j,m

th 30 NATIONAL RADIO SCIENCE CONFERENCE (NRSC 2013) April 16-18, 2013, National Telecommunication Institute, Egypt

C18 :A New Total Variation Based Image Denoising and Deblurring Technique

M F. Fahm/, G. M Abdel Raheem!, us. Mohammed! and 0. F. Fahm/ 1- Department of Electrical Engineering, Assiut University, Assiut, Egypt. 2- Department of Electrical Engineering, Future University, Cairo, Egypt. [email protected]; omar.([email protected] ABSTRACT

Total variation (TV) regularization is popular in image restoration and reconstruction due to its ability to preserve image edges. This paper, describes a new total variation based de-noising scheme. The proposed technique optimally finds the threshold level of the noisy image wavelet decomposition that minimizes the energy of the error between the restored and the noisy image. The minimization algorithm is regularized by including 1st as well as 2nd order derivatives effects of the noisy image, into the minimization scheme. Next, the problem of blind deconvolution of noisy images is addressed. First, the order of the blurring Point Spread Function (PSF), is accurately estimated using a de-noised version of the noisy blurred image. Then, the deconvolution algorithm is modified by including the effects of the 1st as well as 2nd order derivatives of the blurred noisy images into the image update algorithm. Simulation results have shown significant performance improvements of the proposed schemes in both de-noising as well as deblurring noisy image. Keywords: Blind Image Deconvolution, Image de-noising, Image Restoration. I.

Introduction

Image reconstruction techniques have become important tools in computer VISIOn systems and many other applications that require sharp images obtained from noisy and otherwise corrupted ones. At the same time the total variation (Tvr) formulation has proven to provide a good mathematical basis for several basic operations in image reconstruction, such as denoising and deblurring. In the noisy blurred case, the proposed Enhanced Fast Iterative Blind Deconvolution algorithm EFIBD, [1] and the other blind deconvolution techniques like IBD or RL, were found to yield poor image restoration, [2-9]. This is expected as a small perturbation in the blurred noisy image g, may produce unbounded variations in the output, where (1) g=h**f + w h is the point spread function PSF that is responsible for blurring, and is generally unknown,fis the original image while w is the associated zero-mean noise. The solution to this problem is to add a regularization term to the

l

minimization of g

-

gil, Ii = h

* *

j . Many regularization schemes have been proposed,

[lO-12]. Perhaps the

total variation (TV) Ll-norm, is the most widely used, as it ensures that, the restored image has bounded variation rather than being continuous. In this paper, the regularization parameter is chosen by a weighted 1st norm of gradient and hessian matrix of f in the course of iteration. The hessian effect is evaluated the output of a Laplacian filter [13], when driven by the noisy image. The paper starts by describing a new total variation TV de noising scheme. The proposed de-noising scheme selects the optimum threshold T of the noisy image wavelet de composition that minimizes the total variation of the output de-noised image. Next, the proposed denoising scheme is also used to improve the sharpness of the restoration quality by considering the Laplacian of Gaussian 'LoG' filer effect, [13]. Finally, the EFIBD algorithm that performs fast blind image deconvolution in the noiseless case, [14-15] has been modified by including the regularizing Tvr function. Simulation results for image de-noising, image sharpening and image de-blurring; have shown considerable improvements over known techniques for both Gaussian as well as Salt & Pepper noises, with different variances and densities. II. The Total Variation Denoising Scheme

If a signalfis corrupted with an uncorrelated zero-mean noise w, to yield a noisy data g, i.e. g=f+w, then its wavelet coefficient decompositions are given by

(.) is the inner product and Ij/

(g,1j/ ) (1,1j/ ) (w,1j/ ) j,m

j,m

=

j,m

+

j,m

(2)

represents the j'h-scale wavelet bases used. As almost all

speech and images, are nominally baseband signals, the wavelet coefficients

(g,

If/ j,1II

physical signals like

> at fine scales, are mainly

due to noise, and have to be thresholded, Thresholding amounts to keeping coefficients greater than a specific threshold level T, while setting to zero all coefficients less than T. In [16], the threshold level T, is estimated

