4 Image Reconstruction Studies

Transcription

1 APreS-MIDI SCIENCE CASE STUDY 43 4 Image Reconstruction Studies [F. PRZYGODDA, S. WOLF, K.-H. HOFMANN, S. KRAUS, L. MOSONI, TH. RATZKA] The main goal of APRES-MIDI is to perform imaging in the mid-infrared spectral regime with the highest possible resolution. The VLTI with its four 8-meter class telescopes and additional 1.8- meter auxiliary telescopes provides nowadays the best infrastructure for this task. Since the end of 2002, MIDI impressively demonstrated the feasibility of interferometric observations in the mid-infrared in the mid-infrared wavelength range with the VLTI. The two-beam combiner MIDI gives the scientific community already the possibility of high-angular observations, but due to the limited uv-coverage and the lack of phase measurements it is not able to perform imaging. Interpreting MIDI data requires always a-priori information about the structure of the object in form of a model. The parameters of such a model are then determined through MIDI observation. APRES-MIDI will give the opportunity for the combination of at least 3 beams. Therefore, it can overcome the drawbacks of the poor uv-coverage and the lacking phase information of MIDI all at once. With APRES-MIDI, the number of simultaneously measured baselines is 3 and at the same time it can measure a closure phase relation. With the future potential of the VLTI infrastructure which will permit to track even 4 beams, the number of simultaneously measured baselines increases to six and that of closure phase relations to 4. In this section it will be demonstrated, that the amount of interferometric data obtained with 3 or 4 telescopes is enough to perform real imaging by aperture synthesis under realistic observational conditions. 4.1 General Constraints In general, the performance of an optical instrument depends on its transfer function. Any aperture with a finite size will cause a cut-off of high spatial frequencies and hence is limiting the resolution. In case of an interferometer, the highest spatial frequency that can be measured is given by the longest baseline B. The resolution δ depends on the wavelength and can be expressed by the relation δ = λ/b. Regarding the VLTI with its longest baseline of B = 200 meter, the highest possible resolution at 10 micron calculates to δ 10 mas. This is the universal resolution limit of the VLTI for mid-infrared observations. The imaging quality of a system is not only given by the cut-off frequency but also by general properties of its transfer function. In case of incoherent imaging 5, the modulus of the transfer function (MTF) is given by the autocorrelation of the entrance pupil function. For a single dish telescope this function is non-zero in the whole range from zero up to the cut-off frequency. However, the entrance pupil of an interferometer consists of separate apertures. Because of this, the autocorrelation and therefore the MTF is not non-zero for all frequencies. Some information will definitely not be transfered by the interferometer which causes a degeneration of the resulting image. Due to this fact, the quality of an image obtained with aperture synthesis is never as good as that of a single telescope with a diameter equal the longest baseline. The quality depends on the filling factor of the transfer function ( uv-coverage ) and thus strongly on the number, distances and orientation of the used baselines. The influence of the considered baseline configuration will be discussed in Sect Other constraints for aperture synthesis imaging in the mid-infrared are caused by the properties of 5 This is valid for astronomical observations where each point of a given intensity distribution radiates incoherently.

2 44 APreS-MIDI SCIENCE CASE STUDY the spectral regime itself. At 10µm, the sky and instrument background are dominating the measured count rates. Whereas the instrument background can in principle be reduced by cooling, the background of the sky and the telescope is always present. The photon noise of the background flux is affecting the accuracy of the measured signal and therefore the quality of the synthesized image. The described points resolution limit, effect of baseline configuration and background noise are not depending on the design of APRES-MIDI but are caused by the properties of the VLTI s uv-coverage and on physical limits. Main parts of the following discussion are therefore valid for mid-infrared imaging by aperture synthesis with the VLTI in general. 4.2 Simulating APRES-MIDI In addition to the general constraints described above, also other points such as details of the instrumental setup or the performance of the used image reconstruction algorithm have an impact on the image quality. To get an idea about the overall performance of APRES-MIDI, a step-bystep simulation from the observation of a potential target up to the final image reconstruction was done. The single steps of the simulation are built modularly, so that modules can be replaced by others and extensions are possible. The noise has been estimated and considered in the simulations. Fig. 14 shows the flow chart of the simulation. Each of the individual modules will be described in the following sections (Test Image: Sect ; VLTI Simulator: Sect ; Visibilities and Closure Phases: Sect ; Image Reconstruction Techniques: Sect. 4.3; Image Reconstruction Tests: Sect. 4.4). Test Image VLTI Simulator Visibilities + Closure Phases Image Reconstruction Code Reconstructed Image Figure 14: Flow chart describing the simulation of the APRES-MIDI image reconstruction (Sect ) Test Image The simulation starts with an input image which is the basis for the creation of artifical datasets. To test the performance of APRES-MIDI for scientific observations, this input image should be as realistic as possible, which means that it should be an image of a typical APRES-MIDI target. Since

