IMAGE DENOISING BASED ON LAPLACE DISTRIBUTION WITH
LOCAL PARAMETERS IN LAPPED TRANSFORM DOMAIN
Vijay Kumar Nath and Anil Mahanta
Department of Electrical and Electronics Engineering, Indian Institute of Technology Guwahati
Guwahati, Assam, India
Keywords:
Lapped transform, Image denoising, MAP estimator, Laplace distribution, Statistical modeling.
Abstract:
In this paper, we present a new image denoising method based on statistical modeling of Lapped Transform
(LT) coefficients. The lapped transform coefficients are first rearranged into wavelet like structure, then the
rearranged coefficient subband statistics are modeled in a similar way like wavelet coefficients. We pro-
pose to model the rearranged LT coefficients in a subband using Laplace probability density function (pdf)
with local variance. This simple distribution is well able to model the locality and the heavy tailed property of
lapped transform coefficients. A maximum a posteriori (MAP) estimator using the Laplace probability density
function (pdf) with local variance is used for the estimation of noise free lapped transform coefficients. Exper-
imental results show that the proposed low complexity image denoising method outperforms several wavelet
based image denoising techniques and also outperforms two existing LT based image denoising schemes. Our
main contribution in this paper is to use the local Laplace prior for statistical modeling of LT coefficients and
to use MAP estimation procedure with this proposed prior to restore the noisy image LT coefficients.
1 INTRODUCTION
Images are often contaminated by noise in its acquisi-
tion or transmission. The noise mainly arises from
the imaging devices and channels during transmis-
sion. The aim of denoising is to eliminate the noise
while keeping the signal features as much as possible.
In the past several years, considerable work has
been reported on the wavelet based image denois-
ing techniques (Michak et al., 1999)(Fan and Xia,
2001)(Kazubek, 2003) (Eom and Kim, 2004)(Sendur
and Selesnick, 2002)(Rabbani and Vafadust, 2008).
In wavelet based image denoising, one approach is to
design a statistically optimal threshold parameter for
non linear thresholding or shrinkage function like soft
and hard thresholding. Another approach is to esti-
mate the noise free coefficients from the noisy coeffi-
cients with Bayesian estimation methods. If the max-
imum a posteriori (MAP) or minimum mean square
estimation (MMSE) estimator is used for this prob-
lem, the solution requires a priori knowledge about
the distribution of the noise free coefficients. The
shrinkage function is obtained for the corresponding
distribution.
Recently, a few approaches on lapped trans-
form based image denoising (Yang and Nguyen,
2003)(Duval and Nguyen, 2003)(Duval and Nguyen,
2004)(Raghvendra and Bhat, 2006) has been pro-
posed. The motivation of image denoising in lapped
transform (Malvar,1989) domain is that, lapped trans-
forms have good energy compaction and are robust
to oversmoothing. The lapped transforms are or-
thogonal transforms, thus signal and noise statis-
tics can be modeled precisely in the lapped trans-
form domain. Since, the lapped transforms are block
transforms, the lapped orthogonal transform (LOT)
(Malvar, 1989) coefficients are first rearranged into
a wavelet like structure (Malvar, 2000)(Yang and
Nguyen, 2003)(Duval and Nguyen, 2003)(Duval and
Nguyen, 2004)(Xiong et al., 1996), then the rear-
ranged lapped orthogonal transform coefficients sub-
band statistics are modeled in a similar way like
wavelet coefficients. In (Yang and Nguyen, 2003),
Yang et al., set up a maximum a posteriori estima-
tion problem to reduce the compression artifacts and
the additive white gaussian noise in images. In (Du-
val and Nguyen, 2003), Duval et al. proposed to ex-
tend the use of hidden markov tree (HMT) model in
lapped orthogonal transform domain (LOT-HMT). In
(Duval and Nguyen, 2004), Duval et al. further pro-
posed to improve the results of LOT-HMT denois-
ing by combining it with a redundant decomposition.
67
Kumar Nath V. and Mahanta A..
IMAGE DENOISING BASED ON LAPLACE DISTRIBUTION WITH LOCAL PARAMETERS IN LAPPED TRANSFORM DOMAIN.
DOI: 10.5220/0003516900670072
In Proceedings of the International Conference on Signal Processing and Multimedia Applications (SIGMAP-2011), pages 67-72
ISBN: 978-989-8425-72-0
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

