Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Image Analysis with Shapelets

M. den Brokdenbrok@astro.rug.nl

Kapteyn InstituteUniversity of Groningen

Guest lecture for the Applied Signal Processing course, 2008

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Outline

1 Introduction

2 Cartesian Shapelets

3 Polar Shapelets

4 Summary

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Introduction

Post-reduction data analysis important for images

Examples: Fourier transform, wavelets, &c.

Use Shapelets to analyse shape of object

Applications of this technique include gravitational lensingand image simulation.

Technique is fairly young (first article appeared in 2001)

This lecture is based on the articles by Refregier (2003),Refregier & Bacon (2003)

A lot of information can be found athttp://www.astro.caltech.edu/~rjm/shapelets/,including IDL code.

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

One-dimensional basis functions

The goal is to describe a 1-d image (I(x)) as a series of basisfunctions. (cf. Fourier series)The Shapelets dimensionless basis functions are defined asdistorted Gaussians:

φn =(

2nπ12 n!) 1

2Hn(x)e

− x2

2 (1)

In practice, however, these are rescaled to the dimensional basisfunctions:

Bn(x , β) = β12φn(x/β) (2)

So,

I (x) =∑n

InBn(x , β) (3)

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Hermite polynomials

First few Hermite polynomials:

H0(x) = 1

H1(x) = 2x

H2(x) = 4x2 − 2

H3(x) = 8x3 − 12x

H4(x) = 16x4 − 48x2 + 12

Some useful properties:

Hn+1(x) = 2xHn(x)− 2nHn−1(x)H ′n(x) = 2nHn−1(x)

Hn(x) = (−1)nex2 dn

dxne−x

2

= ex2/2

(x − d

dx

)e−x

2/2

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

We can define an inner product between continuous functions:

< f , g >=

∫ ∞−∞

f (x)g(x)dx

The Shapelet basis functions are orthonormal in this innerproduct space:

< Bm,Bn >= δm,n

Since I (x) =∑

n InBn(x , β), we can find the components Insimply by taking the inner product with the right basis function:

Im =< I ,Bm >

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Berry, Hobson & Withington(2004)

However, in practice, theshapelet basisfunctions arediscretized: in this caseorthogonality betweenbasisfunctions is lost. For thisreason, decomposition intoshapelets is usually done by aleast squares fit, or a SingularValue Decomposition.

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

2-Dimensional Basisfunctions

Refregier (2003)

The one-dimensional formalism iseasily extended to 2-D. Eachtwo-dimensional basisfunction is theproduct of two one-dimensionalbasisfunction:

Bnxny (x , y , β) = Bnx (x , β) · Bny (y , β)

This set of 2-D basisfunctions iscomplete (you may prove thatyourself.)

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Example decomposition

In practice, only a finite number of components is used. Highfrequency components start sampling noise, and this is usuallynot what you want. Therefore, a maximum shapelet order isdefined as nx + ny

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Application: weak gravitational lensing

Light of distant sources is bent around object of high mass:gravitational lensing. Three different kinds of lensing

Strong lensing: Great distortions in the light distribution.Multiple images and Einstein rings occur.

Microlensing: High mass object moves before backgroundsource, beaming the light of the background source andincreasing the intensity. Transient effect (used, amongother things, to detect exoplanets)

Weak lensing: the shape distortion can only be detectedby statistically analyzing the shear of a large number ofgalaxies.

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Application: weak gravitational lensing

For weak lensing it is important to have a formalism thatdescribes the shape of the galaxy in a quantitative way. One ofthe most widely used methods is the KSB method (Kaiser,Squires & Broadhurst 1995) This method determines the effectof weak gravitational lensing on the gaussian-weightedquadrupole moment of galaxies. In the shapelets formalism,this corresponds to the n1 + n2 = 2 coefficients.

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Remember: φn =(

2nπ12 n!) 1

2Hn(x)e

− x2

2

Question: Do the Shapelet basisfunctions somehow lookfamiliar to you?

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Remember: φn =(

2nπ12 n!) 1

2Hn(x)e

− x2

2

Question: Do the Shapelet basisfunctions somehow lookfamiliar to you?

Answer: they are also the eigenfunctions of the QuantumHarmonic Oscillator (QHO)

This is very useful, because now we can use the formalismof Quantum Mechanics.

