Blind deconvolution by means of the richardsonlucy algorithm

58J.Opt.Soc.Am.A/Vol.12,No.1/January1995Fish et al.

Blind deconvolution by means of the

Richardson–Lucy algorithm

D.A.Fish,A.M.Brinicombe,and

E.R.Pike

Department of Physics,King’s College London,Strand,London WC2R2LS,UK

J.G.Walker

Department of Electrical and Electronic Engineering,University Park,University of Nottingham,Nottingham NG72RD,UK

Received April7,1994;revised manuscript received August8,1994;accepted August22,1994

A blind deconvolution algorithm based on the Richardson–Lucy deconvolution algorithm is presented.Its

performance in the presence of noise is found to be superior to that of other blind deconvolution algorithms.

Results are presented and compared with results obtained from implementation of a Weiner filter blind

deconvolution algorithm.The algorithm is developed further to incorporate functional forms of the point-

spread function with unknown parameters.In the presence of noise the point-spread function can be

evaluated with1.0%error,and the object can be reconstructed with a quality near that of the deconvolution

process with a known point-spread function.

1.INTRODUCTION

Blind deconvolution is the term given to an image-restoration technique in which complete knowledge of both the point-spread function(PSF)and the object are not available.Ayers and Dainty1proposed a scheme that essentially generalized the Feinup phase retrieval algorithm.2The technique is iterative,and a priori knowledge is limited to the nonnegativity of images.In each iteration one obtains estimates of the object and the PSF by simple inverse filtering.Davey et al.3proposed a similar scheme,but their algorithm assumed further a priori knowledge,i.e.,that the object support was known.In their study a Weiner-type filter was used to obtain estimates of the object and the PSF.This method thus permitted better noise compensation.

In this paper a Weiner filter blind deconvolution algo-rithm is implemented and is compared with a new al-gorithm based on the Richardson–Lucy4,5deconvolution. The Richardson–Lucy algorithm has proved to be robust in the presence of noise;therefore we thought that a blind deconvolution algorithm based on this technique might have advantages over the Ayers–Dainty and the Davey–Lane–Bates algorithms.The results shown here con-firm the high noise tolerance of our new algorithm.

To improve further the performance of this type of algorithm,we incorporated extra a priori knowledge by assuming a functional form for the PSF.It was thought that this method would produce better results because the number of unknowns is reduced from thousands of pixel values to a small number of parameters that describe the PSF.It is likely that blind deconvolution performed in this manner would find use in many areas in which it is not possible to know exactly how an optical system is aberrated but which could be characterized by a few free parameters.One example of such an application is in telescopes in space,where unknown fluctuations of mirrors,which are due to time-varying gravitational fields,do not permit exact knowledge of the PSF.2.BLIND DECONVOLUTION BY THE RICHARDSON–LUCY ALGORITHM

The Richardson–Lucy deconvolution algorithm has be-come popular in the fields of astronomy and medical imag-ing.Initially it was derived from Bayes’s theorem in the early1970’s by Richardson and Lucy.4,5In the early 1980’s it was rederived by Shepp and Vardi6as an al-gorithm to solve positron emission tomography imaging problems,in which Poissonian statistics are dominant. Their method used a maximum-likelihood solution,which was found by use of the expectation maximization algo-rithm of Dempster et al.7The reason for the popularity of the Richardson–Lucy algorithm is its implementation of maximum likelihood and its apparent ability to pro-duce reconstructed images of good quality in the presence of high noise levels.We therefore assumed that a blind form of this algorithm would have the same characteris-tics.A blind deconvolution algorithm similar to the one shown here was also developed by Holmes8by use of the expectation maximization algorithm of Dempster et al.7 We begin with a brief review of the Richardson–Lucy deconvolution method and then present the blind form of the algorithm.The Richardson–Lucy algorithm was developed from Bayes’s theorem.Because it relates con-ditional probabilities the algorithm takes into account statistical fluctuations in the signal and therefore has the ability to reconstruct noisy images.Bayes’s theorem is given by

P?x j y??

P?y j x?P?x?

