Image denoising with block-matching and 3D filtering
Kostadin Dabov, Alessandro Foi, Vladimir Katkovnik, and Karen Egiazarian
Institute of Signal Processing, Tampere University of Technology, Finland
PO BOX 553, 33101 Tampere, Finland
firstname.lastname@tut.fi
ABSTRACT
We present a novel approach to still image denoising based on eective filtering in 3D transform domain bycombining sliding-window transform processing with block-matching. We process blocks within the image in a
sliding manner and utilize the block-matching concept by searching for blocks which are similar to the currently
processed one. The matched blocks are stacked together to form a 3D array and due to the similarity between
them, the data in the array exhibit high level of correlation. We exploit this correlation by applying a 3D
decorrelating unitary transform and eectively attenuate the noise by shrinkage of the transform coe!cients.
The subsequent inverse 3D transform yields estimates of all matched blocks. After repeating this procedure forall image blocks in sliding manner, the final estimate is computed as weighed average of all overlapping blockestimates. A fast and e!cient algorithm implementing the proposed approach is developed. The experimental
results show that the proposed method delivers state-of-art denoising performance, both in terms of objective
criteria and visual quality.
Keywords: image denoising, block-matching, 3D transforms

1. INTRODUCTION
Much of the recent research on image denoising has been focused on methods that reduce noise in transformdomain. Starting with the milestone work of Donoho,1, 2 many of the later techniques3—7 performed denoising inwavelet transform domain. Of these methods, the most successful proved to be the ones4, 5, 7 based on rather sophisticated modeling of the noise impact on the transform coe!cients of overcomplete multiscale decompositions.
Not limited to multiscale techniques, the overcomplete representations have traditionally played a significant rolein improving the restoration abilities of even the most basic transform-based methods. This is manifested by
the sliding-window transform denoising,8, 9 where the basic idea is to successively denoise overlapping blocks
by coe!cient shrinkage in local 2D transform domain (e.g. DCT, DFT, etc.). Although the transform-based
approaches deliver very good overall performance in terms of objective criteria, they fail to preserve details which
are not suitably represented by the used transform and often introduce artifacts that are characteristic of thistransform.
A dierent denoising strategy based on non-local estimation appeared recently,10, 11 where a pixel of the true
image is estimated from regions which are found similar to the region centered at the estimated pixel. Thesemethods, unlike the transform-based ones, introduce very few artifacts in the estimates but often oversmooth
image details. Based on an elaborate adaptive weighting scheme, the exemplar-based denoising10 appears to be
the best of them and achieves results competitive to the ones produced by the best transform-based techniques.
The concept of employing similar data patches from dierent locations is popular in the video processing fieldunder the term of “block-matching”, where it is used to improve the coding e!ciency by exploiting similarity
among blocks which follow the motion of objects in consecutive frames. Traditionally, block-matching has
found successful application in conjunction with transform-based techniques. Such applications include video
compression (MPEG standards) and also video denoising,12 where noise is attenuated in 3D DCT domain.
We propose an original image denoising method based on eective filtering in 3D transform domain bycombining sliding-window transform processing with block-matching. We undertake the block-matching concept
for a single noisy image; as we process image blocks in a sliding manner, we search for blocks that exhibit similarity
to the currently-processed one. The matched blocks are stacked together to form a 3D array. In this manner,
Image Processing: Algorithms and Systems, Neural Networks, and Machine Learning
edited by E.R. Dougherty, J.T. Astola, K.O. Egiazarian, N.M. Nasrabadi, S.A. Rizvi
Proc. of SPIE-IS&T Electronic Imaging, SPIE Vol. 6064, 606414, © 2006 SPIE-IS&T · 0277-786X/06/$15
SPIE-IS&T/ Vol. 6064 606414-1

