An adaptive bandwidth nonlocal means image denoising in wavelet domain
EURASIP Journal on Image and Video Processing
An adaptive bandwidth nonlocal means image denoising in wavelet domain
Su Jeong You 0
Nam Ik Cho 0
0 Department of Electrical and Computer Engineering, INMC, Seoul National University , Gwanak-ro, Gwanak-gu, Seoul, 151-744 , Korea
This paper proposes a new wavelet domain denoising algorithm. In the results of conventional wavelet domain denoising methods, ringing artifacts or wavelet-shaped noises are sometimes observed due to thresholding of small but important coefficients or due to generation of large coefficients in flat areas. In this paper, nonlocal means filtering is applied to each subband of wavelet decomposition, which can keep small coefficients and does not generate unwanted large coefficients. Since the performance of nonlocal means filtering depends on the appropriate kernel bandwidth, we also propose a method to find global and local kernel bandwidth for each subband. In comparison with conventional methods, the proposed method shows lower PSNR than BM3D when pseudo white Gaussian noise is added, but higher PSNR than the spatial nonlocal means filtering and wavelet thresholding methods. For the mixture noise or Poisson noise, which may better explain the real noise from camera sensors, the proposed method shows better or comparable results than the state-of-the-art methods. Also, it is believed that the proposed method shows better subjective quality for the noisy images captured in the low-illumination conditions.
Introduction
Denoising is one of the fundamental image processing
problems and thus has been studied for a long time.
To name a few of the existing methods that are related
with our work and the state-of-the-art methods, there are
wavelet shrinkage methods [
1,2
], a total variation
minimization [3], a prior probability modeling [
4
], nonlocal
means filtering [
5
], and BM3D [
6
]. Among these, the
BM3D generally shows the highest PSNR when the noise
is additive white Gaussian.
In the case of wavelet domain thresholding methods
[
1,2
], an image is transformed into the wavelet domain,
and the coefficients in each subband are suppressed
by hard or soft thresholding. The advantage of wavelet
shrinkage methods is that they require not much
computations while providing pleasing results. The
probabilistic wavelet coefficient modeling method [4] fits the
neighborhoods of coefficients as Gaussian scale mixture
(GSM) model and applies the Bayesian least squares (BLS)
technique to adjust the coefficients. Although wavelet
shrinkage methods and BLS-GSM provide relatively high
PSNR improvement, shrinking or modifying wavelet
coefficients sometimes bring ringing or wavelet-shaped
artifacts. For example, wavelet transformation of a step edge
generates small coefficients up to the highest subbands.
Hence, when the small coefficients are removed by
thresholding and are inverse transformed, then ringing artifacts
arise due to the loss of high frequencies. In the case
of probabilistic wavelet coefficient modeling, unwanted
coefficients can be generated in the homogeneous region,
which result in wavelet-shaped artifacts in the spatial
domain. Another popular denoising method is the
nonlocal means filtering [
5
], which substitutes a noisy pixel by
the weighted sum of neighborhood pixels. The weights are
determined based on the kernel density estimation, which
can be regarded as a Nadaraya-Watson estimator, i.e., a
kind of local constant regression [
7
]. In other words, the
smooth kernel estimate in the nonlocal means approach
is a sum of bumps placed on the data points. The
kernel function determines the shape of the bumps, and the
‘smoothing parameter’ or ‘bandwidth’ controls the degree
of smoothness. In [
8
], an automatic bandwidth selection
method was proposed based on the reduction of entropy
of image patterns, and the global bandwidth was applied
to the overall area of image. However, it is noted that
narrower kernels are suitable for the complex regions,
whereas larger kernels would be better for more sparse
areas. Hence, it is important to find an appropriate
bandwidth according to the local characteristics, which is not
an easy task. One of the main factors that strongly
influence the local properties of the image is the noise statistics
in the neighborhood, and thus, the bandwidth needs to be
adaptively determined according to the local noise
variance. In summary, we need to estimate the local noise
statistics for finding an appropriate bandwidth for the
given region. There are many methods for estimating the
variance of white additive noise in images, but they cannot
be used for the images with non-uniform noise variance.
In this consideration, the estimation of local noise
statistics is necessary to find the appropriate bandwidth for the
given area.
In this paper, inspired by the performance of nonlocal
means filtering method in keeping the structures of the
image while suppressing the noise, we attempt to apply
the n (...truncated)