robert lupton princeton university

SDSS Workshop at UNAM: photo

Robert Lupton Princeton University

Mexico City, 27-28th of February, 2006

Outline

Image Processing

•Vital Statistics

•A quick tour

•Removal of CCD artifacts

•PSF modelling

•CR rejection

•Sky subtraction

•Object detection

•Merging bands

•Atlas images

•Deblending

•Flags

Outputs

•PSF mags and aperture corrections

•Petrosian quantities

•Adaptive moments

•Model fitting

•Star/Galaxy separation

Vital Statistics

The SDSS camera generates data at about 4.6Mby/s, cov-ering about 98.9 deg2/hr (but only 19.8 deg2/hr per band)with an exposure time per band of about 55s.

The camera’s footprint consists of 6 long strips (separatedby about 11arcmin), each 13.5 arcmin wide and up to 120◦

long.

The sky is scanned past the filters in the order r, i, u, z,taking 4.9 minutes to pass from the center of the r to thecenter of the g detector.

These strips are cut into a set of frames of size 2048× 1361pixels (13.5× 9 arcmin2)

The frames taken of the same patch of sky in the five bandsare assembled to form a frame.

We return on a later night to take a second set of stripswith the telescope offset so that the second set of datafills the gaps between the first; the two strips together arereferred to as a stripe

SDSS raw data consists of 16-bit unsigned numbers, andphoto doesn’t convert to float. A consequence of this isthat SDSS output files are 16-bit, and have a 1000DN ‘softbias’ added.

A quick tour

Removal of CCD artifacts.


•Bad columns


•Bad columns

•Saturation trails


•Bad columns


•Exposed serial register


•Bad columns


•Exposed serial register

•Scattered light

PSF modelling

Is SDSS Data Band Limited?

The amplitude of the PSF at the Nyquist limit for a PSFnormalised to unit flux. The bottom axis is in pixels; thetop axis is in arcsec, assuming 0.4 arcsec/pixel.

The KL basis images for frame 756-z6-700, using a histogram-equalised stretch.

The estimated PSF for 35 positionsin frame 756-z6-700, using a linear stretch.

CR rejection

Cosmic Ray Pixels are identified because

CR rejection


They lead to gradients that exceed the band limit;

I − c ∗N(I) > P (d)(I + cN(I))

)In practice we multiply P (d) by some fiddle factor, c2 < 1;we use c = 3.0, c2 = 0.8.

CR rejection


They lead to gradients that exceed the band limit;

I − c ∗N(I) > P (d)(I + cN(I))

)In practice we multiply P (d) by some fiddle factor, c2 < 1;we use c = 3.0, c2 = 0.8.

We find about 160 cosmic rays per field in all bands exceptg, where we find about 330 and z where there are about 135;in total we thus find about as many cosmic rays as objectsin the 5 frames that make up a field.

Sky subtraction

Sky is estimated in 256× 256 pixel regions (∼ 100′′ × 100′′),with centers every 128 pixels. photo then linearily interpo-lated to every pixel

Sky subtraction


This means that photo doesn’t use local sky estimates forobjects.

Sky subtraction



•Sky estimate is essentially noiseless

Sky subtraction




•Good performance for extended objects

Sky subtraction





•Potential problems with rapidly varying backgrounds

Sky subtraction






The sky levels are wrong by 0.05 – 0.1 DN, which is im-portant for e.g. u-band Petrosian (aperture) fluxes.

Sky subtraction






The 100” scale is too small for even 16th (?) magnitudegalaxies

Sky subtraction






We should be removing models of objects before estimatingthe sky.

Object detection

Object detection

•Smooth with the PSF

Object detection


•Detect groups of connected pixels above a threshold. Suchgroups of pixels are known as ‘objects’.

Object detection



•Grow each object approximately isotropically by an amountequal to approximately the FWHM of the PSF.

Object detection




We then remove detected objects, bin the data 2 × 2 andrepeat (then do the same binning 4× 4).

