Engineering, 2013, 5, 522-526 Published Online October 2013 (
Copyright © 2013 SciRes. ENG
Two Modifications of Weight Calculation of the Non-Local
Means Denoising Method
Musab Elkheir Salih, Xuming Zhang*, Mingyue Ding
School of Life Science and Technology, Huazhong University of Science and Technology, Wuhan, China
Email: *
Received 2013
The non-local means (NLM) denoising method replaces each pixel by the weighted average of pixels with the sur-
rounding neighborhoods. In this paper we employ a cosine weighting function in stead of the original exponential func -
tion to improve the efficiency of the NLM denoising method. The cosine function outperforms in the high level noise
more than low level noise. To increase the performance more in the low level noise we calculate the neighborhood si-
milarity weights in a lower-dimensional subspace using singular value decomposition (SVD). Experimental compari-
sons between the proposed modifications against the original NLM algorithm demonstrate its superior denoising per-
formance in terms of peak signal to noise ratio (PSNR) and histogram, using various test images corrupted by additive
white Gaussian noise (AWGN).
Keywords: Non-Local Means; Singular Value Decomposition; Weight Calculation
1. Introduction
Image denoising is one of the most important techniques
and necessary preprocessing steps in image processing
and computer vision. Denoising is to remove unwanted
noise to enhance and restore the original image. Many
methods for image denoising have been suggested, an d a
review of them can be found in [1]. Among them that
paper also proposes the NLM denoising method, which
gives state-of-the-art results. This method replaces each
pixel with a weighted average of other pixels with similar
neighborhoods. The basic idea is that images contain re-
peated structures, and ave raging t hem wil l reduce the nois e
[2]. This new concept for image denoising is popular in
other image processing areas, such as texture synthesis,
where a new pixel is synthesized as the weighted average
of known image pixels with similar neighborhoods [3-5].
The issue of measuring similarity between pixels, or
patches is a hot topic in many applications of image
processing, machine learning, computer vision, i nfo rma -
tion retrieval and document clustering. Document clus-
tering is the measuring of similarity between documents.
In this the document is mapped into the vector space, and
similarity is measured by the angles between the vectors
using cosine function. The smaller the angle between two
documents, the larger the cosine of the angle is and the
higher the similarity is; and vice versa [6]. So the cosine
is a decaying function of the angle between two docu-
The original NLM filter uses the expon ential decaying
function as a weighting function for the similarity mea-
surement and weight calculation. In compare to cosine
function the exponential function decays faster than co-
sine function and this will reduce the effect of averaging
of the NLM filter. Because of this, we proposed a NLM
denoising method using the cosine function. The outper-
formance of cosine function mainly appears in high level
noise. To increase the performance more in the low level
noise we combine the cosine weighting function and
SVD. The SVD belongs to a class of dimensionality re-
duction techniques that deal with the uncovering of latent
data structures and often associate noise to the least im-
portant components. Dimensionality reduction is a noise
reduction process. Removing noise before calculate the
similarities between patches around the pixels will en-
hance the weight calculation and increase the perfor-
mance of NLM algorithm.
The rest of this paper deals about the following. Sec-
tion II introduces the classical NLM algorithm. In section
III, the proposed method followed to modify the classical
NLM algorithm is introduced. In this, we use a cosine
weighting function for the weight calculation, then com-
bine the cosine function with SVD in the low level noise.
The results & discussions for four test images at various
noise levels are shown in Section IV. Section V con-
cludes the paper and proposed future work for more im-
provement to the NLM denoising method.
*Corresponding a uthor.
Copyright © 2013 SciRes. ENG
2. Non-Local Means Filter
The NLM method [1,7] estimates the intensity of each
pixel x in the noisy image u by a weighted average of all
of the pixel intensities in the image (as a convention, we
will refer to the pixel being denoised at any given time as
the “pixel of interest” (POI), denoted x, and denote all
other pixels as y). The weights w(x, y) reflect the proba-
bility that the POI (x) has the same intensity value as the
pixel is being compared to (y). This probability is based
on the similarity between the neighborhoods around x
and y.
The weighting function can be considered as a de-
creasing function depending on the similarity of the
patches around the POI. If a particular local difference
has a large magnitude then the value of w(x, y) will be
small and therefore that measurement will have little
effect on the output image. A small neighborhood, or
“patch”, around each pixel is used to compute the L2
norm. The weighting factor w(x, y), is then a normalized
weighted Gaussian function of this L2 norm. Consider a
discrete noisy image u = f + n, in which n is AWGN.
