Alpha Serpentis - Revision history

en>Jelle Gouw: /* References */

2014-10-28T11:40:43Z

References

← Older revision		Revision as of 13:40, 28 October 2014
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	~~{{Distinguish\|Modified Richardson iteration}}~~		The individual who wrote the article is known as Jayson Hirano and he totally digs that name. One of the issues she enjoys most is canoeing and she's been performing it for fairly a whilst. Mississippi is exactly where his home is. Office supervising is my profession.<br><br>Feel free to visit my page ... psychic readers, [http://kjhkkb.net/xe/notice/374835 why not look here],
	The ~~'''Richardson–Lucy algorithm''', also~~ known as ~~'''Lucy–Richardson [[deconvolution]]''', is an [[iterative procedure]] for recovering a [[latent image]]~~ that ~~has been [[convolution\|blurred]] by a known [[point spread function]].<ref~~ name~~=Richardson1972>{{cite journal~~
	~~\| author = Richardson, William Hadley~~
	~~\| title = Bayesian-Based Iterative Method of Image Restoration~~
	~~\| year = 1972~~
	~~\| journal = [[JOSA]]~~
	~~\| volume = 62~~
	~~\| issue = 1~~
	~~\| pages = 55–59~~
	~~\| url = http://www~~.~~opticsinfobase.org/abstract.cfm?id=54565~~
	~~\| doi = 10.1364/JOSA.62.000055~~
	~~}}</ref><ref name=Lucy1974>{{cite journal~~
	~~\| author = Lucy, L. B.~~
	~~\| year = 1974~~
	~~\| title = An iterative technique for the rectification~~ of ~~observed distributions~~
	~~\| journal = Astronomical Journal~~
	~~\| volume = 79~~
	~~\| issue = 6~~
	~~\| pages = 745–754~~
	~~\| doi = 10.1086/111605~~
	~~\|bibcode = 1974AJ.....79..745L }}</ref>~~

	~~Pixels in~~ the ~~observed image can be represented in terms of the point spread function and the latent image as~~
	~~:<math> d_{i} = \sum_{j} p_{ij} u_{j}\,</math>~~
	~~where <math>p_{ij}</math> is the point spread function (the fraction of light coming from true location <math>j</math> that~~ is ~~observed at position <math>i</math>), <math>u_{j}</math> is the pixel value at location <math>j</math> in the latent image,~~ and <math>d_{i}</math> is the observed value at pixel location <math>i</math>. The statistics are performed under the assumption that <math>u_j</math> are [[Poisson distribution\|Poisson distributed]], which is appropriate for [[photon noise]] in the data.

	The basic idea is to calculate the [[Maximum likelihood\|most likely]] <math>u_j</math> given the observed <math>d_i</math> and known <math>p_{ij}</math>. This leads to an equation for <math>u_j</math> which can be solved iteratively according to
	~~:<math>u_{j}^{(t+1)} = u_j^{(t)} \sum_{i} \frac{d_{i}}{c_{i}}p_{ij}</math>~~
	~~where~~
	~~:<math>c_{i} = \sum_{j} p_{ij} u_{j}^{(t)}.</math>~~

	~~It has~~ been ~~shown empirically that if this iteration converges,~~ it ~~converges to the maximum likelihood solution~~ for ~~<math>u_j</math>.<ref name=Shepp1982>{{citation~~
	~~\| author = Shepp, L. A.~~
	~~\| coauthors = Vardi, Y.~~
	~~\| year = 1982~~
	~~\| title = Maximum Likelihood Reconstruction for Emission Tomography~~
	~~\| journal = IEEE Transactions on Medical Imaging~~
	~~\| volume = 1~~
	~~\| pages = 113~~
	~~\| doi = 10.1109/TMI.1982.4307558~~
	~~}}</ref>~~

	~~This can also be written more generally (for more dimensions) in terms of [[convolution]],<ref name=Fish1995>{{citation~~
	~~\| author = Fish D. A.,~~
	~~\| coauthors = Brinicombe A. M., Pike E. R~~.~~, and Walker J. G.~~
	~~\| year = 1995~~
	~~\| title = Blind deconvolution by means of the Richardson–Lucy algorithm~~
	~~\| journal = [[Journal of the Optical Society of America A]]~~
	~~\| volume = 12~~
	~~\| issue = 1~~
	~~\| pages = 58–65~~
	~~\| url = http://www.math.ufl.edu/~bamair/fish95.pdf~~
	~~\|bibcode = 1995JOSAA..12...58F \|doi = 10.1364/JOSAA.12.000058 }}</ref>~~
	~~:<math>u^{(t+1)} = u^{(t)}\cdot\left(\frac{d}{u^{(t)}\otimes p}\otimes \hat{p}\right)</math>~~
	~~where the division and multiplication are element wise, and <math>\hat{p}</math>~~ is ~~the flipped point spread function, such that~~
	~~:<math>\hat{p}_{nm} = p_{(i-n)(j-m)}, 0\le n,m \le i,j </math>~~