Object detection




We then remove detected objects, bin the data 2 × 2 andrepeat (then do the same binning 4× 4).

This operation is carried out independently in the 5 SDSSbands, and the objects detected in each band are also mergedtogether.

Merging bands

An astrophysical object is defined by:

Merging bands


•The Union of the pixel-masks in the 5 bands

Merging bands



•The Union of the local maxima (peaks) detected in the 5bands

Merging bands



•The Union of the local maxima (peaks) detected in the 5bands

Note that this means that an object’s properties are mea-sured in all 5 bands, even if it’s undetected in some

Atlas images.

Deblending

Deblending

NGC3613

Deblending

NGC2857

Deblending

NGC5774 and NGC5775

Deblending

NGC4643

Deblending

NGC442

The problem of deblending stars is well defined; the imageis made up of a set of δ-functions convolved with a knownPSF, φ:


I = S +∑r

Frδ(x− xr)⊗ φ + n

(I : observed intensity; S : sky level; δ: delta-function;Fr: flux in rth star; φ : PSF; n : noise)


I = S +∑r



All that we have to do is solve a minimisation problem in3r + 1 unknowns.


I = S +∑r



All that we have to do is solve a minimisation problem in3r + 1 unknowns.

Writing efficient, robust, accurate code may not be trivial.

Galaxies are harder.

Galaxies are harder.

Something that looks like the superposition of three galax-ies may well be just that, but without extra information (e.g.redshifts) we cannot be sure that it isn’t simply a messyblobby irregular galaxy that happens to have three peaks —or even a large elliptical galaxy that’s being viewed througha particularily perverse dust cloud.

A 1-D Toy Problem

A 1-D Toy Problem

Let us consider a simple 1-dimensional problem, a ‘star’and two ‘galaxies’.

The procedure is simple:


•Find all the peaks in the image I. Each is associated witha ‘child’ object.


•Find all the peaks in the image I. Each is associated witha ‘child’ object.

•Define a ‘template’ Tr from each peak. This is the imageformed by comparing pairs of pixels symmetrically placedabout the peak of the rth object, and replacing both bythe lower of the two.

•Assume that we can write I =∑

r w(r)T (r), and solve forthe weights in a least-squares sense.

•Assume that we can write I =∑

r w(r)T (r), and solve forthe weights in a least-squares sense.

•For each pixel with intensity Ii, share the flux between thechildren:

C(r)i =

w(r)T (r)∑w(r)T (r)

Ii

Does this work in practice?


In the mosaic the bot-tom left hand panel isthe object being de-blended, and the oth-ers are individual chil-dren (one per tem-plate)

Outputs

PSF mags and aperture corrections

The Maximum Likelihood Estimator (MLE) for an object’sflux is:

fMLE ≡∑

i PiOi/σ2i∑

i P 2i /σ2

i

Outputs

PSF mags and aperture corrections

The Maximum Likelihood Estimator (MLE) for an object’sflux is:

fMLE ≡∑

i PiOi/σ2i∑

i P 2i /σ2

i

photo actually uses

fMLE′ ≡∑

i PiOi∑i P 2

i

For stars, P is of course the point spread function, so theseMLE fluxes are often referred to as PSF fluxes. For a Gaus-sian PSF G(α) the PSF flux has variance

4πα2n2

if the noise due to the object itself can be neglected; in otherwords it has the same noise as an aperture of radius 2α oronly about 0.8 arcsec in 1 arcsecond seeing.

Seeing corrections to PSF magnitudes

We correct for varying seeing by employing our PSF model.



•Reconstruct the PSF at our object and measure its PSFflux fMLE′,KL




•Measure some large aperture flux faper,KL for the recon-structed PSF.




•Measure some large aperture flux faper,KL for the recon-structed PSF.

•Estimate our real object’s ‘PSF’ magnitude as

fPSF ≡ fMLE′ ×faper,KL

fMLE′,KL.