Raghvendra et al. (Raghvendra and Bhat, 2006) pro-
posed to model the LT coefficients using mixture of
Laplace distributions which does not use local param-
eters. The pdf’s without local parameters are not good
in capturing the spatial clustering property of LT coef-
ficients. The spatial clustering property shows that if a
lapped transform coefficient is large, then its adjacent
coefficients are also more likely to be large. The LT
coefficients which represent edges or other important
signal features tends to cluster locally in a subband,
like wavelet coefficients. The pdf’s with local param-
eters can better exploit the local statistics and captures
the intrascale dependencies of the LT coefficients.
In this paper, we model the LT coefficients using a
simple Laplace pdf with local variance. The proposed
model is well capable of capturing the clustering and
the heavy tailed property of LT coefficients. A MAP
estimator which employs the Laplace pdf with local
variance is used to estimate the noise free LT coef-
ficients. The paper is organized as follows. In Sec-
tion 2, an introduction on lapped transforms and its
wavelet like representation is presented. In Section
3, we explain the proposed image denoising scheme
based on local Laplace prior. In Section 4, the perfor-
mance of proposed scheme is evaluated and is com-
pared with other image denoising schemes. Finally,
the concluding remarks are given in Section 5.
2 LAPPED TRANSFORMS
The lapped orthogonal transform (LOT) (Malvar,
1989)(Malvar, 1992) has been proposed to overcome
the blocking artifacts of the DCT and has increased
coding gain. The lapped transforms has extended ba-
sis functions which overlaps across the block bound-
aries. In lapped transforms, the input signal length is
two times its output signal length.
L = 2M (1)
where M is the output signal length and L is the input
signal length. The initial LOT matrix P which may
not be necessarily optimal is given by
P =
1
2

D
e
−D
o
D
e
−D
o
J(D
e
−D
o
) −J(D
e
−D
o
)

(2)
where D
e
and D
o
are the M x M/2 matrices containing
the even and odd DCT functions respectively and J is
the counter identity matrix. The optimal LOT matrix
(Malvar, 1989) is given by
P
0
= PZ (3)
for an optimal Z. The covariance matrix of LOT coef-
ficients is given by
R
0
= Z
′
P
′
R
xx
PZ (4)
where R
xx
is the given signal covariance matrix (Mal-
var, 1989).
R
xx
=








1 ρ ρ
2
.... ρ
L
ρ 1 ρ .... ρ
L−1
. . .
. . .
ρ
L−1
.... ρ 1 ρ
ρ
L
... ρ
2
ρ 1








(5)
We assume the signal model to be first order markov
model with ρ = 0.95. From equation no. (4), when
columns of Z are the eigen vectors of P
′
R
xx
P that is R
0
is diagonal, the transform coding gain (Malvar, 1989)
is maximized. The LOT matrix P
0
is optimal for such
Z. The lapped transforms can be viewed as critically
sampled multirate filter banks.
Like block DCT coefficients (Xiong et al., 1996),
the LT coefficients also can be rearranged in a wavelet
like structure with N = log
2
M decomposition levels.
Fig.1 shows the rearrangement of LOT (M=8) coeffi-
cients into a 3 level wavelet like pyramid structure for
the Barbara image. The image shown in Fig. 1(b) is
composed of 8x8 blocks and the image in Fig. 1(c) is
its wavelet like representation.
(a) (b)
(c)
Figure 1: (a) Barbara image (b) LOT (M=8) block decom-
position. (c) Rearrangement into 3-level wavelet like pyra-
mid structure.
3 PROPOSED DENOISING
METHOD
Many wavelet based image denoising schemes
(Michak et al., 1999)(Fan and Xia, 2001)(Kazubek,
SIGMAP 2011 - International Conference on Signal Processing and Multimedia Applications
68