Z

P?y j x?P?x?d x

,(1)

where P?y j x?is the conditional probability of an event y,given event x.P?x?is the probability of an event x, and P?x j y?is the inverse conditional probability,i.e.,the probability of event x,given event y.The probability P?x?can be identified as the object distribution f?x?;the

0740-3232/95/010058-08$06.00?1995Optical Society of America

Fish et al.Vol.12,No.1/January1995/J.Opt.Soc.Am.A

59

Fig.1.Blind deconvolution based on the Richardson–Lucy algorithm.

conditional probability P?y j x?can be identified as the PSF centered at x,i.e.,g?y,x?;and the probability P?y?can be identified as the degraded image or convolution c?y?.This inverse relation permits the derivation of the iterative algorithm

f i11?x??Z g?y,x?c?y?d y

Z

g?y,z?f i?z?d z

f i?x?,(2)

where i is the iteration number.If an isoplanatic condi-tion exists,then Eq.(2)can be written in terms of con-volutions:

f i11?x???∑c?x?

f i

∏

≠g?2x?

?

f i?x?,(3)

where≠is the convolution operation.The PSF g?x?is

known,so one finds the object f?x?by iterating Eq.(3)

until convergence.An initial guess is required for the

object f0?x?to start the algorithm.Then,in subsequent iterations,because of the form of the algorithm,large de-

viations in the guess from the true object are lost rapidly

in initial iterations,whereas detail is added more slowly

in subsequent iterations.Advantages of this algorithm

include a nonnegativity constraint if the initial guess

f0?x?$0.Also,energy is conserved as the iteration pro-ceeds,which is easily seen by integration of both sides of Eq.(2)over x.

In the blind form of this algorithm two of these deconvo-

lution steps are required.At the k th blind iteration it is

assumed that the object is known from the k21iteration.

The PSF g k?x?is then calculated for a specified number of Richardson–Lucy iterations,as in Eq.(4)below,where the i index represents the Richardson–Lucy iteration. This equation is essentially an inverse of Eq.(3),inas-much as the object and the PSF have reverse roles,and it calculates the PSF from the object.Then f k?x?is calcu-lated for the same number of Richardson–Lucy iterations. This is done with the PSF evaluated from the full itera-tion of Eq.(4).In this case the iteration is performed in the normal manner of Eq.(3),as shown in Eq.(5)below. The degraded image is again given as c?x?in both Eqs.(4) and(5).The loop is repeated as required.One writes

g i11k?x??

?∑c?x?

g i k k21

∏

≠f k21?2x?

?

g i k?x?,(4) f i11k?x??

?∑c?x?

f i k?x?≠

g k?x?

∏

≠g k?2x?

?

f i k?x?.(5)

The above equations are shown in one dimension; the extension for two-dimensional images is straight-forward.Initial guesses are made for the object f00?x?and the PSF g00?x?,and an algorithm loop of the form shown in Fig.1is performed.No positivity constraints are required because the above equations always ensure positivity.The algorithm is different from the Holmes8 algorithm,as only two Richardson–Lucy iterations are performed within one blind iteration,one for an object evaluation and one for the PSF evaluation.It was found that the simulated images used did not perform well with this type of iteration but that when the number of Richardson–Lucy iterations within one blind itera-tion was increased to approximately ten a much better performance was obtained.

To test this algorithm against another blind deconvo-lution algorithm for comparison of performance purposes the Davey et al.3blind deconvolution algorithm with a Weiner filter was implemented.In the implementation used here the support constraint used by Davey et al.was not used because no support constraint was used for the Richardson–Lucy blind deconvolution algorithm.

A convolution was created from a Gaussian to model the PSF and a cross(the object);these can be seen in Fig.2.All the images are64364pixels.Photon noise was added to the image by generation of a random num-ber lying on a Poisson distribution with the mean of the pixel value of the noiseless image,and the numbers that were generated for all the pixels then formed the noisy image.It was found that,with this type of image and ap-proximately1.5%noise(where the percentage value is the

(a)(b)

(c)

Fig.2.(a)Simulated object,(b)Gaussian PSF,and(c)their convolution with1.5%Poissonian noise.

