- #1
Daniel Petka
- 147
- 16
- Homework Statement
- Consider a laser line position estimation by fitting using the Least Square Method (LSM) and prove (or disprove) that it can be considered as a convolution with some function and finding the center by looking for the maximum (zero‐crossing by the derivative). What is the smoothing function?
The Least Square Method (LSM) is defined as:
$$\sum_i[S(x_i)-F(x_i,;a,b,...)]^2=min,$$
where the fitting function is:
$$F(x;y_0,A,x_0,w)=y_0+A\cdot g(x-x_c,w)$$
The fit program will adjust all parameters, but we
are interested only for ##x_c##.
Hint: change sums to integrals in LSM description!
- Relevant Equations
- fitting function: ##F(x;y_0,A,x_0,w)=y_0+A\cdot g(x-x_c,w)##
convolution: ##f(x)=\int S(x-y)K(y)dy##
Least Squares Method: ##\sum_i[S(x_i)-F(x_i,;a,b,...)]^2=min##
I started by converting the LSM from sum to integral form:
$$f(x_c) = \sum_i[S(x_i)-F(x_i,;a,b,...)]^2 to f(x_c) = \int( S(x) - F(x-x_c)^2 dx$$
Since we are not interested in the other parameters (like offset), I assumed that they are fitted correctly and thus ignored them, turning ##F(x-x_c)## directly to ##g(x-x_c)##.
Then I expanded the binomial formula as following:
$$\int S(x)^2 - 2S(x)F(x-x_c) + g(x-x_c)^2 dx$$
And used the linearity of the integral to isolate the part of the equation that doesn't depend on x_0:
$$ f(x_c) = \int S(x)^2 dx + \int 2S(x)g(x-x_c) + g(x-x_c)^2 dx$$
Hence, we have a constant q that isn't affected by the convolution:
$$ f(x_c) = q + \int 2S(x)g(x-x_c) + g(x-x_c)^2 dx$$
The middle term is a convolution og the 2 functions. My idea was to disprove that a Kernel exists, because there is a term that doesn't depend on ##x_c##, but this logic doesn't make any sense after thinking about it. I am completely stuck at this point, since I can neither prove nor disprove that the kernel function exists. Any help would be highly appreciated!
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