The Large Binocular Telescope: a laboratory for image reconstruction M. Bertero DISI – Universita’ di Genova Pisa – October 12, 2005

The Large Binocular Telescope:

a laboratory for image reconstruction

M. Bertero

DISI – Universita’ di Genova

Pisa – October 12, 2005

Joint work with:

- Barbara Anconelli - Genova

- Patrizia Boccacci - Genova

- Marcel Carbillet - Nice

- Serge Correia - Potsdam

-.Gabriele Desiderà - Genova

- Henri Lanteri - Nice

Outline

- The Large Binocular Telescope (LBT)

- Restoration of LBT images

- The standard approach

- Correction for boundary effects

- Super-resolution

- A general approach

- Objects with high dynamic range

- Concluding remarks

The Large Binocular Telescope (LBT) – 1/7

The LBT, under construction on the top of Mount Graham, Arizona, is a new-conception telescope, consisting of:

- two 8.4 m mirrors, with the centers at a distance of 14.4 m;

- a very advanced multi-conjugate AO system (adaptive secondaries, pyramid sensors, etc.);

One of the instruments will be a Fizeau interferometer for the combination of the images of the two mirrors.

http://lbtwww.arcetri.astro.it

First mirror already installed and alluminized

Second mirror transported from the Mirror Lab to Mount Graham

First interferometric light : 2007 – 2008 (?)

The Large Binocular Telescope (LBT) - 2/7

The structure pre-assembled in Milan, June 2001

The enclosure on the top of Mount Graham, Arizona

The Large Binocular Telescope (LBT) - 3/7

LINC/NIRVANA (Lbt Interferometric Camera / Near-IR-Visible Adaptive iNterfermeter for Atronomy) is the German-Italian beam combiner for LBT.

Will combine the radiation from the two 8.4 m primary mirrors in a “Fizeau” mode, allowing true imagery.

Will operate at wavelengths between 0.6 and 2.4 microns.

The first camera will cover 10 arcsec with a pixel size of about 5 mas in K-band (2048 x 2048 pixels).

Multiconjugate AO correction is over a field of about 2 arcmin so that even larger images will be possible in the future (24000 x 24000 ?).

The Large Binocular Telescope (LBT) – 4/7

In case of perfect circular mirrors, perfect interferometer, no atmosphere, the Point Spread Function (PSF) of LBT is given by (we assume that the telescope is aligned in the -direction):

m 14.4 centers mirrorsbetween distance ,

m 8.4mirrors theofdiameter ,

; || , )(cos||

|)|(2),(

2

1

222

2

2

1

112

1

BB

DD

JK

First zero in the -direction: 1 = ;

First zero in the -direction: 1 = 1.22 /D .

The Large Binocular Telescope – 5/7

- plane u,v - plane


Imaging with LINC/NIRVANA : with a few observations at different parallactic angles and suitable reconstruction algorithm, the image,

as seen by a 22.8-m telescope, can be reconstructed !…

22.8 m

8.4 m


* =

incoherent sum

NO !

* =

* =

Restoration of LBT images – 1/4

In the case of LINC/NIRVANA the image restoration problem consists in estimating a unique high-resolution image from p images gj, corresponding to p different orientations of the baseline, being given the corresponding PSFs Kj and backgrounds bj. Therefore it is a problem of multiple images deconvolution. Accuracy and resolution are strongly related to the coverage in the u,v plane.

In astronomy the most frequently used restoration method is an iterative method known as RL-method (Richardson, 1972; Lucy, 1974); in the case of deconvolution problems, it coincides with the ML-EM method (Shepp and Vardi, 1982), proposed in tomography.


Model for the p images acquired by LINC/NIRVANA CCD camera (based on Snyder et al, 1993): for j=1,....,p

(n)r (n)g (n)g (n)g jbj,sj,j

gj,s(n) = number of photons coming from the source; Poisson process with expected value (A jf)(n) = (K j*f)(n) (K j =PSF of the j-th baseline orientation, f(n’) = expected value of the number of photons emitted at pixel n’);

gj,b(n) = number of photons due background emission, etc; Poisson process with (constant) expected value bj ;

rj(n) = read-out noise due to the amplifier; realization of an independent Gaussian process with expected value c and standard deviation s .

