HOW IT's DONE - IMPLEMENTATION
The background light level is classified by Bortle level, and settles the maximum exposure time to achieve an IU value of around 50. The Bortle level roughly relates to relative brightness.
Brightness = IU(8bit) * Lens block * Filter block / ISO / exposure time
Location Bortle Bkg 1/ISOsecs Rel Bkg B/Noise Sony B/Noise QHY cooled
Grand Tetons 1 0.006 1 5
Junction 2-3 0.001-0.0028 3.3 3
Wimberly 4 0.0033 5.5 2.5 3.3 (low I)
Austin 5 0.0036 6.0 5
Line filter reduces intensity 180x, background 1.7x
Cameras Type Chip Pixel Pixel ADC Pixel Diagonal Tech(1). Noise(2) IU Introduced Backfocus
um bit count mm RT -15C
Nikon 1000 RGB IMX206 1.2 4000 12 16M 5 EX 120 2018
Sony 7a RGB IMX235 8.4 3600 14 12M 43 EX 5 2014 18mm
QHY 183 Mono IMX183 2.4 5339 12 20M 15 EX-R 4 1 2017 17.5mm
Svbony 305 RGB IMX290 2.9 1500 12 2M 6 EX-R 11.5mm
Canon E mount 44mm
Pentax M42 45mm
(1) EX = Exmor. EX-R = Exmor - R. Exmor R is a back-illuminated version of Sony's CMOS image sensor. Exmor R was announced by Sony on 11 June 2008. Sony has newly developed a unique photo-diode structure and on-chip lens optimized for back-illuminated structures, that achieves a higher sensitivity of +6dB and a lower random noise of -2dB without light by reducing noise, dark current and defect pixels compared to the conventional front-illuminated structure, 8 dB translates to a 6x improvement.
(2) Noise at around ISO3200 20sec exposure. G30 on QHY = 3200 ISO.
Svbony Capture in SharpCap .FITS preferable to NINA.
Gain in Sharp cap is 10x NINA Gain 300 = ISO 3200.
Save as RAW16 FITS format
PIPP -Debayer in 2 places YES , White balance YES BGGR pattern Output Split RGB NO
Stack in Autostakkert COO align
PS 3x resample.
Topaz OOF V blurry
Stack in DSS with FITS specified in load. Output as TIFF and you get full color image.
30 sec Dark has 4(8bit) FWHM noise - comparable to QHY which makes sense.
Orion ED80 configurations
400mm focal length 80mm aperture f6 lens 1.2 asecs resolution.
Focus assembly only works if focus adjust is horizontal and on bottom, with roller bearings on top. Works well with Alt/Az stage. On a GEM EQ stage, need to lock focus during moves and then rotate focus assembly BEFORE unlocking and refocus.
400mm- focus 2cm + Field F + T2 to NEX + Sony7as Camera limited resolution 400mm RGB 3.5 asecs/pix unusable undersample
400mm- focus 5.2cm + Field F + 38mmT2 + Svb Diffraction/pixel match 400mm RGB 1.1 asecs/pix undersample
400mm- focus 5.2cm + Field F + Filter wheel + QHY First line filter images 400mm HSL 1.1 asecs/pix
400mm-focus 4cm + 2.5" to 1.5" + 1.5" to 5x + T2 to NEX + Sony7as 2000mm RGB 0.8 asecs/pix undersample
400mm-focus 1cm + 2.5" to 1.5" + 1.5" to 5x +38mm T2 + Svb 2000mm RGB 0.25 asecs/pix adequate sampling
400mm-focus 1cm + 2.5" to 1.5" + 1.5" to 5x +85mm T2 + Svb (10x mag) 4000mm RGB 0.12 asecs/pix good sampling
Celestron 6 configurations
To check collimation, place star in middle of field using eyepiece. Defocus and obstruction should be symmetric in the defocus ring.
Move finger shadow to find mirror screw closest to narrowest ring, if in between use opposite screw
Move star to edge of field in direction of narrowest ring.
Adjust screw 1/6 of a turn to move star back to center of field.
If to loose, tighten other two screws, if too tight loosen the other 2.
