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Optik 124 (2013) 2224– 2228

Contents lists available at SciVerse ScienceDirect

Optik

j o ur nal hom epage: www.elsev ier .de / i j leo

mage deblurring based on the estimation of PSF parameters and theost-processing

ong Min Leea, Jeong Ho Leea, Ki Tae Parkb, Young Shik Moona,∗

Department of Computer Science and Engineering, Hanyang University, Ansan, Gyeonggi-do 426-791, Republic of KoreaGeneral Education Curriculum Center, Hanyang University, Ansan, Gyeonggi-do 426-791, Republic of Korea

a r t i c l e i n f o

rticle history:eceived 9 February 2012ccepted 23 June 2012

a b s t r a c t

Two parameters of a point spread function (PSF) are the length and angle of the blur kernel. The deblurringperformance is affected by the estimation accuracy of the two PSF parameters. In this paper, we propose amethod of the PSF parameters estimation and the image reconstruction by using the periodicity of motion

eywords:mage reconstructioneblurring

blurred images in the frequency domain and propose the reconstruction method using the proposed post-processing for reducing ringing artifacts. Experimental results show that the proposed method estimatesthe PSF parameters more accurately than the existing ones.

© 2012 Elsevier GmbH. All rights reserved.

oint spread functioneconvolution

. Introduction

Blurred images can be obtained by camera shaking or objectotion [1,2]. This damaged image is called a motion blurred image.

motion blur is one of the major causes of image deteriora-ion. The motion blurred image is presented by a convolution of

latent image with the Point Spread Function (PSF) [3]. Recently,he estimation method of PSF parameters considering the peri-dicity of motion blurred images in the frequency domain haseen often used [4–7]. However, since these methods may causeata loss by transformation of the Fourier spectrum of the blurred

mage or by binarization of the spectrum, the accuracy of theSF parameters estimation can be degraded. Because the errorsropagate in the iterative reconstruction process, the error ofhe PSF parameters estimation affects the quality of the recon-tructed image. Therefore, we propose an estimation methodf the correct PSF parameters without modifying the spectrumata in order to solve the drawbacks of conventional methodsaused by data loss. Fig. 1 shows the framework of the proposedeblurring method. The outline of the paper is as follows. In Sec-ion 2, we explain a linear motion blur model. In Section 3, weescribe a method for estimating PSF parameters. In Sections 4

nd 5, we explain a reconstruction method. In Section 6, we showhe experimental results. Our conclusion is summarized in Sec-ion 7.

∗ Corresponding author. Tel.: +82 31 400 5196; fax: +82 31 419 1162.E-mail addresses: jmlee@visionlab.or.kr (J.M. Lee), jhlee@cse.hanyang.ac.kr (J.H.

ee), parkkt@cse.hanyang.ac.kr (K.T. Park), ysmoon@cse.hanyang.ac.kr (Y.S. Moon).

030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved.ttp://dx.doi.org/10.1016/j.ijleo.2012.06.067

2. Linear motion blur model

Linear motion blurred images are generated from the convolu-tion of the latent image with the PSF as shown in Eq. (1).

g(x, y) = f (x, y) ∗ h(x, y) (1)

h(x, y) = 1L

∏L

(x cos � + y sin �) (2)

∏L

(u) =

⎧⎪⎨⎪⎩

1 if |u| ≤ L

2

0 if |u| >L

2

(2)

Here, g(x, y), f(x, y), and h(x, y) represent the blurred image, thelatent image, and the PSF, respectively. Generally the PSF describesthe impulse response of a focused optical system to a point source.Eq. (2) is the PSF that consists of two parameters such as a blurlength L and a blur angle �. In order to estimate the periodicity of alinear motion blurred image in the frequency domain, we calculatethe Fourier spectrum of the blurred image.

3. Estimation PSF parameters

3.1. Angle estimation

The angle of a motion blur is estimated by measuring the

direction of the periodically dark region in the Fourier transformof the linear motion blurred image. In this paper, in order to detectthe correct blur direction, we use a method for detecting the pointsnear to zero on u and v axes. Fig. 2 shows how to find the nearest

J.M. Lee et al. / Optik 124 (2013) 2224– 2228 2225

Fig. 1. Framework of the proposed method.

Fig. 2. The estimation process of a blur angle and length: (a) Fourier spectrum, (b)Origin region of spectrum, (c) Minimum value of v axis, and (d) Gradient distributionof v axis. (For interpretation of the references to color in the text, the reader isreferred to the web version of the article.)

Fig. 3. Edge map from the reconstructed result of 200 iteration: (a) Edge of 200iteration, (b) Erosion of (a), and (c) Edge map.

points to zero on each axis. In order to estimate the periodicityof the Fourier spectrum in Fig. 2(a), we examine the values of uand v axes. As shown in Fig. 2(c), the gradient variations of thevalues of each axis are irregular. Therefore, in order to find thepoints for discriminating the periodicity automatically, we usea method for not detecting the transition points but measuringthe gradient variations. When we select a minimum value amongthree adjacent points along the v axis in Fig. 2(c), the section of theminimum values in Fig. 2(d) is in red circle in Fig. 2(a) and (c). Thispoint is the minimum point on v axis. We calculate the angle � byusing the property of a right triangle as Eq. (3).