	~~In problems~~ where ~~the point spread function <math>p_{ij}</math>~~ is ~~dependent on one or more unknown parameters, the Richardson–Lucy algorithm cannot be used~~. ~~A later and more general class of algorithms, the [[expectation-maximization algorithm]]s,<ref>A.P. Dempster, N~~.M. Laird, D.B. Rubin, 1977, ''[http://groups.csail.mit.edu/drl/journal_club/papers/DempsterEMAlgorithm77.pdf Maximum likelihood from incomplete data via the EM algorithm]'', J. Royal Stat. Soc. Ser. B, '''39''' (1), pp. 1–38<~~/ref~~> ~~have been applied to this type of problem with great success~~

	~~==Implementation==~~
	~~For the two dimensional case this can be implemented in MATLAB, to estimate the latent greyscale image from a known blurred image and point spread function:~~
	<~~source lang="MatLab"~~>
	~~function latent_est = RL_deconvolution(observed, psf, iterations)~~
	% to ~~utilise the conv2 function we must make sure the inputs are double~~
	~~observed = double(observed);~~
	~~psf = double(psf);~~
	~~% initial estimate is arbitrary - uniform 50% grey works fine~~
	~~latent_est = 0~~.~~5*ones(size(observed));~~
	~~% create an inverse psf~~
	~~psf_hat = psf(end:-1:1,end:-1:1);~~
	~~% iterate towards ML estimate for the latent image~~
	~~for i= 1:iterations~~
	~~est_conv = conv2(latent_est,psf,'same');~~
	~~relative_blur = observed~~.~~/est_conv;~~
	~~error_est = conv2(relative_blur,psf_hat~~,~~'same');~~
	~~latent_est = latent_est.* error_est;~~
	~~end~~
	~~</source>~~
	~~The MATLAB image processing toolbox has an implementation in the function ''deconvlucy'' as well as a~~ [http://~~www.mathworks~~.~~com/help~~/~~images~~/~~examples~~/~~deblurring-images-using-the-lucy-richardson-algorithm.html demo] on its usage.~~

	~~==References==~~
	~~<references/>~~

	~~{{DEFAULTSORT:Richardson-Lucy deconvolution}}~~
	~~[[Category:Image processing]]~~
	~~[[Category:Estimation theory]~~]

en>Monkbot: /* References */Fix CS1 deprecated date parameter errors

2014-01-13T19:10:30Z

References: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 21:10, 13 January 2014
Line 1:		Line 1:
	The author's [http://~~alles-herunterladen~~.de/~~excellent-advice-~~for~~-picking-~~the~~-ideal-hobby~~/ ~~online psychic reading~~] ~~name~~ is ~~Christy~~. ~~Since he was 18 he's been working as~~ an ~~information officer but he ideas on altering it~~. ~~As a woman what she truly likes is style and she's~~ been ~~doing~~ it for ~~fairly a whilst~~. ~~Mississippi~~ is the ~~only place I~~'~~ve been residing in but~~ [http://~~conniecolin~~.~~com~~/xe/~~community~~/~~24580 authentic psychic readings~~] ~~I will~~ have to ~~move~~ in a ~~year or two.~~<br>~~<br>Feel free~~ to ~~surf to my web page~~: ~~real psychic~~ ([http://www.~~weddingwall~~.com~~.au~~/~~groups~~/~~easy~~-~~advice~~-~~for~~-~~successful~~-~~personal~~-~~development~~-~~today/ www~~.~~weddingwall~~.~~com.au~~])		{{Distinguish\|Modified Richardson iteration}}
			The '''Richardson–Lucy algorithm''', also known as '''Lucy–Richardson [[deconvolution]]''', is an [[iterative procedure]] for recovering a [[latent image]] that has been [[convolution\|blurred]] by a known [[point spread function]].<ref name=Richardson1972>{{cite journal
			\| author = Richardson, William Hadley
			\| title = Bayesian-Based Iterative Method of Image Restoration
			\| year = 1972
			\| journal = [[JOSA]]
			\| volume = 62
			\| issue = 1
			\| pages = 55–59
			\| url = http://www.opticsinfobase.org/abstract.cfm?id=54565
			\| doi = 10.1364/JOSA.62.000055
			}}</ref><ref name=Lucy1974>{{cite journal
			\| author = Lucy, L. B.
			\| year = 1974
			\| title = An iterative technique for the rectification of observed distributions
			\| journal = Astronomical Journal
			\| volume = 79
			\| issue = 6
			\| pages = 745–754
			\| doi = 10.1086/111605
			\|bibcode = 1974AJ.....79..745L }}</ref>

			Pixels in the observed image can be represented in terms of the point spread function and the latent image as
			:<math> d_{i} = \sum_{j} p_{ij} u_{j}\,</math>
			where <math>p_{ij}</math> is the point spread function (the fraction of light coming from true location <math>j</math> that is observed at position <math>i</math>), <math>u_{j}</math> is the pixel value at location <math>j</math> in the latent image, and <math>d_{i}</math> is the observed value at pixel location <math>i</math>. The statistics are performed under the assumption that <math>u_j</math> are [[Poisson distribution\|Poisson distributed]], which is appropriate for [[photon noise]] in the data.

