From multiple images to catalogs

Transcription

1 Lecture 14 From multiple images to catalogs Image reconstruction Optimal co-addition Sampling-reconstruction-resampling Resolving faint galaxies Automated object detection Photometric catalogs

2 Deep CCD imaging Raw image C, additive systematics image B. De-biased flat-field image F. Form the data image D: Image calibration: D(x, y) = [C(x, y) B(x, y)] / F(x, y) Var(D) = [ σ 2 0 +<D > / Gain] F(x,y) could be super-flat. Co-addition of many images: Register flat-fielded shift-and-stare images Map each image onto a reference image. Warp if necessary, so all stars are reqistered to fractional pixels. Simple nearest-neighbor schemes create correlated noise. Sinc filter avoids this. Co-add by median averaging (but there are even better ways!)

3 Imaging and photometry Raw image C, additive systematics image B. De-biased flat-field image F. Form the data image D: Co-add by median filter: Noise model: D( x, y) Var( D) = mediand( x, = 2 D /Gain σ + < 0 > y) over N images / N Digital data: DN = digitized (Gain * Intensity) Find objects: Sort pixels by DN Set thresholds Decrease toward sky level. x

4 Imaging and photometry CCDs and CMOS imagers are pixelated i.e. they sample the image

5 Two methods of co-adding Simple median using fractional pixel interpolation weights Drizzle resampling onto higher resolution grid with sinc interpolation Rather than shifting and adding the original pixels with interpolation, drizzle shrinks each pixel into a smaller footprint (called a "drop") and then places them onto the finely sampled reconstruction grid (hence the name "drizzle").

6 resampling If you know the signal shape, you can determine the value at every location. In practice, we sample the underlying function. Pixels! First let's consider functions which are sampled. For many imaging applications, there is a physical process (e.g., seeing) which limits the amount of high spatial frequency information. If there is an upper frequency cutoff in the power spectrum of the objects being observed, it is possible to recover the full function from samples of it if the samples are sufficiently fine. The Nyquist sampling theorem says that if you have a band-limited function, you can recover the function if you sample at spatial frequencies over twice the frequency at which the power spectrum goes to zero. For example, if you say seeing wipes out scales less than 0.2 arcsec, you need to sample at 0.1 arcsec to recover the full function. If you can recover the function, then you can sample it at different locations. You can do this using the sampling theorem by something known as sinc interpolation. This works by recovering the original function; the process is done by filtering by a box function in transform space, which is equivalent to convolution with a sinc function in real space. However, even at ``critical'' sampling, the binned function is not equivalent to the sampled function. So in fact, sinc interpolation may not be accurate unless you are better than critically sampled. If you are undersampled, it fails miserably (aliasing). Near critical sampling, it can lead to non- flux conserving interpolation.

7 Ideal Reconstruction A band-limited function can be exactly reconstructed by convolution with the appropriate kernel.

8 Shannon says By hypothesis f has a Fourier transform, F bandlimited to 1/2T. Pixelate: Multiply by Comb function) with separation of T -1/(2T) 1/(2T) means convolve the Fourier transform with the Fourier Transform of Comb, but this is just another Comb function with separation of 1/T which just relplicates the Fourier Transform. 5T but that s equivalent to convolving in space with Fourier transform of boxcar, sinc(). So we just need to use sinc interpolation to get back original image. So mutiplying by a boxcar gets back original F

9 Possible kernels Exact area sampling Drizzle redistribution Boxcar smoothing with resampling Step pyramid kernel with different fraction of flux in each box Adjusting the size of the rectangle changes how the algorithm considers the flux in the pixel to be distributed.

10 A big driver in commercial applications is rendering of text.

11 Sampling Features Features in Lanczos resampling. NN LI Lcz2 Lcz3 Lcz4 Moire patterns in linearly interpolated resampled images

12 Automated photometry Search each line in an image for pixels above some threshold. Bayesian prior: convolve with PSF or expected object kernel Use a moving average updated sky as threshold Define objects as connected above-threshold convolved pixels in x,y Deblend overlapping objects For each resulting object, measure the integrated flux with a kernel, and measure various intensity moments Generate a list (catalog) of detected objects, and their photometry Example software: FOCAS, Sextractor, DoPhot

13 Define apertures Aperture photometry round / square / partial pixels allowed Sum counts in aperture: F ˆ = i D i Var( ˆ F ) = PROBLEM: sky Var(D i ) i Noise increases with aperture size So use small aperture to maximise S/N Calibrate bias using bright stars.

14 Better: PSF fitting PSF = point spread function = image of star. Use analytic model, e.g.: Gaussian: DoPHOT: P(x, y) = exp 1 x σ + y2 2 x σ y P(x, y) = 1 z β z β z 6 6 z = 1 2 x 2 σ + 2xy 2 x + y2 2 σ xy σ y β 4 = β 6 = 1 gives truncated series of Gaussian.