Prime focus imaging C6 0.7 asecs resolution
C6 + C6 to T2 converter + T2 to Sony E + Sony7as 1500mm RGB f10 1.0 asecs/pix 0.7x sample
C6 + C6 to T2 converter + 38mm T2 tube + QHY 1500mm HSO f10 0.35 asecs/pix 2x sample
C6 + C6 to T2 converter + T2 to C +Svbony 1500mm RGB f10 0.35 asecs/pix
C6 to 1.25" converter + 1.25 to eyepiece + eyepiece 5x T2 + T2 to Sony E +Sony 7as 7500mm RGB 0.2 asecs/pix 3.5x adequate sampling
C6 to 1.25" converter + 1.25 to eyepiece + eyepiece 5x T2 + T2 to C +Svbony 7500mm RGB 0.07asecs/pix 2amin field 10x good sampling
RASA C6 with Hyperstar 6 V4 with filter 2 asecs resolution Bf 39.8mm M42
C6 + Hyperstar M42 + M42 to T2 + QHY 300mm HSO f2 1.7 asecs/pix sample OK for abb. limited
C6 + Hyperstar M42 + M42 to C +Svbony 300mm RGB f2 1.7 asecs/pix
Screws with white washers on HyperStar rotate camera as desired, tighten screws.
Long screws are PUSH, short are PULL. Push one, pull another see if collimation improves. 1/8 turn at a time.
Use cable tidy to id direction.
Measured resolutions summarized below are consistent with these limitations.
Focal length Aperture f number Mount Back Focus Camera Resolution PIxel res
mm mm mm arcsecs arcsecs
8mm f3.5 Canon EF 44mm Sony7as 120 250
20mm f2.0 Sony E 18mm Sony7as 60 100
35mm f3.5 Pentax M42 44mm Sony7as 40
100mm 50mm f2.8 Sony E 18mm Sony7as 14 20
100mm f2.0 Canon EF 44mm QHY 20 6
300mm 150mm f2.0 C6 RASA T2 55mm QHY/Svb 2 1.7 1.2x sample
400mm 60mm f6.5 Pentax M42 44mm Svb 2
480mm 80mm f7.0 Orion T2 55mm QHY/Svb 1.1 1.4
539mm 70mm f6.5 Fixed Nikon 1000 1.4 0.7 12amin 2.0x sample 6x pix digitization
1500mm 150mm f10 C6 T2 55mm QHY/Svb 0.7 0.4 1.7x sample observable pixellation
7500mm 150mm 5x T2 Svb 0.7 0.07 2amin 10x sample
Measured resolution of Orion ED 80 -400 mm lens 0.8 arc secs edge resolution 0.14 pixel resolution, 1/2 Rayleigh resolution of 1.7 arc secs.
Sony effective pixel resolution is 1.5 x actual due to RGB color pixels.
Photographing the Milky Way requires largest possible field and f number, a 20mm focal length f2 full field lens allows a full sky mosaic with around 40 images 94 degrees per field. A 20 sec untracked exposure at 2000 ISO on a astro mod camera images stars down to magnitude 10, and many nebulae. A 8mm fisheye lens mirror compatible lens will allow filters with a 2.5 um pixel camera (QHY or ASI), will give 50-75 degree field of view and enchanced color contrast Milky Way.
Basic nebula = 30s ISO2000 with 5x to reduce noise. For low intensity nebula of whole MW add 3x 105s @ ISO6400, f/2. or 10x more exposure, needs v dark skies such as Grand Staircase Escalante NP and scanning stage for 20mm lens.
The nebula within the Milky Way are some of the most visually interesting features with sizes from several degrees around Orion, to 10's of minutes. The longest reasonably priced f2 refractive lens is 100 mm with an aperture of 60 mm and an abberation limited resolution of 15 arcsecs. Lenses are designed for a full frame sensor, most of which have insufficient pixel count to match the lens resolution. Reduced field sensors (1/2 frame) can resolve closer to the limit of the optics, so a nominal "100 mm" f2 lens with 1/2 frame sensor has a field equivalent to a 200mm with 7 arc secs per pixel resolution. A larger (400 mm) focal length f2 telescope is available with a Schmidt Cassegrain telescope in Rowe Ackerman configuration where the secondary mirror is replaced by correction optics and imager so the focal length is solely determined by the main mirror. Resolution with the correction optics is 2 arcsecs, paired with 1/2 frame sensor. Based on a 20 sec exposure, crude alignment to within 1 degree is fine.
Using HSO 8nm line filters the intensity is 1/100 compared to RGB filters that have a 150 nm bandwidth.
