar edge intensities as well as pixel intensities. Also, when it is highly probable that a pixel is on the edge, it needs to be less affected by the neighboring pixels. These ideas are encoded into a new guidance term in addition to Eq.(4) as

where \({\sigma _{h}^{2}}(p)\) is the pixel dependent bandwidth, and *h*(*p*) is the intensity of the pixel *p* in the histogram-equalized image of *L*
_{1} which will be explained later in more detail. Comparing this kernel with that of the original BLF in Eq. (4), the third term is our proposal which adaptively controls the weights near the edge areas. The adaptive bandwidth for the BLF has already been considered in [9], where the *σ*
_{
r
} is adjusted along with an offset parameter by the optimization method with some training images. Unlike this previous adaptive BLF, our method is quite a simple algorithm which adjusts *σ*
_{
h
} in the new kernel depending on whether the pixel is on the edge or not.

In summary, our method employs a new guidance image *h*(*p*) in the manner of joint bilateral filtering [3], for reducing the weights on the edge pixels and vice versa. For this, we let the bandwidth in the new term to be pixel dependent, i.e., the pixel difference in the high band (∥*h*(*p*)−*h*(*q*)∥) is considered in weight control. Precisely, our method adjusts *σ*
_{
h
}(*p*) to \(2\sqrt {2}\sigma \) or \(4\sqrt {2}\sigma \) depending on the edge strength of given image.

*L*

_{2}, which is denoted as \(\hat {L_{2}}\). For determining whether a pixel is an edge pixel or not, we apply the Laplacian of Gaussian filter and then thresholding. Specifically, we convolve \(\hat {L_{2}}\) with the kernel defined as

*%*of the mean value are considered the edge pixels. This gives an edge map

*E*(

*p*) which is 1 when the pixel

*p*belongs to edge pixels, and 0 if not. For simplicity, the edge map is obtained from the approximate intensity component (image of (

*R*+

*G*+

*B*)/3), and this edge map is applied to all of color components equally. With this edge map, the kernel bandwidth is determined as

It can be seen that the kernel bandwidth is small when the pixel is on the edge, so that the neighboring pixels less contribute to the averaging and thus the edge intensities are less changed. Conversely, the pixels in the flat areas are more strongly filtered than the edge pixels. It is worth to mention that we use ∥*h*(*p*)−*h*(*q*)∥ (histogram equalized intensities of *L*
_{1} into the range [0,255]) instead of ∥*L*
_{1}(*p*)−*L*
_{1}(*q*)∥, because *L*
_{1}(*p*) can have negative value and its dynamic range is large. Denoting the output of proposed BLF of *L*
_{1} as \(\hat {L}_{1}\), the final denoised image is obtained as \(\hat {L}_{1} + \hat {L}_{2}\).

Throughout the experiments, it is found that the low-band (*L*
_{2}) filtering with a variety of parameter changes does not much affect the overall performance. Hence, we apply just the original BLF with *σ*
_{
r
}=*σ* for the low-band filtering, and we have focused on the kernel design for the high-frequency subband *L*
_{1}. Also, when comparing the results between the adaptive bandwidth and non-adaptive bandwidth (when *σ*
_{
h
} is fixed), the gain by the adaptive scheme is not significant (under 0.1dB PSNR gain) because the edge area is small compared to others.

*λ*=1 corresponds to the proposed Laplacian subband filtering explained above, and 0≤

*λ*<1 gives the edge enhanced results to be explained in the next section.

### 3.3 Image enhancement with the Laplacian subband denoising

**x**is the original image,

**v**is the noise, and

**y**is the observed noisy image, the processed image is constrained to have similar gradient histogram as

**x**. In [12], considering the histogram of gradients of

**y**as the discretization of the pdf of gradient distribution of

**y**, the gradient histogram of the original image

**x**is found by solving

*H*

_{ x },

*H*

_{ y }, and

*H*

_{ v }are the gradient histograms of

**x**,

**y**, and

**v**respectively, ⊗ is the convolution operator, and

*c*·

*R*(

*H*

_{ x }) is a regularization term. For solving this problem,

*H*

_{ y }is estimated from the observed data and

*H*

_{ v }is modeled as a hyper-Laplacian distribution as [12]:

where *k* is normalization factor.

