Fast Auto-Focus = Method and=20 Software for CCD-based Telescopes [1]

By Larry Weber and = Steve=20 Brady

Introduction

Focusing CCD-based = telescopes is one of the drudgeries astronomers constantly face. With the recent advent of = relatively low=20 cost, motorized position-feedback focusing hardware, comes an = opportunity to=20 fully automate the focusing operation. =20 This paper presents a new method and software for automatically = focusing=20 CCD based telescope systems. = It is=20 highly suitable for unattended all-night robotic-telescope operations = such as=20 asteroid searches, astrometry, photometry, imaging, etc. The only = requirements=20 are a CCD based telescope, a commercially available position-feedback=20 motorized-focuser [2] and CCD camera control software = [3].

Automated focusing = systems=20 demand solution to a wide range of challenging technical problems and=20 specifications. Ideally, = any=20 automated focus method should:

Have accuracy = equal to or=20 better than the manual mode of focusing that it = replaces=20

Be fast so that = valuable=20 observing time is not lost=20
Be robust so = that the user=20 can expect it to arrive at the correct focus even in marginal = conditions such=20 as thin clouds or strong sky glow

Be capable of=20 accommodating a wide range of initial out of focus star diameters = (important=20 with filters of differing optical thickness in a filter wheel) =

To facilitate full automation, the software = should have a=20 standard ASCOM interface [4] that allows the user to write simple = scripts that=20 can coordinate the focusing operation with the other telescope = operations. Ideally this means that once = the system=20 and software are properly configured, then the user only needs to press = one=20 focus button to initiate a successful focus operation. We believe our method and = software meets=20 all of these requirements.

Degree = of Focus=20 Metric

To=20 meet the above requirements an accurate and robust focus metric is = needed. Traditionally, astronomers = have used a=20 number of techniques to characterize the Point Spread Function (PSF) of = a star.=20 One common method is a gaussian function fit to the PSF. Unfortunately, = severely=20 out of focus stars frequently have the shape of an annulus or disk, = which does=20 not match the shape of the gaussian accurately. We have found the Full Width = Half=20 Maximum (FWHM) metric used to quantify the width of the star PSF to work = most of=20 the time but not with sufficient accuracy for auto-focusing. This is due to variations in = seeing=20 which cause local peaks in the unfocused star PSF and severely degrades = the=20 calculation of the maximum when assessing the FWHM. =

We=20 have based our auto-focus algorithm on a metric named the Half Flux = Diameter=20 (HFD). The HFD is defined = as the=20 diameter of a circle that is centered on the unfocused star image in = which half=20 of the total star flux is inside the circle and half is outside. The HFD gives a single number, = in units=20 of CCD pixels, that is relatively insensitive to variations in seeing, = star=20 flux, thin clouds or sky background glow. =20 We have found this one metric to be accurate over a very wide = range of=20 unfocused star diameters and flux intensities because the HFD is = determined by=20 integrating all of the flux from the unfocused star area. HFD does not suffer from the = problems=20 found with gaussian fits or FWHM.

Note that in our = earlier=20 version of this paper, [1] we defined our degree of focus metric as a = Half Flux=20 Radius (HFR). The HFD = metric has=20 exactly the same concept as the HFR metric with the obvious=20 identity:

=20 HFD =3D 2 * HFR.

We=20 have adopted HFD, at the recommendation of Douglas B. George. This is for user convenience = since, for=20 small star disks, the measured HFD value is quite similar to the more = familiar=20 FWHM of the star.

V Curve=20 Plot

A Plot of the = measured HFD=20 vs. the precision focus position gives a very stable V shaped curve = where the=20 Best Focus Position is at the apex of the V. The sample to the left shows = actual data=20 that was automatically measured on an 8 inch LX-200, ST-8E system during = good=20 seeing conditions near sea level. =20 The horizontal axis is the focuser position and the vertical axis = is the=20 HFD. This V Curve is an = accurate=20 quantitative measure of the telescope optical convergence cone and beam=20 waist. The measured left = and right=20 sides of the V Curve are delightfully linear with slopes that depend = only on the=20 hardware characteristics such as the optics f-number, the CCD pixel size = and the=20 focuser gear ratios. We = have found=20 these slopes to be remarkably constant over temperature and=20 time.

