by Nikolai V. Shokhirev
Basics of Indirect Measurements
- Measurements scheme
- Linear systems
- System parameters and Measured parameters
- Integral Form
- "Easy" solution
- Ill posed problems
- Model kernel
- Some conclusions
- Incomplete measurements
Singular Value Decomposition
Analysis of accuracy and resolution
If some unit of measure can be applied directly, then this type of measurement is called direct measurement. Example:
Human eyes determine the distance from the different angles of the each eye line of sight:
This is an example of indirect measurement. The determination of the distance L requires additional processing:
Very often we measure not what we need but what we can, and try to estimate the quantity of interest. For example, we can measure the properties of light reflected by a surface and estimate the surface roughness:
We also prefer to detect a harmless radio signal in magnetic resonance imaging rather than really slice a body:
The obvious areas of indirect measurements are very small or very large or remote objects. In general, this is the measurement of the objects that are inaccessible for direct measurements.
This is a schematic presentation of the process of measurement:
Usually the systems are assumed to be "Linear". It means that the total signal (response) from the system is the sum of the signals from its subsystems. In particular, the signal from the subsystems with the same parameters is proportional to the number of such subsystems (Linear dependence on the number).
Mathematically the measured signal from a linear system can be presented in the following form
Here is the number of x-th subsystems, is the signal from this subsystem as a function of the scanning parameter y (e.g. the dependence of intensity in the receiving coil on input pulse duration).
The system parameters here are denoted as x. For example, it can be a particle radius in air pollution control measurements or three coordinates (a vector) of the point in human body in MRI.
Here for simplicity we restrict ourselves with one scanning parameter and one system parameter (in other words, both parameters are scalar).
Actually, this is an equation with unknown function f and the right-hand side
This equation is known as the Fredholm integral equation of the first kind, and
is called the kernel of this equation.
The kernel is a mathematical description of the method of measurement. The interval of system parameter variation
[a, b] and the measurement interval [c, d] are the part of the measurement description as well.
For the first sight, the above equations can be easy resolved. If the signals from each subsystem are known in the set of discrete points ym, then we can always adjust the linear parameters so that the combined signal fits the measured signal . Actually, this is a set of linear equations:
Here is the matrix of the above system, is the unknown vector, is the r.h.s. vector. The integral equation is just a very big set of linear equations.
However the computational practice shows that the solution of this system is unstable and very sensitive to experimental errors.
Unfortunately, in general, the above equations belong to the case of so-called ill posed or ill stated problems. These problems have at least one of the following features:
The Fredholm integral equation of the first kind can have all of the above features. Of course, it depends on the kernel (and the intervals [a, b] and [c, d]).
The best kernel is -function:. The -function is the extreme case of the very narrow large peak so that
The integral equation reduces to
In fact, this is a mathematical model of direct measurements.
The -function can be approximated by a rectangular with the width w and the height 1/w so that . Another popular approximation is the Gaussian function:
This is its graphical representation:
|Gaussian peak G(x)||Gaussian kernel G(y-x)|
The worst kernel is a constant, K(y,x) = c :
The constant kernel K(y,x) = 1
It allows determination only of the area S under the curve f(x):
The kernels corresponding to real measurements are far from the both extremes. They are not necessarily symmetrical because they connect the parameters of different nature.
Let us consider the measurements which mathematical model has the following simple kernel:
This is a very broad peak:
The model kernel
It can be directly shown that for any f(x)
The zero contribution means that such components of the function cannot be detected and, consequently, cannot be restored even for absolutely precise measurement. In other words, this is incomplete measurement and for the extraction of the "lost" information we need some additional measurements. This is quite natural: it is difficult to imagine some universal experiment, which measures everything.
©Nikolai Shokhirev, 2001-2009