we induce high correlation along the dimension of the array in which the blocks are stacked. We exploit thiscorrelation by applying a 3D decorrelating unitary transform which produces a sparse representation of the truesignal in 3D transform domain. E!cient noise attenuation is done by applying a shrinkage operator (e.g. hardthresholding or Wiener filtering) on the transform coe!cients. This results in improved denoising performanceand eective detail preservation in the local estimates of the matched blocks, which are reconstructed by aninverse 3D transform of the filtered coe!cients. After processing all blocks, the final estimate is the weighted
average of all overlapping local block-estimates. Because of overcompleteness which is due to the overlap, weavoid blocking artifacts and further improve the estimation ability.
Although the proposed approach is general with respect to the type of noise, for simplicity of exposition, we
restrict our attention to the problem of attenuating additive white Gaussian noise (AWGN).
The basic approach and its extension to Wiener filtering are presented in Sections 2 and 3, respectively.
An e!cient algorithm which implements the proposed approach is developed in Section 4. Finally, Section 5 isdevoted to demonstration and discussion of experimental results.

2. DENOISING BY SHRINKAGE IN 3D TRANSFORM DOMAIN WITHBLOCK-MATCHING
Let us introduce the observation model and notation used throughout the paper. We consider noisy observationsz : X $ R of the form z (x) = y (x) +  (x), where x 5 X is
 a 2D spatial coordinate that belongs to the image
domain X  Z2 , y is the true image, and  (x)  N 0, 2 is white Gaussian noise of variance 2 . By Zx wedenote a block of fixed size N1 × N1 extracted from z, which has z (x) as its upper-left element; alternatively, we
say that Zx is located at x. With ye we designate the final estimate of the true image.
Let us state the used assumptions. We assume that some of the blocks (of fixed size N1 ×N1 ) of the true image
exhibit mutual correlation. We also assume that the selected unitary transform is able to represent sparsely theseblocks. However, the diversity of such blocks in natural images often makes the latter assumption unsatisfied
in 2D transform domain and fulfilled only in 3D transform domain due to the correlation introduced by blockmatching. The standard deviation  of the AWGN can be accurately estimated (e.g.1 ), therefore we assume its
a-priori knowledge.

2.1. Local Estimates
We successively process all overlapping blocks of fixed size in a sliding manner, where "process" stands for theconsecutive application of block-matching and denoising in local 3D transform domain. For the sub-subsections
to follow, we fix the currently processed block as ZxR , where xR 5 X, and denominate it as "reference block".
2.1.1. Block-matching
Block-matching is employed to find blocks that exhibit high correlation to ZxR . Because its accuracy is significantly impaired by the presence of noise, we utilize a block-similarity measure which performs a coarse initial
denoising in local 2D transform domain. Hence, we define a block-distance measure (inversely proportional tosimilarity) as
 



t
t


1 
2
2
T2D (Zx1 ) , thr2D  2 log (N1 ) 
T2D (Zx2 ) , thr2D  2 log (N1 ) (1)d (Zx1 , Zx2 ) = N1  ,2

where x1 , x2 5 X, T2D is a 2D linear unitary transform operator (e.g. DCT, DFT, etc.), is a hard-thresholdoperator, thr2D is fixed threshold parameter, and n·n2 denotes the L2 -norm. Naturally, is defined as, if || > thr(, thr ) =0, otherwise.
The result of the block-matching is a set SxR  X of the coordinates of the blocks that are similar to ZxR
according to our d-distance (1); thus, SxR is defined as
SxR = {x 5 X | d (ZxR , Zx ) <  match } ,SPIE-IS&T/ Vol. 6064 606414-2(2)

Figure 1. Fragments of Lena, House, Boats and Barbara corrupted by AWGN of  = 15. For each fragment blockmatching is illustrated by showing a reference block marked with ’R’ and a few of its matched ones.

where  match is the maximum d-distance for which two blocks are considered similar. Obviously d (ZxR , ZxR ) = 0,
which implies that |SxR |  1, where |SxR | denotes the cardinality of SxR .
The matching procedure in presence of noise is demonstrated on Figure 1, where we show a few reference
blocks and the ones matched as similar to them.
2.1.2. Denoising in 3D transform domain
We stack the matched noisy blocks Zx5SxR (ordering them by increasing d-distance to ZxR ) to form a 3D arrayof size N1 × N1 × |SxR |, which is denoted by ZSxR . We apply a unitary 3D transform T3D on ZSxR in order
to attain sparse representation of the true signal. The noise is attenuated by hard-thresholding the transform
1
yields a 3D array of reconstructed estimates
coe!cients. Subsequently, the inverse transform operator T3D

 
t


2)
e S = T 1
Z
,

Y

2
log
(N
,
(3)
T
3D
SxR
thr3D
xR
1
3D
e S comprises of |Sx | stacked local block estimates
where thr3D is a fixed threshold parameter. The array Y
xR
R
xR
e
Yx5Sx of the true image blocks located at x 5 SxR . We define a weight for these local estimates asR$ xR =

