How to Extract Data from an Integral

[ecastro]

Consider this form:

$A = \int B\left(x\right) C\left(x\right) dx$

I have the values for $A$ and $C\left(x\right)$ (a value of $C$ per value of $x$), is there a way that I can extract $B\left(x\right)$ numerically or analytically?

Thank you in advance.

 

[Ssnow]

A question, is $A$ a number or a function $A=A(x)$, in other words the integration is definite on an interval or is indefinite?

 

[HallsofIvy]

Given any B(x) there are an infinite number of different functions, C(x), that will give the same A.

 

[the_wolfman]

Consider this form:

$A = \int B\left(x\right) C\left(x\right) dx$

I have the values for $A$ and $C\left(x\right)$ (a value of $C$ per value of $x$), is there a way that I can extract $B\left(x\right)$ numerically or analytically?

Thank you in advance.

Yes, equations of this for are called integral equations. There are numerous methods that have been developed to solve integral equations. However, it's not always possible to find a unique solution.

 

[zinq]

I will assume that the integral in the original post is a definite integral, or else the questions won't make sense.

This is analogous to the discrete case, where you have number B_j and C_j, for 1 ≤ j ≤ n and a number A such that

A = ∑ B_j C_j​

where the sum is over j = 1, 2, 3, ..., n.

Consider the simplest non-trivial case, where n = 2:

A = B₁ C₁ + B₂ C₂,​

where you know A and the C_j, and are trying to determine the B₁ and the B₂.

This gives you one linear equation in two unknowns. As we know from algebra, we need two linear equations (and independent ones at that) to determine two unknowns. It does, of course, present a constraint on the B₁ and B₂ — but that is not enough to solve for them.

This not only gives an idea of why the original integral equation cannot be solved for B(x), but this can even be translated into a corresponding integral equation, with the C(x) function defined as a step function, constant on each of two consecutive unit-length intervals. This discrete case shows that even if B(x) were likewise required to be a step function constant on each of those unit intervals, you still wouldn't be able to solve for it.

 

[ecastro]

A question, is $A$ a number or a function $A=A(x)$, in other words the integration is definite on an interval or is indefinite?

The variable $A$ is a number for this case. I apologise if I haven't mentioned that the integral is definite.

Yes, equations of this for are called integral equations. There are numerous methods that have been developed to solve integral equations. However, it's not always possible to find a unique solution.

I'm trying to search these methods with Google, but I always find sites on how to solve a particular integral equation, not to extract a function from it.

I will assume that the integral in the original post is a definite integral, or else the questions won't make sense.

This is analogous to the discrete case, where you have number B_j and C_j, for 1 ≤ j ≤ n and a number A such that

A = ∑ B_j C_j​

where the sum is over j = 1, 2, 3, ..., n.

Consider the simplest non-trivial case, where n = 2:

A = B₁ C₁ + B₂ C₂,​

where you know A and the C_j, and are trying to determine the B₁ and the B₂.

This gives you one linear equation in two unknowns. As we know from algebra, we need two linear equations (and independent ones at that) to determine two unknowns. It does, of course, present a constraint on the B₁ and B₂ — but that is not enough to solve for them.

This not only gives an idea of why the original integral equation cannot be solved for B(x), but this can even be translated into a corresponding integral equation, with the C(x) function defined as a step function, constant on each of two consecutive unit-length intervals. This discrete case shows that even if B(x) were likewise required to be a step function constant on each of those unit intervals, you still wouldn't be able to solve for it.

Is Singular Value Decomposition method valid for this case? I just happen to read on this technique and I do not know if it is applicable for this problem.

In point of fact, I have encountered this form of equation in image filters, where $C\left(x\right)$ is the filter and $x$ is the wavelength. I have a filtered image, which is $A$, and I want to extract the unfiltered image which is $B\left(x\right)$. Do you have any thoughts on how to solve this kind of problem?

 

[zinq]

I'm no expert on image reconstruction, but I can say that the math problem as you have described it cannot be solved uniquely for the function B(x) — not even close.

Possibly if you have more information about B(x) and C(x), something more intelligent can be said.

 

[Samy_A]

I'm no expert on image reconstruction, but I can say that the math problem as you have described it cannot be solved uniquely for the function B(x) — not even close.

Yes, and "not even close" is an understatement.
If the equation you are trying to solve is $$A=\int_{a}^{b} B(x)C(x) dx \ \ (1)$$
any function $D$ satisfying $\displaystyle \int_{a}^{b} D(x)C(x) dx \neq 0$ is a solution up to a constant, as $$B(x)=\frac{A}{\int_{a}^{b} D(x)C(x) dx} D(x)$$ will be a solution to $(1)$.

 

[Svein]

In Hilbert space terminology, your problem can be written as $A=<B, C>$, where < , > is the scalar product of that particular space (in this case the scalar product is the integral). This means that, for example, $<B+H, C>=<B, C>+<H, C>$. From this simple fact follows that whenever $<H, C>=0$, $A = <B, C>+<H, C> = <B+H, C>$. Therefore, there can be no single solution for $B$.

An interesting question might be: find the "minimal" solution for B. This implies that B is some scalar multiple of C, say λ⋅C. This again means that $A=<\lambda\cdot C, C>=\lambda<C, C>$. Since you know C(x), you can calculate $<C, C>=\int_{a}^{b}C^{2}(x)dx$ and then finding λ is trivial.

 

[OrangeDog]

If you assume it is a type of function, such as a polynomial or exponential, then maybe it is possible with enough equations for the unknowns.