This composite flux has the noise properties of an MLEestimate, combined with an aperture flux’s insensitivity toPSF variation. In practice, we have used ‘aperture 5’ with aradius of 7.5 pixels, 3 arcsec.

This composite flux has the noise properties of an MLEestimate, combined with an aperture flux’s insensitivity toPSF variation. In practice, we have used ‘aperture 5’ with aradius of 7.5 pixels, 3 arcsec.

Finally, we use the composite stellar profile to further cor-rect the magnitude of the previous paragraph to an evenlarger radius, ‘aperture 7’, with a radius of 18.6 pixels, 7.4arcsec.

Petrosian quantities

The Petrosian ratio is given by

R(r) ≡∫ 1.25r0.8r P (r)2πr dr

(1.252 − 0.82)∫ r0 P (r)2πr dr

;



R(r) ≡∫ 1.25r0.8r P (r)2πr dr

(1.252 − 0.82)∫ r0 P (r)2πr dr

;

The Petrosian radius RP is defined by R(RP ) = f1 (1 ≥ f1 ≥0). Clearly R(0) = 1, and for most forms of an object’s radialprofile R(∞) = 0, so almost all objects will have at least oneRP .



R(r) ≡∫ 1.25r0.8r P (r)2πr dr

(1.252 − 0.82)∫ r0 P (r)2πr dr

;


The Petrosian flux FP is defined as the flux within f4×RP ;in all bands the RP used is that measured in the r′ band.



R(r) ≡∫ 1.25r0.8r P (r)2πr dr

(1.252 − 0.82)∫ r0 P (r)2πr dr

;


The Petrosian flux FP is defined as the flux within f4×RP ;in all bands the RP used is that measured in the r′ band.

We also calculate two concentration parameters R50 andR90, the radii containing 50% and 90% of FP .

Adaptive moments

The ‘adaptive moments’ of an object are the 2nd momentsof the best-fit elliptical Gaussian.

Adaptive moments

The ‘adaptive moments’ of an object are the 2nd momentsof the best-fit elliptical Gaussian.

Adaptive moments are also calculated for the best-estimateof the PSF at the position of each object.

Model fitting

photo fits three models:

Model fitting


•A PSF

Model fitting


•A PSF

•A pure exponential disk (truncated beyond 3 re in such away that the profile goes to zero with zero derivative at 4re) convolved with the PSF,

Model fitting


•A PSF


•A deVaucouleurs profile (truncated beyond 7 re in such away that the profile goes to zero with zero derivative at 8re) convolved with the PSF.

Model fitting


•A PSF


•A deVaucouleurs profile (truncated beyond 7 re in such away that the profile goes to zero with zero derivative at 8re) convolved with the PSF.

Each of these galaxy models is specified by four parameters:the central intensity I0, the effective radius re, the axis ratioa/b, and the position angle of the major axis, φ.

We take a number of steps to improve performance:


•We fit the models to the extracted cell profile



•We model the PSF as a sum of Gaussians and a residualtable R:

PSF = αG(σ) + β (G(τ) + bG(cτ)) + R

where b and c are fixed (we adopt 0.1 and 3 respectively).



•We model the PSF as a sum of Gaussians and a residualtable R:

PSF = αG(σ) + β (G(τ) + bG(cτ)) + R

where b and c are fixed (we adopt 0.1 and 3 respectively).

•We compute galaxy models of each type for a range of(re, a/b, φ), convolve each with a set of PSFs of the formsG(σ) and G(τ) + bG(cτ) for a set of values of σ and τ ,extract their profiles, and save the results to disk.

Rather then do the full PSF-convolution, we write:

model = model0 ⊗ PSFKL ≈ model0 ⊗ PSFtable + R

Model Magnitudes

Once (I0, re, a/b) are known for a model of a given class(PSF, exponential, or deVaucouleurs) we can easily calculatethe total flux; we refer to this as a ‘model’ flux.

Model Magnitudes

Once (I0, re, a/b) are known for a model of a given class(PSF, exponential, or deVaucouleurs) we can easily calculatethe total flux; we refer to this as a ‘model’ flux.