The NLM filter is written as
where Nd(y) represents the square patch of size (2d + 1)
× (2d + 1) centered at x, and W is a normalizing term,
W(x) =
( )
. The parameter h will be referred
to as the filter parameter that controls the decay of the
exponential expression in the weighting scheme. This
parameter is typically controlled manually in the algo-
rithm. Choosing a very small h leads to noisy results
identical to the input, while very large h gives an overly-
smoothed image [8].
3. Methodology
3.1. Weight Calculation Using Cosine Function
The L2 norm of the difference between two pixels was
used as input to the cosine function instead of the expo-
nential function. Figure 1(a) demonstrates the behavior
of the two functions to an arbitrary input. The exponen-
tial function was decayed faster than cosine function,
which lead to finishing the weight to zero value instantly
for dissimilar patches, or to a very small weight values
even for the little bit dissimilar patches. This fast decay
reduced the effect of averaging and smoothing of the
NLM filter. On the other side, the cosine function’s de-
cay was slower than exponential function, and so it gave
a considerable weight values for the dissimilar or the
little bit dissimilar patch es instead of zero or small values.
Because of this, our proposed method was based on us-
ing the cosine function instead of the exponential one.
For more investigation of different decay behaviors of
the two decreasing functions, we calculated the weight
corresponding to th e dissimilar ity measurement o f a ll 7 ×
7 patches in 13 × 13 search window around the pixel (20,
20) in the left top smooth sky background region of the
house image corrupted by white Gaussian noise, with
standard deviation (σ) = 40, shown in Figures 1(b). It
appears that cosine function has given higher suitable
weight values while the exponential function has given
small weight values.
This is the major advantage of using cosine function
and results shown in Figures 1(c), (d) proved this. This
advantage has an important effect in the weight calcula-
tion of the NLM denoising algorithm, so the cosine func-
tion was more robust weighting function, and we used it
in NLM algorithm denoising for an efficient i mpl eme n -
tation of this algorithm.
3.2. Weight Calculation in the Reduced
Dimensional Space Using Singular Value
Decompositio n
In 1965 G. Golub and W. Kahan introduced the SVD as a
decomposition technique for calcu lating the singular val-
ues, pseudo-inverse and rank of a matrix [9]. The tech-
nique decomposes a matrix A into three new matrices
U is a matrix whose columns are the eigenvectors of
the AAT matrix. These are termed the left eigenvectors.
S is a matrix whose diagonal elements are the singular
values of A. These are ordered in decreasing order along
Figure 1. (a) Exponential and cosine weighting function; (b)
Dissimilarity measurement of the pixel (20, 20) of house
image; (c ) and ( d) We ight c alcu latio n usi ng ex pone ntial f unc-
tion and cosine function corresponding to similarity mea-
surement respectively.
( )( )()
fx,yw x,y uy
( )( )( )
( )
( )
( )
, exp
wxyuN xuNy
Wx h
= −
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the diago na l of S, i.e. s1 > s2 > s3 > ··· sn.
V is a matrix whose columns are the eigenvectors of
the ATA matrix. These are termed the right eigenvectors.
When computing the SVD of a matrix is desirable to
reduce its dimensions by keeping its first k singular val-
This process is termed dimensionality reduction, and
AK is referred to as the rank k Approximation of A, or the
Reduced SVDof A. If we eliminate dimensions by
keeping the three largest singular values, this is a rank 3
approximation [10]. The top k singular values are se-
lected as a mean for developing a latent semanticsre-
presentation of A that is now free from noisy dimensions.
This “latent semanticsrepresentation is a specific data
structure in low-dimensional space in which documents,
terms and queries are embedded and compared. This
hidden or “latent” data structure is masked by noisy di-
mensions and becomes evident after the SVD. We re-
placed the distances
( )
( )
( )
( )
uN xuNy
where Pd represents the projections of Nd onto the low-
er-dimensional space determined by the SVD. If Nd is a
particular row in A, then the weights for Nd are just the
corresponding row in U multiplied by diagonal elements
of S. The weights for Nd are referred to as projection of
Nd into the k-dimensional space. Better denoising is ob-
tained when similarity between pixels is computed using
the dim e ns i on reduction introduced by the SVD.