3 APreS-MIDI SCIENCE CASE STUDY 45 Figure 15: Left column: Mid-infrared image at 10.5µm of a T Tauri disk without a gap. 2nd column: Disk with a gap from reaching from 2 AU to 3 AU. 3rd column: Disk with an inner radius of 4 AU (inner disk hole). Each row shows the disk under an inclination of 0 o, 30 o, and 60 o, respectively. Two images on the right: Example image (inner hole with a radius of 4 AU, seen under an inclination of 60 o ) and the same image convolved with the PSF corresponding to a 200 meter single-dish aperture. For this example a distance of 140 pc was assumed. (The square root of the brightness is shown to emphase faint parts.) APRES-MIDI will be the first instrument for imaging in the mid-infrared with a resolution in the order of a few milliarcseconds, there are no observed images avaliable yet. Therefore, the input image must be simulated under realistic astrophysical conditions. The radiative transfer code MC3D (Wolf et al. 1999, Wolf 2003) was used to produce images of proto-planetary disks which represent one of the main group of objects for mid-infrared observations (see Sect. 2.1). In the following we use a model for a circumstellar disk as it was derived for the famous Butterfly Star in Taurus - a circumstellar disk in edge-on orientation (Wolf et al. 2003). In order to demonstrate the capability of APRES-MIDI to investigate small-scale structures in these disks, being indications for planet formation, we consider the question whether or not gaps caused by giant planets and inner disk holes due to the fast dust evolution in the inner region of such disks could be observed. The 10µm reemission image of this disk around a 0.5 M T Tauri star is shown under different inclination angles in Fig. 15. For the first test simulations, the image of a disk with an inner radius of 4 AU seen under 60 o (lower right image) was chosen. This image shows several interesting details which allow a good estimation of the quality of the reconstruction. Next to the central source also the disk s hot upper and lower surface is visible in form of a double line 6. Furthermore, this image shows a clear asymmetry with regards to the point symmetry to the center. This property is important for checking the reconstruction performance related to the closure phase. Only asymmetric image features cause a non-zero phase in the Fourier spectrum which then can produce a non-zero closure phase. 6 This detail results from the temperature and density distribution at the inner rim of the disk.

4 46 APreS-MIDI SCIENCE CASE STUDY Object Declination - 30 deg Wavelength 10.5 µm Filter width µm Start of Observation -4.5 hours of meridian End of Observation 4.5 hours of meridian Time for one Data Point 1 hour Detector Integration Time 100 ms Total Integration Time for one Data Point 15 min Table 2: Used parameters for the APRES-MIDI simulation with SimVLTI3 to simulate one night VLTI Simulation Software - SimVLTI3 The test image is the basis for the calculation of visibilities and closure phases with the software package SimVLTI. Since the most recent official version (V ) allows to simulate only a twoway beam combiner, the package had to be extended to simulate three-beam combinations including the calculation of closure phases. Furthermore, the locations of all AT stations and the output of interferometric data in the OI-FITS format 7 were integrated in the software package. With these modifications the software now allows the simulation of interferometric observations with APRES- MIDI. In addition to the input image and the considered UT or AT configurations, further parameters have to be defined. Table 2 gives a summary of the used values. These values describe the simulation of one night of observations. In the case of the usage of ATs, a complete observation of one target will consist of several nights, each night with a different AT configuration. This approach will allow to obtain a sufficient uv-coverage. Simulations of up to 7 nights with 3 telescopes and 5 nights with 4 telescopes will be described in detail in Sect Computing Visibilities and Closure Phases As outlined earlier, the modified version of SimVLTI (hereafter called SimVLTI3) allows the calculation of visibilities and closure phases for snapshots with 3 telescopes. Such a dataset can also be expressed in form of a bispectrum element. This four-dimensional complex quantity can be calculated as follows. Starting with an N N array o(x, y) containing the intensity distribution of the input image, the first step is to calculate its two-dimensional Fourier transform O(u, v) = F[o(x, y)]. (9) According to the Van Cittert-Zernike theorem the observable visibility V is given by the normalized modulus of the Fourier transform V (u,v) = O(u,v) O(0,0). (10) For each simulated snapshot SimVLTI3 picks up the corresponding uv-coordinates (u 12,v 12 ), (u 23,v 23 ), (u 13,v 13 ) according to the 3 projected baselines given by the telescope coordinates, the hour angle 7 The output in this standard for calibrated optical and infrared interferometric data ensures an easy data export to different reconstruction codes. More about OI-FITS under jsy1001/exchange/.

