2-D Dual-Tree Wavelet Based Local Adaptive Image Denoising.pdf by sushaifj

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									      2-D Dual-Tree Wavelet Based Local Adaptive Image Denoising

                   M. Borhani                                                 V. Sedghi
         Sharif University of Technology                         Amir-Kabir University of Technology
             Borhani@ee.sharif.edu                                      S8123232@aut.ac.ir



Abstract: We propose a new wavelet-based method           representation using a set of analyzing functions
for image denoising that applies the two dimensions       that are dilations and translations of a few
Dual Tree wavelet Transform and adapt thereshold in       functions.
each local area to obtain better denoised image.          The wavelet transform comes in several forms. The
Empirical results show that new approach is better than   critically-sampled form of the wavelet transform
older algorithms. In this paper, the structure of Dual    provides the most compact representation;
Tree wavelet transform (DTDWT) is modified and
                                                          however, it has several limitations. For example, it
Comparison results are also presented.
                                                          lacks the shift-invariance property, and in multiple
                                                          dimensions it does a poor job of distinguishing
Keywords: Image Denising, Local Adaptive                  orientations, which is important in image
threshold, Wavelet transforms.                            processing. For these reasons, it turns out that for
                                                          some applications improvements can be obtained
1 Introduction                                            by using an expansive wavelet transform in place
                                                          of a critically-sampled one. (An expansive
In many applications, image denoising is used to          transform is one that converts an N-point signal
produce good estimates of the original image from         into M coefficients with M > N .) There are
noisy observations. The restored image should             several kinds of expansive DWTs; here we
contain less noise than the observations while still      describe and provide an implementation of the
keep sharp transitions (i.e. edges).                      dual-tree complex discrete wavelet transform.
Wavelet transform, due to its excellent localization      The dual-tree complex wavelet transform
property, has rapidly become an indispensable             overcomes these limitations - it is nearly shift-
signal and image processing tool for a variety of         invariant and is oriented in 2D [8]. The 2D dual-
applications, including compression and denoising         tree wavelet transform produces six sub bands at
[1,2,4,5,6,7]. Wavelet thresholding (first proposed       each scale, each of which is strongly oriented at
by Donoho [4,5,6]) is a signal estimation technique       distinct angles.
that exploits the capabilities of wavelet transform       The paper is organized as follows. In Section 2, the
for signal denoising and has recently received            wavelet theresholding is discussed. In Section 3,
extensive research attentions. It removes noise by        we present the one dimension Dual Tree wavelet
killing coefficients that are insignificant relative to   transform. The two dimensions Dual Tree wavelet
some threshold, and turns out to be simple and            transform algorithm is presented in Section 4. In
effective. Wavelet thresholding solution given by         Section 6, the image denoising algorithm is
Donoho has also proven to be asymptotically               presented. Section 5 presents the simulation results
optimal in a minimax MSE (mean squared error)             comparing the two different Dual Tree schemes
sense over a variety of smoothness spaces [2,6].          and final conclusions are drawn in Section 7.
For many natural signals, the wavelet transform is
a more effective tool than the Fourier transform.
The wavelet transform provides a multiresolution
2 Wavelet Theresholding                                  The transform is 2-times expansive because for an
                                                         N-point signal it gives 2N DWT coefficients. If the
Wavelet thresholding for image denoising involves        filters in the upper and lower DWTs are the same,
two steps: 1) taking the wavelet transform of an         then no advantage is gained. However, if the filters
image (i.e., calculating the wavelet coefficients); 2)   are designed is a specific way, then the sub band
discarding (setting to zero) the coefficients with       signals of the upper DWT can be interpreted as the
relatively small or insignificant magnitudes. By         real part of a complex wavelet transform, and sub
discarding small coefficients one actually discards      band signals of the lower DWT can be interpreted
wavelet basis functions which have coefficients          as the imaginary part. Equivalently, for specially
below a certain threshold. The denoised signal is        designed sets of filters, the wavelet associated with
obtained via inverse wavelet transform of the kept       the upper DWT can be an approximate Hilbert
coefficients. One global threshold derived by            transform of the wavelet associated with the lower
Donoho [5, 6].                                           DWT. When designed in this way, the dual-tree
In summary, application of wavelet thresholding          complex DWT is nearly shift-invariant, in contrast
for image denoising involves the following steps:        with the critically-sampled DWT. Moreover, the
1) Define a structure by appropriate importance          dual-tree complex DWT can be used to implement
ordering of all wavelet basis functions. The             2D wavelet transforms where each wavelet is
original wavelet thresholding technique is               oriented, which is especially useful for image
equivalent to specifying a structure that uses only a    processing. The dual-tree complex DWT
magnitude ordering of the wavelet coefficients.          outperforms the critically-sampled DWT for
Obviously, this is not the best way of ordering the      applications      like    image    denoising     and
coefficients. A better tree structure is presented in    enhancement.
this paper.                                              The analysis and synthesis filters are well
2) Estimate the prediction risk for each set of          discussed by [9]. Kingsbury provide a perfect
wavelet functions formed in the structure.               reconstruction analysis and synthesis filterbanks
                                                         for image denoising with critically-sampled for
3 1-D Dual-Tree Wavelet Transform                        discrete wavelet shrinkage. The perfect
                                                         reconstruction property of the dual-tree complex
It turns out that, for some applications of the          DWT is verified in the following example. We
discrete wavelet transform, improvements can be          create a random input signal x of length 512, apply
obtained by using an expansive wavelet transform         the dual-tree complex DWT and its inverse. We
in place of a critically-sampled one. There are          then show that the reconstructed signal y is equal
several kinds of expansive DWTs; here we                 to the original signal x by computing the maximum
describe the dual-tree complex discrete wavelet          value of |x-y|.
transform.
The dual-tree complex DWT of a signal x is
implemented using two critically-sampled DWTs
in parallel on the same data, as shown in the figure.