� = tan−1 OB

OA(3)

3.2. Length estimation

The other PSF parameter L is calculated by Eq. (4) consideringthe � that is obtained by Eq. (3). In Eq. (4), d is a distance betweenspectrum lines, N is an image size, and L is a blur length. As shownin Fig. 2(c), d is calculated by definition of cosine.

L = N

d(4)

d = OB cos �

2226 J.M. Lee et al. / Optik 124 (2013) 2224– 2228

Fig. 4. The intensity distribution of test images.

3

Fctiwmooahtlt

4

rra

Fig. 6. The experimental results of image reconstruction without the post-processing: (a) Original image, (b) Blurred image, (c) Proposed method, and (d)Existing method.

TT

Fig. 5. Performance comparison with the conventional methods.

.3. Reduction of estimation complexity

In order to estimate the correct PSF parameters, we use theourier spectrum in the frequency domain [8]. However, a lot ofomputation time is required to transform the observed imageo the frequency domain. Consequently, the larger the size of anmage, the more increase of the time for the conversion. Therefore,

e propose a new method for reducing the time of parameter esti-ation. To this end, we downscale the observed image to a quarter

f its original size and apply the Discrete Fourier Transform (DFT)n the resized image. Through the estimation method mentionedbove, we can obtain the PSF parameters of the same angle and aalf length of the original one. It has been shown by experimentshat the length of the PSF parameters is linearly proportional to theength of the side of an image. By this estimation technique, theime complexity of estimating the PSF parameters can be reduced.

. Image deconvolution

Deconvolution is the method to remove motion blur andeconstruct a blurred image. The representative method of image

econstruction is Richardson–Lucy (RL) iterative deconvolutionlgorithm [9].

able 1he results of PSF parameter estimation.

Real Estimation Error

Parameter Angle (degree) Length (pixel) Angle (degree) Length (pixel) Angle (degree) Length (pixel)

Cameraman 30.00 10.00 29.62 10.15 0.38 0.15Lena 51.00 22.00 50.96 21.97 0.04 0.03Airplane 43.00 15.00 43.39 15.21 0.39 0.21

J.M. Lee et al. / Optik 124 (2013) 2224– 2228 2227

Table 2Performance comparison with the conventional methods.

Proposed method Moghaddam and Jamzad [11] Zhang and Xu [12]

Parameter Angle (degree) Length (pixel) Angle (degree) Length (pixel) Angle (degree) Length (pixel)

Minimum error 0.0 0.0 0.0 0.0 0.0 0.0Maximum error 1.2 0.8 2.0 1.9 3.0 2.0Average error 0.4 0.2 0.6 0.9 0.9 1.5Standard deviation 0.3 0.2 0.7 0.4 0.7 0.6

Table 3Similarity comparison among the reconstructed results.

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u

c

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20 iterations 200 iterations Proposed method

PSNR(dB) 25.83 28.37 32.70

Pixels in the observed image can be represented in terms of theoint spread function and the latent image as Eq. (5).

i =∑

j

pijuj (5)

here pij is the point spread function, uj is the pixel value at position in the latent image, and di is the observed value at position i. Theasic idea of [9] is to calculate the most similar uj when the observedi and known pij are given. In Eq. (6), if this iteration converges, itas been empirically shown to converge to the maximum likelihoodolution for uj [10]. In our experiments, the reconstructed image isenerated by performing 200 iterations of RL deconvolution.

(t+1)j

= u(t)j

∑i

di

cipij (6)

i =∑

j

u(t)j

pij

. Post-processing for reducing ringing artifacts

Even though increasing the number of iterations of RL algorithmesults in better accuracy of the reconstructed image, the ringingrtifacts may occur. Moreover, even if we know the correct PSF,he ringing artifacts near strong edges will be appeared. Therefore,n this paper, we apply the proposed post-processing for reduc-ng the ringing artifacts. When we observe a blurred image and

reconstructed image, the intensity variations of the two imagesre almost same. However, there are many differences around thedges.

The proposed post-processing method uses the modification ofhe color values according to the efficiency of the divided regions.irstly, we obtain an edge map of the reconstructed image gen-rated by Sobel edge detector as shown in Fig. 3. We apply therosion to the edge map in order to remove the values caused byhe ringing artifacts. Then, we apply the dilation to the erodeddge map and we identify the edge regions in the reconstructedmage. The erosion and the dilation are the two basic morpholog-cal operations. For enhancing the reconstructed image, we utilizehe blurred image, the edge map, and the reconstructed image. Weivide the reconstructed image into three parts: smooth regions,dge regions, and complex texture regions. The ringing artifactsppearing in the smooth regions can be considered more seriously.or the smooth regions, we reduce the color value transitions ofhe reconstructed image. The result is better than the smoothing

mage. In the edge regions, we estimate the color variations aroundhe edges by using the orthogonal gradient of the edge regions.hen, we compensate the false transition of the edge regions causedy the ringing artifacts. For the complex texture regions, we leave

Fig. 7. The experimental results of image reconstruction with the proposed post-processing: (a) Original image, (b) 20 iterations, (c) 200 iterations, and (d) Proposedmethod generated by post-processing.

