Boltzmann Machines and Denoising Autoencoders for Image Denoising
Authors: KyungHyun Cho
ICLR 2013

Abstract: Image denoising based on a probabilistic model of local image patches has been employed by various researchers, and recently a deep (denoising) autoencoder has been proposed by Burger et al. [2012] and Xie et al. [2012] as a good model for this. In this paper, we propose that another popular family of models in the field of deep learning, called Boltzmann machines, can perform image denoising as well as, or in certain cases of high level of noise, better than denoising autoencoders. We empirically evaluate the two models on three different sets of images with different types and levels of noise. Throughout the experiments we also examine the effect of the depth of the models. The experiments confirmed our claim and revealed that the performance can be improved by adding more hidden layers, especially when the level of noise is high.

arXiv:1301.3468v6  [stat.ML]  4 Mar 2013
Boltzmann Machines and Denoising Autoencoders
for Image Denoising
KyungHyun Cho
Aalto University School of Science
Department of Information and Computer Science
Espoo, Finland
kyunghyun.cho@aalto.fi
Abstract
Image denoising based on a probabilistic model of local imag e patches has been
employed by various researchers, and recently a deep (denoising) autoencoder has
been proposed by Burger et al. [2012] and Xie et al. [2012] as a good model for
this. In this paper, we propose that another popular family of models in the ﬁeld of
deep learning , called Boltzmann machines, can perform image denoising as well
as, or in certain cases of high level of noise, better than den oising autoencoders.
We empirically evaluate the two models on three different se ts of images with
different types and levels of noise. Throughout the experim ents we also examine
the effect of the depth of the models. The experiments conﬁrm ed our claim and
revealed that the performance can be improved by adding more hidden layers,
especially when the level of noise is high.
1 Introduction
Numerous approaches based on machine learning have been pro posed for image denoising tasks
over time. A dominant approach has been to perform denoisingbased on local statistics over a whole
image. For instance, a method denoises each small image patch extracted from a whole noisy image
and reconstructs the clean image from the denoised patches. Under this approach, it is possible to
use raw pixels of each image patch [see, e.g. Hyv¨ arinen et al., 1999] or the representations in another
domain, for instance, in wavelet domain [see, e.g. Portilla et al., 2003].
In the case of using raw pixels, sparse coding has been a method of choice. Hyv¨ arinen et al. [1999]
proposed to use independent component analysis (ICA) to est imate a dictionary of sparse elements
and compute the sparse code of image patches. Subsequently, a shrinkage nonlinear function is
applied to the estimated sparse code elements to suppress th ose elements with small absolute mag-
nitude. This sparse code elements are used to reconstruct a n oise-free image patch. More recently,
Elad and Aharon [2006] also showed that sparse overcomplete representation is useful in denoising
images. Some researchers claimed that better denoising per formance can be achieved by using a
variant of sparse coding methods [see, e.g. Shang and Huang, 2005, Lu et al., 2011].
In essence, these approaches build a probabilistic model of natural image patches using a layer of
sparse latent variables. The posterior distribution of eac h noisy patch is either exactly computed or
estimated, and the noise-free patch is reconstructed as an e xpectation of a conditional distribution
over the posterior distribution.
Based on this interpretation, some researchers have propos ed very recently to utilize a (probabilis-
tic) model that has more than one layers of latent variables for image denoising. Burger et al. [2012]
showed that a deep multi-layer perceptron that learns a mapp ing from a noisy image patch to its
corresponding clean version, can perform as good as the stat e-of-the-art denoising methods. Simi-
larly, Xie et al. [2012] proposed a variant of a stacked denoi sing autoencoder [Vincent et al., 2010]
1
that is more effective in image denoising. They were also abl e to show that the denoising approach
based on deep neural networks performed as good as, or someti mes better than, the conventional
state-of-the-art methods.
Along this line of research, we aim to propose a yet another ty pe of deep neural net-
works for image denoising, in this paper. A Gaussian-Bernou lli restricted Boltzmann ma-
chines (GRBM) [Hinton and Salakhutdinov, 2006] and deep Bol tzmann machines (GDBM)
[Salakhutdinov and Hinton, 2009, Cho et al., 2011b] are empi rically shown to perform well in im-
age denoising, compared to stacked denoising autoencoders . Furthermore, we extensively evalu-
ate the effect of the number of hidden layers of both Boltzman n machine-based deep models and
autoencoder-based ones. The empirical evaluation is conducted using different noise types and lev-
els on three different sets of images.
2 Deep Neural Networks
We start by brieﬂy describing Boltzmann machines and denoising autoencoders which have become
increasingly popular in the ﬁeld of machine learning.
2.1 Boltzmann Machines
Originally proposed in 1980s, a Boltzmann machine (BM) [Ack ley et al., 1985] and especially its
structural constrained version, a restricted Boltzmann machine (RBM) [Smolensky, 1986] have be-
come increasingly important in machine learning since [Hin ton and Salakhutdinov, 2006] showed
that a powerful deep neural network can be trained easily by s tacking RBMs on top of each other.
More recently, another variant of a BM, called a deep Boltzma nn machine (DBM), has been pro-
posed and shown to outperform other conventional machine le arning methods in many tasks [see,
e.g. Salakhutdinov and Hinton, 2009].
We ﬁrst describe a Gaussian-Bernoulli DBM (GDBM) that has L layers of binary hidden units and
a single layer of Gaussian visible units. A GDBM is deﬁned by its energy function
−E(v, h |θ) =
∑
i
−(vi −bi)2
2σ2 +
∑
i,j
vi
σ2 h(1)
j wi,j +
∑
j
h(1)
j c(1)
j +
L∑
l=2