Summary of the main features of astronomical images

- Bandlimiting (the support of the FT in the {u,v}-plane is bounded)

- Background due to sky emission (to be estimated)

- Contamination by both Poisson and Gaussian noise (whose statistical parameters can be estimated)

- High Dynamic range (very often orders of magnitudes between the brightness of different objects in the field of view)


Properties to be satisfied by the restored images:

- Nonnegativity (assuming an estimate of the background)

- Conservation of the total flux

- Correct estimation of the relative local fluxes (relative magnitude of stars: photometry)

- Correct estimation of the relative angular separation (astrometry)


The standard approach – 1/5

In the case of images dominated by photon noise, the read-out noise can be neglected and the likelihood function is given by:

p

1j j

(n)gjj

n

b] f)(n)[(A -

! (n)g

]b f)(n)[(Ae f)|P(g

j

j

To maximize this function is equivalent to minimize the Kullback-Leibler directed distance or Csiszár I-divergence measure of the discrepancy between the detected images g j and the computed images A j f + b j (j=1,...,p):

. (n)gbf)(n)(A bf)(n)(A

(n)gln (n)g )g ; Af(J

1jjj

jj

j

nj0

p

j


If we denote by AjT the transposed matrices, the standard EM-

method becomes:

- initialize with f(0) >= 0;

- given f(k), compute f(k+1) by:

. 1

1

1

j(k)

j

jTj

p

j

(k))(k

bfA

gAf

pf

Again it is assumed that each PSF is normalized in such a way that the sum of its pixel values is one.

The method is slow and computationally heavy.


The iterates converge, for any initial guess, to a minimum point of the I-divergence; the minimum may not be unique and may be corrupted by strong noise propagation.

Each iterate is non-negative.

The total flux is correctly reproduced (exactly in the case of zero background).

In the case of sparse and resolved objects (systems of isolated stars at an angular distance greater than the diffraction limit) the photometric (relative magnitudes) and astrometric (relative angular separations) parameters are correctly given.

In the case of diffuse objects the iterates have a semi-convergent behaviour.


The analogy between LBT-images and projections in tomography suggests the extension to LBT (Bertero and Boccacci, 2000) of the OSEM (Ordered Subsets Expectation Maximization) method (Hudson and Larkin, 1994):

.constraintflux

thesatisfiesit way that asuch in erenormaliz -

;

:compute and mod 1set ,given -

,constraintflux thesatisfying ,0 with initialize -

1

1

0

)(k

j(k)

j

jTj

(k))(k

(k)

)(

f

bfA

gAff

p kj f

f

The computational cost per iteration is reduced by a factor of about 4/(3p+1).

original objectimage withPSF @ 60 deg.

0.7”

reconstruction

Reconstruction of a diffuse object


Correction for boundary effects – 1/4

When the object extends beyond the boundary of the FOV, the FFT-based methods are equivalent to use periodic boundary conditions; hence they introduce discontinuities at the boundary, which, in the deconvolved image, produce Gibbs oscillations, also called ripples.

A simple method has been proposed, which consists in reconstructing the object over a domain broader than the FOV, hence without introducing artificial boundary conditions.

One can apply the Richardson-Lucy (or OSEM) method to this problem; the use of FFT is still possible using arrays 2Nx2N and extending the detected images by zero padding.

The results are quite satisfactory.


Scheme of the approach::

bottom left:

the array of the RLM -factors

bottom right:

the windowed weights, defining the reconstruction region


The modified OSEM algorithm is as follows (the bar denotes 2Nx2N arrays:

.constraintflux


;


elsewhere; , 0 ; , /1

),...,1( n compute


1

,1

,,

0

)(k

j(k)

j

jTj

(k)jR

)(k

(k)

jRjjR

STjj

)(

f

bfA

gAfwf

p kj f

nwRnnnw

pjMAn

f

Correction for boundary effects - 4/4

Ideal PSFs SR=70% SR=26%

OSEM

New method

• In astronomy the term denotes any method able to provides a resolution better than the diffraction limit;

• if is the wavelenght the limit is: /D in the case of a single mirror with diameter D;

- /B in the case of LBT with maximum baseline B.

• the amount of super-resolution is controlled by:– the angular Space – Bandwidth product (SBP), given

by the ratio between the angular size of the object and the diffraction limit;

– Signal-to-noise ratio (SNR): the amount of super-resolution increases with increasing SNR

Super-resolution – 1/6


The standard restoration method implements the constraint of non-negativity as well as a constraint on the total flux of the object and therefore it already provides a moderate amount of super-resolution.

However the crucial constraint is on the support of the object. It can be implemented by using a ’’localization’’ property of the algorithm: due to the ’’multiplicative’’ structure of the algorithm, if the initial guess is zero in one pixel, all the iterates are zero in that pixel.