(a) (b)
Figure 2: (a) The Gaussian and Laplacian pdf fitted to the histogram of the rearranged LT coefficients in a particular subband
of Lena (512x512) image. (b) The histogram, Gaussian and Laplace pdf in the log domain.
2003)(Eom and Kim, 2004) assume local distribu-
tion of the transform coefficients to be Gaussian with
spatially varying variance and employ linear mini-
mum mean square error (MMSE) estimator locally
to restore the noisy coefficients. The Laplace pri-
ors are also widely used for the statistical modeling
of wavelet transform coefficients. Recently, Rabbani
(Rabbani, 2009) proposed to use the local Laplace
prior for modeling of Steerable Pyramid coefficients.
Fig. 2(a) shows the histogram of a specific subband of
rearranged LOT coefficients for Lena image. The his-
togram of coefficients in each subband of rearranged
LOT coefficients has a sharp peak around zero and its
tails decays to zero much slower than the Gaussian
pdf. Fig. 2(a) also indicates that Laplace distribu-
tion better fits the histogram of data. The same issue
can be better studied from the image in Fig. 2(b) ,
where the histogram, Gaussian and Laplace pdf’s are
plotted in the log domain. The image in Fig. 2(b)
shows that Gaussian pdf fails to match with the his-
togram, particularly in tails. In this paper, we propose
to model the rearranged LT coefficients in a subband
using Laplace pdf with local variance. The MAP es-
timator using this local Laplace prior is used for the
estimation of noise free LT coefficients. Fig. 3 shows
the block diagram of the proposed LT based image
denoising method.
We assume that the image is corrupted by additive
white Gaussian noise with variance σ
2
n
. The orthogo-
nal LT coefficients of the noisy image are given by
y(k) = x(k) + n(k) (6)
where x(k) denotes the clean LT coefficients and n(k)
is additive white Gaussian noise. When we use a
MAP estimator to estimate x(k) from the noisy ob-
servation y(k), we have (Rabbani and Vafadust, 2008)
ˆx(k) = argmax
x(k)
P
x(k)|y(k)
(x(k)|y(k)) (7)
The above equation can also be easily written as
ˆx(k) = argmax
x(k)

P
n
(y(k) −x(k))P
x(k)
(x(k))

(8)
In this paper, we have assumed the noise to be zero
mean gaussian with variance σ
2
n
p
n
(n(k)) =
1
√
2πσ
n
exp

−
n
2
(k)
2σ
2
n

(9)
Substituting Eq. no. (9) in Eq. no. (8), we have
ˆx(k) = argmax
x(k)

−
(y(k) −x(k))
2
(2σ
2
n
)
+ f(x(k))

(10)
where f(x
k
) = log

P
x(k)
(x(k))

. Thus, the MAP esti-
mate of x(k) is achieved by setting the derivative with
respect to ˆx(k) equals to zero
y(k) − ˆx(k)
σ
2
n
+ f
′
( ˆx(k)) = 0 (11)
In this paper, we propose to model the LT coefficients
using Laplace pdf with local variance, thus
P
x(k)
(x(k)) =
1
σ
x
(k)
√
2
exp −
√
2|x(k)|
σ
x
(k)
!
(12)
For this particular case, we have
f(x(k)) = −log

σ
x
(k)
√
2

−
√
2|x(k)|
σ
x
(k)
(13)
We have
f
′
(x(k)) = −
√
2
σ
x
(k)
sign(x(k)) (14)
So,
y(k) = ˆx(k) +
√
2σ
2
n
σ
x
(k)
sign(ˆx(k)) (15)
The above equation can also be written as
ˆx(k) = sign(y(k)) |y(k)|−
√
2σ
2
n
σ
x
(k)
!
+
(16)
From the above equation, (b)
+
can be defined as
(b)
+
=

