psf interpolation - cosmostatwl.cosmostat.org/gentile_psf_interpolation_nice_2013.pdf · 7...

Marc Gentile, Frederic Courbin(June 2013)

PSF Interpolation

GREAT10 as a “test bench”

1

Laboratoire d'AstrophysiqueEcole Polytechnique Fédérale de Lausanne

Switzerlandhttp://lastro.epfl.ch

Wednesday, June 5, 13

• The PSF interpolation problem

• Interpolating GREAT10 Star Challenge PSFs

• Main results

2

Laboratoire d'AstrophysiqueEcole Polytechnique Fédérale de Lausanne

Switzerland


The PSF: a major source of systematics

3

• The spread function (PSF) is a major source of systematic errors in weak lensing studies

• First-order systematics1. Convolution with the PSF alters apparent galaxy size & ellipticity2. PSF varies across the field of view: aberrations, stability, ... PSF correction must be applied at galaxy positions ... but PSF only ∼known at star positions and varies over time/space

• Accurate shear measurement thus demands to1. Interpolate the PSF at the positions of galaxies2. Correct galaxy shapes from PSF effects at those positions


The PSF: a major source of systematics

3

• The spread function (PSF) is a major source of systematic errors in weak lensing studies

• First-order systematics1. Convolution with the PSF alters apparent galaxy size & ellipticity2. PSF varies across the field of view: aberrations, stability, ... PSF correction must be applied at galaxy positions ... but PSF only ∼known at star positions and varies over time/space

• Accurate shear measurement thus demands to1. Interpolate the PSF at the positions of galaxies2. Correct galaxy shapes from PSF effects at those positions1. Interpolate the PSF at the positions of galaxies


4

The PSF: a major source of systematics• The PSF typically varies across the field of view

- Telescope optics, imperfect stability of the telescope/camera- Atmospheric effects (from ground)- Other mechanical and thermal effects

SDSS STRIPE 82 DLS (Jee et al. 2012)


The PSF interpolation problem

• Cannot be ignored even with a perfect PSF correction- This prompted the launch of the GREAT10 Star Challenge

• Two main difficulties1. Find an appropriate PSF model

2. Find the appropriate spatial interpolation algorithm

5

... and reconstruct the PSF at specific locations


PSF modeling: main approaches

• Interpolation of pixel intensities

• Realistic parametric PSF profile representation- Moffat, Airy, ...

• Decomposition in terms of basis functions- Analytical: Shapelets, other types of polynomials, ...

(Refregier, 2003, Bernstein & Jarvis, 2002, ...)

- Non-analytical: extract basis from the data itselfe.g. PCA (Jarvis & Jain, 2004, Jee & Tyson., 2011, ...)

• Model the physical effects causing PSF variation - as done, e.g., for the HST ACS camera

6


7

Interpolation algorithms: conventional approach

• Traditional method: least-square polynomial fitting- Assume PSF varies smoothly across the field- Fit a low-order polynomial surface to the data- The quantities fitted depend on the PSF model

• But polynomials have well-known shortcomings- Cannot capture small or abrupt variations- Increasing the polynomial order does not help

• GREAT10 Star Challenge: an opportunity to- Evaluate accuracy of existing interpolation schemes- Explore other PSF interpolation algorithms/approaches


8

The GREAT10 Star Challenge

• 1300 spatially-varying, noisy star fields in 26 sets

• Challenge: reconstruct the PSF at 1000 specific positions within each image

- Moffat, Airy profiles- FWHM: 1.5, 3, 6 pixels- Turbulent/Non-Turbulent- Optical aberrations or not- Density: 500, 1000, 2000- Spikes: none, +, *


9

The GREAT10 Star Challenge

Non-Turbulent PSF field Kolmogorov Turbulence model


Approach for the GREAT10 Star Challenge

PSF Prediction Pipeline

10

1. Fit input PSFs to elliptical Moffat Profiles

2. Interpolate Moffat parameters at asked positions + validate results (cross-validation + jacknifing)

3. Reconstruct PSFs from interpolated parameterse1 e2

1.

2.

3.