60J.Opt.Soc.Am.A/Vol.12,No.1/January1995Fish et al.

(a)

(b)

Fig.3.Blind deconvolution by the Weiner filter algorithm. Reconstructions of the object(left)and the PSF(right)with(a) zero noise and(b)1.5%noise.

standard deviation divided by the intensity at the bright-est point in the image),good reconstructions could be ob-tained.Figure3shows reconstructions by means of the blind Weiner filter algorithm;Fig.3(a)shows the noise-less case,in which the images shown are the best-error object(left)and the PSF(right)after400iterations.The term best error refers to the least error between the origi-nal convolution and the convolution of the reconstructed object and the PSF.Figure3(a)shows the case of1.5% noise;the reconstructions have deteriorated,but the cross is still distinguishable.Noise levels much above this fig-ure resulted in unrecognizable reconstructions.

The blind Richardson–Lucy algorithm performed far better on the same image.In Fig.4(a)images are shown with1.5%and10.0%noise(left and right,respectively). Figures4(b)and4(c)show reconstructions for both these cases,respectively.It can immediately be seen that the performance of this algorithm is far superior to the pre-vious algorithm.Good reconstructions are obtained at both noise levels.The algorithm was applied to many other images for which Gaussian PSF’s were used,and it was found that as long as the blurring of the PSF was not too severe then reasonable reconstructions could gener-ally be obtained,in some cases with noise levels as high as15%.

3.SEMIBLIND DECONVOLUTION

As mentioned in Section1,further a priori information could be incorporated by assuming knowledge of the form of the PSF.In a real situation it may be known that a telescope suffers from spherical aberration,but be-cause of time-varying factors such as the changing gravi-tational field that exists around a telescope in orbit the extent of this aberration may not be known.This situ-ation would reduce the number of unknown variables in the deconvolution from perhaps thousands of pixel values to one or two unknown constants.We have termed this approach semiblind deconvolution.

A.Weiner Semiblind Algorithm

This algorithm used the blind deconvolution with a Weiner filter as its basis.The only part of the algo-rithm altered was the image-plane constraints on the PSF.In the blind algorithm the constraint was just nonnegativity.This constraint was replaced by a least-squares-fitting procedure.Initially convolutions were created with Gaussians,so Gaussians of varying widths were compared with the evaluated PSF.The Gauss-ian giving the least error in fitting was then chosen as the PSF,and the next object guess was evaluated with this PSF.

To illustrate how well this algorithm performed in the absence of noise,Fig.5shows the reconstructions of the object and the PSF at every iteration.In this particular case the images are all1283128pixels.This algorithm converged within three iterations and produced a perfect reconstruction of the satellite object.When we tried to

(a)

(b)

(c)

Fig.4.Blind deconvolution by the Richardson–Lucy algorithm.

(a)Convolutions with1.5%(left)and10.0%(right)noise.(b) Reconstructions of the object(left)and the PSF(right)at the 1.5%noise level.(c)Reconstructions of the object(left)and the PSF(right)at the10.0%noise level.

Fish et al.Vol.12,No.1/January1995/J.Opt.Soc.Am.A61

(a)

(b)

(c)

(d)

Fig.5.Semiblind deconvolution by a Weiner filter-based algorithm.(a)True object(left),random starting guess of the object(center), and noiseless convolution(right).(b)Object(left)and PSF(right)from the first iteration.(c)Object(left)and PSF(right)from the second iteration.(d)Object(left)and PSF(right)from the third iteration.

reconstruct with noisy images,however,the algorithm always converged on the delta-function solution,i.e.,a Gaussian of smallest possible width was evaluated as the PSF.Even with noise values less than0.1%the algorithm performed poorly.We therefore tried using the Richardson–Lucy algorithm.

B.Semiblind Deconvolution by the

Richardson–Lucy Algorithm

The semiblind form of the algorithm took as its basis the blind algorithm.A number of blind iterations were performed,and then a least-squares fit on the function evaluated as the PSF was found.A PSF was then created with the fitting parameters,and then another series of blind iterations was performed,with this PSF being used as the starting point.This procedure was then repeated for a specified number of iterations.Ini-tially,simple one-variable PSF forms were chosen,i.e., Gaussians of unknown width.