1
Nhar ,

if Nhar  1otherwise,

1,

(4)

where Nhar is the number of non-zero transform coe!cients after hard-thresholding. Observe that 2 Nhar ise S . Thus, sparser decompositions of ZS result in less noisy estimates whichequal to the total variance of YxRxR
are awarded greater weights by (4).

2.2. Estimate AggregationxR
After processing all reference blocks, we have a set of local block estimates Yex5S, ;xR 5 X (and their correxR
sponding weights $xR , ;xR 5 X), which constitute an overcomplete representation of the estimated image due tothe overlap between the blocks. It is worth mentioning that a few local block estimates might be located at thesame coordinate (e.g. Yexxba and Yexxbb are both located at xb but obtained while processing the reference blocks at
xa and xb , respectively). Let YexxmR (x) be an estimate of y (x), where x, xR 5 X, and xm 5 SxR . We zero-extend
YexxmR (x) outside its square support in order to simplify the formulation. The final estimate ye is computed as aweighted average of all local ones as given bySS$xR YexxmR (x)
xR 5X
x 5S
S m xR, ;x 5 X,(5)ye(x) = S
xR 5X
xm 5Sx $ xR "xm (x)R

W

Equality holds only if the matched blocks that build ZSxR are non-overlapping; otherwise, a certain amount ofcorrelation is introduced in the noise.

SPIE-IS&T/ Vol. 6064 606414-3

where "xm : X $ {0, 1} is the characteristic function of the square support of a block located at xm 5 X.

One can expect substantially overcomplete representation of the signal in regions where a block is matchedto many others. On the other hand, if a match is not found for a given reference block, the method reduces todenoising in 2D transform domain. Thus, the overcomplete nature of the method is highly dependent on theblock-matching and therefore also on the particular noisy image.

3. WIENER FILTER EXTENSION
Provided that an estimate of the true image is available (e.g. it can be obtained from the method given in theprevious section), we can construct an empirical Wiener filter as a natural extension of the above thresholding
technique. Because it follows the same approach, we only give the few fundamental modifications that are requiredfor its development and thus omitting repetition of the concept. Let us denote the initial image estimate bye : X $ R. In accordance with our established notation, Ex designates a square block of fixed size N1 × N1 ,extracted from e and located at x 5 X.

3.1. Modification to Block-Matching
In order to improve the accuracy of block-matching, it is performed within the initial estimate e rather than the
noisy image. Accordingly, we replace the thresholding-based d-distance measure from (1) with the normalized
L2 -norm of the dierence of two blocks with subtracted means. Hence, the definition (2) of SxR becomes

 

(6)
SxR = x 5 X | N11  ExR  ExR  Ex  Ex 2 <  match ,where ExR and Ex are the mean values of the blocks ExR and Ex , respectively. The mean subtraction allows for
improved matching of blocks with similar structures but dierent mean values.

3.2. Modification to Denoising in 3D Transform Domain
The linear Wiener filter replaces the nonlinear hard-thresholding operator. The attenuating coe!cients for the
Wiener filter are computed in 3D transform domain as


2


T3D ESxR ,
WSxR = 2T3D ESxR  + 2
where ESxR is a 3D array built by stacking the matched blocks Ex5SxR (in the same manner as ZSxR is built bystacking Zx5SxR ). We filter the 3D array of noisy observations ZSxR in T3D -transform domain by an elementwise
multiplication with WSxR . The subsequent inverse transform givese Sx = T 1 WSx T3D ZSx
,
Y
3D
R
R
R

(7)

e S comprises of stacked local block estimates Ye xR
where Y
xR
x5SxR of the true image blocks located at the matchedlocations x 5 SxR . As in (4), the weight assigned to the estimates is inversely proportional to the total variancee S and defined as
of Y
xR
41
xR | 
N1 [
N 1 |S
2
[
[
 F

E
$ xR = CWSxR (i, j, t) D .
3

i=1 j=1 t=1

SPIE-IS&T/ Vol. 6064 606414-4(8)

Figure 2. Flowchart for denoising by hard-thresholding in 3D transform domain with block-matching.