The=20 software automatically fits straight lines to the data between the user=20 specified high and low HFD limits. =20 It also measures the straight-line slopes, position intercepts = and the=20 position intercept difference (shown in the lower section). The telescope-CCD-focuser = system can be=20 completely characterized by measuring the three parameters, the Left = Slope, the=20 Right Slope and the Position Intercept Difference. Once the system is = characterized with=20 accurate values for these three parameters, the best focus position can = be=20 predicted from a well-sampled HFD measurement at a known position and = the=20 additional knowledge of which side of Best Focus the system is on. Note, the value of the = Position at Best=20 Focus shown is only of transient interest and is not useful for = characterizing=20 the system.

HFD Auto-Focus Strategy

To begin, the user must characterize the=20 telescope-CCD-focuser system by automatically measuring the V Curve and=20 extracting the three parameters. = In=20 principle, the Best Focus Position can be determined with a measured = target star=20 HFD, the position of the focuser and knowledge of which side of Best = Focus the=20 focuser is on. In = practice more=20 than one HFD measurement is made and the results are averaged to = increase the=20 signal to noise ratio making the Best Focus determination more = accurate.

We=20 have written a user-friendly MS Windows application to automatically = focus the=20 telescope. In a typical run the telescope can be routinely brought to = focus in=20 only one minute with a LX-200, ST-8E system. The process = is:

A non-precision = move of=20 the focuser is made to a point that is known to be sufficiently far = from Best=20 Focus (user selected inside or outside of Best = Focus)=20
A full frame 3x3 = binned=20 image is taken with a minimal exposure time=20
The brightest = star is=20 found, the position determined and the HFD measured=20
The target star = is=20 sub-framed and a second exposure taken (download time is minimized due = to the=20 smaller frame size)=20
The focuser = position is=20 then precision moved to a user specified Near Focus Position which is=20 determined from target star HFD and the V Curve slope=20 constant=20
An average HFD = is=20 determined from a small number of rapid sub frame exposures which = reduces the=20 influence of random variables such as seeing=20
The Best Focus = Position is=20 then determined and the stepper motor automatically moves the focuser = to this=20 Position

This technique = allows=20 accurate determination of the best focus position by moving the focuser = in one=20 direction which eliminates issues such as backlash and mirror flop, = which are=20 found on many telescope focus mechanisms.

This strategy = determines=20 Best Focus based on the HFD of the one brightest star on the CCD. We have found this one star = strategy to=20 be valuable since this star can be centered on the CCD chip to reduce = the=20 influence of off axis optical aberrations such as field curvature and = coma. With robotic telescope = operation, it is=20 relatively easy to find a sufficiently bright star for this = strategy. We also allow the user to = manually=20 select a single target star in a crowded star = field.

Currently, the = gating time=20 factor in the auto-focus process is the slow download time of the CCD = and not=20 exposure time. We have = found that=20 if a target star is selected in the 1^st to 7^th = magnitude=20 range, exposure times for can be kept very short and still achieve equal = or=20 better performance than longer exposures. =20 For the ST-8E camera we typically use the minimum 0.11 sec and = have good=20 results with the faster 0.05 sec shutter speeds of the Apogee AM16 = camera.=20 Depending on the users optical system, camera, etc. dimmer stars can be = used=20 with the exposure time adjusted accordingly.

Exposure = Sequence

This HFD Focus = method can be=20 best illustrated by walking through a typical exposure sequence. The first step is to take a = full frame=20 3X3 binned exposure in order to initially find the brightest star. The image from this exposure = is shown=20 below.

