Difference between revisions of "Tikhonov.m"
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{{DISPLAYTITLE:tikhonov.m}} __NOTOC__ | {{DISPLAYTITLE:tikhonov.m}} __NOTOC__ | ||
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Tikhonov regularised solution to K*x=y with a positivity constraint on x using regularised Newton-Raphson method. | Tikhonov regularised solution to K*x=y with a positivity constraint on x using regularised Newton-Raphson method. | ||
==Syntax== | ==Syntax== | ||
| − | [x,err,reg]=tikhonov(K,D,KtK,DtD,H,y,lambda) | + | [x,err,reg]=tikhonov(K,D,KtK,DtD,H,y,lambda) |
==Arguments== | ==Arguments== | ||
| − | K - kernel matrix, may be complex, may be non-square | + | K - kernel matrix, may be complex, may be non-square |
D - regularisation matrix, leave empty to use finite | D - regularisation matrix, leave empty to use finite | ||
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==Outputs== | ==Outputs== | ||
| − | x - a real vector, a minimum (subject to positivity) | + | x - a real vector, a minimum (subject to positivity) |
of norm(K*x-y,2)^2+lambda*norm(D*x,2)^2 | of norm(K*x-y,2)^2+lambda*norm(D*x,2)^2 | ||
| − | + | ||
err - error signal norm(K*x-y,2)^2 | err - error signal norm(K*x-y,2)^2 | ||
| − | + | ||
reg - regularisation signal norm(D*x,2)^2 | reg - regularisation signal norm(D*x,2)^2 | ||
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==Notes== | ==Notes== | ||
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For best numerical performance, scale K to have approximately unit 2-norm, and y to have approximately unit 1-norm. | For best numerical performance, scale K to have approximately unit 2-norm, and y to have approximately unit 1-norm. | ||
==See also== | ==See also== | ||
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[[Import,_export,_and_visualisation#Miscellaneous_data_processing|Miscellaneous data processing]] | [[Import,_export,_and_visualisation#Miscellaneous_data_processing|Miscellaneous data processing]] | ||
Revision as of 15:50, 5 April 2026
Tikhonov regularised solution to K*x=y with a positivity constraint on x using regularised Newton-Raphson method.
Syntax
[x,err,reg]=tikhonov(K,D,KtK,DtD,H,y,lambda)
Arguments
K - kernel matrix, may be complex, may be non-square
D - regularisation matrix, leave empty to use finite
difference second derivative matrix
KtK - K'*K, for repeated calls it may be faster to pre-
compute this quantity, leave empty otherwise
DtD - D'*D, for repeated calls it may be faster to pre-
compute this quantity, leave empty otherwise
H - Tikhonov Hessian 2*real(KtK+lambda*DtD), for re-
peated calls it may be faster to precompute this
quantity, leave empty otherwise
y - a column vector, may be complex
lambda - Tikhonov regularisation parameter
Outputs
x - a real vector, a minimum (subject to positivity)
of norm(K*x-y,2)^2+lambda*norm(D*x,2)^2
err - error signal norm(K*x-y,2)^2
reg - regularisation signal norm(D*x,2)^2
Notes
For best numerical performance, scale K to have approximately unit 2-norm, and y to have approximately unit 1-norm.
See also
Version 2.9, authors: Anupama Acharya, Ilya Kuprov