Interpolation knowing true Moffat parameters⇒ No error related to PSF modeling (fitting, reconstruction)

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• Tested five algorithms independently of the PSF model

Interpolation algorithms tested on GREAT10 data

25

Method Description Range Exact Stochastic

Polyfit Polynomial least-square fitting Global N N

B-splines Bivariate regression splines Global N N

IDW Inverse distance weighting Local Y NRBF Radial basis function Local Y NKriging Ordinary Kriging Local Y Y


12

• “Global” versus “Local” interpolators- Global: accounts for all available observations- Local: only considers neighboring observations,

takes advantage of spatial autocorrelation

Interpolation algorithms tested on GREAT10 data

• “Exact” versus “Approximate” interpolators- Exact: predictions at target point coincide with observations- Approximate: estimates differ at targets (smoothing effect)

• “Stochastic” versus “Deterministic” interpolators- Stochastic: assumes underlying probabilistic model

=> yield error estimates- Deterministic: no error estimate produced


“Local” PSF Interpolation Methods

• Interpolator only considers immediate neighbors• Accounts for existing auto-correlation in the data

13

z(x0) =NX

i=1

�i z(xi)

- Weight ! i is method-specific- e.g. IDW:

M. Gentile, F. Courbin and G. Meylan: Point Spread Function interpolation

1995). The TPS interpolating spline is obtained by minimizingan energy function of the form (3)

S ( f ,↵) =NX

i=1

kz(si) � f (si)k2 + ↵ Jm(g) dsi (4)

The most common choice of m is 2 with J2 of the form

J2(g) =Z

D

n⇣@2g@x2

⌘2+ 2⇣ @2g@x @y

⌘2+⇣@2g@y2

⌘2odx dy (5)

where the roughness function g(x, y) is given by

g(s) = a0 + a1x + a2y +NX

1

�i �(s � si) (6)

� being the radial basis function (RBF): �(x, y) = d2i ln(di)

with euclidean distance di =p

(x � xi)2 + (y � yi)2. The �i areweighting factors.

Constructive approach of spline interpolation

Interpolating splines obtained through such a scheme are of-ten referred to as least squares splines (Dierckx 1980; Hayes &Halliday 1994; Dierckx 1995). For such splines, goodness of fitis measured through a least squares criterion, as in the variationalapproach, but smoothing is implemented in a di↵erent way: inthe variational solution, the number and positions of knots arenot varied, and the approximating spline is obtained by mini-mizing an energy function. On the other hand, in the constructiveapproach, one still tries to find the smoothest spline that is alsoclosest to the observation points. But this is achieved by optimiz-ing the number and placement of the knots and finding the cor-responding coe�cients c in the B-Spline basis. This is measuredby a so-called smoothing norm G(c). Thus, the approximatingspline arises as the minimization of

S ( f ,↵) =NX

i=1

kz(si) � f (si)k2 + ↵G(c) (7)

using the same notation as in (3). An example of knot placementstrategy is to increase the number of knots (i.e. reduce the inter-knot distance) in areas where the surface to fit varies faster ormore abruptly. By the way, we note that minimization is not ob-tained by increasing the degree of the spline (which is kept low,typically 3).

Whatever the approach followed for obtaining a suitable in-terpolating spline, spline interpolation is essentially global, ap-proximate and deterministic, as it involves all available observa-tions points, makes use of smoothing and does not provide anyestimation on interpolation errors. The interpolation can how-ever be made exact by setting the smoothing coe�cient to zero.Also, for smoothing splines (variational approach) a techniquecalled generalized cross-validation (GCV) (Craven & Wahba1978; Wahba 1990; Hutchinson & Gessler 1994) allows to au-tomatically choose, in expression (4), suitable parameters for ↵and m for minimizing cross-validation residuals. Otherwise, onecan always use standard cross-validation or Jackknifing to opti-mize the choice of input parameters (see Sect. 4.3).

The most frequently used splines for interpolation are cubicsplines, which are made of polynomials pieces of degree atmost 3 that are twice continuously di↵erentiable. Experience hasshown that in most applications, using splines of higher degree

Table 3. Spline interpolation: Pros and cons

Spline Interpolation

ProsAble to capture both broad and detailed featuresThe tradeo↵ between goodness and roughness of fit can beadjusted through smoothing

Cons

Overall smoothness may still be too highKeep a tendency to oscillateNo estimation of interpolation errors in most algorithmsPotentially less e�cient to compute than local interpolationalgorithms

seldom yields any advantage. As we saw earlier, splines avoidthe pitfalls of polynomial fitting while being much more flexible,which allows them, despite their low degree, to capture finer-grained details. The method assumes the existence of measure-ment errors in the data and those can be handled by adjusting theamount of smoothing.