			The basic idea is to calculate the [[Maximum likelihood\|most likely]] <math>u_j</math> given the observed <math>d_i</math> and known <math>p_{ij}</math>. This leads to an equation for <math>u_j</math> which can be solved iteratively according to
			:<math>u_{j}^{(t+1)} = u_j^{(t)} \sum_{i} \frac{d_{i}}{c_{i}}p_{ij}</math>
			where
			:<math>c_{i} = \sum_{j} p_{ij} u_{j}^{(t)}.</math>

			It has been shown empirically that if this iteration converges, it converges to the maximum likelihood solution for <math>u_j</math>.<ref name=Shepp1982>{{citation
			\| author = Shepp, L. A.
			\| coauthors = Vardi, Y.
			\| year = 1982
			\| title = Maximum Likelihood Reconstruction for Emission Tomography
			\| journal = IEEE Transactions on Medical Imaging
			\| volume = 1
			\| pages = 113
			\| doi = 10.1109/TMI.1982.4307558
			}}</ref>

			This can also be written more generally (for more dimensions) in terms of [[convolution]],<ref name=Fish1995>{{citation
			\| author = Fish D. A.,
			\| coauthors = Brinicombe A. M., Pike E. R., and Walker J. G.
			\| year = 1995
			\| title = Blind deconvolution by means of the Richardson–Lucy algorithm
			\| journal = [[Journal of the Optical Society of America A]]
			\| volume = 12
			\| issue = 1
			\| pages = 58–65
			\| url = http://www.math.ufl.edu/~bamair/fish95.pdf
			\|bibcode = 1995JOSAA..12...58F \|doi = 10.1364/JOSAA.12.000058 }}</ref>
			:<math>u^{(t+1)} = u^{(t)}\cdot\left(\frac{d}{u^{(t)}\otimes p}\otimes \hat{p}\right)</math>
			where the division and multiplication are element wise, and <math>\hat{p}</math> is the flipped point spread function, such that
			:<math>\hat{p}_{nm} = p_{(i-n)(j-m)}, 0\le n,m \le i,j </math>

			In problems where the point spread function <math>p_{ij}</math> is dependent on one or more unknown parameters, the Richardson–Lucy algorithm cannot be used. A later and more general class of algorithms, the [[expectation-maximization algorithm]]s,<ref>A.P. Dempster, N.M. Laird, D.B. Rubin, 1977, ''[http://groups.csail.mit.edu/drl/journal_club/papers/DempsterEMAlgorithm77.pdf Maximum likelihood from incomplete data via the EM algorithm]'', J. Royal Stat. Soc. Ser. B, '''39''' (1), pp. 1–38</ref> have been applied to this type of problem with great success

			==Implementation==
			For the two dimensional case this can be implemented in MATLAB, to estimate the latent greyscale image from a known blurred image and point spread function:
			<source lang="MatLab">
			function latent_est = RL_deconvolution(observed, psf, iterations)
			% to utilise the conv2 function we must make sure the inputs are double
			observed = double(observed);
			psf = double(psf);
			% initial estimate is arbitrary - uniform 50% grey works fine
			latent_est = 0.5*ones(size(observed));
			% create an inverse psf
			psf_hat = psf(end:-1:1,end:-1:1);
			% iterate towards ML estimate for the latent image
			for i= 1:iterations
			est_conv = conv2(latent_est,psf,'same');
			relative_blur = observed./est_conv;
			error_est = conv2(relative_blur,psf_hat,'same');
			latent_est = latent_est.* error_est;
			end
			</source>
			The MATLAB image processing toolbox has an implementation in the function ''deconvlucy'' as well as a [http://www.mathworks.com/help/images/examples/deblurring-images-using-the-lucy-richardson-algorithm.html demo] on its usage.

			==References==
			<references/>

			{{DEFAULTSORT:Richardson-Lucy deconvolution}}
			[[Category:Image processing]]
			[[Category:Estimation theory]]

en>LaaknorBot: Robot: Adding pt:Alpha Serpentis

2012-08-14T02:06:20Z

Robot: Adding pt:Alpha Serpentis

New page

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