0, b < 0
b, otherwise
(17)
IMAGE DENOISING BASED ON LAPLACE DISTRIBUTION WITH LOCAL PARAMETERS IN LAPPED
TRANSFORM DOMAIN
69

Lapped Transform

Rearrangement of
coefficients into
octave-like
decomposition
Rearrangement
of coefficients
into block
decomposition
Inverse Lapped
Transform



2
2
( ) ( ),
( )
n
x
x k SOFTLAMAP y k
k
σ
σ
§ ·
=
¨ ¸
¨ ¸
© ¹


( )
x k

( )
y k

Estimation of
local variance

( )
x
k
σ

Noisy image
Output
image
Figure 3: Block diagram of the proposed LT based image denoising method based on local Laplace prior.
If the SOFTLAMAP operator (Rabbani and Vafadust,
2008)(Rabbani, 2009) is defined as
SOFTLAMAP(p, η) = sign(p)(|p|−η)
+
(18)
Now, Eq. no. (16) can be written as
ˆx(k) = SOFTLAMAP y(k),
√
2σ
2
n
σ
x
(k)
!
(19)
For each noisy LOT coefficient, an estimate of σ
2
x
(k)
is formed based on its local neighborhood Z(k). In
this paper, we use a square window Z(k) centered at
y(k). The estimate for σ
2
x
(k) is given as (Michak et al.,
1999)
ˆ
σ
2
x
(k) = max 0,
1
M
∑
j∈Z(k)
y
2
( j)−σ
2
n
!
(20)
where M is the number of coefficients in Z(k).
4 EXPERIMENTAL RESULTS
We tested our algorithm on a large number of test
images, but report results only for Lena, Barbara,
Boat and Fingerprint. Table 1 shows the peak sig-
nal to noise ratio (PSNR) comparison between the
proposed method and two existing LT based im-
age denoising schemes. The PSNR values of pro-
posed method are averaged over six runs. The pro-
posed low complexity image denoising scheme based
on local Laplace prior outperforms both the LOT-
HMT (Duval and Nguyen, 2003) method which is
based on complex hidden markov tree model and
also the LLMDM (Lapped Laplace Mixture Distri-
bution Model) method (Raghvendra and Bhat, 2006)
which uses computationally expensive Laplace mix-
ture model (with out local parameters). Table 2
shows the PSNR comparison between the proposed
method (M=8 and M=16) and several wavelet (or-
thogonal discrete wavelet transform) based image de-
noising schemes. The PSNR results for (Bhuiyan
et al., 2008) are obtained using Bayesian MMSE esti-
mator only. The authors of (Bhuiyan et al., 2008) have
reported results only for Lena, Barbara and Boat im-
ages, therefore we compare our results with (Bhuiyan
et al., 2008) method only for these three images.
The proposed LT domain method using local Laplace
prior is referred to as LOT-Lap. We call the imple-
mentation of local Laplace prior in wavelet domain
as WT-Lap (Sendur and Selesnick, 2002; Rabbani
and Vafadust, 2008; Rabbani, 2009). An orthogo-
nal wavelet transform with four levels of decompo-
sition and Daubechies length-8 wavelet is used for
the implementation of WT-Lap, (Michak et al., 1999)
and (Chang et al., 2000) algorithms. The proposed
method consistently outperforms (Chang et al., 2000),
(Bhuiyan et al., 2008) and WT-Lap algorithms in
terms of PSNR for all the test images. The LOT-Lap
even outperforms the wavelet domain local Wiener
filtering method for all the tested images at almost all
the tested noise levels. The PSNR results are encour-
aging especially for highly textured images. Fig. 4
shows the visual results. The proposed method pro-
vides images with less distortions in the smooth re-
gions and near the edges.
5 CONCLUSIONS
In this paper, we model the rearranged LOT coeffi-
cients in each subband using Laplace pdf with local
variance. The experimental results show that the
proposed low complexity image denoising scheme
outperforms all the previously published results
on LT based image denoising schemes and several
wavelet based image denoising schemes. The pro-
posed denoising scheme shows encouraging results
for test images with high amount of textures. The
experimental results indicates that the local Laplace
pdf is more appropriate model than the models used
earlier for modeling the rearranged LT coefficients.
The Laplace mixture model with local parameters
may further improve the results but at increased
computational cost.
SIGMAP 2011 - International Conference on Signal Processing and Multimedia Applications
70

