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Appendix A: Removal of detector artifacts from MIDI acquisition images
As Fig. A.1a shows, noticeable vertical stripes are present in the skysubtracted frames of the MIDI acquisition data of L_{2} Pup. These detector artifacts presumably result from the high brightness of L_{2} Pup (the central region within 0.̋1 is saturated). No vertical stripes appear in the images of the PSF reference stars. The intensity of the vertical stripes is different in the upper
and lower regions with respect to the star, and it therefore is necessary to remove the stripes in the upper and lower regions separately. In each region, we removed the stripes as follows. First, for each column affected by the vertical stripes, we computed the median of the pixel values from the rows sufficiently far away from the star. Then we subtracted this median from all pixels in the column. We carried out this procedure for all columns that were affected by the stripes. Figure A.1 demonstrates that the image is nearly free from the stripes after this procedure.
Fig. A.1
Removal of the detector artifacts in the MIDI acquisition images of L_{2} Pup. a) One of the skysubtracted frame of L_{2} Pup, showing vertical stripes in the columns near the center. b) Same frame after the removal of the vertical stripes. The color scale in the diffraction core is saturated to clearly show the stripes. 

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Appendix B: Image reconstruction
Fig. B.1
Bestfit model consisting of a uniformdisk central star and an elliptical Gaussian shown in the same manner as Fig. 6. 

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The observed visibilities plotted in Fig. 3a show a steep drop at low spatial frequencies ≲3 × 10^{6} rad^{1} (=baselines shorter than 6 m), which suggests a very extended component. The visibilities observed at spatial frequencies from ~6 × 10^{6} rad^{1} to 10^{7} rad^{1} (baselines from 13 to 22 m) may appear reminiscent of the first and the second visibility lobe expected from a uniform disk or limbdarkened disk. However, the visibility from a uniform disk without an extended component is 0.13 (or lower for a limbdarkened disk) in the extrema of the second visibility lobe, much lower than the observed values of ~0.3. With an extended component as revealed by the speckle data, the visibility would be even lower. We first attempted to explain these observed data using geometrical models. While simple geometrical models may not fit the data completely, they are useful for characterizing the approximate geometry of the object and can also be used as an initial model for the image reconstruction.
Appendix B.1: Geometrical model: uniform disk + elliptical Gaussian
We tried to fit the data with a geometrical model consisting of a uniformdisklike central star and an elliptical Gaussian. The free parameters are the uniformdisk diameter of the central star (φ_{⋆}), the fractional flux contribution of the central star f_{⋆}, the widths of the elliptical Gaussian along the major and minor axes σ_{major} and σ_{minor} (the elliptical Gaussian is given by e^{− ((x/σmajor)2 + (y/σminor)2)}), and the position angle of its major axis PA (measured from North to East). We searched for the bestfit model by varying φ_{⋆} = 8 ... 22 (mas) with Δφ_{⋆} = 2 (mas), f_{⋆} = 0.1 ... 0.7 with Δf_{⋆} = 0.05, σ_{major} = 30 ... 100 (mas) with Δσ_{major} = 10 (mas), σ_{minor} = 10 ... 50 (mas) with Δσ_{minor} = 10 (mas), and PA = 70° ... 100° with ΔPA = 5° (the PA was limited to this range from the elongation of the image reconstructed from the speckle data alone). The bestfit model, which is plotted in Fig. B.1, is characterized by φ_{⋆} = 12 mas, f_{⋆} = 0.3, σ_{major} = 50 mas, σ_{minor} = 20 mas, and PA = 95° with a reduced χ^{2} of 27.2. Figure B.1 reveals that the bestfit model cannot reproduce the observed visibilities at spatial frequencies of (0.6−2.0) × 10^{7} rad^{1} (baseline length = 13–44 m). Most of the observed visibilities at these spatial frequencies are noticeably higher than predicted by the model. This means that there is some sharp structure that is not seen in the smooth Gaussian model.
Appendix B.2: Geometrical model: uniform disk + elliptical ring
We then tried a model consisting of the uniformdisklike central star and an elliptical ring, because a ring gives rise to visibilities much higher than a uniform disk or limbdarkened disk at long baselines. The free parameters are the uniformdisk diameter of the central star φ_{⋆}, the fractional flux contribution of the central star f_{⋆}, the semimajor and semiminor axes of the elliptical ring (R_{major} and R_{minor}, respectively), and its position angle PA. The width of the ring was set to be 10% of its radius at each position angle. We searched for the bestfit model by varying φ_{⋆} = 8 ... 22 (mas) with Δφ_{⋆} = 2 (mas), f_{⋆} = 0.1 ... 0.6 with Δf_{⋆} = 0.05, R_{major} = 30 ... 100 (mas) with ΔR_{major} = 10 (mas), R_{minor} = 10 ... 50 (mas) with ΔR_{major} = 10 (mas), and PA = 70° ... 100° with
ΔPA = 5°. Figure B.2 shows a comparison of the bestfit ring model with the observed data. This model is characterized by φ_{⋆} = 16 mas, f_{⋆} = 0.6, R_{major} = 60 mas, R_{minor} = 40 mas, and PA = 95° with the reduced χ^{2} = 54.8. While the visibilities at (0.6−1) × 10^{7} rad^{1} from this model is as high as or even higher than the observed data, the fit to the speckle data and the data at longer baselines is worse than the above star + Gaussian model as shown by the worse reduced χ^{2}.
Fig. B.2
Bestfit model consisting of a uniformdisk central star and an elliptical ring shown in the same manner as Fig. 6. 