                                         h0 (n)   ↓2
                        h0 (n)   ↓2
      h0 (n)     ↓2                      h1(n)    ↓2
                        h1(n)    ↓2
       h1(n)     ↓2                      g0(n)    ↓2
                         g0(n)   ↓2
         g0(n)   ↓2                      g1(n)    ↓2
                        g1(n)    ↓2
        g1(n)    ↓2

                                                              Figure 2: ψ (t ) for complex Dual Tree DWT
     Figure 1: Dual-Tree complex DWT structure
4       2-D Dual-Tree Wavelet Transform                 4.2 Complex 2-D Dual-Tree Wavelet Transform

In the wavelet domain, each shift of the image          The complex 2-D dual-tree DWT also gives rise to
corresponds to a different tree of wavelet              wavelets in six distinct directions, however, in this
coefficients.                                           case there are two wavelets in each direction as
One of the advantages of the dual-tree complex          will be illustrated below. In each direction, one of
wavelet transform is that it can be used to             the two wavelets can be interpreted as the real part
implement 2D wavelet transforms that are more           of a complex-valued 2D wavelet, while the other
selective with respect to orientation than is the       wavelet can be interpreted as the imaginary part of
separable 2D DWT.                                       a complex-valued 2D wavelet. Because the
There are two versions of the 2D dual-tree wavelet      complex version has twice as many wavelets as the
transform: the real 2-D dual-tree DWT is 2-times        real version of the transform, the complex version
expansive, while the complex 2-D dual-tree DWT          is 4-times expansive. The complex 2-D dual-tree is
is 4-times expansive. Both types have wavelets          implemented as four critically-sampled separable
oriented in six distinct directions. We describe the    2-D DWTs operating in parallel. However,
real version first.                                     different filter sets are used along the rows and
                                                        columns. As in the real case, the sum and
4.1 Real 2-D Dual-Tree Wavelet Transform                difference of sub band images is performed to
                                                        obtain the oriented wavelets.
The real 2-D dual-tree DWT of an image x is             The twelve wavelets associated with the real 2D
implemented      using    two      critically-sampled   dual-tree DWT are illustrated in the figure 4 as
separable 2-D DWTs in parallel. Then for each           gray scale images.
pair of sub bands we take the sum and difference.
The six wavelets associated with the real 2D dual-
tree DWT are illustrated in the following figures as
gray scale images.
Note that each of the six wavelets is oriented in a
distinct direction. Unlike the critically-sampled
separable DWT, all of the wavelets are free of
checker board artifact. Each sub band of the 2-D
dual-tree transform corresponds to a specific
orientation.
To show directional wavelets for reduced 2-D
DWT, all of the wavelet coefficients are set to               Figure 4: 2D Dual-Tree complex wavelets
zero, for the exception of one wavelet coefficient
in each of the six sub bands. We then take the          Note that the wavelets are oriented in the same six
inverse wavelet transform. It is shown in figure 3.     directions as those of the real 2-D dual-tree DWT.
                                                        However, here we have two in each direction. If
                                                        the six wavelets displayed on the first (second) row
                                                        are interpreted as the real (imaginary) part of a set
                                                        of six complex wavelets, then the magnitude of the
                                                        six complexes are shown on the third row. As
                                                        shown in the figure, the magnitudes of the complex
                                                        wavelets do not have an oscillatory behavior-
                                                        instead they are bell-shaped envelopes.
                                                        In the reminder of this paper implementation of
                                                        these structure are compared and local adaptive
                                                        algorithm for threshold’s optimization are
                                                        presented.
                                                        Due to space limit, we only show results on 8-bit
                                                        Barbara image in this paper. Figure 10 shows the
    Figure 3: Directional wavelets for real 2-D DWT     comparable denoising results on 512 * 512
                                                        Barbara images corrupted by gaussian white noise
                                                        σ 2 = 40 .
5 Image Denoising                                               Table 1: Analysis and synthesis filters