After the full = frame=20 exposure is made, the software finds the brightest star on the chip, = frames the=20 target star, subtracts the background level and then measures the=20 HFD.

The=20 main user form that is active while the focusing operation proceeds = allows the=20 operator to easily monitor the quality of the focus process. The largest = plot on=20 this form is a vertical bin of the framed region centered on the = star. The vertical lines to the left = and right=20 of the vertical bin curve specify the boundaries of the star detected by = the=20 algorithm. The regions = outside of=20 the boundaries are used to determine the background level of the image, = which is=20 subtracted from the image before the vertical bin curve is=20 plotted.

The=20 small box in the lower left corner shows the location of the target star = on the=20 CCD chip. Note that this directly corresponds to the location of the = star on the=20 full frame image shown above. =20 Ideally the measured star is near the center of the CCD chip so = that off=20 axis aberrations are minimized. =20 This star locator is useful for verifying centering since most of = the=20 exposures have very small frame sizes.

For=20 each exposure taken, the Half Flux Diameter is determined which is used = for=20 setting the next focus position and frame size.

In=20 this sequence, the new focuser position is set to 9087 and a new = exposure=20 taken. Note that the star = size is=20 considerably smaller and that the exposure frame size has been reduced = to fit=20 the star based on the HFD of the previous exposure. Also, the target star HFD is=20 considerably less than the value measured above. This was by design since the = value of=20 9087 was calculated based on the results of the previous=20 exposure.

The=20 formula used to calculate the new position is:

NP=20 =3D OP + (NewHFD - OldHFD) / Slope,

Where: NP is the = new=20 position,

OP is the old=20 position,

NewHFD is the = desired new=20 HFD,

OldHFD is the = measured HFD=20 at the old position

Slope is the V = Curve slope=20 on the side of focus we are operating.

This formula is = simply the=20 equation for a straight line that fits the appropriate side of the V = Curve.=20

When the measured = HFD of the=20 initial exposure is large, the focuser position is adjusted so that the = next=20 exposure will give approximately one half the quantity: original HFD = minus the=20 Near Focus HFD (defined below). = So=20 in the example just above, we went from HFD =3D 24.82 to HFD =3D = 17.41. This half step algorithm is = used to=20 insure that severely out of focus stars will properly converge, in spite = of=20 system problems such as mirror flop.

The=20 Near Focus HFD is a user specified value that should be set close to the = Best=20 Focus position but still on the linear part of the V Curve. In the example presented here = the Near=20 Focus HFD is set to10.

When the measured = HFD is=20 determined to be within a factor of two of the Near Focus HFD, then the = Near=20 Focus HFD position is calculated and the focuser moves to that = position. In this example, the position = of 9087=20 gave a HFD of 17.41, which is within a factor of 2 of the Near Focus HFD = of=20 10. The Near Focus HFD = position of=20 9153 was calculated and was found to have an HFD value of=20 10.01.

At=20 the Near Focus Position, multiple sub-frames are taken (5), and the = average HFD=20 value determined. Telescope tracking errors may cause the star to move = out of=20 the exposure sub-frame window because the sub-frame size is small. This is corrected by = determining the=20 target star centroid following each exposure and positioning the next = sub-frame=20 on that position.

The=20 average HFD value is used to determine the final Best Focus position = (9240). The=20 focuser is moved to the Best Focus position and a final exposure is = taken with=20 the target star HFD measured.

Log Record

Our = software maintains a=20 running Log of all operations with a time stamp for each line, which is = useful=20 for monitoring operations and for debugging scripted robotic = operations. In the example log shown, at = time:

20:56:57 the Focus button was clicked and = RoboFocus=20 was commanded to go to the initial position of 9021. After arriving at the position = it starts=20 the full chip 3x3 binned exposure.

20:57:24 the exposure was completed and = the HFD is=20 measured to be 26.3076 with a total star flux of 14309.