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Appendix B.3: Geometrical model: obscured uniform disk + elliptical Gaussian
We found out that the observed data are much better reproduced if the southern half of the aforementioned uniformdisklike central star + elliptical Gaussian model is obscured. In this model, the intensity of the uniformdisk star + elliptical Gaussian model I_{0}(x,y) (x and y are the coordinates on the sky with the origin at the central star) is modified as follows:
where ε is a parameter to smoothen the obscuration edge. We set ε to be 0.2, which decreases the intensity to zero over ~2 mas in the y direction. The free parameters are the uniformdisk diameter of the central star, the fractional flux contribution of the central star f_{⋆}, the widths of the elliptical Gaussian along the major and minor axes σ_{major} and σ_{minor}, and the position angle of its major axis PA. We searched for the bestfit model by varying φ_{⋆} = 8 ... 22 (mas) with Δφ_{⋆} = 2 (mas), f_{⋆} = 0.2 ... 0.6 with Δf_{⋆} = 0.05, σ_{major} = 30 ... 100 (mas) with Δσ_{major} = 10 (mas), σ_{minor} = 10 ... 50 (mas) with Δσ_{minor} = 10 (mas), and PA = 70° ... 100° with ΔPA = 5°.
The bestfit model is characterized by φ_{⋆} = 20 mas, f_{⋆} = 0.45, σ_{major} = 70 mas, σ_{minor} = 30 mas, and PA = 85°. As Fig. B.3 shows, this model can reproduce the observed visibilities much better than the above two models. The uncertainties in f_{⋆}, σ_{major}, σ_{minor}, and PA are ±0.05, ±10 mas, ±10 mas, and ±5°. The uniformdisk diameter of the central star is in the range between 18 and 20 mas. This agrees with the 17.5 mas derived by K14, given the difference in the data and the model used by them and us. The reduced χ^{2} of this model is 16.2, which is much better than the star + Gaussian or star + ring models, but still much larger than 1. This is because the fit to the visibilities observed at the longest baselines (spatial frequencies higher than ~2 × 10^{7} rad^{1}) and to the observed closure phases are not satisfactory. However, this disagreement can be due to smallscale structures that are not included in our geometrical model, but can be modeled by the image reconstruction.
Fig. B.3
Bestfit model consisting of a halfobscured uniformdisk central star and an elliptical Gaussian shown in the same manner as Fig. 6. 

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Appendix B.4: MiRA parameters
We used the bestfit halfobscured star + elliptical Gaussian model as the initial model. The regularization scheme of the maximum entropy method was adopted, with the prior being the Gaussian with the same widths as the initial model. The degree of regularization was set to μ = 10^{3} (see Thiébaut 2008 for details of the regularization scheme and the definition of μ). We also reconstructed the image with the total variation regularization scheme, but the image shows no noticeable differences.
© ESO, 2015