*L*

_{1}in our subband BLF scheme is also a kind of gradient image, where the above GHP approach can be applied. Applying the Laplacian subband decomposition to Eq. (8), we have the high band relationship as

*H*

_{ r }, we obtain it in a similar manner as Eq. (9), except that the positive and negative coefficients are considered separately in order not to diminish the peaks of coefficients that appear around the edges. To be specific, we obtain

*H*

_{ r }as

where *H*
_{
y,+} is the histogram of positive values in \(\hat {L_{1}}\), *H*
_{
y,−} for the negative values, and *H*
_{
v
} is the histogram of \(\mathcal {L}_{1}({\mathbf v})\) that is modeled as Eq. (10). The range of parameters for solving this problem is set the same as [12], i.e., *κ*∈[0.001,3] and *γ*∈[0.02,1.5]. Then, the histogram of *L*
_{1} is matched to *H*
_{
r
}, which is denoted as \(\hat {L}_{1,matched}\) in Fig. 2 and the edge enhanced image is obtained as \(\hat {L}_{1}=\lambda \cdot \hat {L}_{1,f}+\\(1-\lambda)\cdot \hat {L}_{1,matched}\).

## 4 Experimental results

### 4.1 Experiments on pseudo white and Poisson noise

*σ*or Poisson noise with parameter

*Q*. We compare our subband BLF (SBLF) algorithm with the original BLF [2], multiresolution bilateral filter (MBLF) [10], BLS-GSM [11], NLM [4], and BM3D [5] with the authors’ source codes. According to [10], we set

*σ*

_{ d }=1.8 and

*σ*

_{ r }=2

*σ*. Also, 9×9 windows are used for the original BLF, MBLF, and the proposed method. The multiresolution BLF is implemented in MATLAB, and others are implemented in MATLAB and C/C++ through MATLAB MEX functions, and the codes are run on a PC with an Intel Core i5 CPU and 4 GB RAM.

*σ*=20,30,40,50. PSNR for each of the images and other experimental results are available at http://ispl.snu.ac.kr/~idealgod/SBLF, where our source code and full-resolution images of all the figures in this paper are also available. As shown in Table 2, the proposed method yields better results than BLF and MBLF, and comparable results with the BLS-GSM. When compared with the nonlocal methods, the proposed method shows better PSNR than NLM, but lower than BM3D. However, as shown in the last row of Table 2, the proposed method needs much less computation time than the nonlocal methods, as well as other local methods except the original BLF. Figure 4 is a sample set of restored images, which shows that the proposed SBLF provides better visual quality than other local methods, and comparable quality with BM3D.

Averaged PSNRs for AWGN

| Local self-similarity | Nonlocal self-similarity | ||||
---|---|---|---|---|---|---|

BLF | MBLF | BLS-GSM | SBLF | NLM | BM3D | |

20 | 30.2919 | 30.8692 | 32.4754 | 32.4435 | 32.7027 | 34.6959 |

30 | 27.8154 | 28.3605 | 30.1490 | 30.2798 | 29.9845 | 32.2624 |

40 | 25.9968 | 26.2451 | 28.2862 | 28.5453 | 27.8337 | 30.1545 |

50 | 24.5750 | 24.3909 | 26.7048 | 27.0307 | 25.9953 | 28.6703 |

Time (s) | 0.2788 | 10.2818 | 18.1894 | 0.8729 | 88.7329 | 5.7958 |

*Q*∈{5,10,15}. It can be seen that the results show similar trends as the Gaussian noise case.