20:57:32 a smaller sub-frame is taken and = the HFD=20 value used to calculate the Near Focus position 9166.

20:57:37 Near Focus Exposures begin (5 = taken).

20:57:48 Best Focus Position was = determined to be=20 9253.

20:57:54 RoboFocus moves to the Best Focus = position=20 and a final exposure taken yielding an HFD of 2.9019

20:57:54 the focusing operation was = completed. Total=20 time =3D 57 seconds!

The=20 auto focusing runs shown on the above running Log were conducted in = conditions=20 of variable thin clouds. = The column=20 at the far right of this log shows the total star flux; note that during = the=20 last run, the flux varies from a high of 14309 to a low of 4607. The focus was successful in = spite of=20 more than a 3x factor in flux variation due to thin clouds. Note that on the previous = auto-focus=20 run, that started shortly before the top line of the log at 20:56:29, = the clouds=20 were thinner so that the total star flux went as high as 42509. In spite of this wide = variation in total=20 star flux from run to run, both runs arrived at practically the same = final=20 positions (9254 and 9253) and final HFD values (2.7680 and=20 2.9019)!

HFD Measurement Algorithm

After a framed star exposure is taken, the HFD = measurement=20 algorithm steps are:

1. = Subtract the background level from the image

2. = Find the centroid of the star by a simple weighted = averages=20 method

3. = Determine the radius of each pixel from the = centroid

4. = Sort the pixels in order of increasing radius

5. = Generate an integral of the pixel flux along the = diameter (=3D 2=20 X radius) dimension. This = integral=20 is shown on the user form as a small plot to the left of the Find and = Focus=20 buttons. This plots the = diameter=20 along the horizontal axis and the integrated flux along the vertical = axis. The integral shows zero = integrated flux=20 at the zero diameter and it shows the full star flux at the largest=20 diameter.

6. = Determine the Half Flux Diameter from this integral, = which is=20 simply the diameter where the integrated flux is half of the full star=20 flux. This HFD point is = marked on=20 the flux integral plots with a vertical line.

This algorithm works very well for stars with = large HFD of=20 perhaps 20 or more. = However for=20 smaller HFDs the discrete nature of the square pixels introduces a = significant=20 error that increases as the HFD gets smaller. This is corrected by first = exactly=20 calculating the fractional contribution of pixel flux that is inside the = HFD=20 circle for each pixel that intersects the HFD circle. Then an iterative technique is = used that=20 makes small adjustments to the circle diameter to converge at the true = HFD where=20 half of the total flux is inside the circle and half is outside.

System Profile Data

A foundation of our auto focus algorithm is = accurate=20 knowledge of the parameters that characterize the system. The three parameters are the V = Curve=20 Left Slope, Right Slope and the Position Intercept Difference. These parameters will change = when the=20 f-number of the system changes due to the use of Barlow lenses or focal = reducing=20 lenses. We have provided = a simple=20 data base form, shown here, that records and preserves this data for = each system=20 configuration.

When a user selects a different system = configuration, the=20 System Profile for that unique system is selected. Every time that a V Curve is = completed,=20 the user has the option of saving the measured parameters in the System = Profile=20 database. Each row shown = in the=20 above form is the data from a single V Curve sequence. The values of the Left Slope, = Right=20 Slope and Position Intercept Difference found in the database are = averaged and=20 shown in the System Profile frame in the upper part of the form. The user can elect to include = or exclude=20 V Curve parameters from this average by placing a Y or N in the far left = column=20 of the database. In order = to=20 evaluate the quality of the V Curve data, the number of points used to = measure=20 the slopes for each V Curve side (NPTS) are tabulated as well as the = standard=20 deviation of the straight line fit to the V Curves for both the left and = the=20 right sides of the V Curve. = When=20 the user clicks the Update button, the averaged parameter values are = used to=20 update the actual parameters used to perform the Focus operation.