                                                        Analysis      Analysis       Synthesis       Synthesis
One technique for denoising is wavelet
                                                        filter 1st     filters        filter 1st       filters
thresholding (or "shrinkage"). When we
                                                            0         -0.0112          0.0112             0
decompose data using the wavelet transform, we              0          0.0112          0.0112             0
use filters that act as averaging filters, and others   -0.0884        0.0884         -0.0884         -0.0884
that produce details. Some of the resulting wavelet      0.0884        0.0884          0.0884         -0.0884
coefficients correspond to details in the data set       0.6959       -0.6959          0.6959          0.6959
(high frequency sub-bands). If the details are           0.6959        0.6959          0.6959         -0.6959
small, they might be omitted without substantially       0.0884       -0.0884          0.0884          0.0884
affecting the main features of the data set. The idea   -0.0884       -0.0884         -0.0884          0.0884
of thresholding is to set all high frequency sub-        0.0112           0               0            0.0112
band coefficients that are less than a particular        0.0112           0               0           -0.0112
threshold to zero. These coefficients are used in an
inverse wavelet transformation to reconstruct the
data set [7].
This program has two parameters, one for noise
signal and the other for threshold point. A sample
noise signal is shown below, whose dimension is
512 x 512. We first take the forward DWT over 4
scales (J=4). Then a denoising method called soft
thresholding is applied to wavelet coefficients
through all scales and sub bands. Function sets
coefficients with values less than the threshold(T)
to 0, then substracts T from the non-zero
coefficients. After soft thresholding, we take
inverse wavelet transform.
The following example shows how to convert an
image to double data type, how to creat a noise
signal and display the denoised image. Note that
we use a threshold value of 35, which is the
optimal threshold point for this case. We will
introduce how to find the optimal threshold value                         Fig.5. noisy image
in the later part of this section.
From the resulting image, we can see the denoising
capability of separable 2-D DWT. Now we want to
improve the effect by using complex 2-D dual-tree
DWT. The optimal threshold value for this method
is 40.
We can see that complex 2-D dual-tree method
removes more noise signal than separable 2-D
method does. Actually, 2-D dual-tree method
outperforms separable method. This can be proved
by figure 7. It illustrates the denoising capability
for three different methods with complex 2-D dual-
tree method be the best, followed by real 2-D dual-
tree method and separable method. Also, it tells us
where the optimal threshold points locate. For each
method, applying the optimal threshold point
yields the mininum RMS error. Therefore, a
threshold producing the minimum RMS error is the
optimal one.
                                                        Fig. 6. Denoised image (by complex 2-D dual-tree DWT)
            22
                                                                   Standard 2D              using a Bayesian estimation approach beginning
            20
                       RMS error V.S. Threshold Pt.
                                                                   Real 2D dual
                                                                   Complex 2D dual
                                                                                            with the new bivariate non-Gaussian model. The
                                                                                            plot is illustrated in the figure below.
            18


            16                                                                              6.1 Local Adaptive Image Denoising
RMS error




            14
                                                                                            Using the bivariate shrinkage function above,
            12
                                                                                            developed an effective and low complexity locally
                                                                                            adaptive image denoising algorithm in [7]. This
            10                                                                              shrinkage function requires the prior knowledge of
            8
                                                                                            the noise variance and the signal variance for each
                                                                                            wavelet coefficient. Therefore the algorithm first
            6
                 0     5     10       15    20     25     30       35    40      45    50
                                                                                            estimates these parameters (see [7] for the recipe of
                                                 Threshold pt.                              the estimation rules).
                                                                                            Briefly, the algorithm is summarized as follows:
                      Figure 7: rms error V.S. threshold points                             1.       Calculate the noise variance.
                                                                                            2.       For each wavelet coefficient.
6 Bivariate Shrinkage                                            Functions            for   3.       Calculate signal variance
Wavelet Based Denoising                                                                     4.       Estimate each coefficient
We have proposed a new simple non-Gaussian
bivariate probability distribution function to model
the statistics of wavelet coefficients of natural                                           3
images. The model captures the dependence
                                                                                            2
between a wavelet coefficient and its parent. Using
Bayesian estimation theory we derive from this                                              1

model a simple non-linear shrinkage function for                                            0
wavelet denoising, which generalizes the soft                                               -1
thresholding approach of Donoho and Johnstone.
                                                                                            -2
The new shrinkage function, which depends on
both the coefficient and its parent, yields improved                                        -3
                                                                                             4
results for wavelet-based image denoising.                                                       2                                                   4
Let w2 represent the parent of w1 ( w2 is the                                                          0                                         2
                                                                                                                                         0
wavelet coefficient at the same spatial position                                                             -2                 -2
                                                                                                                   -4   -4
as w1 , but at the next coarser scale). Then
 y = w + n where w = ( w1 , w2 ) , y = ( y1 , y 2 )                                                  Figure 8: A bivariate shrinkage function.

