NEW TECHNIQUES FOR HIGH-CONTRAST IMAGING WITH ADI: THE ACORNS-ADI SEEDS DATA REDUCTION PIPELINE^*

As configured for SEEDS, HiCIAO's H2RG detector reads out data in 32 channels at a pixel rate of 100 kHz with correlated double sampling (CDS). Reading out all 2048 × 2048 pixels thus takes about 1.3 s. H2RG detectors also feature a non-destructive read mode, so that up-the-ramp sampling could be implemented in the future on longer exposures. As discussed by Moseley et al. (2010), each of the 32 readout channels has its own reference voltage, which appears as a bias—a non-zero count level corresponding to zero intensity—in raw HiCIAO data. Superimposed on this is a time-varying reference voltage that is largely shared between the 32 channels. To calibrate the count level corresponding to zero intensity, we therefore need to fit for 32 stable reference voltages and at least one function. The H2RG detector provides four rows of pixels at each detector edge, for a total of 32,704 reference pixels (the "R" in H2RG). These pixels are not light-sensitive, but are subject to the same reference voltages as the rest of the array, and can be used to estimate the pixel-by-pixel bias. For some observations, SEEDS has taken data with the guide window capability (the "G" in H2RG), reading out only a subarray of the detector. In this mode, there is a single readout channel (and reference voltage) and there are no reference pixels.

For SEEDS data taken in the normal 32-channel readout mode, the 512 reference pixels at the ends of the channel are sufficient to determine the stable reference voltages. The H2RG read noise is very nearly Gaussian, so we use a sigma-reject technique to calculate their offset from zero. ACORNS-ADI iteratively rejects 3σ outliers, takes the mean of the remaining reference pixels, and subtracts this value from all of the pixels in each readout channel. This calibration is good to approximately the read noise over $\sqrt{512}$ (the number of reference pixels), or about 1 e⁻ per frame. The first row of Table 1 shows the read noise as measured in a series of 30 dark frames taken in 2010 December with no bias corrections. The difference between the root variance within a readout channel (27 e⁻) and over the entire array (52 e⁻) is almost completely removed by a mean subtraction using only the 512 reference pixels, as shown in the second row of Table 1.

The time-varying reference voltage, largely shared among the 32 readout channels, is more difficult to subtract. Moseley et al. (2010) describe two techniques. One of these relies on reference pixels interspersed throughout the detector array, which would be difficult to implement in an imaging survey like SEEDS. The other technique saves the reference voltage in place of one of the 32 channels of science pixels. Appropriate weighting of the Fourier components of this reference voltage then provides a good estimate of the bias to be subtracted from each readout channel.

The H2RG detector has 1/f noise that extends from the frame rate, ∼1 Hz, to a knee at ∼3 kHz, where the noise becomes uncorrelated and the optimal weighting of the Fourier components falls to zero (Moseley et al. 2010). SEEDS does not currently save the reference voltage, which would require the sacrifice of 1/32 of the FOV; we have therefore estimated the possible improvement in read noise by applying a high-pass filter, removing all read noise with a frequency ≲ 3 kHz from each channel. Removing all of the low-frequency noise reduces the total read noise by about 10%, from 27 e⁻ to 25 e⁻ (fourth row of Table 1).

Without the reference voltage, ACORNS-ADI implements two techniques to estimate and subtract the reference voltage; the user must select which to use. The first technique uses the reference pixels at the edges of channels 1 and 32, which provide eight measurements of the reference voltage every 64 pixels. We first remove outliers with sigma-rejection, and then subtract the convolution of the time series of reference pixels with a masked Gaussian. The Gaussian is normalized to unit area, and is zero where no reference voltage is available. We choose its width to optimize the reference subtraction as follows.

While the H2RG detector has a 1/f noise that extends up to a knee at ∼3 kHz, sparse sampling (and read noise) limits our ability to measure the high frequency noise. As shown in Table 1, a perfect suppression of the noise up to ∼3 kHz would decrease the read noise by ∼10%, or the variance by just under 20%. However, a poor estimate of the reference voltage can increase the noise. We can then estimate our fractional suppression of the read noise as

$$\begin{equation} \frac{0.2}{\ln 3000} \int _{1\,\rm Hz}^{\nu _{\rm max}} \frac{d\nu }{\nu } - \frac{1}{N} , \end{equation} \tag{ 1 }$$

where N is the effective number of pixels used to estimate the reference voltage at each point, ν_max∝N⁻¹, and ln 3000 is the value of the integral with ν_max = 3 kHz. Maximizing Equation (1), we find that N ∼ 40. Since only one in eight pixels has a corresponding reference pixel, this is equivalent to an effective smoothing length of ∼300 science pixels. We therefore cannot suppress a noise of ≳300 Hz, and achieve a ∼5% reduction in read noise (third row of Table 1) rather than a ∼10% reduction (fourth row). Like Moseley et al. (2010), we optimally suppress low-frequency read noise by scaling the reference signal; we find a best-fit scaling of 0.87.