Averaged PSNRs for Poisson noise

| Local self-similarity | Nonlocal self-similarity | ||||
---|---|---|---|---|---|---|

BLF | MBLF | BLS-GSM | SBLF | NLM | BM3D | |

5 | 30.4115 | 30.4539 | 30.3704 | 31.8086 | 30.2686 | 33.5917 |

10 | 28.6911 | 28.5554 | 29.0285 | 30.0963 | 28.2948 | 31.7019 |

15 | 27.5984 | 27.3157 | 28.2461 | 28.9788 | 27.2667 | 30.4961 |

Time(s) | 0.3078 | 10.8430 | 16.8132 | 0.9421 | 21.5664 | 6.9882 |

### 4.2 Experiments on mixed noise

*σ*=10, and Table 6 presents the results for 10 % impulse noise + Poisson noise with

*Q*=10. It can be seen that the proposed method shows comparable or sometimes better PSNR than the BM3D. The reason for these results seems that there are not much similar patches for the nonlocal methods when there are randomly distributed impulsive noises.

PSNRs for mixed noise (20 % impulse noise)

Image | Local self-similarity | Nonlocal self-similarity | ||||
---|---|---|---|---|---|---|

BLF | MBLF | BLS-GSM | SBLF | NLM | BM3D | |

1 | 24.7155 | 22.3321 | 24.1887 | 25.0244 | 23.6376 | 24.9905 |

2 | 22.7271 | 22.3321 | 23.1560 | 23.5464 | 22.8560 | 24.9531 |

3 | 24.1931 | 23.5109 | 23.2620 | 25.1243 | 23.5753 | 23.2513 |

4 | 24.5633 | 23.5109 | 24.3265 | 25.6176 | 24.3059 | 24.8292 |

5 | 24.2419 | 23.0465 | 23.4489 | 24.9681 | 23.7099 | 25.9463 |

6 | 26.8504 | 22.8714 | 26.7785 | 27.3136 | 24.0875 | 24.9552 |

7 | 24.7198 | 24.5795 | 23.9426 | 25.3261 | 25.9881 | 27.2638 |

8 | 22.8777 | 24.5795 | 22.8013 | 24.2091 | 24.5127 | 27.1690 |

9 | 21.8124 | 23.2055 | 21.4086 | 22.6783 | 23.2277 | 25.4997 |

10 | 24.2438 | 23.2055 | 22.9440 | 24.7666 | 22.3793 | 23.8574 |

11 | 23.4927 | 22.1039 | 22.7081 | 24.1268 | 21.1000 | 22.6501 |

12 | 24.4074 | 22.1039 | 23.8970 | 25.3700 | 22.3230 | 24.5317 |

13 | 23.3632 | 21.2694 | 22.7738 | 23.9723 | 22.3291 | 24.0320 |

14 | 23.5229 | 22.8539 | 23.0868 | 24.0552 | 22.7604 | 25.3259 |

15 | 25.8834 | 22.8539 | 25.2027 | 26.6016 | 21.0988 | 25.2427 |

16 | 21.2509 | 21.9349 | 20.6449 | 21.9225 | 23.5069 | 23.5867 |

17 | 24.0441 | 24.5466 | 24.4997 | 24.7115 | 22.8812 | 23.9613 |

Avg. | 23.9359 | 22.9906 | 23.4747 | 24.6667 | 23.1929 | 24.8262 |

Avg. time | 0.2767 | 9.3545 | 16.8914 | 0.9955 | 38.2350 | 9.0230 |

PSNRs for mixed noise (20 % impulse noise + Gaussian noise *σ*=10)

Image | Local self-similarity | Nonlocal self-similarity | ||||
---|---|---|---|---|---|---|