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228 J.M. Lee et al. / Opti

he reconstructed results without changes. After performing theroposed post-processing, we can acquire the final result image.s shown in Fig. 4, the final reconstructed result is better than

he other results without the post-processing. We reduce the coloralue transitions of the deblurred image so that the ringing artifactsear strong edges can be decreased.

. Experimental result

.1. Accuracy of PSF parameter estimation

We have applied the proposed method of PSF parameter estima-ion to various images, such as Lena, Cameraman, and Airplane. Ourxperiments were carried out under various conditions of motionlur and the results of the estimated PSF parameters are shown

n Table 1. We compare our algorithm with the conventional onesn terms of the errors of parameter estimation. We use the samemages of the existing methods [11,12] and measure the errorsf parameter estimation under the random blur conditions. Com-ared with the existing methods for image deblurring, the proposedethod is more accurate for estimating the PSF parameters as

hown in Table 2 and Fig. 5.

.2. Performance of image reconstruction

In this section, the experiment shows the relation betweenhe accuracy of parameter estimation and quality of the imageeconstruction. Because there may be the propagation of errorsn the iterative deconvolution process, it need to accurately esti-

ate the PSF parameters for the image reconstruction. As shownn Fig. 6, the experimental result shows that the proposed methodffectively reduce the ringing artifacts and the quality of theeconstructed image has been improved, compared to the existingethod [11].

.3. Final result with the post-processing

The experiment shows the results of the post-processing. Ashown in Fig. 7, the final result with the proposed post-processings more similar to the original image and most ringing artifactsas been reduced. Table 3 shows the similarity of the original

mage and the reconstructed image based on the peak signal-to-oise ratio (PSNR). Our method has improved the experimentalerformance by 15% compare to the result without the proposedost-processing.

[

(2013) 2224– 2228

7. Conclusion

In this paper, we have proposed the estimation method of thePSF parameters � and L for reconstructing a linear motion blurredimage and the reconstruction method using the proposed post-processing. The proposed method has accurately estimated byconsidering the periodicity of a Fourier spectrum in the blurredimage. In addition, the performance of reconstruction has beenimproved by applying the proposed post-processing method forreducing the ringing artifacts near the edges. Compared withthe conventional approaches for image reconstruction, the pro-posed method shows better performance for linear motion blurredimages.

Acknowledgement

This work was supported by National Research Foundation ofKorea Grant funded by the Korean Government (2009-0077434).

References

[1] K. Sato, S. Ishizuka, A. Nikami, M. Saot, Control techniques for optical imagestabilizing system, IEEE Trans. Consum. Electron. 39 (3) (1993) 461–466.

[2] M. Oshima, T. Hayashi, S. Fujioka, T. Inaji, H. Mitani, J. Kajino, K. Ikeda, K.Komoda, VHS camcorder with electronic image stabilizer, IEEE Trans. Consum.Electron. 35 (4) (1989) 749–758.

[3] M. Tanaka, K. Yoneji, M. Okutomi, Motion blur parameter identification from alinearly blurred image, in: Proceeding on International Conference ConsumerElectronics, Las Vegas, USA, 2007, pp. 1–2.

[4] M. Cannon, Blind deconvolution of spatially invariant image blur with phase,IEEE Trans. Acoust. Speech Signal Process. 24 (1) (1976) 58–63.

[5] R. Fergus, B. Singh, A. Hertzmann, S.T. Roweis, W.T. Freeman, Removing camerashake from a single photograph, ACM Trans. Graph. 25 (3) (2006) 787–794.

[6] L. Yuan, J. Sun, L. Quan, H.-Y. Shum, Image deblurring with blurred/noisy imagepairs, ACM Trans. Graph. 26 (3) (2007), Article 1.

[7] S. Chalkov, N. Meshalkina, C.S. Kim, Post-processing algorithm for reducingringing artifacts in deblurred images, in: International Technical Conference onCircuits/Systems, Computers and Communications, Shimonoseki, Japan, 2008,pp. 1193–1196.

[8] J.H. Lee, K.T. Park, Y.S. Moon, Image deblurring by using the estimation of PSFparameters for image devices, in: Proceeding on International Conference Con-sumer Electronics, Las Vegas, USA, 2010, pp. 1–2.

[9] W.H. Richardson, Bayesian-based iterative method of image restoration, J. Opt.Soc. Am. 62 (1) (1972) 55–60.

10] L.A. Shepp, Y. Vardi, Maximum likelihood reconstruction for emission tomo-graphy, IEEE Trans. Med. Imaging 1 (2) (1982) 113–122.

11] M.E. Moghaddam, M. Jamzad, Linear motion blur parameter estimation in noisy

images using fuzzy sets and power spectrum, EURASIP J. Adv. Signal Process.2007 (1) (2007) 1–8.

12] T.T. Zhang, G. Xu, Identification of motion blurred parameter by using T-normoperator, in: International Conference on Machine Learning and Cybernetics,Hong Kong, 2007, pp. 1606–1610.