In some cases the results for this algorithm showed remarkable noise tolerance.In Fig.6results are shown for semiblind deconvolution on a series of point sources.

62J.Opt.Soc.Am.A/Vol.12,No.1/January1995Fish et al.

(a)

(b)

Fig.6.Semiblind deconvolution by the Richardson–Lucy-based algorithm.(a)Object(left)and convolution(right)with20.0% noise.(b)Reconstruction of the object(left)and the fitted Gauss-ian PSF(right).

(a)(b)

(c)

https://www.wendangku.net/doc/9315763501.html,parison of Richardson–Lucy semiblind deconvolu-tion with standard deconvolution algorithms.(a)Reconstruc-tion by semiblind deconvolution with a0.1-pixel step width.(b) Reconstruction by Fourier regularization.(c)Reconstruction by the Richardson–Lucy algorithm.

The image contained approximately20.0%noise.The reconstruction shown is good,considering the noise level.

The algorithm was also tried on the noisy image of the cross used earlier for the pure blind deconvolution re-search.Although the PSF was fitted in each iteration with a Gaussian of the correct size,the results were not good;in fact,the pure blind deconvolution results were better.Therefore it was decided that a Gaussian fitting process should be performed after the blind deconvolu-tion and then a specified number of Richardson–Lucy it-erations performed with the guessed Gaussian.The step width in Gaussian fitting was obviously important:with the step width of1pixel for the Gaussian radius at the 1?e height,the correct PSF width of3pixels was guessed.

(a)

(b)

Fig.8.Many-variable semiblind deconvolution.(a)Object (left)and PSF(right).(b)Convolution with1.0%noise.

(a)

(b)

Fig.9.Reconstructions of the image shown in Fig.8.(a) Richardson–Lucy deconvolution after1000iterations.(b) Semiblind deconvolution after15iterations:object(left)and PSF(right).

Fish et al .

Vol.12,No.1/January 1995/J.Opt.Soc.Am.A

63

(a)

(b)

Fig.10.Error graphs for the 1.0%noise image shown in Fig.8.(a)Fitting parameters A 2,C 1,C 2with iteration number.(b)Percentage error in the PSF with iteration number.

At a step width of 0.1pixel the Gaussian width obtained was 3.2pixels.The results for both cases are similar and are shown in Fig.7(a)for the 0.1-pixel case.

The results show that the slight error made in finding the width of the Gaussian PSF does not make the re-constructions significantly worse.This is probably due to the high level of noise on the image,which results in the loss of a large amount of information.To show the impressiveness of these results straightforward deconvo-lutions with a Weiner filter and the Richardson–Lucy al-gorithm are shown in Figs.7(b)and 7(c).It can be seen that the semiblind deconvolution results are comparable with the usual methods of deconvolution.

To extend this research to cases in which it may be used in realistic situations,more than one fitting variable may be needed to describe the PSF accurately.To test this possibility,a simple PSF was created that had the functional form

y ?r ??X k ∑A k r 2exp ?1.0?C k 21B k ∏exp μ2r 2C k 2?

,

(6)