If the object is a star we’d like the ‘model’ magnitude toequal that measured any other way. Our KL expansion al-lows us to reconstruct the PSF at the position of our object,and to determine the ratio of its PSF and model fluxes; mul-tiplying our model flux by this ratio then achieves our goal.

The SDSS has photometry in 5 bands, and two ways of cal-culating model magnitudes present themselves: either usingthe best-fit model in each band, or using the model deter-mined in some fixed band to calculate the magnitude in allbands. Both approaches have their virtues.


Fitting a model in each band gives the best estimate of thetotal flux in that band, but does not in general give the bestestimate of the object’s colour as the flux contains errors dueto both the photon noise in the image and to uncertaintiesin the model parameters.


Fitting a model in each band gives the best estimate of thetotal flux in that band, but does not in general give the bestestimate of the object’s colour as the flux contains errors dueto both the photon noise in the image and to uncertaintiesin the model parameters.

Using the model from a fixed band generates better colours,but if the structure of the object is substantially different indifferent bands the magnitudes may be incorrect.

Composite Model Magnitudes

Once we have models deV(I0, re, a/b) and exp(I0, re, a/b) fitto the data in any band, we can ask for the linear combina-tion of these two models which provides the best fit:

cmodel(I0, re, a/b) ≡ gdeV deV(I0, re, a/b) + gexpexp(I0, re, a/b).

Let us call the ratio gdeV /gexp ‘fdeV ’; it is restricted to lie inthe range 0—1. One application of fdeV is to estimate thetotal magnitude of the galaxy in a way that is less sensitive tothe departures of the true profile from either of our idealisedforms:

fluxcmodel ≡ fdeV fluxdeV + (1− fdeV )fluxexp

Star/Galaxy separation

SDSS uses the difference between the PSF and model mag-nitudes as a star/galaxy classifier.

Star/Galaxy separation

SDSS uses the difference between the PSF and model mag-nitudes as a star/galaxy classifier.

Some single-band objects are classified as cosmic rays.

Flags

Name Bit DescriptionCANONICAL CENTER 1 Measure Objects

used canonicalcentre, not theone determinedin this band

BRIGHT 2 Object’s proper-ties were mea-sured in BRIGHTpass

EDGE 3 Object is tooclose to edgeof frame to bemeasured

Name Bit DescriptionBLENDED 4 Object is a blendCHILD 5 Object is a childPEAKCENTER 6 The quoted cen-

ter is the positionof the peak pixel

NODEBLEND 7 No deblendingwas attempted

NOPROFILE 8 Object was toosmall to estimatea profile

NOPETRO 9 No Petrosianradius could bemeasured

Name Bit DescriptionMANYPETRO 10 Object has more

than one Pet-rosian radius

NOPETRO BIG 11 No Petrosian ra-dius could be es-timated as objectis too big

DEBLEND TOO MANY PEAKS 12 Object has toomany peaks todeblend

CR 13 Object’s foot-print containsat least one CRpixel

Name Bit DescriptionMANYR50 13 Object has more

than one Pet-rosian 50% ra-dius

MANYR90 15 Object has morethan one Pet-rosian 90% ra-dius

BAD RADIAL 16 Radial profile in-cludes some lowS/N points

Name Bit DescriptionINCOMPLETE PROFILE 17 Object is within

the Petrosian ra-dius of the edgeof the frame

INTERP 18 Object containsinterpolated pix-els

SATUR 19 Object containssaturated pixels

NOTCHECKED 20 Object containsNOTCHECKEDpixels

SUBTRACTED 21 Object had wingssubtracted

Name Bit DescriptionNOSTOKES 22 Object has

no measured‘stokes’ shapeparameters

BADSKY 23 The sky level atthe position ofthe object thatthe object’s peakpixel is negative