Measured=20 Specifications

As=20 mentioned above, we typically achieve a perfect focus one minute after = the focus=20 operation is initiated for a ST-8E CCD camera. This value is strongly = dependent on how=20 far the star is initially out of focus. =20 Stars farther from focus obviously take longer due to focuser = move time=20 and may also require additional exposures.

Stars that are not = too far=20 from best focus can range between 7^th and 1^st = magnitude=20 for an 8 inch LX-200 and a ST-8E camera using 0.11-second = exposures. We have made similar = measurements on a=20 custom built 10-inch Newtonian that utilizes a precision secondary = mirror=20 focuser. The high = magnitude limit=20 is determined by the low signal to noise ratio that gives lower accuracy = HFD=20 values. Of course dimmer = target=20 stars can be used with longer exposure times. The low magnitude limit is due = to the=20 saturation of the star on the CCD. =20 This limit is strongly dependent on the user selected Near Focus=20 Position. Absence of CCD = saturation=20 is most critical for the Near Focus exposures since they are used to = determine=20 the Best Focus Position. = The user=20 can reliably focus stars even brighter than 1^st magnitude by=20 adjusting the Near Focus Position so that the HFD values at the Near = Focus=20 Position are sufficiently large to avoid = saturation.

The=20 algorithm can accommodate out-of-focus star images that have diameters = as large=20 as 90% of the height of the CCD chip. =20 The algorithm that determines the background level imposes this=20 limit. If the background = level was=20 somehow known and properly subtracted, then the HFD algorithm could = accommodate=20 even larger star image diameters.

ASCOM Object

The=20 software includes an ASCOM object [4] that can be called by user written = scripts=20 to control the focus operations along with other ASCOM controlled = telescope=20 operations [5,6]. = The object=20 contains the following Methods and Properties:

Methods:=20

Focus &nbs= p; =20 Starts the Focus operation which continues until perfect focus is = achieved

FindStar =20 Finds the brightest star on chip, centers in small frame and = calculates=20 HFD

SingleExpose Takes single = exposure in=20 small frame and calculates HFD

Properties:

HalfFluxDiameter =20 (read only) = The=20 most recently measured Half Flux Diameter

Position &nbs= p; =20 (read only) = The=20 current RoboFocus Position

StarXCenter &nbs= p; =20 (read write) =20 The CCD chip X coordinate of the star = center

StarYCenter &nbs= p; =20 (read write) =20 The CCD chip Y coordinate of the star = center

SingleExposeFrameWidth =20 (read write) Width = in pixels=20 of SingleExpose frame

Software Availability

The=20 software described here is available in executable application form as = freeware=20 from the authors. It is = currently=20 at beta test status. It = is written=20 in Microsoft Visual Basic 6.0. = To=20 obtain the executable installation package = contact:

Larry Weber larryweber@idsi.net =20 or

Steve Brady sebrady@adelphia.net = .

Software downloads = are=20 available at http://www.focusmax.org/=20 .

References

[1] This paper is an updated = version of our=20 paper of the same title originally presented at the 2001 Minor Planet=20 Amateur/Professional Workshop, pp.104-113, Tucson = AZ.

[2] Technical Innovations = RoboFocus stepper=20 motor control. See http://www.homedome.com/=

[3] Diffraction Limited MaxIm DL = CCD camera=20 control. See http://www.cyanogen.com/=

[4]=20 Astronomy Common Object = Model=20 (ASCOM). See=20 http://www.ascom-standards.org

[5]=20 Robert B. Denny, = =93ASCOM: Review of=20 the Technology and Milestones,=94 2001 Minor Planet Amateur/Professional = Workshop,=20 pp.88-92, Tucson AZ.

[6]=20 Jeffrey S. Medkeff, = =93Small Robotic=20 Observatories: Operations, Deployment and Future Direction,=94 2001 = Minor Planet=20 Amateur/Professional Workshop, pp.93-103, Tucson = AZ.