15 Optimal extraction of stellar flux If P(x,y) is known, scale this to fit the skysubtracted star image: Iterate ˆ F = i P i (D i sky) / σ i 2 2 P i2 / σ i i F σ 2 i = σ ˆ P i + sky Gain Var( ˆ F ) = i 1 P i2 /σ i 2

16 PSF fitting We want to reconstruct F but must also find MODEL PARAMETERS: Centroid:x 0, y 0 Shape parameters:σ x, σ y, σ xy, β 4, β 6, etc. How to find centroid: Rough: Refine: x 0 ~ 1 N i (D i sky)x i 0 = (D i sky)(x i x 0 )P(x i x 0 ) i x 0 x (x i -x 0 )P(x i -x 0 ) x

17 Shape parameters: second moments of intensity Same area under fitted curves: σ x2 ~ σ y 2 ~ σ xy ~ Orthogonality: (D xy sky) (x x 0 ) 2 x,y (D xy sky) (y y 0 ) 2 x,y Centroid and shape parameters are roughly orthogonal to area, so small errors won t affect photometry true only for totally isolated stars! (D xy sky) (x x 0 )(y y 0 ) x,y x 0 x x 0 x x 0 x

18 Better: use optimal filter 2 σ x 2 σ y ~ ~ x, y x, y ( D ( D xy xy sky ) G sky ) G xy xy ( x ( y x y 0 0 ) ) 2 2 σ xy Apodization : ~ ( D x, y sky ) G ( x Given a prior for the signal G, the second moments have highest S/N and are more immune to outlying flux from other objects or bright pixels xy xy x 0 )( y y 0 ) x 0 x If stellar photometry, G is PSF. If galaxy, G is estimate of galaxy profile.

19 Automated photometry of crowded fields e.g. DoPHOT, FOCAS, Sextractor Set threshold, find objects. Many objects, often overlapping Make object list. Store parameters F, x 0, y 0, σ x, σ y,... Classify objects based on size/shape parameters: star double star galaxy: bigger than PSF, not round blemish, cosmic: <PSF, saturated σ-clip thresholds set to divide categories Flag cosmic rays, saturated pixels; ignore flagged pixels σ 2 faint galaxies stars cosmics mag = -2.5 log F bright

20 Automated photometry -- continued Find mean PSF shape parameters by weighted averages over all objects classified as stars May be functions of position and/or brightness Scale PSF to measure star fluxes. Subtract all nearby objects before scaling. Perform aperture photometry if desired ITERATE!

21 Problems with crowded-field photometry Light pollution residual PSF structure near bright objects Crowding errors/incompleteness/misclassification use Monte carlo tests inject fake stars or other objects at random positions see what fraction are recovered correctly with what errors in flux Sky levels PSF wings and large numbers of faint stars merge to create bumpy pseudo-sky Solution: use local sky level, e.g. median of pixels in ring around object Fit polynomial or spline to local sky levels Boost error bars accordingly

22 Galaxy images Design and fit appropriate models e.g. Ellipse fitting (structure of elliptical galaxies) Aim: find ellipses that fit image contours at specified brightnesses Sample data around a trial ellipse Fit truncated sin/cos series: f (θ) = A + Bsinθ + C cosθ θ 0 2π + Dsin 2θ + E sin 2θ A: If mean contour level too high/low, expand/shrink ellipse B, C: If ellipse not centred, adjust x 0, y 0 D, E: If shape/orientation not correct, adjust ellipticity/position angle. Plot ellipse parameters vs size or contour level to quantify radial brightness profile, twists, shifts, etc

23 Galaxy photometry completeness Histogram of pixel values Consider N images with sky mean μ and rms noise σ Significance R = y 1/2

24 Galaxy photometry completeness Consider N images with sky mean μ and rms noise σ Χ 2 image Significance R = y 1/2

25 Subaru telescope Point spread function: 0.7 arcsec full width at half maximum

26 resolving galaxies A given galaxy at high redshift (more distant) should appear smaller. But two effects oppose this: cosmological angle-redshift relation, and greater star formation in the past (higher surface brightness). Here are lots of galaxy surface brightness vs radius (arcsec) in redshift bins from z = for apparent mag. At a surface brightness of 28 i mag/sq.arcsec (horizontal dashed line) most galaxies at z<3 are resolved in 0.6 arcsec FWHM seeing (vertical dashed line).

27 Comparing HST with Subaru ACS: 34 min (1 orbit) PSF: 0.1 arcsec (FWHM) 2 arcmin

28 Comparing HST with Subaru Suprime-Cam: 20 min PSF: 0.52 arcsec (FWHM)

29 1981AJ J

30 F4p22 public, N(mx), x=(tot, iso, ap), for B,V,R,z.

31 F4p22 public, N(mx), x=(tot, iso, ap), for B,V,R,z. Highlighted: FLAGS=0

32 Galaxy photometry completeness Number per square arcminute vs log flux DEEP LENS SURVEY Completeness suffers for galaxies at or below the sky noise on an individual image. If the goal is to reconstruct the galaxy counts N(mag) then Monte Carlo simulations can be used to extract that statistically. But if the goal is the photometry of individual galaxies (such as shape) then we can do better: use all the information we have.

33 3-band reconstruction Each pixel in reconstructed image based on probability of same object of resolution equal to the point spread function is present in all 3 bands. reconstruction Band 1 Band 2 Band 3

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