Name Bit DescriptionPETROFAINT 24 The surface

brightness at theposition of atleast one candi-date Petrosianradius was toolow

TOO LARGE 25 Object is toolarge to beprocessed

DEBLENDED AS PSF 26 Object was de-blended as a PSF

Name Bit DescriptionDEBLEND PRUNED 27 Deblender

pruned the listof candidatechildren

ELLIPFAINT 28 The center’sfainter than thedesired ellipticalisophote

BINNED1 29 Object wasdetected in thesmoothed 1 × 1binned image

Name Bit DescriptionBINNED2 30 Object was

detected in thesmoothed 2 × 2binned image

BINNED4 31 Object wasdetected in thesmoothed 4 × 4binned image

MOVED 32 Object mayhave moved(but probablydidn’t; see DE-BLENDED AS MOVING)

Name Bit DescriptionDEBLENDED AS MOVING 33 Object was

deblended as amoving object

NODEBLEND MOVING 34 Object is a re-jected candidateto be deblendedas moving

TOO FEW DETECTIONS 35 Object has toofew detections todeblend as mov-ing

Name Bit DescriptionBAD MOVING FIT 36 Object’s cen-

troids as afunction of timewere inconsistentwith moving at aconstant velocity

STATIONARY 37 The object’smeasured veloc-ity is consistentwith zero

PEAKS TOO CLOSE 38 At least somepeaks were tooclose, and thusmerged

Name Bit DescriptionBINNED CENTER 39 Object was

found to bemore extendedthan a PSFwhile centroid-ing, and theimage was thusbinned to use amore appropriatesmoothing scale

Name Bit DescriptionLOCAL EDGE 40 Object’s center

in at least onband was toonear the edge ofthe frame

BAD COUNTS ERROR 41 The PSF or fibermagnitude’s er-ror is bad or un-known

Name Bit DescriptionBAD MOVING FIT CHILD 42 A potential

child’s fit as amoving objectwas poor, andthe child wasthus taken to bestationary

Name Bit DescriptionDEBLEND UNASSIGNED FLUX 43 A significant

part of thetotal flux wasdivided amongthe childrenusing the specialalgorithm forhandling other-wise unassignedflux

SATUR CENTER 44 Object’s center’sis very near(or includes)saturated pixels

Name Bit DescriptionINTERP CENTER 45 Object’s center’s

is very near(or includes)interpolatedpixels

DEBLENDED AT EDGE 46 Object’s tooclose to theedge to applythe deblend-ing algorithmcleanly, butwas deblendedanyway

Name Bit DescriptionDEBLEND NOPEAK 47 Object had no

detected peak inthis band

PSF FLUX INTERP 48 A significantamount of ob-ject’s PSF flux isinterpolated

TOO FEW GOOD DETECTIONS 49 Object has toofew good detec-tions to be de-blended as mov-ing

Name Bit DescriptionCENTER OFF AIMAGE 50 Object contained

at least one peakwhose center layoff the atlas im-age in some band

DEBLEND DEGENERATE 51 At least one po-tential child hasbeen pruned asbeing too simi-lar to some othertemplate

BRIGHTEST GALAXY CHILD 52 Object is thebrightest childgalaxy in a blend

Name Bit DescriptionCANONICAL BAND 53 This band was

primary (usuallyr)

AMOMENT UNWEIGHTED 54 Object’s ‘adap-tive’ momentsare actuallyunweighted

AMOMENT SHIFT 55 Object’s centermoved too farwhile determin-ing adaptive mo-ments

Name Bit DescriptionAMOMENT MAXITER 56 Too many itera-

tions while deter-mining adaptivemoments

MAYBE CR 57 Object may be acosmic ray

MAYBE EGHOST 58 Object may bean electronicsghost

NOTCHECKED CENTER 59 Object’scenter isNOTCHECKED

Name Bit DescriptionHAS SATUR DN 60 Object’s counts

include DN inbleed trails

DEBLEND PEEPHOLE 61 Deblend wasmodified byoptimiser

SPARE3 62 UnusedSPARE2 63 UnusedSPARE1 64 Unused

The End

robert lupton princeton university

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