4. ALGORITHM
We present an algorithm which employs the hard-thresholding approach (from Section 2) to deliver an initial
estimate for the Wiener filtering part (from Section 3) that produces the final estimate. A straightforward
implementation of this general approach is computationally demanding. Thus, in order to realize a practicaland e!cient algorithm, we impose constraints and exploit certain expedients. In this section we introduce these
aspects and develop an e!cient implementation of the proposed approach.
The choice of the transforms T2D and T3D is governed by their energy compaction (sparsity) ability for noisefree image blocks (2D) and stacked blocks (3D), respectively. It is often assumed that neighboring pixels in small
blocks extracted from natural images exhibit high correlation; thus, such blocks can be sparsely represented
by well-established decorrelating transforms, such as the DCT, the DFT, wavelets, etc. From computational
e!ciency point of view, however, very important characteristics are the separability and the availability of fastalgorithms. Hence, the most natural choice for T2D and T3D is a fast separable transform which allows for sparse
representation of the true-image signal in each dimension of the input array.

4.1. E!cient Image Denoising Algorithm with Block-Matching and 3D Filtering
Let us introduce constraints for the complexity of the algorithm. First, we fix the maximum number of matchedblocks by setting an integer N2 to be the upper bound for the cardinality of the sets SxR 5X . Second, we doblock-matching within a local neighborhood of fixed size NS × NS centered about each reference block, insteadof doing it in the whole image. Finally, we use Nstep as a step by which we slide to every next reference block.
(e.g.,
Accordingly, we introduce XR  X as the set of the reference blocks’ coordinates, where |XR |  N|X|
2
step
Nstep = 1 implies XR = X).
In order to reduce the impact of artifacts on the borders of blocks (border eects), we use a Kaiser window
Wwin2D (with a single parameter ) as part of the weights of the local estimates. These artifacts are inherent of

SPIE-IS&T/ Vol. 6064 606414-5

many transforms (e.g. DFT) in presence of sharp intensity dierences across the borders of a block.
Let the input noisy image be of size M × N , thus |X| = M N. We use two buers of the same size–ebu forestimates and wbu for weights–to represent the summations in the numerator and denominator, respectively,in (5). For simplicity, we extend our notation so that ebu (x) denotes a single pixel at coordinate x 5 X andebu x designates a block located at x in ebu (the same notation is to be used for wbu ).
A flowchart of the hard-thresholding part of the algorithm is given in Figure 2 (but we do not give such for
the Wiener filtering part since it requires only the few changes given in Section 3). Following are the steps ofthe image denoising algorithm with block-matching and 3D filtering.
1. Initialization. Initialize ebu (x) = 0 and wbu (x) = 0, for all x 5 X.
2. Local hard-thresholding estimates. For each xR 5 XR , do the following sub-steps.
(a) Block-matching. Compute SxR as given in Equation (2) but restrict the search to a local neighborhoodof fixed size NS × NS centered about xR . If |SxR | > N2 , then let only the coordinates of the N2 blocks
with smallest d-distance to ZxR remain in SxR and exclude the others.

(b) Denoising by hard-thresholding in local 3D transform domain. Compute the local estimate blocks
xR
Yex5S
and their corresponding weight $ xR as given in (3) and (4), respectively.
xR(c) Aggregation. Scale each reconstructed local block estimate YexxR , where x 5 SxR , by a block of weights
W (xR ) = $xR Wwin2D and accumulate to the estimate buer: ebu x = ebu x + W (xR ) YexxR , for all
x 5 SxR . Accordingly, the weight block is accumulated to same locations as the estimates but in theweights buer: wbu x = wbu x + W (xR ), for all x 5 SxR .
ebug (x)3. Intermediate estimate. Produce the intermediate estimate e (x) = wbug(x) for all x 5 X, which is to be
used as initial estimate for the Wiener counterpart.