4. Experiment Results and Discussions
In the experiment, the size of patch, search window, h
parameter values and rank k Approximation were se-
lected corresponding to the best PSNR value. Our pro-
posed NLM algorithm was applied on four test images
(Lena, Cameraman, Pepper, and House); Using a 256 ×
256 image size for all the images, and a 128 × 128 image
size for Lena image. The test images were corrupted by
AWGN with zero mean at σ = 10 (low noise level), and
40 (high noise level). The results were shown using
PSNR in decibels (dB) and histogram to demonstrate the
superior performance of the proposed method in noise
reduction. We used the PSNR measurement defined as
10log MAX
where, MSE is Mean Square Error, and MAX is maxi-
mum intensity value.
Table 1 lists The PSNR comparison results. Us ing co-
sine function produces better PSNR values when com-
pared to the original NLM algorithms for all the different
noise levels. The average performance is high (0.7054
dB) for the all images corrupted by a high level noise,
while it is low (0.0440 dB) for the all images corrupted
by low level noise.
That means the cosine function outperforms in the
high level noise and this because of the dissimilarity be-
tween patches will increase more than in the low level of
noise the and so the exponential function will give zero
or small weight values while cosine weight function w ill
give them more and suitable weight values. This is useful,
and it enhances the effect of averaging and smoothening
of the NLM filter.
The results s ho w that the most cosine function supe-
riority, where the performance is (1.12 dB), mainly ap-
pears in the noisy (AWGN) house image, (σ = 40). To
increase the average performance for the images cor-
rupted by low level noise, which is low as we mentioned
previously, we combined the SVD with cosine function
to calculate the distances between the pixels in the low
dimensional space rather than the full space. The results
showed an increased accuracy over using the full space.
The average performance increased form (0.0440 dB to
0.2505 dB). This is because the hidden or “latent” data
structure is masked by noisy dimensions and becomes
evident after the SVD. And similarity computed in low-
er-dimensional space becomes more accurate because
SVD remove noisy dimensions.
Although we only applied the SVD on images cor-
rupted by low level noise to increase the performance of
the algorithm; but also it will work for images corrupted
by moderate level noise. Because the noise will increase
more than in the low level of noise but SVD will remove
it and the weight calculation will be more efficiency. Our
method works well for all the images, but it works better
for images (House and Pepper) which have a smooth
region more than images (Lena and Cameraman) which
have more details.
Table 1. Performance of the original nonlocal means de-
noising algorithm and the modified algorithm.
10 40
Exp Cos Cos + SVD Exp Cos
Lena 32.0748 32.0174 32.1233 23.5971 24.1023
Cameraman 33.0014 33.0629 33.3857 25.3136 25.8697
Pepper 33.0034 34.0264 34.1926 25.7601 26.4025
House 35.1249 35.2667 35.4981 27.5251 28.6433
Average 33.5494 33.5934 33.7999 25.5490 26.2544
Performance 0.0440 0.2505 0.7054
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Figure 2. Histograms of the: (a) Original image (b). Noisy image. (c) and (d) Denoised image by nonlocal means denoising algo-
rithm using exponential and cosine function respectiv ely.
Figure 2 shows, that using exponential or cosine func-
tions with the NLM filter retrieves a similar histogram
and distinguishes three main peaks of the histogram of
the original image. However, the cosine function in-
creases the sharpness of the peaks and the contrast be-
tween them, because there is a small peak appears in the
histogram of the filtered image using the cosine function
and do not appear when using exponential f unction.
5. Conclusion and Future Work
This work shows that using cosine weighting function
and computing the similarity in lower-dimensional space
increases the effect of averaging and denoising perfor-
mance of the NLM deno ising algorithm, in low and high
level noise especially for smooth images. The next work
is seeking for another and most appropriate weighting
function that will enhance this filter even for images
containing more details. Furthermore, it will be very
suitable if we use the weighting function locally instead
of globally to suit the structure of the image. By selecting
a function preserves the edges for image with high details,
and for a smooth region choosing a function preserves
the smoothness. Also a future work is to find another
robust dimensionality reduction method to remove the
noisy dimensions instead of SVD. This will be a good
direction to improve the efficiency of NLM algorithm.
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