I need to describe the optical aberrations of my set-up with Zernike polynomials. To do this, I want to fit the first 15 polynomials in the following way on my data:

$$\chi^2 = \sum_k\left(\beta_k^x - \sum_{n=2}^M C_n\frac{\partial P_n}{\partial x}\right)^2 + \sum_k\left(\beta_k^y - \sum_{n=2}^M C_n\frac{\partial P_n}{\partial y}\right)^2$$ where $\beta_x$ and $\beta_y$ are my data, $P_n$ are the Zernike polynomials, and $C_n$ are the coefficients. Because the coefficients $C_n$ for both the $x$ and $y$ derivative of the polynomial needs to be the same, I do not think I can use a normal polynomial least square. I need to do this in Labwindows/CVI.

Does anyone have an idea how to approach this problem?


If I read this correctly, you can rewrite your problem as an ordinary least square problem $\left\|Ax-b\right\|$, where A is the stacked Vandermonde-Matrix of the derivatives of the Zernike polynomials and $b$ the stacked measured data.

$$\chi^2 =\left \|\begin{pmatrix} \frac{\partial P_n}{\partial x}\\ \frac{\partial P_n}{\partial y} \end{pmatrix}C-\begin{pmatrix}\beta^x\\\beta^y\end{pmatrix}\right\|^2$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.