∑
j
h(l)
j c(l)
j +
∑
j,k
h(l)
j h(l+1)
k u(l)
j,k

, (1)
where v = [vi]i=1...Nv and h(l) =
[
h(l)
j
]
j=1...Nl
are Nv Gaussian visible units and Nl binary
hidden units in the l-th hidden layer. W = [wi,j] is the set of weights between the visible neurons
and the ﬁrst layer hidden neurons, while U(l) =
[
u(l)
j,k
]
is the set of weights between the l-th and
l + 1-th hidden neurons. σ2 is the shared variance of the conditional distribution of vi given the
hidden units.
With the energy function, a GDBM can assign a probability to e ach state vector x =
[v; h(1); ··· ; h(L)] using a Boltzmann distribution:
p(x |θ) = 1
Z(θ) exp {−E(x |θ)}.
Based on this property the parameters can be learned by maxim izing the log-likelihood
L = ∑ N
n=1 log ∑
h p(v(n), h | θ) given N training samples {v(n)}n=1,...,N, where h =[
h(1); ··· ; h(L)]
.
Although the update rules based on the gradients of the log-l ikelihood function are well de-
ﬁned, it is intractable to exactly compute them. Hence, an ap proach that uses variational
approximation together with Markov chain Monte Carlo (MCMC ) sampling was proposed by
Salakhutdinov and Hinton [2009].
2
It has, however, been found that training a GDBM using this approach starting from randomly initial-
ized parameters is not trivial [Salakhutdinov and Hinton, 2 009, Desjardins et al., 2012, Cho et al.,
2012]. Hence, Salakhutdinov and Hinton [2009] and Cho et al. [2012] proposed pretraining algo-
rithms that can initialize the parameters of DBMs. In this pa per, we use the pretraining algorithm
proposed by Cho et al. [2012].
A Gaussian-Bernoulli RBM (GRBM) is a special case of a GDBM, w here the number of hidden
layers is restricted to one, L = 1. Due to this restriction it is possible to compute the posterior distri-
bution over the hidden units conditioned on the visible unit s exactly and tractably. The conditional
probability of each hidden unit hj = h(1)
j is
p(hj = 1|v, θ) =f
( ∑
i
wij
vi
σ2 + cj
)
. (2)
Hence, the positive part of the gradient, that needs to be app roximated with variational approach
in the case of GDBMs, can be computed exactly and efﬁciently. Only the negative part, which is
computed over the model distribution, still relies on MCMC sampling, or more approximate methods
such as contrastive divergence (CD) [Hinton, 2002].
2.2 Denoising Autoencoders
A denoising autoencoder (DAE) is a special form of multi-lay er perceptron network with 2L −1
hidden layers and L −1 sets of tied weights. A DAE tries to learn a network that recon structs an
input vector optimally by minimizing the following cost function:
N∑
n=1


Wg(1) ◦···◦ g(L−1) ◦f(L−1) ◦···◦ f(1)
(
η(v(n))
)
−v(n)