Table 1: PSNR (in dB) comparison between proposed LOT [M=8] based image denoising scheme and two existing LT based
image denoising schemes.
Lena (512x512)
σ
n
Noisy (Duval and Nguyen, 2003) (Raghvendra and Bhat, 2006) Proposed
7.7 30.42 33.80 35.0 35.32
15.5 24.33 29.60 31.70 32.03
23.1 20.88 27.20 29.80 30.01
33.1 17.72 24.90 28.20 28.29
Barbara (512x512)
σ
n
Noisy (Duval and Nguyen, 2003) (Raghvendra and Bhat, 2006) Proposed
7.7 30.39 33.30 33.70 34.20
15.5 24.34 29.10 29.70 30.23
23.1 20.85 26.60 27.60 28.05
33.1 17.76 24.20 25.70 26.13
Table 2: PSNR (in dB) comparison between proposed LOT [M=8 and M=16] based image denoising scheme and several
wavelet based image denoising schemes.
Lena (512x512)
σ
n
10 15 20 25
Bayes-shrink (Chang et al., 2000) 33.23 31.20 29.95 28.99
LAWML (Michak et al., 1999) 34.12 31.93 30.38 29.18
Ref. (Bhuiyan et al., 2008) 33.89 32.01 30.71 29.75
WT-Lap 34.04 32.05 30.69 29.64
LOT-Lap (M=8) 34.06 32.20 30.82 29.69
LOT-Lap (M=16) 34.09 32.26 30.85 29.76
Barbara (512x512)
σ
n
10 15 20 25
Bayes-shrink (Chang et al., 2000) 31.10 28.72 27.12 25.90
LAWML (Michak et al., 1999) 32.51 30.07 28.38 27.09
Ref. (Bhuiyan et al., 2008) 31.84 29.49 27.89 26.70
WT-Lap 32.19 29.75 28.10 26.89
LOT-Lap (M=8) 32.69 30.43 28.83 27.57
LOT-Lap (M=16) 32.89 30.68 29.18 28.06
Boat (512x512)
σ
n
10 15 20 25
Bayes-shrink (Chang et al., 2000) 31.89 29.76 28.33 27.25
LAWML (Michak et al., 1999) 32.50 30.36 28.86 27.67
Ref. (Bhuiyan et al., 2008) 32.16 30.22 28.83 27.82
WT-Lap 32.30 30.28 28.87 27.77
LOT-Lap (M=8) 32.38 30.35 28.95 27.86
LOT-Lap (M=16) 32.32 30.26 28.94 27.84
Fingerprint (512x512)
σ
n
10 15 20 25
Bayes-shrink (Chang et al., 2000) 30.98 28.73 27.21 26.09
LAWML (Michak et al., 1999) 31.30 28.97 27.36 26.15
WT-Lap 30.77 28.55 27.02 25.86
LOT-Lap (M=8) 31.13 28.93 27.38 26.17
LOT-Lap (M=16) 31.45 29.15 27.55 26.39
IMAGE DENOISING BASED ON LAPLACE DISTRIBUTION WITH LOCAL PARAMETERS IN LAPPED
TRANSFORM DOMAIN
71