5 APreS-MIDI SCIENCE CASE STUDY 47 and the object inclination. The modulus of the complex visibility is calculated and stored in V 12, V 23, and V 13. The Fourier phase ( ) Im(O(u,v)) φ(u, v) = phase(o(u, v)) = atan (11) Re(O(u,v)) is also calculated for each of the 3 baselines and stored in φ 12, φ 23 and φ 13. Since the 3 baselines build up a loop, the coordinate (u 13,v 13 ) can be expressed by ( u 12 u 23, v 12 v 23 ). The bispectrum element B, defined as the product of 3 complex visibilities, is given by B(u 12,u 12,v 23,v 23 ) = O(u 12,v 12 )O(u 23,v 23 )O (u 13,v 13 ) (12) = V 12 V 23 V 31 e i(φ 12+φ 23 +φ 31 ). (13) The sum of phases φ 12 + φ 23 + φ 31 or with other words the phase of the bispectrum element is also known as the closure phase. This observable has the property to be independent of atmospheric phase disturbances and allows to recover the necessary constraints of the Fourier phase information for the image reconstruction. For a realistic simulation of an interferometric observation, measurement errors also have to be considered. For mid-infrared interferometric observations one can distinguish between two main error sources. First, because of the high background radiation, the corresponding photon noise will always be dominant. Second, due to the instability of the atmosphere in the mid-infrared, a typical miscalibration of the measured object visibility is unavoidable (see Sect. 3 for a detailed discussion and quantification of these errors). 4.3 Image Reconstruction Techniques In this section, a brief overview about the applied and tested image reconstruction techniques is provided. In particular, we consider 1. The Building Block method (Sect ), 2. Hybrid Mapping and Self-Calibration (Sect ), and 3. Difference mapping (Sect ). The results of the image reconstruction studies are presented and discussed in Sect Building Block Method The goal of the Building Block method (Hofmann & Weigelt 1993) is the reconstruction of a diffraction-limited image of the object o(x) from the visibilities and closure phases measured by an optical/infrared long-baseline interferometer. From these visibilities and closure phases the object bispectrum is derived, which is the input data of the Building Block method. The object bispectrum is defined as O (3) (u,v) = O(u) O(v) O (u + v) = O(u) O(v) O (u + v) e i[φ(u)+φ(v) φ(u+v)] (14)

6 48 APreS-MIDI SCIENCE CASE STUDY where O(u) and φ(u) are the visibility and phase of the Fourier transform O(u) of the object o(x) at spatial frequency u, respectively. The phase of the bispectrum β(u, v) := φ(u) + φ(v) φ(u + v) is the well-known closure phase of the object. The object intensity distribution of an object o(x) can be described as a sum of many building blocks (e.g. δ-functions). The reconstructed image is approximately o(x) convolved with the diffractionlimited PSF of a single-dish telescope with the maximum projected baseline length of the interferometric measurement as diameter. The Building Block method iteratively produces images o k (x) (k=1,2,...) by adding one building block δ(x x ) at each iteration step k. The goal of the Building Block method is to find an image o k (x) which minimizes the χ 2 function Z Q[o k (x)] := O(3) k (u,v) O(3) (u,v) 2 dudv. (15) σ(u, v) In Eq. 15 O (3) (u,v) is the measured bispectrum built up by the observed visibilities and closure phases of the object. The quantity σ(u,v) denotes the error of O (3) (u,v) calculated from the errors of the closure phases and visibilities and O (3) k (u,v) is the bispectrum of the reconstructed object o k(x) after iteration step k. The new building block for the image o k+1 (x) of the next iteration step k + 1 is positioned at the particular coordinate x = x 0 in o k (x) which leads to an image which minimizes the χ 2 function Q[o k (x) + 1/k δ(x x 0 )]. For a large number k of iterations, the contribution of the new building block 1/k δ(x x 0 ) is very small. Therefore, Q[o k (x) + 1/k δ(x x 0 )] can be approximated by a Taylor expansion: Q[o k (x) + 1/k δ(x x 0 )] Q Q[o k (x)] + 1/k o k (x 0 ) = Z (3) O k Q[o k (x)] + 6/k (u,v) σ 2 (u,v) O k (u) O k (v) e +2πi(u+v) x 0 dudv. (16) As can be seen from Eq. 16, a positive building block is added at the position of the absolute minimum of Q Q o k (x ), and a negative building block at the position of the absolute maximum of o k (x ). Experience shows, that adding positive and negative building blocks simultaneously improves the reconstructions. Negative building blocks are added only if the positivity constrain is not violated. To add more than one positive and one negative building block per iteration step accelerates, of course, the iteration process and also yields better reconstructions. This is done by adding positive Q (negative) building blocks at that positions where the values of o k (x are smaller (larger) than a ) fraction of the value of its absolute minimum (maximum). The reconstructions shown in this report are performed without regularization (see Sect ) Hybrid Mapping and Self-Calibration The radio astronomy community has developed many strategies to reconstruct images from interferometric data. One of the earliest approaches to perform this task for phase-instable interferometers was presented by Cornwell & Wilkinson (1981) with the Hybrid Mapping algorithm.