2
, (3)
where
f(l) = φ(W(l)⊤
h(l−1)) and g(l) = φ(W(l)h(2L−l))
are encoding and decoding functions for l-th layer with a component-wise nonlinearity function φ.
W(l) is the weights between the l-th and l + 1-th layers and is shared by the encoder and decoder.
For notational simplicity, we omit biases to all units.
Unlike an ordinary autoencoder, a DAE explicitly sets some o f the components of an input vector
randomly to zero during learning via η(·) which explicitly adds noise to an input vector. It is usual
to combine two different types of noise when usingη(·), which are additive isotropic Gaussian noise
and masking noise [Vincent et al., 2010]. The ﬁrst type adds a zero-mean Gaussian noise to each
input component, while the masking noise sets a set of randomly chosen input components to zeros.
Then, the DAE is trained to denoise the corrupted input.
Training a DAE is straightforward using backpropagation algorithm which computes the gradient of
the objective function using a chain-rule and dynamic programming. Vincent et al. [2010] proposed
that training a deep DAE becomes easier when the weights of a deep DAE are initialized by greedily
pretraining each layer of a deep DAE as if it were a single-layer DAE. In the following experiments
section, we follow this approach to initialize the weights and subsequently ﬁnetune the network with
the stochastic backpropagation.
3 Image Denoising
There are a number of ways to perform image denoising. In this paper, we are interested in an
approach that relies on local statistics of an image.
As it has been mentioned earlier, a noisy large image can be de noised by denoising small patches
of the image and combining them together. Let us deﬁne a set of N binary matrices Dn ∈Rp×d
that extract a set of small image patches given a large, whole image x ∈Rd, where d = whc is the
product of the widthw, the height h and the number of color channelsc of the image andp is the size
3
of image patches (e.g., n = 64if an 8 ×8 image patch). Then, the denoised image is constructed by
˜x =
( N∑
n=1
D⊤
n rθ (Dnx)
)
⊘
( N∑
n=1
D⊤
n Dn1
)
, (4)
where ⊘is a element-wise division and 1 is a vector of ones. rθ (·) is an image denoising function,
parameterized by θ, that denoises N image patches extracted from the input image x.
Eq. (4) essentially extracts and denoises all possible imag e patches from the input image. Then, it
combines them by taking an average of those overlapping pixels.
There are several ﬂexibilities in constructing a matrix D. The most obvious one is the size of an
image patch. Although there is no standard approach, many previous attempts tend to use patch
sizes as small as 4 ×4 to 17 ×17. Another one, called a stride, is the number of pixels between two
consecutive patches. Taking every possible patch is one opt ion, while one may opt to overlapping
patches by only a few pixels, which would reduce the computational complexity.
One of the popular choices for rθ (·) has been to construct a probabilistic model with a set of
latent variables that describe natural image patches. For i nstance, sparse coding which is, in
essence, a probabilistic model with a single layer of latent variables has been a common choice.
Hyv¨ arinen et al. [1999] used ICA and a nonlinear shrinkage function to compute the sparse code of
an image patch, while Elad and Aharon [2006] used K-SVD to build a sparse code dictionary.
Under this approach denoising can be considered as a two-step reconstruction. Initially, the posterior
distribution over the latent variables is computed, or esti mated, given an image patch. Given the
estimated posterior distribution, the conditional distri bution, or its mean, over the visible units is
computed and used as a denoised image patch.
In the following subsections, we describe how rθ (·) can be implemented in the cases of Boltzmann
machines and denoising autoencoders.
3.1 Boltzmann machines
We consider a BM with a set of Gaussian visible units v that correspond to the pixels of an image
patch and a set of binary hidden units h. Then, the goal of denoising can be written as
p(v |˜v) =
∑
h
p(v |h)p(h |˜v) =Eh|˜v [p(v |h)] , (5)
where ˜v is a noisy input patch. In other words, we ﬁnd a mean of the conditional distribution of the
visible units with respect to the posterior distribution ov er the hidden units given the visible units
ﬁxed to the corrupted input image patch.
However, since taking the expectation over the posterior di stribution is usually not tractable nor
exactly computable, it is often easier to approximate the qu antity. We approximate the marginal
conditional distribution (5) of v given ˜v with
p(v |˜v) ≈p(v |h)Q(h) =p(v |h = µ ),
where µ = EQ(h) [h] and Q(h) is a fully factorial distribution that approximates the pos terior
p(h |˜v). It is usual to use the mean of v under p(v |˜v) as the denoised, reconstructed patch.
Following this approach, given a noisy image patch ˜v a GRBM reconstructs a noise-free patch by
ˆvi =
Nh∑
j=1
wij E [h |˜v] +bi,
where bi is a bias to the i-th visible unit. The conditional distribution over the hid den units can be
computed exactly from Eq. (2).
Unlike a GRBM, the posterior distribution of the hidden unit s of a GDBM is neither tractably
computable nor has an analytical form. Salakhutdinov and Hi nton [2009] proposed to utilize
a variational approximation [Neal and Hinton, 1999] with a f ully-factored distribution Q(h) =
4
∏ L
l=1
∏
j µl
j , where the variational parameters µ(l)
j ’s can be found by the following simple ﬁxed-
point update rule:
µ(l)
j ←f