where r is the radius and the variables A k ,B k ,C k were given the values A 1?0.0,A 2?0.1,B 1?1.0,B 2?0.0,C 1?1.0,C 2?5.0.Then the variables A 2,C 1,C 2were allowed to change their values,so that the PSF was a Gaussian plus a Gaussian times its radius squared.Incorrect values for these variables were introduced into the program,and a PSF was created.Then,as before,a blind deconvolu-tion process evaluated a new object and a new PSF.A PSF with the functional form given above and with free variables A 2,C 1,C 2was fitted to the evaluated PSF by a Levenberg–Marquardt 9nonlinear least-squares-fitting routine.This routine returned new values for A 2,C 1,C 2,and the process was repeated for a specified number of iterations.The simulated object and the PSF used to illustrate this algorithm are shown in Fig.8(a),and a 1.0%noise convo-lution is shown in Fig.8(b)(the object used was the cross shown above).To compare the results of this semiblind deconvolution algorithm a Richardson–Lucy deconvolu-tion was performed with 1000iterations with the known PSF.The result of this process is shown in Fig.9(a)and can be compared with the results of 15iterations of the semiblind deconvolution algorithm shown in Fig.9(b).The results compare well.In Fig.10(a)the variation of the fitting parameters A 2,C 1,C 2with iteration number are shown.The start-ing values introduced into the program were A 2?0.5,C 1?3.0,C 2?7.0,giving a 74.0%error in the PSF.It (a)(b)(c)Fig.11.Reconstructions of an image with 4.0%noise.(a)4.0%noise image.(b)Richardson–Lucy deconvolution after 1000iterations.(c)Semiblind deconvolution after 15iterations:ob-ject (left)and PSF (right).

64J.Opt.Soc.Am.A /Vol.12,No.1/January 1995

Fish et al .can be seen that at this noise level the algorithm is con-verging on the correct values.This convergence is high-lighted by Fig.10(b),which shows the percentage error in the PSF with iteration number.This result is slightly false because the true PSF will not be known in a real situation,but the figure shows the convergent properties of the algorithm,which do not occur in cases such as the Ayers–Dainty,the blind Weiner,and the blind Richardson–Lucy algorithms.In the case of the blind Richardson–Lucy algorithm,the image shown in Fig.8(b)was used for comparison with the results of the semi-blind algorithm.It was found that the algorithm did not converge to the correct values for the fitting parameters,and in fact the algorithm eventually diverged.The final values for the fitting parameters from the semiblind al-gorithm were A 2?0.102,C 1?1.06,C 2?5.03,with an overall error in the PSF evaluation of 1.09%.

At 2.0%,3.0%,and 4.0%noise levels similar results were obtained,and convergence was seen at the correct values of the fitting variables.The results of the 4.0%noise image are shown in Fig.11.This figure shows the Richardson–Lucy deconvolution after 1000iterations

and

(a)

(b)

Fig.12.Error graphs for the 4.0%noise image.(a)Fitting parameters A 2,C 1,C 2with iteration number.(b)Percentage error in the PSF with iteration

number.

(a)(b)Fig.13.Error graphs for a 6.0%noise image.(a)Fitting pa-rameters A 2,C 1,C 2with iteration number.(b)Percentage er-ror in the PSF with iteration number.the semiblind deconvolution after 15iterations.Again the results compare quite well.Figure 12shows the variation in the fitting parameters and the percentage error in the PSF with iteration number.Again conver-gence is evident.When the same image with 6.0%noise was tried,the results were not so good.Figure 13shows the variation of fitting parameter and the percentage er-ror in the PSF,and it can be seen that convergence is reached after eight iterations and that the algorithm then starts to diverge.It therefore appears that the algorithm has a certain noise tolerance.4.CONCLUSIONS A blind deconvolution algorithm has been presented here that is based on the Richardson–Lucy algorithm.The algorithm presented is similar to that presented by Holmes,8but the implementation given here seems to have a more stable performance on the images chosen.The noise tolerance of the present algorithm is also far better than that of algorithms such as the Ayers–Dainty 1and the Weiner filter algorithms,used for comparison purposes in this paper.

Fish et al.Vol.12,No.1/January1995/J.Opt.Soc.Am.A65

In many real situations it may be the case that some knowledge of the PSF can be obtained.Therefore func-tional forms for the PSF’s were chosen with a number of unknown variables.It was found that accurate decon-volutions of a quality near that of a deconvolution with full knowledge of the PSF can be made.It is hoped that this research can be extended to real images with PSF’s containing unknown amounts of aberration,with the al-gorithm evaluating both the aberration coefficients and the object.

ACKNOWLEDGMENTS

The authors are grateful to the U.S.Army Research Office for supporting this research under a project en-titled“Diffraction limited imaging using aberrated optics,”grant DAAL03-92-G-0142.