7 APreS-MIDI SCIENCE CASE STUDY 49 The Algorithm The underlying concept is the self-calibration of the phases using the measured closure phase information and a source model, which is improved iteratively. To start the iteration, a point source model might suffice as initial source model. This was done in our experiments by setting two of the 3 phases to zero. Then, the third phase for each closure phase triangle is given by the closure phase relation. Starting with this simple initial phase model, a first Dirty Map (DM) can be computed by Fourier transforming the sampled Fourier plane. The DM will contain spurious side-lobes and artefacts due to the incomplete sampling. We use CLEAN deconvolution (Högbom 1974) to remove these artefacts introduced by the specific Dirty Beam (DB, also Point Spread Function, PSF). In each CLEAN cycle, the weighted DB is subtracted iteratively from the DM. The weight-factor f for this subtraction is given by a constant gain-factor g times the peak intensity DM max (x max,y max ) of the strongest feature within the map f = g DM max (x max,y max ). By cleaning only positive features within a limited area of the dirty map, the CLEAN deconvolution imposes positivity and confinement constraints on the reconstructed image. The coordinates (x max,y max ) and intensities DM max (x max,y max ) of these Clean Components (CC) are stored within a list. After a constant number of CLEAN steps is performed, the remaining features represent the residual noise map NM. By computing the Fourier transform (FT) of the CC, the new phases are calculated using the closure phase information. This procedure is repeated until convergence is reached (see Fig. 16). Within a last step, the CC are convolved with the Clean Beam (CB). The CB might be approximated by the PSF which would be achieved by a filled aperture of the same diameter as the longest baseline within the array configuration. However, since interferometric arrays including a small number of telescopes, often do not achieve a uniform distributed uv-sampling up to the same baseline lengths for all position angles, we obtained the CB by fitting ellipses to different contour levels of the DB. The fitting parameters for these ellipses are major and minor semi axes and the rotation angle Difference Mapping Difference mapping is another algorithm in the framework of self-calibration methods. For our image reconstruction studies with this algorithm, we use the Caltech Difmap package (Shepherd 1995). Comparable to hybrid-mapping (Sect ), the first step of the difference mapping algorithm is to calibrate the observed phases to a model, taking the measured closure-phase relations into account. Subsequently, CLEANing and self-calibration are repeated iteratively. CLEANing and phase selfcalibration cycles can be executed on the difference map, which is the Fourier-transform of the difference of the measured and CLEANed visibilities. Using the final image as a new starting model usually does not improve the image.

8 50 APreS-MIDI SCIENCE CASE STUDY HYBRID MAPPING Data (Vis, CP) Sampling Model FT repeat until convergence CP FT CLEAN DM NM DB CB New Phases CC * + CM Figure 16: Scheme of the Hybrid Mapping Algorithm: CP=Measured Closure Phase; Vis=Measured Visibilities; DM=Dirty Map; CM=Clean Map; CC=Clean Components; DB=Dirty Beam; CB=Clean Beam; FT=Fourier Transform