Nl− 1∑
i=1
µ(l−1)
i w(l−1)
ij +
Nl+1∑
k=1
µ(l+1)
k w(l)
kj + c(l)
j

, (6)
where f(x) = 1
1+exp{−x} .
Once the variational parameters are converged, a GDBM reconstructs a noise-free patch by
ˆvi =
Nl∑
j=1
wij µ(1)
j + bi.
The convergence of the variational parameters can take too m uch time and may not be suitable in
practice. Hence, in the experiments, we initialize the vari ational parameters by feed-forward prop-
agation using the doubled weights [Salakhutdinov and Hinto n, 2009] and perform the ﬁxed-point
update in Eq. (6) for at most ﬁve iterations only. This turned out to be a good enough compromise
that least sacriﬁces the performance while reducing the computational complexity signiﬁcantly.
3.2 Denoising autoencoders
An encoder part of a DAE can be considered as performing an app roximate inference of a fully-
factorial posterior distribution of top-layer hidden unit s, i.e. a bottleneck, given an input image
patch [Vincent et al., 2010]. Hence, a similar approach to th e one taken by BMs can be used for
DAEs.
Firstly, the variational parameters µ (L) of the fully-factorial posterior distribution Q(h(L)) =∏
j µ(L)
j are computed by
µ (L) = f(L−1) ◦···◦ f(1) (˜v) .
Then, the denoised image patch can be reconstructed by the de coder part of the DAE. This can be
done simply by propagating the variational parameters through the decoding nonlinearity functions
g(l) such that
ˆv = g(1) ◦···◦ g(L−1)
(
µ (L)
)
.
Recently, Burger et al. [2012] and Xie et al. [2012] tried a de ep DAE in this manner to perform
image denoising. Both of them reported that the denoising pe rformance achieved by DAEs is com-
parable, or sometimes favorable, to other conventional ima ge denoising methods such as BM3D
[Dabov et al., 2007], K-SVD [Portilla et al., 2003] and Bayes Least Squares-Gaussian Scale Mix-
ture [Elad and Aharon, 2006].
4 Experiments
In the experiments, we aim to empirically compare the two dom inant approaches of deep learning ,
namely Boltzmann machines and denoising autoencoders, in image denoising tasks.
There are several questions that are of interest to us:
1. Does a model with more hidden layers perform better?
2. How well does a deep model generalize?
3. Which family of deep neural networks is more suitable, Bol tzmann machines or denoising
autoencoders?
In order to answer those questions, we vary the depth of the mo dels (the number of hidden layers),
the level of noise injection, the type of noise–either whiteGaussian additive noise or salt-and-pepper
noise, and the size of image patches. Also, as our interest lies in the generalization capability of the
models, we use a completely separate data set for training th e models and apply the trained models
to three distinct sets of images that have very different properties.
5
(a) Textures
 (b) Aerials
 (c) Miscellaneous
Figure 1: Sample images from the test image sets
4.1 Datasets
We used three sets of images, textures, aerials and miscellaneous, from the USC-SIPI Image
Database1 as test images. Tab. 1 lists the details of the image sets, and Fig. 1 presents six sam-
ple images from the test sets.
These datasets are, in terms of contents and properties of images, very different from each other. For
instance, most of the images in the texture set have highly re petitive patterns that are not present in
the images in the other two sets. Most images in the aerials set have both coarse and ﬁne structures,
for example, a lake and a nearby road, at the same time in a sing le image. Also, the sizes of the
images vary quite a lot across the test sets and across the images in each set.
Set
# of all images # of color images Min. Size Max. Size
Textures 64 0 512 ×512 1024 ×1024
Aerials 38 37 512 ×512 2250 ×2250
Miscellaneous 44 16 256 ×256 1024 ×1024
Table 1: Descriptions of the test image sets.
As we are aiming to evaluate the performance of denoising a ve ry general image, we used a large
separate data set of natural image patches to train the models. We extracted a set of100, 000 random
image patches of sizes4×4, 8×8 and 16×16 from CIFAR-10 dataset [Krizhevsky, 2009]. From each
image of 50, 000 training samples of the CIFAR-10 dataset, two patches from r andomly selected
locations have been collected.
We tried denoising only grayscale images. When an image was in an RGB format, we averaged the
three channels to make the image grayscale.
4.2 Denoising Settings
We tried three different depth settings for both Boltzmann machines and denoising autoencoders; a
single hidden layer, two hidden layers and four hidden layers. The sizes of all hidden layers were set
to have the same number of hidden units, which was the constant factor multiplied by the number of
pixels in an image patch2.
We denote Boltzmann machines with one, two and four hidden la yers by GRBM, GDBM(2) and
GDBM(4), respectively. Denoising autoencoders are denote d by DAE, DAE(2) and DAE(4), re-
spectively. For each model structure, Each model was trained on image patches of sizes 4 ×4, 8 ×8
and 16 ×16.
The GRBMs were trained using the enhanced gradient [Cho et al., 2011c] and persistent contrastive
divergence (PCD) [Tieleman, 2008]. The GDBMs were trained by PCD after initializing the param-
eters with a two-stage pretraining algorithm [Cho et al., 20 12]. DAEs were trained by a stochastic
backpropagation algorithm, and when there were more than on e hidden layers, we pretrained each
layer as a single-layer DAE with sparsity target set to 0.1 [Vincent et al., 2010, Lee et al., 2008].
The details on training procedures are described in Appendix A.
One important difference to the recent work by Xie et al. [201 2] and Burger et al. [2012] is that
the denoising task we consider in this paper is completely blind. No prior knowledge about target
1http://sipi.usc.edu/database/
2We used 5 as suggested by Xie et al. [2012].
6
White noise Salt-and-pepper noise
Aerials
0.4
16
18
20
22
24
26PSNR
Noise Level0.1 0.2 0.4
16
18
20
22
24
26PSNR
Noise Level0.1 0.2
T extures
0.4
16
18
20
22
24
26
28PSNR
Noise Level0.1 0 .2 0.4
16
18
20
22
24
26
28PSNR
Noise Level0.1 0 .2
Misc.
0.4
15
20
25PSNR
Noise Level
0.1 0 .2 0.4
15
20
25PSNR
Noise Level
0.1 0 .2
 