REFERENCES

1.G.R.Ayers and J.C.Dainty,“Iterative blind deconvolution

method and its applications,”Opt.Lett.13,547–549(1988).2.J.R.Feinup,“Phase retrieval algorithms:a comparison,”

Appl.Opt.21,2758–2769(1982).

3. B.L.K.Davey,https://www.wendangku.net/doc/9315763501.html,ne,and R.H.T.Bates,“Blind decon-

volution of noisy complex-valued image,”https://www.wendangku.net/doc/9315763501.html,mun.69, 353–356(1989).

4.W.H.Richardson,“Bayesian-based iterative method of image

restoration,”J.Opt.Soc.Am.62,55–59(1972).

5.L.B.Lucy,“An iterative technique for the rectification of

observed distributions,”Astron.J.79,745–754(1974).

6.L.A.Shepp and Y.Vardi,“Maximum likelihood reconstruc-

tions for emission tomography,”IEEE Trans.Med.Imaging MI-1,113–122(1982).

7. A.P.Dempster,https://www.wendangku.net/doc/9315763501.html,ird,and D.B.Rubin,“Maximum

likelihood from incomplete data via the EM algorithm,”J.R.

Stat.Soc.39,1–38(1977).

8.T.J.Holmes,“Blind deconvolution of quantum-limited inco-

herent imagery:maximum likelihood approach,”J.Opt.Soc.

Am.A9,1052–1061(1992).

9.W.H.Press, B.P.Flannery,J. A.Teukolsky,and W.T.

Vetterling,Numerical Recipes:the Art of Scientific Comput-ing(Cambridge U.Press,Cambridge,1988).

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●I wonder if it’s because I have been at school for so long that I’ve grown so crazy about going home. ●It is because she wasn’t well that she fell far behind her classmates this semester. ●I can well remember that there was a time when I took it for granted that friends should do everything for me. ●In order to make a difference to society, they spent almost all of their spare time in raising money for the charity. ●It’s no pleasure eating at school any longer because the food is not so tasty as that at home. ●He happened to be hit by a new idea when he was walking along the riverbank. ●I wonder if I can cope with stressful situations in life independently. ●It is because I take things for granted that I make so many mistakes. ●The treasure is so rare that a growing number of people are looking for it. ●He picks on the weak mn in order that we may pay attention to him. ●It’s no pleasure being disturbed whena I settle down to my work. ●I can well remember that when I was a child, I always made mistakes on purpose for fun. ●It’s no pleasure accompany her hanging out on the street on such a rainy day. ●I can well remember that there was a time when I threw my whole self into study in order to live up to my parents’ expectation and enter my dream university. ●I can well remember that she stuck with me all the time and helped me regain my confidence during my tough time five years ago. ●It is because he makes it a priority to study that he always gets good grades. ●I wonder if we should abandon this idea because there is no point in doing so. ●I wonder if it was because I ate ice-cream that I had an upset student this morning. ●It is because she refused to die that she became incredibly successful. ●She is so considerate that many of us turn to her for comfort. ●I can well remember that once I underestimated the power of words and hurt my friend. ●He works extremely hard in order to live up to his expectations. ●I happened to see a butterfly settle on the beautiful flower. ●It’s no pleasure making fun of others. ●It was the first time in the new semester that I had burned the midnight oil to study. ●It’s no pleasure taking everything into account when you long to have the relaxing life. ●I wonder if it was because he abandoned himself to despair that he was killed in a car accident when he was driving. ●Jack is always picking on younger children in order to show off his power. ●It is because he always burns the midnight oil that he oversleeps sometimes. ●I happened to find some pictures to do with my grandfather when I was going through the drawer. ●It was because I didn’t dare look at the failure face to face that I failed again. ●I tell my friend that failure is not scary in order that she can rebound from failure. ●I throw my whole self to study in order to pass the final exam. ●It was the first time that I had made a speech in public and enjoyed the thunder of applause. ●Alice happened to be on the street when a UFO landed right in front of her. ●It was the first time that I had kept myself open and talked sincerely with my parents. ●It was a beautiful sunny day. The weather was so comfortable that I settled myself into the

英语句子结构和造句

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