9 APreS-MIDI SCIENCE CASE STUDY Image Reconstruction Tests In this section we present and discuss our studies of the imaging performance of APRES-MIDI in combination with the VLTI. The influence of the chosen baseline configuration, the number of telescopes and nights, the number and accuracy of data points per night, the influence of noise, and the possible gain achieved by combining interferometric measurements (with ATs) with acquisition images obtained with UTs are described UV-Coverage As mentioned above, the imaging performance of APRES-MIDI will strongly depend on the available uv-coverage. In this section we want to investigate different baseline configurations to find out, under which conditions an acceptable image reconstruction is possible. The investigated baseline configurations were chosen in a way, that they could be implemented during realistic observing conditions. In particular it was assumed, that a configuration will not be changed during one observing night. Data sets for different configurations and numbers of observing nights were created and the quality of the reconstructed images was evaluated. The simulation was done for both, the 3 and the 4 telescope mode. In case of simulations with a low number of telescopes or nights (relatively poor uv-coverage) different configurations were tested. In this first test, noise was not considered. The simulation software SimVLTI3 was set up to create exact data points, allowing to estimate a given uv-coverage on the reconstructed images. In other words, this tests the quality of the reconstructed images on the uv-coverage and the performance of the image reconstruction alogorithm only, independent of noise. Tests with the goal to investigate the influence of noisy data points on the reconstruction will follow in Sect As a quantitative indicator of the reconstruction quality, the distance (=deviation) between the original image o(x, y) and the reconstructed image i(x, y) was calculated for each simulation. The distance calculation was performed on basis of the convolved versions of the images. The applied PSF was that of a single-dish with a diameter of the longest baseline used in the given configuration. We specify 3 different definitions for the distance d as follows: Beauty Contest Measure (Lawson et al. 2004): x,y o(x,y) [i(x,y) o(x,y)] 2 d A = x,y o(x,y) (17) Balanced Measure: d B = x,y (o(x,y) + i(x,y)) [i(x,y) o(x,y)] 2 x,y o(x,y) + i(x,y) (18) Squared Difference Measure: d C = x,y [ i(x,y) x,y i(x,y) o(x,y) ] 2. (19) x,y o(x,y) All definitions are based on the squared distance between each pixel of the original image and the corresponding pixel of the reconstructed image. To ensure that both images are overlaid correctly,

10 52 APreS-MIDI SCIENCE CASE STUDY a cross correlation was performed to find the best fitting position. The definitions differ in the way of weighting each pixel. The Beauty Contest Measure d A takes no pixel into account, where the original image o(x, y) is zero. Therefore, artefacts in the surrounding of the reconstructed image (where the original is almost zero) are not considered in the measurement. The Balanced Measure d B and Squared Difference Measure d C avoid this disadvantage and take all parts into account Image Reconstruction Results In this subsection, we discuss the results achieved with the different image reconstruction routines. If not stated differently, the results are based on the 10µm image of a circumstellar disk with an inner hole (radius 4 AU), seen under an inclination of 60 o (see Fig. 15[right]). This object was chosen because of its complex small structures and clear features which - beside a quantitative analysis - allow a direct qualitative comparison of the reconstructed with the original image. [A] Results: Building Block Method The resulting images (see below) show in most of the cases the elongated ring-like structure of the input image. Of course, the quality gains with a higher number of measured uv data points. In two cases the image reconstruction failed. Here, the used baseline configuration was not covering the low frequencies in the uv-plane which turned out to be necessary for the image reconstruction with the Building Block method. The appearance of the reconstructed images obtained with the smallest considered number of data points (3 telescopes and 3 nights) strongly depends on the used baseline configuration. We want to remind the reader, that the tests were performed without noise. The differences in the obtained images are therefore just a result of the uv-coverage. It seems, that here the lower limit of a sufficient uv-coverage is given to get a meaningful image reconstruction. The central point source becomes detectable in datasets with more than 150 visibility points which requires five or more nights in the 3-telescope mode or 3 nights in the 4-telescope mode. The detection of the double line of the inner rim of the disk requires the high number of visibilities and closure phases obtainable with 4 telescopes in at least 3 nights. The importance of closure phase information becomes apparent by comparing 3- and 4-telescope observations with a similar number of visibility points (e.g. datasets 7 Nights x 3 ATs or 7 Nights x 3 ATs compared to 3 Nights x 4 ATs). In the case of the 4-telescope observation, the distance is much smaller compared with a three-telescope because of the higher number of closure phase relations. We can conclude from these first tests, that a meaningful image reconstruction requires 5 to 7 nights of observing time in the three-telescope mode or 3 nights in the 4-telescope mode. For further investigations we decided to use the baseline configuration 7 Nights x 3 ATs. Tab. 3 summarizes the distance measurements for the investigated baseline configurations.