 
DAE DAE(2) DAE(4) GDBM(2) GDBM(4) GRBM
Figure 2: PSNR of grayscale images corrupted by different ty pes and levels of noise. The median
PSNRs over the images in each set together.
images and the type or level of noise was assumed when trainin g the deep neural networks. In
other words, no separate training was done for different typ es or levels of noise injected to the test
images. Unlike this, Xie et al. [2012], for instance, traine d a DAE speciﬁcally for each noise level
by changing η(·) accordingly. Furthermore, the Boltzmann machines that we propose here for image
denoising, do not require any prior knowledge about the level or type of noise.
Two types of noise have been tested; white Gaussian and salt- and-pepper. White Gaussian noise
simply adds zero-mean normal random value with a predeﬁned variance to each image pixel, while
salt-and-pepper noise sets a randomly chosen subset of pixels to either black or white. Furthermore,
three different noise levels (0.1, 0.2 and 0.4) were tested. In the case of white Gaussian noise, they
were used as standard deviations, and in the case of salt-and-pepper noise, they were used as a noise
probability.
After noise was injected, each image was preprocessed by pixel-wise adaptive Wiener ﬁltering [see,
e.g., Sonka et al., 2007], following the approach of Hyv¨ arinen et al. [1999]. The width and height of
the pixel neighborhood were chosen to be small enough (3 ×3) so that it will not remove too much
detail from the input image.
Denoising performance was measured mainly with peak signal-to-noise ratio (PSNR) computed by
−10 log10(ǫ2), where ǫ2 is a mean squared error between an original clean image and th e denoised
one.
4.3 Results and Analysis
In Fig. 2, the performances of all the tested models trained on 8 ×8 image patches are presented3.
The most obvious observation is that the deep neural network s, including both DAEs and BMs,
did not show improvement over their shallow counterparts in the low-noise regime (0.1). However,
the deeper models signiﬁcantly outperformed their corresp onding shallow models as the level of
3Those trained on patches of different sizes showed similar trend, and they are omitted in this paper.
7
injected noise grew. In other words, the power of the deep mod els became more evident as the
injected level of noise grew.
This is supported further by Tab. 2 which shows the performan ce of the models in the high noise
regime (0.4). In all cases, the deeper models, such as DAE(4) , GDBM(2) and GDBM(4), were the
best performing models.
Another notable phenomenon is that the GDBMs tend to lag behind the DAEs, and even the GRBM,
in the low noise regime, except for the textures set. A possib le explanation for this rather poor
performance of the GDBMs in the low noise regime is that the ap proximate inference of the pos-
terior distribution, used in this experiment, might not hav e been good enough. For instance, more
mean-ﬁeld iterations might have improved the overall perfo rmance while dramatically increasing
the computational time, which would not allow GDBMs to be of any practical value. The GDBMs,
however, outperformed, or performed comparably to, the oth er models when the level of injected
noise was higher.
It should be noticed that the performance depended on the typ e of the test images. For instance,
although the images in the aerials set corrupted with salt-and-pepper noise were best denoised by the
DAE with four hidden layers, the GDBMs outperformed the DAE(4) in the case of the textures set.
We emphasize here that the deeper neural networks showed les s performance variance depending
on the type of test images, which suggests better generaliza tion capability of the deeper neural
networks.
Visual inspection of the denoised images provides some more intuition on the performances of the
deep neural networks. In Fig. 3, the denoised images of a sample image from each test image set are
displayed. It shows that BMs tend to emphasize the detailed s tructure of the image, while DAEs,
especially ones with more hidden layers, tend to capture the global structure.
Additionally, we tried the same set of experiments using the models trained on a set of 50, 000
random image patches extracted from the Berkeley Segmentat ion Dataset Martin et al. [2001]. In