4. Local Wiener filtering estimates. Use e as initial estimate. The buers are re-initialized: ebu (x) = 0
and wbu (x) = 0, for all x 5 X. For each xR 5 XR , do the following sub-steps.
(a) Block-matching. Compute SxR as given in (6) but restrict the search to a local neighborhood of fixed
size NS × NS centered about xR . If |SxR | > N2 , then let only the coordinates of the N2 blocks withsmallest distance (as defined in Subsection 3.1) to ExR remain in SxR and exclude the others.
and(b) Denoising by Wiener filtering in local 3D transform domain. The local block estimates Ye xR
their weight $ xR are computed as given in (7) and (8), respectively.

x5SxR(c) Aggregation. It is identical to step 2c.
ebug (x)5. Final estimate. The final estimate is given by ye(x) = wbug
, for all x 5 X.
(x)

4.2. Complexity
The time complexity order of the algorithm as a function of its parameters is given by
#
$

 2


N1 + N2 NS2OT3D (N1 , N1 , N2 )+ O MNO (MN OT2D (N1 , N1 )) + O MN
,
2
2
Nstep
Nstep
where the first two addends are due to block-matching and the third is due to T3D used for denoising and where
OT2D (N1 , N1 ) and OT3D (N1 , N1 , N2 ) denote the complexity orders of the transforms T2D and T3D , respectively.
Both OT2D and OT3D depend on properties of the adopted transforms such as separability and availability offast algorithms. For example, the DFT has an e!cient implementation by means of fast transform Fourier(FFT). The 2D FFT, in particular, has complexity O (N1 N2 log (N1 N2 )) as opposed to O N12 N22 of a custom
non-separable transform. Moreover, an eective trade-o between complexity and denoising performance can beachieved by varying Nstep .

SPIE-IS&T/ Vol. 6064 606414-6

Table 1. Results in output PSNR (dB) of the denoising algorithm with block-matching and filtering in 3D DFT domain.

Image
/ PSNRLena
512 × 512

Barbara512 × 512House
256 × 256

Peppers
256 × 256Boats
512 × 512

Couple512 × 512

Hill
512 × 5125/ 34.15
10/ 28.13
15/ 24.61
20/ 22.11
25/ 20.17
30/ 18.59
35/ 17.25
50/ 14.16
100/ 8.13

38.63
35.83
34.21
33.03
32.08
31.29
30.61
29.08
26.04

38.18
34.87
33.08
31.77
30.75
29.90
29.13
27.51
24.14

39.54
36.37
34.75
33.54
32.67
31.95
31.21
29.65
25.92

37.84
34.38
32.31
30.87
29.80
28.97
28.14
26.46
23.11

37.20
33.79
31.96
30.65
29.68
28.90
28.20
26.71
24.00

37.40
33.88
31.93
30.58
29.57
28.75
28.03
26.46
23.60

37.11
33.57
31.79
30.60
29.74
29.04
28.46
27.21
24.77

5. RESULTS AND DISCUSSION
We present experiments conducted with the algorithm introduced in Section 4, where the transforms T2D and T3D
are the 2D DFT and the 3D DFT, respectively. All results are produced with the same fixed parameters–but
dierent for the hard-thresholding and Wiener filtering parts. For the hard-thresholding, N1 is automaticallyselected in the range 7  N1  13 based on ,  match = 0.233, N2 = 28, Nstep = 4, NS = 73,  = 4, th2D = 0.82,
and th3D = 0.75. For the Wiener filtering, N1 is automatically selected in the range 7  N1  11 based on ,+ 0.0105, N2 = 72, Nstep = 3, NS = 35, and  = 3. In Table 1, we summarize the results of the match = 4000
proposed technique in terms of output peak signal-to-noise ratio (PSNR) in decibels (dB), which is defined as#$
2552
PSNR = 10 log10.
S|X|1 x5X (y (x)  ye (x))2
At http://www.cs.tut.fi/~foi/3D-DFT, we provide a collection of the original and denoised test images thatwere used in our experiments, together with the algorithm implementation (as C++ and MATLAB functions)which produced all reported results. With the mentioned parameters, the execution time of the whole algorithm
is less than 9 seconds for an input image of size 256 × 256 on a 3 GHz Pentium machine.