Milky Way 8mm Canon mount with Canon-T2 filter adaptor and QHY
Nebula in context 100mm Canon mount with with Canon-T2 filter adaptor and QHY
Nebula close up C6 RASA f2 lens combined with QHY cooled camera.
Gain 30 (6400ISO) and 200secs exposure gets to similar signals to Milky Way reference conditions. An additional 10x in exposure time, requires 30 arc min (60 pixel) alignment to the pole star for tracking which implies reasonably careful align.
C6 RASA f2 lens with Svbony camera has using 1.5 degree field 70x20secs @ 6400ISO or equivalent good for Whirlpool, and Leo Triplet. Stars with magnitude 10 just resolved. Most deep space objects are relatively close to pole star axis and make align to axis less critical. RGB and Ha composite image makes for a really good look. Andromeda needs larger field QHY camera with RGB filters or 2 stitched Svbony fields.
LENSES FOR PLANETS
The planets are much higher brightness and much smaller (20-40) arcsecs, so resolution is the key requirement.
C6 Prime with Svbony, ideally with 5x to get lucky image oversample.
For repeat exposure set autotimer = 2*exp time + 10 / delay 5
GEM stage implementation
For long exposures the telescope must scan in sync with the earth. The German EQ (GEM) stage has a Right Ascension axis that must be aligned with earths axis and a Declination axis at right angles.
The RA axis is aligned the earths axis using a camera that is mounted to the RA. The RA is rotated to find the center of rotation. The camera is also used to find the earths rotation axis which is close to the pole star. The Altitude and Azimuth adjustment is use to align the stage axis to earths axis.
The Polemaster camera is sold by QHY for the polar align process, with a 10 degree image field. A cell phone camera is a convenient way to place the polestar in the Polemaster field. A cell phone clamp is used to mount the cell phone to the stage.
Find polestar using cell phone camera app
Center polestar in App window 11x9 degrees wide
Follow instructions to find stage center by moving polemaster mount. Start at -60 then 0 then +60 to eliminate any square error in Polemaster to RA axis.
Check stage axis by lining up red icon and polestar - rotate polemaster arm - 10 polemaster pixel error = 2 arc min.
Check camera align by pointing at polestar and rotating camera theta.
Align stage center to polestar (Rough align) within 4 arcmins - 0.01 arc secs/sec star drift
Earths axis is 44 arc mins from polestar (https://astronomy.stackexchange.com/questions/38469/the-position-of-polestar) 1/3 of the way from polestar to lambda Ursa Minor.
Align stage center to earth axis (Fine align red to green boxes) up to 30 arc secs (single pixel) - 0.001 arc secs/sec drift
AVX use model
Mechanical : Telescope & cell finder mounted to AVX, plate to 400mm + SVBONY guide camera & Polaris finder mount.
Electrical: AVX Handset USB to lap top USB; Polemaster, QHY and SBONY to USB hub to lap top USB2. SBONY guide ST4 to AVX St4.
Power; Mount to Battery 12v, QHY cooler to Battery lighter.
Software : CWI controls AVX mount, NINA controls QHY, PHD2 controls SVBONY & AVX guide, Polemaster controls Polemaster camera.
Align guide and Polaris finder to Telescope using local reference. Polaris is an option.
In a 2 degree field; Polaris, Rotation axis, stage axis all will be visible. Set tracking off, gain down, exposure long and create star trail. Then slew RA to see stage axis.
Align to Stars
2) Star Above. Use phone to identify brightest star overhead. Use laser pointer to align telescope
3) Star near target - repeat.
Use Cell phone to find target, then telescope, keep adding references closer to the target so the key pas works and the align gets better.
Align Stars - bright and directly above
Arcturus in Bootes
Spica in Virgo
Regulus in Leo
Vega in Lys
Altair in Aquarius
Arcturus in Bootes
Altair in Aquarius
Capella in Auriga
Stephans quintet in Andromeda
Sirius in Canis Major
Capella in Auriga
Rigel in Orion
Procyon in Canis Minor
Pollox in Gemini
Betelgeuse in Orion
Aldebran in Taurus
Reboot PC, cycle mount, reconnect USB.
Restart using Last Align so mount does not move.
Check pole align
To measure pole align errors, skip star align , and use QHY with PHD2 in "drift mode".
Celestron PECTool for summing PEC runs.
Mount design RA worm gear with period 480 secs or 8 min cycle, so need at least 24 mins to get a drift trend.