this case, 100 patches from randomly chosen locations from each of 500 images were collected to
form the training set. We obtained the results similar to tho se presented in this paper. The results
are presented in Appendix B.
Method
Aerials Textures Misc.
Wiener 15.7 (0.1) 15.5 (0.6) 15.9 (0.6)
DAE 16.4 (0.2) 16.2 (0.9) 16.6 (0.8)
DAE(2) 17.6 (0.2) 17.1 (1.2) 17.7 (1.1)
DAE(4) 20.8 (0.7) 18.7 (2.8) 20.2 (2.0)
GRBM 19.2 (0.4) 18.0 (1.7) 18.9 (1.5)
GDBM(2) 22.3 (1.4) 18.7 (3.2) 20.1 (2.4)
GDBM(4) 22.1 (1.1) 18.7 (3.0) 20.2 (2.2)
Aerials Textures Misc.
16.3 (0.5) 14.9 (1.3) 15.3 (1.5)
17.1 (0.6) 15.7 (1.4) 16.3 (1.7)
18.1 (0.7) 16.4 (1.7) 17.3 (2.0)
20.1 (1.1) 17.2 (2.8) 19.0 (2.7)
18.9 (0.9) 16.6 (2.1) 17.6 (2.2)
20.3 (1.4) 16.5 (3.0) 17.5 (2.6)
20.3 (1.3) 16.6 (2.9) 17.6 (2.5)
(a) White Gaussian Noise (b) Salt-and-Pepper Noise
Table 2: Performance of the models trained on 4 ×4 image patches when the level of injected noise
was 0.4. Standard deviations are shown inside the parenthes es, and the best performing models are
marked bold.
5 Conclusion
In this paper, we proposed that, in addition to DAEs, Boltzma nn machines, GRBMs and GDBMS,
can also be used for denoising images. Furthermore, we tried to ﬁnd empirical evidence supporting
the use of deep neural networks in image denoising tasks.
Our experiments suggest the following conclusions for the questions raised earlier:
Does a model with more hidden layers perform better?
In the case of DAEs, the experiments clearly show that more hidden layers do improve performance,
especially when the level of noise is high. This does not always apply to BMs, where we found that
the GRBMs outperformed, or performed as well as, the GDBMs in few cases. Regardlessly, in the
high noise regime, it was always beneﬁcial to have more hidden layers.
8
Original Noisy DAE DAE(4) GRBM GDBM(4)
16.79 18.96 18.10 18.96
(a) Textures
16.02 18.59 17.75 19.37
(b) Aerials
17.59 21.34 19.42 18.98
(c) Miscellaneous
Figure 3: Images, corrupted by salt-and-pepper noise with 0.4 noise probability, denoised by various
deep neural networks trained on 8 ×8 image patches. The number below each denoised image is
the PSNR.
How well does a deep model generalize?
The deep neural networks were trained on a completely separa te dataset and were applied to three
test sets with very different image properties. It turned ou t that the performance depended on each
test set, however, with only small differences. Also, the tr end of deeper models performing better
could be observed in almost all cases, again especially with high level of noise. This suggests that
a well-trained deep neural network can perform blind image denoising, where no prior information
about target, noisy images is available, well.
Which family of deep neural networks is more suitable, BMs or DAEs?
The DAE with four hidden layers turned out to be the best perfo rmer, in general, beating GDBMs
with the same number of hidden layers. However, when the level of noise was high, the Boltzmann
machines such as GRBM and GDBM(2) were able to outperform the DAEs, which suggests that
Boltzmann machines are more robust to noise.
One noticeable observation was that the GRBM outperformed, in many cases, the DAE with two
hidden layers which had twice as many parameter. This potent ially suggests that a better inference
of approximate posterior distribution over the hidden unit s might make GDBMs outperform, or
comparable to, DAEs with the same number of hidden layers andunits. More work will be required
in the future to make a deﬁnite answer to this question.
Although it is difﬁcult to make any general conclusion from t he experiments, it was evident that
deep models, regardless of whether they are DAEs or BMs, performed better and were more robust
to the level of noise than their more shallow counterparts. I n the future, it might be appealing
to investigate the possibility of combining multiple deep n eural networks with various depths to
achieve better denoising performance.
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11
A T raining Procedures: Details
Here, we describe the procedures used for training the deep neural networks in the experiments.
A.1 Data Preprocessing
Prior to training a model, we normalized each pixel of the tra ining set such that, across all the
training samples, the mean and variance of each pixel are 0 and 1.