If the domain of the object is known, one can initialize the algorithm with the characteristic function of the domain.

• construction of a mask: matrix with values 0 or 1:– 1 where the image resulting from the first step

is greater than a fixed treshold = given percent of the image maximum value

– 0 elsewhere.

Result of the first step Mask


2. RL restoration initialized with the mask of the domain as estimated in step 1.• smaller number of iterations (250-5000)• can already provide a super-resolved image

but the photometry may not be correct

OS-EM5000 iter


Result of the first step Result of the second step

If the photometry is not correct the first two steps are used to estimate the positions of the two stars

original difference of magnitude

difference of magnitude after step 3

3. solution of a least-squares problem , where the

unknowns are the magnitudes of the stars


we use a circular mask with diameter

number of iterations:

Step 1: 10000

Step2 : 250:5000


Minimum separation limit in the case of binaries with the same magnitude

A general approach – 1/9

LINC/NIRVANA will require a broad spectrum of image restoration methods designed for different purposes:

- quick-look methods, to be routinely used for a first inspection of the observed astronomical object; possibly very efficient even if not very accurate;

- ad hoc methods for the restoration of objecs with specific features (high dynamic range, low photon counting, edges,..)

It should be interesting to have a general approach for the minimization of a broad class of functionals with the constraints of nonnegativity and flux conservation (L1-norm). We investigate the extension of an approach proposed by Lanteri et al. (2001-2002).


Minimization of functionals of the following type:

J(f,g) = J0(Af;g) + JR(f) ,

where the first term is a discrepancy between detected and computed data and the second term a regularization functional, with the additional constraints:

. c f(n) , 0 f(n)n

The constant c can be chosen in such a way that the flux of the restored object is consistent with the fluxes of the detected images (constant backgrounds) :

. b(n)g p

1c

p

1j njj


The approach relies on the Split Gradient Method (Lanteri et al. 2002), i. e. on the following decompositions of the gradients:

(f)V(f)U(f)J

(f;g)V(f;g)U(Af;g)J

V(f;g)U(f;g)J(f;g)

RRRf

of

f

00

where U, V are nonnegative arrays (V positive). It is obvious that these decompositions always exist even if they are not unique. For practical applications their explicit expressions as functions of f and g = {g1, …., gp} are needed.

The approach is basically a descent method.


The basic algorithm is as follows:

.constraintflux


;

:compute ,given -


1

1

0

)(k

ω(k)

ω(k)ω(k)k(k))(k

(k)

)(

f

;gfV

;gfV;gfUf ff

f

f

and are relaxation parameters; is the step in the descent direction: it can be chosen in order to assure nonnegativity and convergence.


In the particular case of step 1, the algorithm takes a very simple form:

; 1

ω

(k)

(k)(k))(k

;gfV

;gfUff

convergence is not guaranteed (even if it has been proved in particular cases) but nonnegativity is automatic !

By taking into account the structure of the discrepancy functional in the multi-images case, one gets:

.

fμ V;gfV

fμ U;gfU

ff

ω

(k)R

p

jj

(k),j

(k)R

p

jj

(k),j

(k))(k

10

10

1


The dependence of the algorithm on the multi-images suggests the following general OS-version:

.constraintflux


;



1

0

01

0

)(k

ω

(k)Rj

(k),j

(k)Rj

(k),j

(k))(k

(k)

)(

f

f Vpμ

;gfV

f Upμ

;gfUff

pk j f

f

In such a way each iteration has the same computational cost of a single-image iteration.


- Poisson noise: 100

j,jjj

jTjj,j f ;g V ,

bfA

g A f ;gU

In the case =0 one obtains the OS-EM algorithm.

- Gauss noise: jjTjj,jj

Tjj,j bfAAf ;g V , g A f ;gU 00

In the case =0 one obtains an OS version of ISRA (Iterative Space Reconstruction Algorithm) with the addition of background contribution.

Two examples:


Possible regularizations:

f ;f V, ff U , ||f||ffJ R(R)

R(R)

R 2

2

1

f ;DDIf Vf, DDf U , f||I-D||fJ TR

TRR 2

2

1

D is a matrix with nonnegative entries. Example: the discrete Laplacian = 4(I-D)

c

ff, V

c

ff, U

nf

nf nffJ

(R)

RR(R)n

R ln1lnln


As an example we obtain an acceleration of the OSEM (Ordered Subsets Expectation Maximization) method (Hudson and Larkin, 1994) derived from Tanaka, 1982:

.constraintflux


;


,constraintflux thesatisfying 0,f with initialize -

1

1

(0)

)(k

ω

j(k)

j

jTj

(k))(k

(k)

f

bfA

gAff

p kj f

Computational cost per iteration reduced by a factor 4/(3p+1); with =2, reduction of the number of iterations by a factor 2.