11 APreS-MIDI SCIENCE CASE STUDY 53 Config. Vis. Clos.ph. d A d B d C 3 Nights x 3 ATs - A Nights x 3 ATs - C Nights x 3 ATs - D Nights x 3 ATs Nights x 3 ATs Nights x 4 ATs - A Nights x 4 ATs Table 3: Distance measurements for the investigated baseline configurations indicating the quality of the image reconstruction. Note: For each configuration, the Fourier transformed image with the uv-tracks is shown (left image) as well as the reconstructed image (unconvolved - middle image; convolved - right image). Configuration: 3 Nights x 3 ATs - A Baselines: B5-D0-J3, B5-B1-D1, J6-A0-J2 Number of Visibilities: 90, Number of Closure Phase Relations: 30 Distance measurements: d A = , d B = , d C = UV-Plane East South Spatial Frequency (arcsec -1 ) Configuration: 3 Nights x 3 ATs - B Baselines: B5-J6-J1, B5-D0-J3, J6-A0-J2 Number of Visibilities: 90, Number of Closure Phase Relations: 30 UV-Plane East South Spatial Frequency (arcsec -1 )

12 54 APreS-MIDI SCIENCE CASE STUDY Configuration: 3 Nights x 3 ATs - C Baselines: B5-D0-J3, B5-B1-D1, B5-M0-G2 Number of Visibilities: 90, Number of Closure Phase Relations: 30 Distance measurements: d A = , d B = , d C = UV-Plane East South Spatial Frequency (arcsec -1 ) Configuration: 3 Nights x 3 ATs - D Baselines: B5-J6-J1, B5-B1-D1, J6-A0-J2 Number of Visibilities: 90, Number of Closure Phase Relations: 30 Distance measurements: d A = , d B = , d C = UV-Plane East South Spatial Frequency (arcsec -1 ) Configuration: 5 Nights x 3 ATs Baselines: B5-J6-J1, B5-D0-J3, B5-B1-D1, B5-M0-G2, J6-A0-J2 Number of Visibilities: 150, Number of Closure Phase Relations: 50 Distance measurements: d A = , d B = , d C = UV-Plane East South Spatial Frequency (arcsec -1 )

13 APreS-MIDI SCIENCE CASE STUDY 55 Configuration: 7 Nights x 3 ATs Baselines: B5-J6-J1, B5-D0-J3, B5-B1-D1, B5-M0-G2, J6-A0-J2, J1-D1-G2, J6-A0-M0 Number of Visibilities: 210, Number of Closure Phase Relations: 70 Distance measurements: d A = , d B = , d C = UV-Plane East South Spatial Frequency (arcsec -1 ) Configuration: 3 Nights x 4 ATs - A Baselines: B5-G1-J3-D0, B5-B1-K0-D1, G2-A0-J2-J6 Number of Visibilities: 180, Number of Closure Phase Relations: 120 Distance measurements: d A = , d B = , d C = UV-Plane East South Spatial Frequency (arcsec -1 ) Configuration: 3 Nights x 4 ATs - B Baselines: B5-J6-J1-A0, B5-G1-J3-D0, G2-A0-J2-J6 Number of Visibilities: 180, Number of Closure Phase Relations: 120 UV-Plane East South Spatial Frequency (arcsec -1 )

14 56 APreS-MIDI SCIENCE CASE STUDY Configuration: 5 Nights x 4 ATs Baselines: B5-J6-J1-A0, B5-G1-J3-D0, B5-B1-K0-D1, B5-M0-G2-J5, G2-A0-J2-J6 Number of Visibilities: 300, Number of Closure Phase Relations: 200 Distance measurements: d A = , d B = , d C = UV-Plane East South Spatial Frequency (arcsec -1 ) [B] Results: Hybrid Mapping and Self-Calibration Our work on data measured with the IOTA interferometer showed that for simple source structures like unresolved binaries, even a very poor uv-plane coverage allows to reconstruct reliable aperture synthesis images (Monnier et al. 2004). Already a slightly more complicated object brightness distribution (e.g. a binary system with two resolved stellar surfaces) requires perceptible higher amounts of observation time to achieve a proper convergence of the Hybrid Mapping algorithm (Kraus 2005). Using the data simulated within the APRES-MIDI study, we reconstructed images for a protostellar disk with an inner hole observed on 3 nights with 4 ATs (see Fig. 17; see Tab. 3 for a detailed description of this configuration). The comparison between the image to the left (without noise) and the image to the right (including noise) demonstrates that in cases of extended emission, the algorithm relies not only on a good sampling of the uv-plane, but also strongly on the noise content of the raw data. Additionally, the resolution achievable within the reconstructed maps is ultimatively limited by the beam size due to CLEAN deconvolution used. Figure 17: Images reconstructed using the conventional hybrid mapping algorithm. Left: without noise content; Right: with 5% statistical noise.