The original mean and variance were discarded after trainin g. During test, we computed the mean
and variance of all image patches from each test image and used them instead.
A.2 Denoising Autoencoders
A single-layer DAE was trained by the stochastic gradient de scent for 200 epochs. A minibatch of
size 128 was used at each update, and a single epoch was equiva lent to one cycle over all training
samples.
The initial learning rate was set to η0 = 0.05 and was decreased over training according to the
following schedule:
ηt = η0
1 + t
5000
.
In order to encourage the sparsity of the hidden units, we used the following regularization term
−λ
N∑
n=1
q∑
j=1
(
ρ −φ
( p∑
i=1
v(n)
i wij + cj
)) 2
,
where λ = 0.1, ρ = 0.1 and p and q are respectively the numbers of visible and hidden units. φ is a
sigmoid function.
Before computing the gradient at each update, we added a white Gaussian noise of standard deviation
0.1 to all components of an input sample and forced randomly chosen 20% of input units to zeros.
The weights of a deep DAE was ﬁrst initialized by layer-wise p retraining. During the pretraining,
each layer was trained as if it were a single-layer DAE, follo wing the same procedure described
above, except that no white Gaussian noise was added for the layers other than the ﬁrst one.
After pretraining, we trained the deep DAE with the stochast ic backpropagation algorithm for 200
epochs using minibatches of size 128. The initial learning r ate was chosen to be η0 = 0.01 and the
learning rate was annealed according to
ηt = η0
1 + t
5000
.
For each denoising autoencoder regardless of its depth, we used a tied set of weights for the encoder
and decoder.
A.3 Restricted Boltzmann Machines
We used the modiﬁed energy function of a Gaussian-Bernoulli RBM (GRBM) proposed by
Cho et al. [2011a], however, with a single σ2 shared across all the visible units. Each GRBM was
trained for 200 epochs, and each update was performed using a minibatch of size 128.
A learning rate was automatically selected by the adaptive learning rate [Cho et al., 2011a] with the
initial learning rate and the upper-bound ﬁxed to 0.001 and 0.001, respectively. After 180 epochs,
we decreased the learning rate according to
η ←η
t ,
where t denotes the number of updates counted after 180 epochs of training.
12
A persistent contrastive divergence (PCD) [Tieleman, 2008] was used, and at each update, a single
Gibbs step was taken for the model samples. Together with PCD , we used the enhanced gradient
[Cho et al., 2013], instead of the standard gradient, at each update.
A.4 Deep Boltzmann Machines
We used the two-stage pretraining algorithm [Cho et al., 201 2] to initialize the parameters of each
DBM. The pretraining algorithm consists of two separate stages.
We utilized the already trained single-layer and two-layer DAEs to compute the activations of the
hidden units in the even-numbered hidden layers of GDBMs. Noseparate, further training was done
for those DAEs in the ﬁrst stage.
In the second stage, the model was trained as an RBM using the coupled adaptive simulated temper-
ing [CAST, Salakhutdinov, 2010] with the base inverse tempe rature set of 0.9 and 50 intermediate
chains between the base and model distributions. At least 50 updates were required to make a swap
between the slow and fast samples.
The initial learning rate was set to η0 = 0.01 and the learning rate was annealed according to
ηt = η0
1 + t
5000
.
Again, the modiﬁed form of an energy function [Cho et al., 2011b] was used with a shared variance
σ2 for all the visible units. However, in this case, we did not use the enhanced gradient.
After pretraining, the GDBMs were further ﬁnetuned using the stochastic gradient method together
with the variational approximation [Salakhutdinov and Hin ton, 2009]. The CAST was again used
with the same hyperparameters. The initial learning rate was set to 0.0005 and the learning rate was
decreased according to the same schedule used during the second stage.
13
White noise Salt-and-pepper noise
Aerials
0.4
16
18
20
22
24
26PSNR
Noise Level0.1 0.2 0.4
16
18
20
22
24
26PSNR
Noise Level0.1 0.2
T extures
0.4
16
18
20
22
24
26
28PSNR
Noise Level0.1 0 .2 0.4
16
18
20
22
24
26
28PSNR
Noise Level0.1 0 .2
Misc.
0.4
15
20
25PSNR
Noise Level
0.1 0 .2 0.4
15
20
25PSNR
Noise Level
0.1 0 .2
 