Nominal PE stage performance +-15 asecs, improved to +-2 corrected.
Baseline test using 480mm lens 1.5 asecs/pix - 30 sec exposure with drift of 0.04 asecs/sec produces 1 pixel error. 5% of the worm cycle. This will be good enough with f2 RASA optics with 2 asecs resolution.
To preselect section of worm gear, UTILITIES/MENU/RECORD/ENTER/ENTER - indexes the worm gear. Target will be somewhere in 2 degree window.
Position directly up - measure Az error using drift feature, northerly drift = error west of pole. Use for high accuracy cyclic RA worm error.
Position to E or W - measure Alt/tilt error using drift feature, northerly drift = error north of pole.
Position on Polaris - pick edge star with 0.15 asecs/sec untracked error. and run for 24 mins to measure RA, PEC and scan rate error over 600 pix with 0.1 asecs/pix resolution with no DEC axis contribution.
Position directly up comparing 2 stars E and W of meridian. flipping RA and DEC sides. Collect images and find difference in RA for stars on equatorial - difference in RA is 2x orthogonality between RA and DEC. http://company7.com/library/techin/orthogonality.html
Analysis of tracking errors gives an idea of underlying causes. Unguided, the DEC axis is fixed so after removing linear drift from pole align and orthogonality, the residual RMS or Standard Deviation is a quantitative measure of the combined seeing and star analysis errors, and their impact on spot size. In most cases seeing dominates, so "seeing limited" star spot size FHWM = 2x RMS or standard deviation. In Austin in summer, the estimated seeing limited spot size = 0.9 asecs FWHM.
For my AVX stage with 400mm lens & Svbony camera , the contributions to unguided RA error over a 480 sec worm gear cycle after careful Polemaster align are;
Raw = 2.6 asecs RMS or +- 8 asecs 3 sigma - consistent with specification
Pole align error contribution = 9.6 asecs p-t-p
Worm gear contribution = 3 asecs p-t-p
Non random 10 sec exposure = 4.5 asec p-t-p 30-50 sec period
After non-random error removal = 0.72 asecs RMS
Seeing = 0.45 asecs RMS based on unguided DEC.
Residual random position error = 0.55 asecs RMS - probably best that can be done !
The resulting uncorrectable spot size from random positioning errors with perfect seeing = 1.1 asecs FWHM.
The most common solution is using a guide telescope to remove both pole align error and any systematic motion errors. Guiding programs such as PHD2 have measured sub pixel noise ( 0.3 pixel RMS) detection of star motion by following the shape of the image of a star. Best performance guiding needs a guided telescope and camera with similar pixel and optical resolution to the imaging telescope and camera.
For my AVX stage guided using 400mm lens with Svbony camera with resolution of 2 asecs FWHM, over 480 sec, with 1 sec exposure.
RA raw error = 0.83 asecs RMS random that produces a star spot size = 1.6 asecs FWHM.
Worm error = 0.3 asecs p-t-p
The guided raw error is slightly larger than the unguided random residual, suggesting that guiding may be over correcting on noise. Guiding corrects systematic errors, attempting to correct random errors will just increase total error. increasing exposure time will tend to average out random position errors.
This also shows guiding within 2x of seeing RMS. The seeing error can be reduced by increasing exposure time averaging out the seeing. A 4x increase in exposure should produce a 2x improvement in noise and guiding. Another strategy is to use multiple stars so that independent random variations in each star average out. Same rule should apply 4 stars should produce 2x noise reduction.
For the AVX stage, the practical limit to best resolution in long exposures is probably FWHM of around 1-2 arc secs. A 30 sec unguided exposure has a good chance of matching guided.
There are alternatives with improved RA accuracy, but their tracking performance is still limited by the alignment to the earths rotation axis. A dedicated RA scanning stage such as the Fornax Lightrack, uses a drive on the end of a long lever to achieve a RA tracking accuracy of +-1 arcsecs over 8 mins. It needs an additional RA & DEC axis stage to point the telescope, so there is no star find capability. A $5000 stage such a Fornax EQ mount has a claimed full cycle error of +- 6 arc secs. The addition of a high accuracy encoder (Telescope Drive Master) can achieve +- 1-2 arc secs.
Two modes; open loop (unguided) and guided.
Unguided; if using CWI make sure that Siderial rate is 100%.