15 APreS-MIDI SCIENCE CASE STUDY 57 [C] Results: Difference Mapping Self-calibration methods, difference and hybrid mapping, need a starting model. This model can determine the final image and it will reflect its characteristics if the observed data are insufficient (e.g. sparse uv-coverage) or the parameters of the image reconstruction are incorrect. Furthermore, the uv-coverage directly effects the number of free model parameters. Using the amplitude and closurephase information, as it will be provided by APRES-MIDI operated in a three-telescope mode (e.g., in combination with FINITO), we study the influence of the starting model on the resulting reconstructed image (see Fig. 18; see also Fig. 15 for the original input image). Investigating the visibility data, important information about the source structure can be extracted, such as the size of the components (from the visibility amplitudes on the different baselines) and asymmetric brightness distribution (from the closure-phases below a particular angular resolution). Simple models (e.g., combinations of point sources, Gaussian brightness distribution, uniform disk, optically thin sphere, ring, and exponential brightness distribution) can be easily considered in Difmap. Here, we consider the 5 Nights x 3 ATs baseline configuration (see Sect [A] for a detailed description of this configuration). We find, that a good fitting two-component model is a combination of a Gaussian distribution and ring components. This model is point-symmetric. In spite of this fact, the final reconstructed image (Fig. 18 bottom right) is in agreement with the input source structure: The upper rim is brighter and broader than the lower. The double rim is slightly resolved. Increasing the number of components allows to fine-tune the model. At the end of the image reconstruction, the fit of the closure phases of the clean components (CC) can be used as a Figure 18: Image reconstruction results from visibility and closure phase without noise for the 5 Nights x 3 ATs baseline configuration. The starting models are: a 20 mas disk and a 40 mas ring component (top left), a 10 mas Gaussian distribution and a 40 mas ring (top right), 3 Gaussian components (bottom left), and a combination of a 60 mas ring and a 40 mas Gaussian components (bottom right). These models are the best-fitting two- and three-component models. The differences of the measured and fitted (clean components) closure phases are up to 60, 40, 40 and 20 degrees, respectively.

16 58 APreS-MIDI SCIENCE CASE STUDY measure of the reliability of the reconstructed image. We want to mention, that we also run test simulations, using a point source as the starting model. However, this model did not allow to reconstruct the complex test image (but only the orientation), but was successfully tested for simpler structures, such as binaries. In the second study within the framework of the difference mapping algorithm, we considered the case that Fourier phases of the target can also be measured (such as by operating APRES-MIDI with PRIMA; Delplancke et al. 2002). Since the performance of PRIMA in the N band is not known in the moment, we apply phase errors. Some tens of degrees of phase errors, depending on the uv-coverage, can be corrected with self-calibration. The model for self-calibration can be derived from the Fourier-transform of the visibility data (dirty map). Even with perfectly measured phases, CLEANing can be executed on the Fourier transform of the visibility data (DM), allowing to correct the effect of Fourier plane sampling deficiencies. Finally, these images can be considered as an estimation of what is possible with difference mapping. With the Fourier phase information, a high-quality image reconstruction with the difference mapping algorithm is possible in more cases than with the Building Block method with closure phase information only. Two examples are shown among our reconstruction results in Fig. 19: The first case is that of the 3 Nights x 3 ATs - B configuration. Here, the Building Block method presumably failed because of the lack of the short baselines and the small number of the visibility points (see Sect [A]; see Fig. 19a). The other is case of that of the 3 Nights x 4 ATs - B configuration (see Fig. 19c for the new result).

17 APreS-MIDI SCIENCE CASE STUDY 59 a) b) c) d) e) Figure 19: Reconstructed images using Fourier phase information instead of the closure phase information. The original image is that of a circumstellar disk around a T Tauri, shown in Fig. 15(right). From top to bottom: a) 3 nights x 3 ATs A and B, b) 3 nights x 3 ATs C and D, c) 3 nights x 4 ATs A and B, d) 5 nights x 3 ATs and 5 nights x 4 ATs, e) 7 nights x 3 ATs and 1 night x 4 UTs. See Tab. 3 for a detailed description of the above configurations.