 
DAE DAE(2) DAE(4) GDBM(2) GDBM(4) GRBM
Figure 4: PSNR of grayscale images corrupted by different ty pes and levels of noise. The median
PSNRs over the images in each set together. The models used for denoising in this case were trained
on the training set constructed from the Berkeley Segmentation Dataset.
B Result Using a T raining Set From Berkeley Segmentation Dat aset
Fig. 4 shows the result obtained by the models trained on the training set constructed from the Berke-
ley Segmentation Dataset. Although we see some minor differences, the overall trend is observed to
be similar to that from the experiment in the main text (see Fig. 2).
Especially in the high-noise regime, the models with more hi dden layers tend to outperform those
with only one or two hidden layers. This agrees well with what we have observed with the models
trained on the training set constructed from the CIFAR-10 dataset.
14


Peer review:
[Review]:
Dear reviewers (bf00) and (9120),

First of all, thank you for your thorough reviews. 

Please, find my response to your comments below. A revision
of the paper that includes the fixes made accordingly will
be available at the arXiv.org tomorrow (Tue, 5 Mar 2013
01:00:00 GMT).

To both reviewers (bf00) and (9120):

Thank you for pointing out the mistakes in some of the
equations. As both of you noticed, there was a problem in
Eq. (4). There should be as many binary matrices $D_n$ as
there are image patches from each test image. This mistake
happened as I was trying to put the procedure into a more
compact mathematical equation. There was no mistake in the
implementation. I have fixed the equation and its
accompanying text description accordingly.

Also, in Eq. (5), the term inside the last expectation
shoudl be p(v | h) instead of p(	ilde{v} | h). Thank you
again for pointing that out. 


To reviewer (bf00):

 - 'The experiments could be not easily reproduced'
   I have added the detailed configurations used for
   training each model as an appendix. 

 - 'models were not compared with any state-of-the-art
   denoising method'
   The aim of the paper was to propose an alternative deep
   neural network model that might be used in place of
   denoising autoencoders which were recently proposed to
   excel in the task of image denoising. However, I agree
   that the comparison with other approaches would make the
   paper more interesting.

 - 'should describe how to construct training set in detail'
   I have added how the training set was constructed.

 - 'high-resolution natural image data sets could be more
   proper'
   I fully agreed with you and thank you for the suggestion.
   I have run the same set of experiment using the training
   set constructed from the Berkeley Segmentation Benchmark
   (BSD-500). The results closely resemble those presented
   already in the paper, and the overall trend did not
   change. I have appended the new figure (same format as
   Fig.  2) obtained using the new training set in the
   appendix.


To reviewer (9120):

 - 'Proper layer-sizes cross validation should be performed'
   I fully agree with you. One most important thing that
   be checked, in my opinion, is the performance of
   single-layer models having the same number of hidden
   units as multi-layer models (e.g., GRBM with 640 and 1280
   hidden units trained on 8x8 patches). I will run the
   experiment, and if time permits, will add the results in
   the paper.


 - 'In eq of hat{v}_i'
   Thank you for pointing it out. I have mistakenly put
   p(h|	ilde{v}), implicitly assuming a case of RBM with
   binary hidden units, where p(h|	ilde{v}) coincides with
   E[h|	ilde{v}]. However, for a general BM, you are
   correct and I have fixed it accordingly.