Objects with high dynamic range – 1/5 A model of young binary star (simulation based on observations of the

GG Tau system)

Objects with high dynamic range – 2/5

One possible approach can be provided by an ‘‘adaptive’’ penalization of the I-divergence (Geman and Mac Clure, 1987) :

. (n)f

(n)f

2)f(J

22

2

2R

n

Penalization is obtained by means of a semi-convex functional (the Hessian is bounded from below); the result is a ‘‘Tikhonov regularization’’ in regions where f(n) is small with respect to and ‘‘no regularization’’ in regions where f(n) is large with respect to ; in these regions the EM algorithm is dominating. Uniqueness of the minimum is not guaranteed. The previous iterative algorithm becomes very simple.

In the case of single image the algorithm is as follows


.constraintflux


; 1

:compute and mod 1set given -

;constraintflux thesatisfying,0 with initialize -

1

1

222

0

)(k

(k)j

Tj

(k)

(k))(k

(k)

(k)(k)

(k)

)(

f

bfA

gA

cp

ff

] η)[(f

f c

p kj f

f

Objects with high dynamic range – 4/5 Iterates of the OSEM method (linear scale)

Without Regularization

With Regularization

K=100 K finale


Ideal PSFs SR=70% SR=26%

Without Reg.

With Reg.

Concluding remarks

AIRY , version 3.0: http://dirac.disi.unige.it

A sample of open problems:

- Specific methods for the reconstruction of faint objects; in these cases the photon noise may be comparable to the (Gaussian) read-out noise so that both Poisson and Gauss statistics must be taken into account.- Development of wavelet based methods taking into account the different angular scales of astronomical objects.- Development of powerful acceleration methods, for instance along the lines of the Biggs-Andrews approach (MATLAB).- Domain decomposition methods for large scale images or space-variant PSFs.

http://dirac.disi.unige.it/

http://dirac.disi.unige.it/

ReferencesLBT site: http://lbtwww.arcetri.astro.it

[1] M. Bertero and P. Boccacci, 1998, Introduction to Inverse Problems in Imaging, IOP, Bristol[2] M. Bertero, and P. Boccacci, 2000, Application of the OS-EM method to the restoration of LBT images, Astron. Astrophys. Suppl. Ser., 144, 181-186[3] M. Bertero, and P. Boccacci, 2000, Image restoration methods for the Large Binocular Telescope, Astron. Astrophys. Suppl. Ser., 147, 323-332[4] S. Correia, M. Carbillet, P. Boccacci, M. Bertero, and L. Fini, 2002, Restoration of interferometric images –I. The software package AIRY, Astron. Astrophys., 387, 733-743[5] M. Carbillet, S. Correia, P. Boccacci, and M. Bertero, 2002, Restoration of interferometric images –II. The case-study of the Large Binocular Telescope, Astron. Astrophys., 387, 744-757

http://lbtwww.arcetri.astro.it/

[6] D. L. Snyder, A. M. Hammoud, and R. L. White, 1993, Image recovery from data acquired with a charged-coupled -device camera, 1993, J. Opt. Soc. Am., A-10, 1014-1023[7] W. H. Richardson, 1972, Bayesian-based iterative method of image restoration, J. Opt. Soc. Am., 62, 55-59[8] L. Lucy, 1974, An iterative technique for the rectification of observed distribution, Astron. J., 79, 745-754[9] L. A. Shepp and Y. Vardi, 1982, Maximum likelihood reconstruction for emission tomography, IEEE Trans. Med. Imaging, 1, 113-122[10] A. R. De Pierro, 1987, On the convergence of the iterative image space reconstruction algorithm for volume ECT, IEEE Trans. Med. Imaging, 6, 124-125[11] H. Lanteri, M. Roche, and C. Aime, 2002, Penalized maximum likelihood image restoration with positivity constraints: multiplicative algorithms, Inverse Problems, 18, 1397-1419

Documents

The Large Binocular Telescope: a laboratory for image reconstruction M. Bertero DISI – Universita’ di Genova Pisa – October 12, 2005