On the minus side, cubic or higher degree splines are some-times criticized for producing an interpolation that is ”toosmooth”. They also keep a tendency to oscillate (although thiscan be controlled unlike with standard polynomials). In addition,the final spline estimate is influenced by the number and place-ment of knots, which confers some arbitrariness to the method,depending on the approach and algorithm used. This can be aproblem since there is, in general, no built-in mechanism forquantifying interpolation errors. Lastly, spline interpolation is aglobal method and performance may su↵er on large datasets. Asummary of the main strengths and weaknesses of spline inter-polation is given in Table 3.

3.4. inverse distance weighting

Inverse distance weighting (IDW) (Shepard 1968) is one of theoldest spatial interpolation method but also one of the most com-monly used. The estimated value z(x0) at a target point x is givenby Eq. (2) where the weights �i are of the form:

�i =1

d �(x0, xi)

.

NX

i=1

1d �(x0, xi)

� � 0NX

i=1

�i = 1 (8)

In the above expression, d(x0, xi) is the distance between pointsx0 and xi, � is a power parameter and N is the number of pointsfound in some neighborhood around the target point x0. Scalingthe weights �i so that they sum to unity ensures the estimation isunbiased.

The rationale behind this formula is that data points near thetarget points carry a larger weight than those further away. Theweighting power � determines how fast the weights tend to zeroas the distance d(x0, xi) increases. That is, as � is increased, thepredictions become more similar to the closest observations andpeaks in the interpolation surface becomes sharper. In this sense,the � parameter controls the degree of smoothing desired in theinterpolation.

Power parameters between 1 and 4 are typically chosen andthe most popular choice is � = 2, which gives the inverse-distance-squared interpolator. IDW is referred to as ”moving av-erage” when � = 0 and ”linear interpolation” when � = 1.

For a more detailed discussion on the e↵ect of the powerparameter �, see e.g. Laslett et al. (1987); Burrough (1988);Brus et al. (1996); Collins & Bolstad (1996). Another wayto control the smoothness of the interpolation is to vary the

6

M. Gentile, F. Courbin and G. Meylan: Point Spread Function interpolation

1995). The TPS interpolating spline is obtained by minimizingan energy function of the form (3)

S ( f ,↵) =NX

i=1

kz(si) � f (si)k2 + ↵ Jm(g) dsi (4)

The most common choice of m is 2 with J2 of the form

J2(g) =Z

D

n⇣@2g@x2

⌘2+ 2⇣ @2g@x @y

⌘2+⇣@2g@y2

⌘2odx dy (5)

where the roughness function g(x, y) is given by

g(s) = a0 + a1x + a2y +NX

1

�i �(s � si) (6)

� being the radial basis function (RBF): �(x, y) = d2i ln(di)

with euclidean distance di =p

(x � xi)2 + (y � yi)2. The �i areweighting factors.

Constructive approach of spline interpolation

Interpolating splines obtained through such a scheme are of-ten referred to as least squares splines (Dierckx 1980; Hayes &Halliday 1994; Dierckx 1995). For such splines, goodness of fitis measured through a least squares criterion, as in the variationalapproach, but smoothing is implemented in a di↵erent way: inthe variational solution, the number and positions of knots arenot varied, and the approximating spline is obtained by mini-mizing an energy function. On the other hand, in the constructiveapproach, one still tries to find the smoothest spline that is alsoclosest to the observation points. But this is achieved by optimiz-ing the number and placement of the knots and finding the cor-responding coe�cients c in the B-Spline basis. This is measuredby a so-called smoothing norm G(c). Thus, the approximatingspline arises as the minimization of

S ( f ,↵) =NX

i=1

kz(si) � f (si)k2 + ↵G(c) (7)

using the same notation as in (3). An example of knot placementstrategy is to increase the number of knots (i.e. reduce the inter-knot distance) in areas where the surface to fit varies faster ormore abruptly. By the way, we note that minimization is not ob-tained by increasing the degree of the spline (which is kept low,typically 3).