978-1-4673-6222-11131$31.00 ©2013 IEEE

280


assuming that the associated noise is Gaussian. Simulation results have also verified that this estimation may not perform as well, for other types of noise. T is evaluated in terms of the associated noise variance, i.e. T a loge , N is the length of the noisy signal. The variance 0-2 is estimated either from the median M =

of / g, '1/ \ j,

(J2N ) m) at the finest j

scale as

0-

=

E(M)

or more accurately through estimating the probability

0.6745

distribution function pdf, of the noise that mainly occupies in the finest wavelet coefficient de-composition of the noisy signal/image. Their amplitude distribution is first computed using classical histogram estimation techniques. Next, this estimation is smoothed up using Bspline pdl estimation, as described in [17]. In this paper, the threshold level T, is optimally chosen to minimize some weighted functions of the total variation function � of the noisy image, [10-12]. The total variation function of a vector X, is defined as

�=

LIX(n+l)-X(n)1 . For a matrix 1', the total variation amounts to evaluating the derivatives along the n

y1 (m,n),yj (m,n) .The total variation along the & axis, is defined as the LI first norm of y1 (m,n),yj (m,n ) , respectively. Denote these norms by horizontal and vertical axis. Denote these derivatives by

Ax

paper, we take

Ao =

)

A

l

+ A

J, as a measure of the total variation. Thus, minimizing

Ao,

x

Ay.

and y

In this

amounts to noise

reduction, as adjacent image pixels are correlated to each other, while noise is not. To improve noise reduction capabilities, we include second order derivative to the minimization scheme, as a multi-variable functionl(x), can be expanded as

I(x) = I(xo)+ V(j(x))L1x+iv2(j(x))L1x2+..

2 J is the hessian matrix. Now, as first and higher derivatives are sensitive to noise, then minimizing amounts to noise minimization.

1'1721 1

1

(3)

1'171 I , '172 I I

is computed as the LI-norm of the noisy image, when filtered by a

Laplacian filter. So, the proposed de-noising scheme amounts to, choosing the threshold level T that minimizes

�=IV/I+lv2 1

1 +llg- III

(4) Matlab subroutine, as well as the pdf-based de-noising technique. 2-level 'bior4.4' wavelet decomposition is used for threshold determination. PDF is estimated by smoothing 128 bins histogram of the coefficients of the finest wavelet packets, using 3-level cubic Bspline wavelets, [17]. Fig. (1), shows the PSNR and edge error energies of some noisy images, corrupted with zero-mean Gaussian noise with different variances, as well as Salt & Pepper noise with different densities. The edge error energy qr =

I Edg(jj ) -Edg(I)11 , Edg(ji) is

the

edge

of

the

lossless

image.

smaller qr

The

I

the

sharper is the image. These results indicate the following points: 1. The Median-based de-noising technique is inadequate for de-noising Salt & Pepper noise. 2. The performance of the proposed TVR technique offers significant PSNR improvements, as well as sharper denoised image. Figure (2), compares the quality of the de-noised images, of these three schemes. To end this section, the sharpness of the de image can be enhanced by boosting the energies of the edges. To achieve this, we propose filtering the image by a Laplacian filter, as the LClllooCll responds only to the fine detail in the im�e. In order to reduce the sensitivities of 2nd order derivatives to noise, the filtered image is processed by a lowpass Gaussian filter. This results in the Laplacian of Gaussian kernel, Log filter.. If the original im�e is adda::J to the Log filtera::J im�e, enhCllcing of fine details Gal be ochiENa::J. Thus, if the de-nois:rl im�a! S, is processed by Log filter " to yield an edge image Sd, then sharpening can be obtained by constructing the image

l

l

Ssh =S\+aSd· Again, the parameter a , is determined through minimizing q=IVSShl+ v2SSh .

Fig. (3), illustrates the performance of the PSNR and error edge energy of the TVR and TVR sharpened images when corrupted with Gaussian as well as Salt & Pepper with different variances and densities respectively. This figure indicates that the proposed sharpened technique indeed yields sharper images. The followings table illustrates the PSNR improvements of Sharpened TVR over Median, PDF and TVR denoised images when the images are corrupted with Gaussian as well as Salt and Pepper noises.