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Application: Fourier transform

Because we know that the Basisfunctions are theEigenfunctions of the QHO, the Fourier transform of aBasisfunction is straightforward:

The dimensionless 1-D equation of the Q.H.O looks likeĤΨ = (x̂2 + p̂2)Ψ = E Ψ

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Position space

In position space, the base kets used are the position kets:

x̂ |x ′ >= x ′|x ′ >

which are orthonormal:

< x ′′|x ′ >= δ(x ′′ − x ′)

Any physical state can be expanded in these base kets:

|α >=∫

dx ′|x ′ >< x ′|α >

Here we recogize < x ′|α > as ψα(x ′) and hence< β|α >=

∫dx ′ψβ

∗(x ′)ψα(x′)

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Momentum space

For momentum space, we can do something similar:

p̂|p′ >= p′|p′ >

The momentumspace base kets are also orthonormal:

< p′′|p′ >= δ(p′′ − p′)

|α >=∫

dp′|p′ >< p′|α >

Using momentum as generator of translations, one can derive:

p̂|α > =∫

dx ′|x ′ >(−i~ ∂

∂x ′< x ′|α >

)< x ′|p̂|α > = −i~ ∂

∂x ′< x ′|α >

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Relation between position/momentum space

From the above, we know:

< x ′|p̂|p′ >= p′ < x ′|p′ >= −i~ ∂∂x ′

< x ′|p′ >

From this, the relation between position and momentum spacefollows:

< x ′|p′ >= Aeip′x′

~

Normalization requires that A = 1√2π

and hence:

|Φ >=∫

dp|p >< p|Φ >=∫

dp|p > φ(p)

φ(x) =< x |Φ >=∫

dp < x |p > φ(p) =∫

dp1√2π

eip′x′

~ φ(p)

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Application: Fourier transform

Because we know that the Basisfunctions are theEigenfunctions of the QHO, the Fourier transform of aBasisfunction is straightforward:

The dimensionless 1-D equation of the Q.H.O looks likeĤΨ = (x̂2 + p̂2)Ψ = E Ψ

In x-space we have (x2 − ∂2∂x2

)Ψ(x) = E Ψ(x)

Transforming this equation to momentum space:(p2 − ∂2

∂p2)Ψ(p) = E Ψ(p)

From the symmetry between x and p, we deduce that theFourier transform of the basisfunction should be, up to amultiplicative constant, equal to the same basisfunction.

It turns out that φ̃(k) = inφn(x)

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Fourier transform of an image

The decomposition of a 2d image yields a vector withshapelets coefficients and a scale parameter (β.)

The fourier transform of this image consists of thecomponents of this vecor, multiplied with flying factors ofi , and with scale 1/β

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Convolution

It is however more interesting to look at convolution of animage:

h(x) =

∫d2x f (x− x′)g(x′)

Each image can be decomposed into shapelets with scalesα, β, γ and coefficients fn =< n, α|f >,gn =< n, β|g >,hn =< n, γ|h >.Convolution is bilinear (remember, in Fourier space you have asimple product of two functions). Therefore, the shapeletcoefficients of a convolved image should look like:

hn =∑m,l

Cn,m,lfmgl

Cn,m,l(α, β, γ) is a 2d convolution tensor. The coefficients ofthis tensor can be calculated analytically (a rather lengthyexpression though.)

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Deconvolution

Smoothing an image in Shapelet formalism is just matrixmultiplication. Therefore, if the smoothing kernel is known, animage can be deconvolved by multiplying with the inversematrix.Lots of applications require PSF deconvolution

weak lensing

micro lensing

photometry

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Application: simple astro/photometric expressions

Using the basisfunctions, we can compute certain photo- andastrometric quantities fairly easy.

1 Total flux: F = π12β∑even

n1,n2 212(2−n1−n2)

( n1n1/2

) 12 fn1,n2

2 First moment: xf = πβ2F−1

∑oddn1

∑evenn2

(n1 +

1)12 2

12(2−n1−n2)

( n1+1(n1+1)/2

) 12( n2n2/2

) 12 fn1,n2

3 Rms radius: r2f = πβ2F−1

∑oddn1

∑evenn2

(n1 + 1)12 (1 + n1 +

n2)212(4−n1−n2)

( n1n1/2

) 12( n2n2/2

) 12 fn1,n2

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Polar shapelets

In 1d, the shapelet basisfunctions are the eigenfunctions of

Ĥ = â†â +1

2

where ↠and â are the creation and annihilation operators,with:

↠= x̂ − i p̂â = x̂ + i p̂[â†, â

]= 1

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

We can define 2d angular momentum operators as follows:

L̂l = âl†âl

L̂r = âl†âr

âl =1√2

(â1 + i â2)

âr =1√2

(â1 − i â2)

The Hamiltonian may thus be written as:

Ĥ = a†l al + a†r ar + 1 = N̂l + N̂r + 1

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Polar shapelets

The operators N̂r and N̂l form a complete set of observables(you can work out the commutation relations yourself). Theeigenfunctions,

N̂l |nl , nr >= nl |nl , nr >,

are the basisfunctions for polar shapelets. The eigenfunctionscan be defined in terms of r and θ:

χn,m(r , θ, β) =−1(n−|m|)/2

β|m|+1

[((n − |m|)/2)!π((n + |m|)/2)!

] 12

×

r |m|L|m|(n−|m|)/2

r2

β2e− r

2

2β2 e−imθ

Lqp is the associated Laguerre polynomial.

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Polar Shapelets

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Polar Shapelets

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Application: morphological classification

Galaxies come in different types, p.e. spirals, ellipticals.Classification is always a problem, because it is somewhatsubjective. The authors below try to classify galaxies usingShapelets.Recipe:

1 Select suitable sample of galaxies from SDSS

2 Put galaxies at same redshift (by rebinning)

3 Convolve galaxies to common PSF

4 Perform Principal Components Analysis of shapeletcoefficients.

(Kelly & McKay, astro-ph:/307395)

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Application: morphological classification

9 dimensions needed

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Application: morphological classification

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Application: image simulation

There exists some packages for image simulation (p.e. IRAF’sartdata), but most packages only provide symmetric profiles.The creation of bars, spiral arms &c is more difficult.Recipe:

1 Draw large number of galaxies from sample (in this case,HDF).

2 Determine the distribution of shapelet coefficients

3 Pick shapelet coefficients at random from this distribution

(Massey, Refregier, Conselice & Bacon, astro-ph/0301449)

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Application: image simulation

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Application: image simulation

Image simulation with Shapelets has another nicety:Gravitationally shearing an image is can be done analytically.First, let’s look at a general image transformation, usinginfinitesimal displacement:

x− > x′ = (1 + Ψ)x + �

Ψ =

(κ+ γ1 γ2 − ργ2 + ρ κ− γ1

)Assuming all quantities are small, we can describe thetransformed image as:

f ′ ≈ (1 + ρR̂ + �T̂ + κK̂ + γi Ŝi )f

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Application: image simulation

One of the beautiful things of the Shapelet formalism is, thatthese transformation matrices are expressible in creation andannihilation operators.

R̂ = −i (x̂1p̂2 − x̂2p̂1) = â1â†2 − â†1â2

K̂ = −i (x̂1p̂1 + x̂2p̂2) = 1 +1

2

(â†21 + â

†22 − â

21 − â22

)Ŝ1 = −i (x̂1p̂1 − x̂2p̂2) =

1

2

(â†21 − â

†22 − â

21 + â

22

)Ŝ2 = −i (x̂1p̂2 + x̂2p̂1) = â†1â

†2 − â1â2

T̂j = −i p̂j =1√2

(â†j − âj), j = 1, 2.

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

The infinitesimal transformations are easily converted to finitedisplacements, by taking the exponent of the matrix.This method has for example been used for STEP 2 (ShearTesting Programme), where a set of simulated sheared galaxieswas created. The purpose of this was recovering the shear ofthese galaxies, to test how well weak lensing methods perform.

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

Summary

Shapelets are a way to decompose (astronomical) imagesin a set of orthogonal basisfunctions.

The basisfunctions come in two tastes: cartesian and polarshapelets

The basisfunctions have nice properties:1 Analytical expressions for flux, first, second moment2 Analytical convolution/deconvolution (calculationally easy)

Applications include:1 Weak gravitational lensing2 Accurate photometry3 Image compression4 Image simulation5 Morphological classification

Shapelets

Mark denBrok

Introduction

CartesianShapelets

PolarShapelets

Summary

Appendix

For FurtherReading

For Further Reading I

Refregier A. 2003, MNRAS 338 35

Massey R. & Refregier A. 2005, MNRAS 363, 197

Refregier A. & Bacon D. 2003, MNRAS 338 48

Massey R., Refregier A., Conselice C. & Bacon D. 2004,MNRAS 348 214

Kelly B. & McKay T. 2004, AJ, 127, 625

Massey R. et al. 2006, MNRAS submitted

IntroductionCartesian ShapeletsPolar ShapeletsSummaryAppendix