BLF | MBLF | BLS-GSM | SBLF | NLM | BM3D | |

1 | 25.1272 | 22.1891 | 24.1444 | 24.8254 | 23.5478 | 24.9216 |

2 | 23.3915 | 21.0752 | 22.8482 | 23.2027 | 23.3204 | 22.9711 |

3 | 24.3678 | 23.4815 | 23.2735 | 24.9511 | 24.2775 | 24.7898 |

4 | 24.7087 | 23.0925 | 24.3643 | 25.6365 | 23.7272 | 25.9443 |

5 | 24.5624 | 22.8267 | 23.4414 | 24.9404 | 24.0864 | 24.9208 |

6 | 27.1773 | 24.5983 | 26.7542 | 27.3291 | 25.9194 | 27.2793 |

7 | 25.0360 | 22.3752 | 24.0196 | 25.3808 | 24.5317 | 25.5305 |

8 | 23.5980 | 23.2134 | 22.7675 | 24.0974 | 22.3253 | 23.8019 |

9 | 22.3192 | 19.7029 | 21.3331 | 22.6534 | 21.0883 | 22.6279 |

10 | 24.6008 | 22.1769 | 22.9668 | 24.8366 | 22.4003 | 24.5818 |

11 | 24.0335 | 21.3128 | 22.7450 | 24.1277 | 22.4418 | 24.0698 |

12 | 24.5797 | 22.9088 | 23.9241 | 25.4467 | 22.8593 | 25.3869 |

13 | 24.1103 | 22.0438 | 22.8545 | 24.1076 | 23.5837 | 23.7107 |

14 | 24.0712 | 21.5176 | 23.0955 | 24.1099 | 22.8613 | 24.0113 |

15 | 25.9574 | 24.5380 | 25.1631 | 26.5531 | 24.5009 | 26.3651 |

16 | 21.7938 | 19.3832 | 20.7904 | 21.9789 | 20.7998 | 21.8605 |

17 | 24.8058 | 22.4147 | 24.4745 | 24.7011 | 24.3419 | 24.3968 |

Avg. | 24.3671 | 22.2853 | 23.4682 | 24.6399 | 23.3302 | 24.5394 |

Avg. time | 0.2779 | 10.3789 | 16.7026 | 0.9277 | 32.0858 | 8.3840 |

PSNRs for mixed noise (10 % impulse noise + Poisson noise *Q*=10)

Image | Local self-similarity | Nonlocal self-similarity | ||||
---|---|---|---|---|---|---|