281


Table (1) PSNR improvements Cameraman

Noisy

Median Based

PDF Based

Opt. Tvr

Sharpend Tvr

19.4

20.97

22.65

23.10

23.00

Noisy

Median Based

PDF Based

Opt. Tvr

Sharpened Tvr

19.02

21.71

24.51

24.32

24.33

14.75

Gaussian

Salt & Pepper

Salt & Pepper

Lena

22.77

14.67

Gaussian Noise

21,20

21.13

21.74

22.84

22.78

21.79

23.02

III. Blind Deconvolution of Noisy Images

In this section, we will show how the above de-noising techniques, can be implemented in the blind deconvolution process of noisy images. To start, let the original image f of size M x N is blurred by an unknown PSF h, of size] x K, then the blurred received noisy image g is given by

g(m,n)

=

J-I

K-I

I I h(k,j) f(m - k,n - j)+ w(m,n)

(5)

k=O j=O

( ) is the associated zero-mean additive noise. For simplicity, let M =N, J=K=Np. Using circular convolution properties [10-11], overlap is avoided if each row vector off, g and w is padded by zeros to make its length equals

w m,n

Mo

M+Np-l

=

=

No. So, if

1, g & 111

represent MaNa column vectors formed by stacking the rows of the extended

padded matrices, Eqn. (5) can be expressed in terms of the block circulant matrix H, (size MaNa x MaNa), as

g= H!+w', ' H=

HO HMo-I Hil Ho

HI H2

HNo-1 HNo-2

Ho

he(j,O)

he(j,No -1)

he(j,l)

he (j,I)

he (j,O)

h.(j,2)

he(j,No -2)

he(j,NO-I)

Hj-

. .. .

(6)

he(j,O)

Eqn. (6) indicates that, even in case of prior knowledge of hand w, the recovery off, apart from requiring a huge amount of computations for ordinary sized images, it may result in an unbounded solution. Several Iterative Blind Deconvolution techniques [6-8], have been proposed to solve this problem in the noiseless case, using 2-D FFT. However, the success of these algorithms in restoring the original images depends on the precise estimation of the PSF order. In [9-11], an Enhanced Fast Iterative Blind Deconvolution algorithm EFIBD is proposed, that can accurately estimate the PSF order. It can be summarized as follows: 1- De-noise the received blurred image using any of the techniques proposed in sec. II, to get g( m,n ) . 2-

For an assigned order, estimate j

&

h, using any iterative blind deconvolution algorithm, like

RL

or IBD

algorithms. Next, evaluate

34-

g = j **h , e( m,n ) = g( m,n ) - g( m,n ) . Decompose ( n) using I-level wavelet decomposition. In this paper, the 'bior4.4' wavelet family is used. e m,

Evaluate the energy of the HH sub-band, for every prescribed PSF order. The exact order is the one that gives the minimum detail energy.

Having accurately estimated the PSF order, the Enhanced Fast Iterative Blind Deconvolution algorithm EFIBD, described in [10-11], and proved to have a superior deconvolution performance in the noiseless case, is now modified to deal with noisy blurred images. The modification is achieved by including a weighted total variation function into the minimization scheme. The proposed Total Variation-based Enhanced Fast Iterative Blind Deconvolution algorithm TV-EFIBD, is described as follows: th hk . Evaluate H(k) from as in Eqn. (6). Obtain the total variation Ao of 1. For the k estimate, evaluate g

k

=

* *

lk

gk ( m,n ) . Filter gk ( m,n) by a Laplacian filter and denote the 1st norm of its output by Alp and its energy by

Elp'

2. Evaluate the error ,A, g g - g (k) . Reshape =

j, g , g

& 11 g into a column vector form. 282


3.

Sort

I�gl in ascending order, using the Matlab command [V,lJ= sort��gl). Subsequently, sort H(k) in

the form Hs =H(k)(Is'!J.

I� g I

4.

F or a prescribed number M,., pick the largest deviations of

5.

Partition Hs as H =

6.

�(k) Only update!Mx responsible for the Mx largest deviations of �gM

s

[

]

Hll H\2 H2\ H22 t Mx

. Denote by I':l. g M = � g (end - M x x

+1

:

end) .

(7)

l J

through minimizing the objective (8)

a

is a weighing factor. Any unconstrained minimization routine can be used to minimize this function. The

updated image is sharpened by filtering by a Laplacian of Gaussian filter 'LoG', as described at sec. III. This concludes the image restoration cycle. 7.

l

l

Updating h (k)proceeds similarly through minimizing the energy of g- g: using the updated image. This update can easily be achieved through minimizing the objective function

8.

r;

=

2 (m, n), e g - Ii * *j .

LLe m

This concludes the PSF update. Go to (1) and reiterate until convergence is achieved. This is checked by ensuring

n

lj