FoFi – A free, automatic telescope focus finder software

At the time of publication this project is not even in ALPHA state. However, I feel now is the right time to publish it since it reached a state where at least the source code could be useful to others. You can find the source code on github here.

The software aims to support the amateur astronomer (and especially astrophotographer) with one of the most critical but also most annoying tasks: Finding the best focus position for the camera. The main goal is to provide a free and easy to use software that just does the job – automatically.

With a given configuration it should also be possible to execute “FoFi” from the command-line without requiring any user interaction. This way you can include a call to the Focus Finder into a script. This might be useful if the entire observation process should be automated and you want to re-focus from time to time to compensate the temperature drift.

Finding perfect focus with V-Curve fitting based on a hyperbolic function

What is a V-Curve?

A V-Curve is the result of moving the focus of a telescope from an outside focus position into the focus and then further into the same direction out of focus again. This way a V-Curve comes into existence where the focus position is on the x-axis and the focus measure on the y-axis (right figure). Probably not surprising this curve is called V-Curve since it is similar to a V. There are different ways to measure the focus – for example the “Half-Flux Diameter” (HFD) and the “Full width at half maximum” (FWHM). In this case the I use the HFD measure.

Night sky image processing – Part 7: Automatic Star Recognizer

In the parts 1-6 of my “Night sky image processing” Series I wrote about the different stages of night sky image processing. In this article I want to put all the pieces together and develop an “automatic star recognizer” which takes an astro-image as input and outputs a list of the recognized stars with their HFD and FWHM values.

The star recognizer processing pipeline

Basically the processing pipeline is as follows:

ROI = Region of interest

In order to achieve this I use most of the concepts I have looked into earlier:

As input I used this FITS image for testing.

Night sky image processing – Part 6: Measuring the Half Flux Diameter (HFD) of a star – A Simple C++ implementation

In Part 5 of my “Night sky image processing” Series I wrote about measuring the FWHM value of a star using curve fitting. Another measure for the star focus is the Half Flux Diameter (HFD). It was invented by Larry Weber and Steve Brady. The main two arguments for using the HFD is robustness and less computational effort compared to the FWHM approach.

There is another article about the HFD available here. Another short definition of the HFD I found here. The original paper from Larry Weber and Steve Bradley is available here.

Definition of the HFD?

Let’s start with the definition: “The HFD is defined as the diameter of a circle that is centered on the unfocused star image in which half of the total star flux is inside the circle and half is outside.”

In a mathematical fashion this looks like this:

$$\sum\limits_{i=0}^{N} V_i \cdot (d_i – HFR) = 0 \Leftrightarrow HFR = \frac{\sum\limits_{i=0}^{N} V_i \cdot d_i}{\sum\limits_{i=0}^{N} V_i}$$

where:

• $V_i$ is the pixel value minus the mean background value (!)
• $d_i$ is the distance from the centroid to each pixel
• $N$ is the number of pixels in the outer circle
• $HFR$ is the Half Flux Radius for which the sum becomes $0$

Night sky image processing – Part 5: Measuring FWHM of a star using curve fitting – A Simple C++ implementation

In Part 4 of my “Night sky image processing” Series I wrote about star centroid determination with sub-pixel accuracy. Based on this centroid position the FWHM determination of the star takes place. The FWHM (Full Width Half Maximum) value is a measurement for the star width. In this part I write about the so called curve-fitting which is helpful to determine the FWHM value from such an image. Note that for the following calculation and implementation I do not consider the sub-pixel determination of the centroid.

Determination of the FWHM

Let’s say we want to determine the FWHM in x-direction (red line). There is of course also one in y-direction. Basically, what happens is:

1. Extract the pixel line through the centroid in x direction (those gray-level values usually form a Gaussian distribution: $$y(t)|_{b, p, c, w} = b + p \cdot e^{-\frac{1}{2} \cdot \big(\frac{t – c}{w}\big)^2}$$ We define a “data-point” as (x,y)=(pixel x-position, pixel gray-level value).
2. Based on those (x,y) data-points determine the 4 parameters of a Gaussian curve so that the Gaussian curve and the data-points fit “as good as possible”. As fitting algorithm we use the so called “Levenberg-Marquart” algorithm. It tries to minimize the quadratic error between the data-points and the curve.
3. The result is a Gaussian curve i.e. a parameter set which describes the curve (c=center, p=peak, b=base and the w=mean width). One of those parameters – the mean width is the FWHM value.