Improved regularized solution of the inverse problem in turbidimetric measurements

Janusz Mroczka; Damian Szczuczyński

doi:10.1364/AO.49.004591

Applied Optics
Vol. 49,
Issue 24,
pp. 4591-4603
(2010)
•https://doi.org/10.1364/AO.49.004591

Improved regularized solution of the inverse problem in turbidimetric measurements

Janusz Mroczka and Damian Szczuczyński

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Janusz Mroczka and Damian Szczuczyński, "Improved regularized solution of the inverse problem in turbidimetric measurements," Appl. Opt. 49, 4591-4603 (2010)

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About this Article
History
- Original Manuscript: February 23, 2010
- Revised Manuscript: June 4, 2010
- Manuscript Accepted: July 7, 2010
- Published: August 16, 2010
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 5, Iss. 13

Abstract

We present results of simulation research on the constrained regularized least-squares (RLS) solution of the ill-conditioned inverse problem in turbidimetric measurements. The problem is formulated in terms of the discretized Fredholm integral equation of the first kind. The inverse problem in turbidimetric measurements consists in determining particle size distribution (PSD) function of particulate system on the basis of turbidimetric measurements. The desired PSD should satisfy two constraints: non negativity of PSD values and normalization of PSD to unity when integrated over the whole range of particle size. Incorporating the constraints into the RLS method leads to the constrained regularized least-squares (CRLS) method, which is realized by means of an active set algorithm of quadratic programming. Results of simulation research prove that the CRLS method performs considerably better with reconstruction of PSD than the RLS method in terms of better fidelity and smaller uncertainty.

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	Inverse Method
	L—Identity Matrix		L—First Derivative		L—Second Derivative		L—Third Derivative
p (%)	RLS	CRLS	RLS	CRLS	RLS	CRLS	RLS	CRLS
0.0	$2.67 \times 10^{- 18}$	$3.22 \times 10^{- 18}$	$4.32 \times 10^{- 18}$	$4.65 \times 10^{- 18}$	$4.67 \times 10^{- 18}$	$4.57 \times 10^{- 18}$	$4.72 \times 10^{- 18}$	$4.63 \times 10^{- 18}$
2.5	0.000443	0.000545	0.000351	0.000395	0.000951	0.000264	0.085627	0.001455
5.0	0.000604	0.000714	0.001508	0.000886	0.014054	0.000812	0.241890	0.002354
7.5	0.000712	0.000763	0.043188	0.001516	0.043476	0.001521	0.450080	0.002335

	Inverse Method
	L—Identity Matrix		L—First Derivative		L—Second Derivative		L—Third Derivative
p (%)	RLS	CRLS	RLS	CRLS	RLS	CRLS	RLS	CRLS
0.0	0.002173	0.002017	0.000906	0.000538	0.000980	0.000595	0.001053	0.000444
2.5	0.001115	0.000671	0.001069	0.000645	0.001149	0.000754	0.012059	0.000331
5.0	0.001194	0.000817	0.001392	0.001007	0.001785	0.000686	0.020425	0.000849
7.5	0.001293	0.001023	0.004439	0.001170	0.006477	0.000504	0.033867	0.000900

	Inverse Method
	L—Identity Matrix		L—First Derivative		L—Second Derivative		L—Third Derivative
p (%)	RLS	CRLS	RLS	CRLS	RLS	CRLS	RLS	CRLS
0.0	$2.90 \times 10^{- 18}$	$3.14 \times 10^{- 18}$	$4.06 \times 10^{- 18}$	$4.13 \times 10^{- 18}$	$4.91 \times 10^{- 18}$	$4.31 \times 10^{- 18}$	$4.75 \times 10^{- 18}$	$4.22 \times 10^{- 18}$
2.5	0.000443	0.000545	0.000351	0.000395	0.000951	0.000264	0.085627	0.001455
5.0	0.000612	0.000731	0.001277	0.000842	0.003640	0.000746	0.258470	0.002053
7.5	0.000852	0.000879	0.030492	0.001388	0.037157	0.001446	0.204980	0.002412

	Inverse Method
	L—Identity Matrix		L—First Derivative		L—Second Derivative		L—Third Derivative
p (%)	RLS	CRLS	RLS	CRLS	RLS	CRLS	RLS	CRLS
0.0	0.002229	0.002040	0.000796	0.000790	0.000812	0.000810	0.001118	0.000858
2.5	0.001173	0.000801	0.001110	0.000833	0.001033	0.000918	0.005639	0.001133
5.0	0.001211	0.000856	0.001323	0.001012	0.001138	0.000963	0.024739	0.001378
7.5	0.001231	0.000928	0.003163	0.001208	0.001730	0.001059	0.038134	0.001551

	Inverse Method
	L—Identity Matrix		L—First Derivative		L—Second Derivative		L—Third Derivative
p (%)	RLS	CRLS	RLS	CRLS	RLS	CRLS	RLS	CRLS
0.0	$2.67 \times 10^{- 18}$	$3.22 \times 10^{- 18}$	$4.32 \times 10^{- 18}$	$4.65 \times 10^{- 18}$	$4.67 \times 10^{- 18}$	$4.57 \times 10^{- 18}$	$4.72 \times 10^{- 18}$	$4.63 \times 10^{- 18}$
2.5	0.000443	0.000545	0.000351	0.000395	0.000951	0.000264	0.085627	0.001455
5.0	0.000604	0.000714	0.001508	0.000886	0.014054	0.000812	0.241890	0.002354
7.5	0.000712	0.000763	0.043188	0.001516	0.043476	0.001521	0.450080	0.002335

Abstract

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Applied Optics