Residual error tests; hand controlled 0.04 asecs/sec, Supports 30 sec exposure with Orion ED 80 with QHY - 1 asecs/pix.
Off axis guiding will not work for RASA f2 configuration.
On axis guiding - use the Orion ED, Pentax 400mm or 100mm lens as a piggy back guide scope on the Celestron.
Use a mono uncooled camera. LH uses Orion 60mm guider 240 mm focal length 32 oz.
400 mm pentax lens / M42 to T2 / 20 mm T2 /T2 to C/ SVBONY SV305 Pro Telescope Camera, 2MP USB3.0 Astronomy Camera, 1.25 inch Astronomy Guiding Camera 2 degrees 2 asecs/pix. - Mount to shared plate with Telescope. Vixen bar to base of shared plate.
100 mm lens - QHY as finder/guider 11 degrees, 7 asecs/pix.
Polemaster as a finder 11 degree field - 30 asecs res.
AXT test results March 22
Unguided 480 sec
Measurement DEC noise 0.45 asec RMS = 1/3 pixel
DEC was at neutral balance point showed 7 asec of backlash
Drift 0.02 asec/sec at limit for Polemaster
Residual RA RMS 2.6 asec
RA PEC after 120 sec rolling average = 3 asec p-t-p, similar to pole drift, error is low when PEC is in opposite direction to pole drift.
30 sec exposure RA 1 asec RMS - high res
120 sec exposure RA 2 asec RMS - f2 RASA PEC will help
Error after linear = 0.45 asecs - measures seeing and detection effects
Error raw = 2.6 asecs RMS
Error after linear = 1.6 asecs RMS = +- 5 asecs 3 sigma
Worm contribution = 3 asecs p-t-p
Error after worm = 0.77 asecs RMS
Subgroup 10 shows non random = 4.5 asec p-t-p 30-50 sec period
Error after sub 10 = 0.72 asecs RMS - no obvious non randomness left.
Contributions - Seeing 0.45 asecs RMS, random position 0.55 asecs RMS
Unguided 30 sec for high res (1 asec / pix), 120 sec for f2 RASA - add PEC
Guided 480 sec - 1 sec period.
RA = 0.83 asecs RMS mean zero - 1.6 asecs FWHM.
Subgroup 10 shows random variation (10 sec exposure)
Worm contribution = 0.3 asecs p-t-p
Good enough for RASA and Galaxies.
At 1 sec exposure - 10x reduction in systematic, 15% increase in random.
Need to increase the exposure time.
Place DEC out of neutral balance to avoid backlash steps.
Make sure that camera axes match the RA - DEC directions... better calibration
in the case of standard deviation, the mean is removed out from observations, but in root mean square the mean is not removed. however in the case of noise where the mean is zero, the two concept are the same.
For 400mm lens aperture 70 mm aperture Dawes resolution = 0.7 asecs = FWHM
Align results July 2022
C6 Hyperstar w. QHY 183 RT = 1.7 asecs/pix. Drift = 0.1 asecs/sec or 30 sec unguided limit. Background G30 (3200 ISO), 6 sec exp, B = 61 IU(8) FWHM = 4 IU(8). B/N = 15 & 1.4E-3 iso-1 sec-1. Smallest star FWHM = 3.4 asecs with poor (1.5 asecs dither) seeing and observable mis-collimation. 11Mag stars are saturated S/B = 4. Need to correct collimation. Basically performing to spec !
Fornax use model
Fornax with 3 axis Manfrotto stage. R axis on RA plane. Tilt 1 provides DEC axis. Tilt 2 fixed at 90.
Orthogonality error limits scanning to 0.1 asecs/sec. Accepting a 2 pixel scan error:
100 mm lens and Sony 18 asecs/pix so up to 360 sec
100 mm lens and QHY 7 asecs/pix so up to 140 sec, sets limit for line filter imaging.
480 mm lens and Sony 4 asecs/pix so up to 80 sec, works fine with 30 sec max in the Sony.
Milky way 3D.
If needed remove stars from nebula - copy nebula, B&W, contrast to isolate stars in nebula, expand 2 pix, delete any stars that are part of nebula, make mask, shift nebula, copy from shift, paste into original nebula, blur to smooth.
Make nebula transparent.
Make foreground stars - original image, contrast adjust to isolate a few bright stars, delete any associated with nebula.
