by Nikolai V. Shokhirev
Basics of Indirect Measurements
Singular Value Decomposition
Analysis of accuracy and resolution
Contents
- Introduction
- Definitions
- 3D-space Example
- Mapping of Spaces
- Interpretation of Mapping
- Use of singular functions
- Experimental noise
- Reconstruction criterion
- Units of Information
- Accuracy and Resolution
- Reliable Reconstruction Region
- Accuracy and Interval of Measurements
- References
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:
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}_{ }is the (u_{1} , u_{2})-plane (blue). The subspace F_{2} is the u_{3} -axis and not the rest of the space.
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 f do not produce the measuring signal in this experiment (instrument). This just reflects an incompleteness of this experiment. From the reconstruction viewpoint this is the source of instability of the inverse problem.
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.
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 g_{n} above the noise level should be taken into account. It also limits the set of functions u_{n} which can be used for reconstruction. The number of singular functions involved in reconstruction is the actual number of the units of information available for this instrument and noise level.
The number of functions and their shape gives the answer for the following important questions
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:
©Nikolai Shokhirev, 2001-2009