- Basics of Indirect Measurements

- Singular Value Decomposition

- Analysis of accuracy and resolution

- Implementation

- Singular Value Decomposition

- Analysis of accuracy and resolution

- Implementation

The basic equation is the integral equation of the first kind:

(1)

The integral operator

(2)

connects two in general **different** spaces: the space of
reconstructing functions *f(x)* and the space of measuring functions *g(x)*.

Let us define:

*U*is a set of all functions (vectors) {*u*} (the complete basis set)._{ n}*U*_{1 }is a set of functions*u*corresponding to non-zero_{ n }*U*_{2}is the rest of basis functions*u*_{ n}- V is the complete set of functions {
*v*}_{n} *V*_{1}=*K U*_{1}(is a set of functions*v*corresponding to non-zero )_{n}*V*_{2}is the rest of basis functions*v*_{n}*F*is the space spanned by the set of vectors*U.*In other words, any vector*f*in*U*is a linear combination of all basis functions:

*f*=*r*_{1}*u*_{1}+*r*_{2}*u*_{2}+*r*_{3 }*u*_{3}+ . . .- The set of vectors
*U*_{1}spans a subspace*F*_{1}of the space*F* - The set
*U*_{2}*F*_{2} - The basis set
*V*forms the space of measuring functions*G*:*f*=*s*_{1}*v*_{1}+*s*_{2}*v*_{2}+*s*_{3 }*v*_{3}+ . . - The subspace
*G*_{1}is spanned by the set*V*_{1 } - The subspace
*G*_{2}is spanned by the set*V*_{2}

The three-dimensional space has only three basis vectors: *U*
= {*u*_{1} , *u*_{2} , *u*_{3}}. Let *U*_{1}
= {*u*_{1} , *u*_{2}} then *u*_{2} = { *u*_{3}}.
The space *F* is the whole 3D-space. The subspace *F _{1}*

In this example
the space *G* is also formed by the set of three basis vectors *V*
= {*v*_{1} , *v*_{2} , *v*_{3}}. The subspace
*G*_{1 }is
the (*v*_{1} , *v*_{2})-plane (green). The subspace *F*_{2}
consists of the *u*_{3} -axis.

F-space |
G-space |

The operator (2) transforms the F-space in the following way

It maps the whole space *F* and the subspace *F*_{1}
to *G*_{1}. It maps the subspace *F*_{2} to 0.

The last of the above equations means that the *F _{2}*
components of a reconstructing function

The *F*_{1} components produce a signal which is a
function from the subspace *G*_{1}. This signal can be used for
reconstruction of the unknown function. It can be done by means of
pseudo-inverse operator. Although it is not necessarily the best **practical**
method of reconstruction.

There are **no** components in
the *F*-space that produce a signal in the *G*_{2} subspace.
If the right-hand side of the equation (1) contains components from the *G*_{2}
subspace, then the equation does not have a solution.

The following picture summarize the interpretation

The singular functions (vectors) are natural bases for the integral operator (1) or for a given method of measurement.

- The basis set
*U*_{1}is used for the construction of unknown function*f*. Only this part of the reconstructing function can be determined from the experiment. However, the*U*_{2}components can be used for additional shaping of the result, e.g. to make it smooth. - The basis set
*V*_{1}should be used for a natural filtering of measuring functions. All the*V*_{2}components of a signal must be filtered out before processing.

In practice some random noise makes contribution to a signal:

It causes errors in the coefficients of reconstructed function:

The noise amplification factor can be defined as

**Any** ** f(x) that reproduces g(y) within an
experimental errors is a solution of the
equation (1). **

The above criterion further restricts the set of functions within *G _{1}*:
only the components with the

The number of functions and their shape gives the answer for the following important questions

- What is the accuracy of reconstruction?
- What is the resolution?
- What is the region where the unknown function can be reliably reconstructed?

The accuracy is the largest amplitude of the rejected functions. The resolution is the shortest half-wavelength of the functions that can be used for reconstruction.

Typical singular function behavior |

Extremely useful characteristics which is usually ignored, is Reliable
Reconstruction Region, **RRR**. It can be defined as a region where are
situated the nodes (zeros) of the functions *u _{n}* used for
reconstruction. In the above picture the RRR is approximately [0.5, 10].

The accuracy of measurements and the interval of measurements affect the overall accuracy, resolution and Reliable Reconstruction Region. Analytical calculations and numerical experiments (see Implementation) show the following:

- The
**accuracy of measurements**increases the number of singular function and primarily increases**accuracy and resolution of reconstruction** - The extension of the
**interval of measurements**modifies the integral operator (by increasing the interval [c,d], see Basics of Indirect Measurements) and increases the**Reliable Reconstruction Region**(e.g. the depth of detection, or interval of sizes).

**V.S.Bashurova, K.P.Koutsenogii, A.Yu.Pusep, N.V.Shokhirev**. Determination of atmospheric aerosol size distribution functions from screen diffusion battery data: Mathematical aspects.., v.22, p.373-388, 1991.>**J.Aerosol Sci****A.Yu.Pusep, V.S.Bashurova, N.V.Shokhirev A.I.Burshtein**. The development of the software for NMR-tomography of underground water. Tech. Report of the Institute of Chemical Kinetics and Combustion. Academy of Sciences of the USSR, Novosibirsk, 1991.**N.V.Shokhirev, L.A.Rapatskii, A.M.Raitsimring**. Electron-ion pair distribution function reconstructed from radiation-chemical and photochemical experiments.., v.105, p.117-126, 1986.**Chem.Phys****A.Yu.Pusep, N.V.Shokhirev.**Determination of particle-size distribution functions in aerosols from measurements by diffusion batteries.., v.48, p.108-113, 1986.**Colloid J**-
**N.V.Shokhirev, V.V.Konovalov, A.Yu.Pusep, and A.M.Raitsimring**. Recovering the e-aq distribution in Photoemission.., v.20, p.316-321, 1984.**Theor.Exper.Chem** **A.Yu.Pusep, N.V.Shokhirev**. Application of a singular expansion for analyzing spectroscopic inverse problems.., v.57, p.482-486, 1984.**Opt.Spectrosc**

© Nikolai Shokhirev, 2001 - 2017

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