Whatever the approach followed for obtaining a suitable in-terpolating spline, spline interpolation is essentially global, ap-proximate and deterministic, as it involves all available observa-tions points, makes use of smoothing and does not provide anyestimation on interpolation errors. The interpolation can how-ever be made exact by setting the smoothing coe�cient to zero.Also, for smoothing splines (variational approach) a techniquecalled generalized cross-validation (GCV) (Craven & Wahba1978; Wahba 1990; Hutchinson & Gessler 1994) allows to au-tomatically choose, in expression (4), suitable parameters for ↵and m for minimizing cross-validation residuals. Otherwise, onecan always use standard cross-validation or Jackknifing to opti-mize the choice of input parameters (see Sect. 4.3).

The most frequently used splines for interpolation are cubicsplines, which are made of polynomials pieces of degree atmost 3 that are twice continuously di↵erentiable. Experience hasshown that in most applications, using splines of higher degree

Table 3. Spline interpolation: Pros and cons

Spline Interpolation

ProsAble to capture both broad and detailed featuresThe tradeo↵ between goodness and roughness of fit can beadjusted through smoothing

Cons

Overall smoothness may still be too highKeep a tendency to oscillateNo estimation of interpolation errors in most algorithmsPotentially less e�cient to compute than local interpolationalgorithms

seldom yields any advantage. As we saw earlier, splines avoidthe pitfalls of polynomial fitting while being much more flexible,which allows them, despite their low degree, to capture finer-grained details. The method assumes the existence of measure-ment errors in the data and those can be handled by adjusting theamount of smoothing.

On the minus side, cubic or higher degree splines are some-times criticized for producing an interpolation that is ”toosmooth”. They also keep a tendency to oscillate (although thiscan be controlled unlike with standard polynomials). In addition,the final spline estimate is influenced by the number and place-ment of knots, which confers some arbitrariness to the method,depending on the approach and algorithm used. This can be aproblem since there is, in general, no built-in mechanism forquantifying interpolation errors. Lastly, spline interpolation is aglobal method and performance may su↵er on large datasets. Asummary of the main strengths and weaknesses of spline inter-polation is given in Table 3.

3.4. inverse distance weighting

Inverse distance weighting (IDW) (Shepard 1968) is one of theoldest spatial interpolation method but also one of the most com-monly used. The estimated value z(x0) at a target point x is givenby Eq. (2) where the weights �i are of the form:

�i =1

d �(x0, xi)

.

NX

i=1

1d �(x0, xi)

� � 0NX

i=1

�i = 1 (8)

In the above expression, d(x0, xi) is the distance between pointsx0 and xi, � is a power parameter and N is the number of pointsfound in some neighborhood around the target point x0. Scalingthe weights �i so that they sum to unity ensures the estimation isunbiased.

The rationale behind this formula is that data points near thetarget points carry a larger weight than those further away. Theweighting power � determines how fast the weights tend to zeroas the distance d(x0, xi) increases. That is, as � is increased, thepredictions become more similar to the closest observations andpeaks in the interpolation surface becomes sharper. In this sense,the � parameter controls the degree of smoothing desired in theinterpolation.

Power parameters between 1 and 4 are typically chosen andthe most popular choice is � = 2, which gives the inverse-distance-squared interpolator. IDW is referred to as ”moving av-erage” when � = 0 and ”linear interpolation” when � = 1.

For a more detailed discussion on the e↵ect of the powerparameter �, see e.g. Laslett et al. (1987); Burrough (1988);Brus et al. (1996); Collins & Bolstad (1996). Another wayto control the smoothness of the interpolation is to vary the

6


Summary of PSF interpolation results

• RBF best overall• Main factors affecting accuracy

- Turbulence, PSF size, star density• Kriging then RBF/splines most accurate on non-turbulent PSF• Global methods least accurate on turbulent PSF• Local methods more sensitive to low/high densities• PSF size estimate degrade when FWHM below 3 pixels

14

Bearing in mind that

• These results have been obtained on simulated data• Star images are isolated, bright enough and well sampled• Spatial variation of the PSF may favor local interpolators• The GREAT10 turbulence model may lack realismSee: Interpolating point spread function anisotropy, Gentile et al., A&A, 2013


psf interpolation - cosmostatwl.cosmostat.org/gentile_psf_interpolation_nice_2013.pdf · 7...

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