BLF | MBLF | BLS-GSM | SBLF | NLM | BM3D | |

Image | BLF | MBLF | BLS-GSM | SBLF | NLM | BM3D |

1 | 28.5342 | 25.4233 | 27.4656 | 28.3195 | 27.6470 | 29.1604 |

2 | 25.8080 | 23.6747 | 25.0903 | 25.3725 | 26.0011 | 25.4205 |

3 | 25.2271 | 25.3809 | 24.9301 | 26.0284 | 25.6082 | 26.7212 |

4 | 25.2890 | 24.7387 | 25.7451 | 26.8534 | 26.2451 | 28.4774 |

5 | 25.9956 | 24.7914 | 25.1994 | 26.5095 | 26.3872 | 27.7189 |

6 | 29.0505 | 26.8902 | 28.9912 | 29.4322 | 28.8034 | 30.5975 |

7 | 26.5713 | 24.6991 | 26.4335 | 27.3299 | 27.0361 | 28.9383 |

8 | 23.1844 | 22.8339 | 22.4087 | 23.4954 | 23.3715 | 23.5693 |

9 | 24.4265 | 22.6400 | 24.2504 | 25.6088 | 24.9965 | 26.2820 |

10 | 26.3400 | 24.4213 | 25.6495 | 27.1205 | 25.2780 | 27.6961 |

11 | 26.3146 | 24.2364 | 25.2243 | 26.8433 | 25.8733 | 27.6886 |

12 | 24.9814 | 23.8919 | 24.7310 | 26.1163 | 25.0271 | 27.1157 |

13 | 27.3758 | 24.9263 | 26.2754 | 27.4453 | 27.0570 | 27.8019 |

14 | 26.9052 | 24.4638 | 26.1080 | 27.2045 | 26.7907 | 28.0307 |

15 | 26.3950 | 25.4183 | 25.7355 | 27.0892 | 26.1018 | 27.8844 |

16 | 24.0633 | 22.1646 | 23.4939 | 25.2126 | 24.3040 | 25.7068 |

17 | 27.5090 | 26.3678 | 27.7605 | 27.4656 | 27.5485 | 27.8591 |

Avg. | 26.1159 | 24.5272 | 25.6172 | 26.6733 | 26.1221 | 27.4511 |

Avg. time | 0.2777 | 10.3959 | 16.2441 | 1.1010 | 33.1900 | 9.4430 |

It is worth to note that all the algorithms need (estimated) noise variance as the input, for controlling the filter parameters. In the case of above simulated noises, we know the noise variance and use it for the kernel parameters. However, in the case of mixed noise and the real noise in the following subsection, we cannot know whether the estimate noise variance (by any of estimation methods) is accurate or not. Hence, we try many experiments with the input variance in the range of [10,70] and choose the best one for each of the algorithms.

### 4.3 Experiments with real noise

PSNRs for real noise

Image | Local self-similarity | Nonlocal self-similarity | ||||
---|---|---|---|---|---|---|

BLF | MBLF | BLS-GSM | SBLF | NLM | BM3D | |

1 | 32.5736 | 31.6800 | 33.0025 | 33.5443 | 34.1830 | 34.4310 |

2 | 36.5380 | 35.5875 | 37.0479 | 37.2656 | 37.8776 | 38.4951 |

3 | 22.6169 | 22.7788 | 22.2161 | 22.8347 | 22.5071 | 22.6712 |

Avg. | 30.5762 | 30.0154 | 30.7555 | 31.5482 | 31.5226 | 31.8658 |

### 4.4 Noisy image enhancement

*λ*=0.3. For enhancing the noisy images, a plausible method would be to denoise the image first and then apply the conventional image enhancement methods. Since the proposed method is based on the BLF, the comparison is performed with the schemes that apply BLF first and then enhance the image with [6] or [8]. Figures 7 and 8 show these comparisons, where (a) is the original image, (b) is the noisy one, (c) is the result of sequentially applying BLF denoising and high band amplification, (d) is the BLF followed by edge aware local Laplacian filtering [6], (e) is the result of sequentially applying BLF denoising and guided filtering [8], (f) is the result of denoising via TEID [12], (g) is the result of ABF [9], and (h) is the output of proposed algorithm. The figures show that the proposed method effectively suppresses the noise while enhancing the texture and edges. In the case of [12] (Fig. 7 f and Fig. 8 f), it can be seen that the noise is well removed while “preserving” the textures. On the other hand, the results in Fig. 7 h and Fig. 8 h show that the proposed method enhances the texture area (especially feather areas and patterns around the eyes), because the proposed scheme with

*λ*<1 adds the matched high frequency components to the denoised high band.

## 5 Conclusions

We have proposed an image denoising method based on the Laplacian subband decomposition and BLF. The input image is decomposed into two subbands by the Laplacian of Gaussian, and the BLF is applied to each of the subbands with appropriate filtering kernel and parameter. The experiments show that the proposed method increases PSNR compared to the original BLF and other multi-resolution filtering methods. For the real noisy images, the proposed method also yields comparable results to the non-local similarity methods such as BM3D and NLM, while requiring less computation time. Since the proposed method is based on the Laplacian decomposition, the edge enhancement can also be efficiently achieved along with the denoising.

## Declarations

### Acknowledgements

This work was supported in part by Samsung Electronics, and in part by the Ministry of Science, ICT and Future Planning, Korea, through the Information Technology Research Center support Program supervised by the National IT Industry Promotion Agency under Grant NIPA-2014-H0301- 14-1019.

## Authors’ Affiliations

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