Deep Space look
To create realistic deep sky images without blocking stars:
1) Copy master image - set 50% opaque - and move so that that stars pairs are distinct - set transparency 100% opaque.
2) On original - select stars using Astropanel/Astro/Hot pixel/Harder filter.
3) On filter - Create mask using magic wand - adjust tolerance on background - Contract 1 pixel.
4) Copy filter on moved master to select local backgrounds to each star
5) Reorder layers so copied filter is added to original - Merge layers
6) Select galaxies-feather - invert-Blur to eliminate residual variation
7) Apply levels by moving zero to obtain almost black background - do not overdo !
TARGETS - Whirlpool , Leo triplet
Exposure 1/125 ISO 3200 PP7. Continuous pics with hand controller - 40 or more frames. Lucky stack at 10%.
Remove background - point / Tolerance 60 Feather 10
HOS filter image
Filter changer -
30s crude align
120s needs fine align.
Bias - readout noise - lens cap and short exposure
Dark - system noise - lens cap and same exposure time as lights (images)
Flats - vignette - white tee shirt over lap top screen.
Dark flats -noise on flat - lens cap with same exposure time as Flats
Collect at least 10 long exposures, with dark flats. 1/4 and 1/10 times for HDR.
Stack images in Deep Sky Stacker - group the different filters and exposure times, stack repeated images using a common reference selected from best H image. Take non repeating images, use same reference frame, check images in turn and save. Use separate groups for different image sets that DO NOT include reference.
In PS assemble HSL layers, change mode to RGB, align, equalize core color area, paste into RGB layers. reset black level, separately merge HDR color images, then merge color and L shorter exposure times.
f2 30secs ISO2000 RGB is benchmark for good nebula =
Filter increases exposure 500x, reduces bkg 6x.
G30 is ISO 6000 so f2 240s G30 HOS should resolve M8.
Example Rosette Nebula HOO image f2 120secs x 4 frames per channel. In Wimberly S/N for Nebula 2.5:1, a 2x improvement with line filters for a M5.5 nebula, background 4 AU. Eagle nebula (pillars of creation M6.0 should increase exposure time to 200 secs. The nebula had a intensity of 10 AU, so darker Bortle or longer exposure for dimmer objects.
Example Orion Nebula using 400mm f7 (1/10x) 120s G30 HOS filter = partial Orion nebula bkg 7, nebula edge 50. effectively 1/10 of full nebula. 4 deg field.
Example Elephants Trunk Nebula M5.6
Telescope: TEC-140 (F7)
Camera: SBIG ST-8300M
Mount: AP900 GTO
HA : 18x30 minutes (binned 1x1)
SII :18x30 minutes (binned 1x1)
OIII : 18x30 minutes (binned 1x1)
Useful for smaller galaxies (<1 degree) and excellent seeing using 400mm with QHY for resolution 1.4 asecs/pix. limited to 10 sec exposure time, 30 sec for moderate seeing.
Align and stack multiple exposures in Deep Sky Stacker, set brightness at mid point. In PS, copy / paste RGB channels, adjust dark and mid point. Copy in L as new layer, change layer combine from Normal to Luminosity.
Sony7a astro mod without filter
Use curves - reduce white contrast - reduce red contrast - increase green slightly - reduce saturation.
Use scanning stage to track and create a time lapse video. How to deal with transition in and out of totality.
Use Hyperlapse on phone to cover whole transition.
Take multiple images- blend using mean to HDR
Sharpen by taking radial blur and subtracting original - do it twice.
3.1. Pseudocode for MGN 1. Replace spurious negative pixels with zero or local mean/median.
2. Create Gaussian kernel of width wi . Kernel elements should sum to unity.
3. Convolve image with kernel to create local mean image B ⊗ kw.
4. Calculate difference between image and the local mean image, square the difference, and convolve with kernel. Square root the resulting image to give ‘local standard deviation’ image σw (Equation (2)).
5. Calculate normalized image Ci by subtracting the local mean image and dividing by the local standard deviation image (Equation (1)). Store the result.
6. Apply arctan transformation on Ci to give C 0 i .
7. Repeat 2-6 with the different kernel widths wi .
8. Take mean, or weighted mean if preferred, of the C 0 i to give a weighted mean locally-normalised image.
9. Calculate a global gamma-transformed image C 0 g by applying Equation (4).
10. Sum the weighted mean locally-normalised image with the global normalized image C 0 g , with appropriate weight h (as Equation (5)).