In Figure 3, we compare the output PSNR of our method with the reported ones of three6, 7, 10 state-of-art
techniques known to the authors as best. However, for standard deviations 30 and 35 we could neither find nor
reproduce the results of both the FSP+TUP 7 and the exemplar-based10 techniques, thus they are omitted.
In Figure 4, we show noisy ( = 35) House image and the corresponding denoised one. For this test
image, similarity among neighboring blocks is easy to perceive in the uniform regions and in the regular-shaped
structures. Hence, those details are well-preserved in our estimate. It is worth referring to Figure 1, where
block-matching is illustrated for a fragment of House.

Pairs of noisy ( = 35) and denoised Lena and Hill images are shown in Figures 5 and 6, respectively. The
enlarged fragments in each figure help to demonstrate the good quality of the denoised images in terms of faithfuldetail preservation (stripes on the hat in Lena and the pattern on the roof in Hill).
We show fragments of noisy ( = 50) and denoised Lena, Barbara, Couple, and Boats images in Figure 7. For
this relatively high level of noise, there are very few disturbing artifacts and the proposed technique attains goodpreservation of: sharp details (the table legs in Barbara and the poles in Boats), smooth regions (the cheeks of
Lena and the suit of the man in Couple), and oscillatory patterns (the table cover in Barbara). A fragment of
Couple corrupted by noise of various standard deviations is presented in Figure 8.
In order to demonstrate the capability of the proposed method to preserve textures, we show fragments ofheavily noisy ( = 100) and denoised Barbara in Figure 9. Although the true signal is almost completely buried
under noise, the stripes on the clothes are faithfully restored in the estimate.

SPIE-IS&T/ Vol. 6064 606414-7

Noise standard deviation

Noise standard deviation(a) Barbara(b) Lena

Noise standard deviation

Noise standard deviation(c) Peppers(d) House

Figure 3. Output PSNR as a function of the standard deviation for Barbara (a), Lena (b), Peppers (c), and House (d).
The notation is: proposed method (squares), FSP+TUP 7 (circles), BLS-GSM 6 (stars), and exemplar-based10 (triangles).

SPIE-IS&T/ Vol. 6064 606414-8

Figure 4. On the left are a noisy ( = 35) House and two enlarged fragments from it; on the right are the denoised
image (PSNR 31=21 dB) and the corresponding fragments.

We conclude by remarking that the proposed method outperforms–in terms of objective criteria–all techniques known to us. Moreover, our estimates retain good visual quality even for relatively high levels of noise.
Our current research extends the presented approach by the adoption of variable-sized blocks and shapeadaptive transforms,13 thus further improving the adaptivity to the structures of the underlying image. Also,
application of the technique to more general restoration problems is being considered.

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Figure 5. On the left are noisy ( = 35) Lena and two enlarged fragments from it; on the right are the denoised image(PSNR 30=61 dB) and the corresponding fragments.

Figure 6. On the left are noisy ( = 35) Hill and two fragments from it; on the right are the denoised image (PSNR
28=46 dB) and the corresponding fragments from it.

SPIE-IS&T/ Vol. 6064 606414-10

(a) Lena ( = 50, PSNR 29=08 dB)(b) Barbara ( = 50, PSNR 27=51 dB)(c) Couple ( = 50, PSNR 26=46 dB)(d) Boats ( = 50, PSNR 26=71 dB)

Figure 7. Fragments of noisy ( = 50) and denoised test images.

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2006.

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(a)  = 25, PSNR 29=57 dB(b)  = 50, PSNR 26=46 dB(c)  = 75, PSNR 24=74 dB(d)  = 100, PSNR 23=60 dB

Figure 8. Pairs of fragments of noisy and denoised Couple for standard deviations: 25 (a), 50 (b), 75 (c), and 100 (d).

Figure 9. Fragments of noisy ( = 100) and denoised (PSNR 24=14 dB) Barbara.

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