(a) (b) (c)
(d) (e) (f)
Figure 4: A fragment of Barbara image, (a) Original im-
age, (b) Noisy image (SSIM (Wang et al., 2004) =0.7770),
(c) Image denoised by WT-Lap method (SSIM=0.8949),
(d) Image denoised by LAWML (Michak et al., 1999)
(SSIM=0.9001), (e) Image denoised by LOT-Lap (M=8)
(SSIM=0.8974), (f) Image denoised by LOT-Lap (M=16)
(SSIM=0.9011).
REFERENCES
Bhuiyan, M. I. H., Ahmad, M. O., and Swamy, M. N. S.
(2008). Wavelet-based image denoising with the nor-
mal inverse gaussian prior and linear mmse estimator.
IET Image Processing, 2(4):203–217.
Chang, S., Yu, B., and Vetterli, M. (2000). Adaptive
wavelet thresholding for image denoising and com-
pression. IEEE Transactions on Image Processing,,
9:15321546.
Duval, L. and Nguyen, T. Q. (2003). Lapped transform do-
main denoising using hidden markov trees. In IEEE
International Conference on Image Processing, vol-
ume 1, pages 125–128.
Duval, L. and Nguyen, T. Q. (2004). Hidden markov tree
image denoising with redundant lapped transforms. In
IEEE International Conference on Acoustics, Speech
and Signal Processing, volume 3, pages 193–196.
Eom, I. K. and Kim, Y. S. (2004). Wavelet-based denoising
with nearly arbitrarily shaped windows. IEEE Signal
Processing Letters, 11(12):937–940.
Fan, G. and Xia, X. G. (2001). Image denoising using lo-
cal contexual hidden markov model in the wavelet do-
main. IEEE Signal Processing Letters, 8(5):125–128.
Kazubek, M. (2003). Wavelet domain image denoising by
thresholding and wiener filtering. IEEE Signal Pro-
cessing Letters, 10(11):324–326.
Malvar, H. S. (1989). The lot : Transform coding without
blocking effects. IEEE Transactions on Accoustics,
Speech and Signal Processing, 37(4):553–559.
Malvar, H. S. (1992). Signal Processing with Lapped Trans-
forms. Norwood, MA : Artech House.
Malvar, H. S. (2000). Fast progressive image coding with-
out wavelets. In Data Compression Conference, pages
243–252.
Michak, M. K., Kozintsev, I., and Ramchandran, K. (1999).
Low complexity image denoising based on statistical
modelling of wavelet coefficient. IEEE Signal Pro-
cessing Letters, 6(12):300–303.
Rabbani, H. (2009). Image denoising in steerable pyramid
domain based on local laplace prior. Pattern Recogni-
tion, 42:2181–2193.
Rabbani, H. and Vafadust, M. (2008). Image / video denos-
ing based on a mixture of laplace distributions with lo-
cal parameters in multidimensional complex wavelet
domain. Signal Processing, 88(11):158–173.
Raghvendra, B. S. and Bhat, P. S. (2006). Image denosing
using mixture distributions with lapped transforms. In
National Conference of Communications, pages 217–
220.
Sendur, L. and Selesnick, I. W. (2002). Bivariate shrinkage
functions for wavelet-based denoising. IEEE Trans-
actions on Signal Processing, 50:27442756.
Wang, Z., Bovik, A. C., Sheikh, H. R., and Simoncelli, E. P.
(2004). Image quality assessment : From error vis-
ibility to structural similarity. IEEE Transactions on
Image processing, 13(4):600–612.
Xiong, Z., Guleryuz, O. G., and Orchard, M. T. (1996). A
dct based embedded image coder. IEEE Signal Pro-
cessing Letters, 3(11):289–290.
Yang, S. and Nguyen, T. Q. (2003). Denoising in the
lapped transform domain. In IEEE International Con-
ference on Acoustics, Speech and Signal Processing,
volume 6, pages 173–176.
SIGMAP 2011 - International Conference on Signal Processing and Multimedia Applications
72