18 60 APreS-MIDI SCIENCE CASE STUDY Noise Tests Fig. 20 shows the result of the image reconstruction with the building block method for the disk image considered in Sect. 4.4, but now under the assumption of noisy data (5% error; 7 nights x 3 ATs = 210 visibilities; 70 closure phases). For each of the 5 independent simulations, a different random seed was used. The noisy artefacts in the surrounding of the object reach an intensity of up to 15% of the maximum intensity in the image. When taking only the central structure into account, the noise results in a relative error in the reconstructed images of about 10% represented, by the dashed line in Fig. 21. Figure 20: Simulations with a relative error of 5%. Figure 21: Horizontal cut through the center of the five noisy images. In the upper plot the mean and the 1-σ-errors are displayed. In the lower panel the relative error is shown. With these good results at hand one may think of another observing strategy: taking more visibility and closure phase measurements in the same time, while accepting a lower signal-to-noise ratio. For Fig. 22, the reconstruction was based on an almost doubled number of simulated measurements (399 visibilities and 133 closure phases) when compared with Fig. 21. The noise in the simulated data was increased by a factor 2 to reflect the reduced time available per measurement (30 min). The reconstructions from the larger number of noisier data points show no significant difference with

19 APreS-MIDI SCIENCE CASE STUDY 61 Figure 22: Horizontal cut through five images reconstructed from data with a noise of 7%. In the upper plot the mean and the 1-σ-errors are displayed. In the lower panel the relative error is visualized. respect to the reconstructions from the slow, but preciser measurements simulated before. This leaves us with a wide variety of available observing strategies that can be optimized with respect to the technical requirements.

20 62 APreS-MIDI SCIENCE CASE STUDY Combination of Single Dish UT Information with VLTI Data Figure 23: The bottom row shows the reconstruction when taking the informations of a single-dish image into account. In the upper row the recontructions without the additional informations are shown. Images taken with the UTs either during the acquisition process or in addition to the interferometric measurements with the UTS or ATs may serve as additional sources for visibilities and closure phases. The images represent perfectly covered uv-planes with baselines comparable to the shortest AT baselines available. In Fig. 23, an example is shown, how the additional data influences the quality of the reconstruction obtained from interferometric data taken with 4 UTs (100 closure phases, 150 squared visibilities). In addition, 64 closure phases and 96 squared visbilities with a signal-to-noise ratio 50 within a pupil diameter of 8 m have been simulated.

21 APreS-MIDI SCIENCE CASE STUDY Conclusion: Image Reconstruction Tests We have tested 3 different image reconstruction methods. All of them allow to reconstruct the main features of the considered disk model, based on an uv-coverage which can be achieved in a reasonable observing time. In particular, 3-7 nights of observations with 3 ATs (at varying locations) will result in an uv-coverage which is sufficient in order to reconstruct images which allow to address the science questions outlined in Sect. 2. As expected, the quality of the reconstructed images improves significantly if a) Fourier phases instead of closure phase relations are considered and b) 4 instead of 3 telescopes will used, such as it will be possible with APRES-MIDI in combination with PRIMA. We find that the building block method (Sect ) produces images with the highest quality. It is able to handle also noisy data without a significant degradation of the reconstructed images. These images allow to discuss and further investigate the inner structure of circumstellar disks and similarly structured objects. When single-dish images taken with the UTs can be included, an improvement of the reconstructed images has been proven by our preliminary tests. References [1] Cornwell, T. J., Wilkinson, P. N. 1981, MNRAS, 196, 1067 [2] Delplancke, F., Leveque, S.A., Kervella, P., Glindemann, A., D Arcio, L. 2002, SPIE 4006, 365 [3] Hofmann, K.-H., Weigelt, G. 1993, A&A, 278, 328 [4] Högbom, J. A. 1974, AAPS, 15, 417 [5] Lawson, P.R., Cotton, W.D., Hummel, C.A., Monnier, J.D., Zhao, M., et al. 2004, AAS, 205, # [6] Monnier, J. D., Traub, W. A., Schloerb, F. P., Millan-Gabet, R., Berger, J.-P., et al. 2004, ApJ, 602, L57 [7] Kraus, S., Schloerb, F.P., Traub, W.A., Carleton, N.P., Lacasse, M., et al. 2005, submitted to AJ [8] Shepherd, M.C. 1995, ASP Conf. Ser. 125 [9] Wolf, S. 2003, Comp. Phys. Comm., 150, 99 [10] Wolf, S., Henning, Th., Stecklum, B. 1999, A&A, 349, 839 [11] Wolf, S., Padgett, D.L., Stapelfeldt, K.R. 2003, ApJ, 588, 373