and n = (n1 , n 2 ) . The noise values n1 , n 2 are iid                                     7 Empirical Results
zero-mean Gaussian with variance σ n . Based on
                                     2


the empirical histograms we have computed, we                                               The     following    sections    describe   the
propose the following non-Gaussian bivariate pdf                                            implementation of this algorithm using both the
                                                                                            separable DWT and the complex dual-tree DWT.
                              3                       3                                     Two different approaches are compare in figure
  p w ( w) =                          . exp(−                 w12 + w2 )
                                                                     2
                                                                                      (1)
                           2πσ    2
                                                 σ                                          10-12. Figure 10 is a noisy image of Barbara.
With this pdf, w1 and w2 are uncorrelated, but not                                          Separable DWT applied to figure 10 to obtain results
                                                                                            that is shown in figure 11. Result of Dual Tree DWT
independent. The MAP estimator of w1 yields the                                             was drown in figure 12. One can compare two
following bivariate shrinkage function                                                      approaches and find quality of different structures.
                                           3σ n
                                              2                                             Comparison of signal to noise ratio and Quality are
                     ( y12 + y 2 −
                               2
                                                   )                                        given in Table 2.
w1 =
ˆ                                          σ           . y1                           (2)
                               y12 + y 2
                                       2


For this bivariate shrinkage function, the smaller
the parent value, the greater the shrinkage. This is
consistent with other models, but here it is derived
                                                       Table 2: Comparison of S/N and Quality

                                             Noise        Separable   Quality    DTDWT      Quality
                                             level        S/N (dB)               S/N (dB)
                                             10/255       32.1778     very       33.2962    Very
                                                                      good                  good
                                             40/255       24.8396     good       26.2060    Very
                                                                                            good
                                             50/255       23.9009     good       25.2147    Good
                                             70/255       22.7343     not bad    23.8492    Good
                                             80/255       22.2854     not bad    23.3051    not bad
                                             90/255       21.9698     bad        22.9180    not bad
                                             140/255      20.7283     bad        21.4548    bad
                                             190/255      19.8390     very bad   20.5095    bad

                                             8 Conclusions
                                             Two dimensions Dual Tree wavelet Transform are
                                             proposed and local adaptive theresholding
       Figure 10: First: Noisy image         algorithm is presented. Empirical results show that
                                             new approach is better than older algorithms.

                                             References
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                                                 Wavelet Thresholding for Image Denoising",
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                                                 1997
                                             [2] A. Chambolle, R. A. DeVore, N-Y Lee and B.
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                                                 processing: variational problems, compression
                                                 and noise removal through wavelet shrinkage”,
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                                                 319-335, 1998
                                             [3] V. Cherkassky and F. Mulier, Learning from
                                                 Data: Concepts, Theory and Methods, Wiley
                                                 Interscience, 1998
                                             [4] D. L. Donoho, "Wavelet Thresholding and
Figure 11: Denoised image by Separable DWT       W.V.D.: A 10- minute Tour", Int. Conf. on
                                                 Wavelets and Applications, Toulouse, France,
                                                 June 1992
                                             [5] D. L. Donoho and I. M. Johnstone, "Ideal
                                                 spatial adaptation via wavelet thresholding",
                                                 Biometrika, vol. 81, pp. 425-455, 1994
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                                                 Thresholding", IEEE Trans. Information
                                                 Theory, vol. 41 May 1995
                                             [7] L. Sendur, I.W. Selesnick, "Bivariate shrinkage
                                                 with local variance estimation", IEEE Signal
                                                 Processing Letters, 9(12), 438-441, Dec 2002.
                                             [8] N. G. Kingsbury. "Complex wavelets for shift
                                                 invariant analysis and filtering of signals",
                                                 Applied and Computational Harmonic
                                                 Analysis, 10(3):234-253, May 2002.
                                             [9] N. G. Kingsbury, "Image processing with
                                                 complex wavelets”, Phil. Trans. Royal Society
                                                 London A, September 1999.
  Figure 12: Denoised image by Dual Tree

								
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