ACORNS-ADI also implements a technique to remove correlated read noise using some or all of the science pixels. To give an accurate estimate of the bias, these pixels must be uniformly illuminated. SEEDS generally observes a saturated central source whose seeing halo extends out to several hundred pixels in radius (several arcseconds); pixels beyond this halo can be used to better estimate the high frequency components of the reference voltage. ACORNS-ADI estimates the (uniform) illumination using the difference between the reference and science pixels, and takes the median of each set of up to 32 simultaneous readouts. Using all of the science pixels, this procedure reduces the read noise by ∼10% over the perfect subtraction of all of the read noise of frequency ≲3 kHz (last line of Table 1), much more than the ∼2% expected from self-subtraction. The result is nearly as good when using only half of the science pixels, those, at least, 800 pixels from the center (fifth line of Table 1). This level, ∼23 e⁻, indicates the read noise that is not shared between readout channels. We recommend using unilluminated science pixels for read noise suppression if they are available. We do not recommend a bias correction using only reference pixels unless the field is crowded.

For a typical exposure of ∼1–10 s, the dark current is ∼0.05–0.5 e⁻, a factor of ∼50–500 lower than the read noise. Pixel-to-pixel variations in the dark current will be still lower. A perfect suppression of the dark current would reduce the read noise by ≲0.01%; we therefore do not attempt to correct for it, but treat the dark current as part of the background.

HiCIAO shares the problem of "hot," indium-contaminated pixels with other H2RG detectors. As a result, ∼2% of HiCIAO's pixels are unusable. These hot pixels are stable from night-to-night, though because the detector slowly degrades with time, new hot pixel maps must be made periodically. ACORNS-ADI corrects these pixels by taking the median of all uncontaminated pixels in a 5 × 5 box centered on the hot pixel. Because the data are oversampled, with a PSF core that is typically ∼6 pixels in diameter, this will not significantly bias the intensity. We mask hot pixels throughout the bias calculation described above.

Because of the short exposure times in most SEEDS images, cosmic rays are rarely a problem. We do implement an algorithm to reject isolated discrepant pixels or clusters of pixels similar to the one used by Lafrenière et al. (2007b). We aggressively smooth the image with a large median filter, identify pixels that are above a certain threshold in the difference between the original and the smoothed maps, and mask them.

Flat-field images are stable to ∼2% from month to month, and are even more stable from night to night. We therefore construct one flat-field for each observing run from nine dithered dome flats. We correct the bias of each frame with the method described in Section 3.1 using only the reference pixels, and then median-combine the nine dithered images. ACORNS-ADI divides each science frame by this master flat after performing the bias correction. We do not attempt to correct for the detector's nonlinearity as it approaches saturation.

The field distortion is the difference between positions on the detector and positions on-sky. We model the distortion using a third-order two-dimensional (2D) polynomial in pixel coordinates relative to the center of the FOV; the distortion at the image center is thus zero by definition. We use Hubble Space Telescope images of the globular clusters M5 and M15 for our reference images, extracting stellar positions using SExtractor (Bertin & Arnouts 2010). We use the same tool to extract stellar positions in HiCIAO images. We then fit for the polynomial coefficients, using Markov Chain Monte Carlo (Press et al. 2007) to minimize the difference between Hubble positions and corrected HiCIAO positions. This technique gives best-fit values and errors on all parameters. On a good night, we measure the horizontal and vertical pixel scales to 0.01%, and the direction of true north to a precision of ∼0005; poor conditions degrade our precision by a factor of up to ∼5. Finally, we use bilinear interpolation to transform our image to the new coordinate system, which has a pixel scale of 9.5 mas, similar to the pixel scale of the raw data. The distortion correction is described in more detail by Suzuki et al. (2010).

Our corrected images appear to be free of systematics, with the distribution of offsets between Hubble and HiCIAO positions being Gaussian in both the horizontal and vertical coordinates, and random over the FOV. The orientation of true north has been stable to ∼003 since the SEEDS survey began in 2009, but the plate scale has changed by up to 2% due to small changes in the optical setup and a new camera lens which was installed in 2011 April. Apart from changes in the overall plate scale, the field distortion has been extremely stable over the duration of the SEEDS survey. It is dominated by a ∼3% difference between the horizontal and vertical pixel scales and a ∼03 offset between the vertical axis on the detector and the celestial north.

To accurately centroid images, we need to build a model of the HiCIAO PSF. Unfortunately, HiCIAO's AO system must be re-tuned at the beginning of each observation, and its PSF varies accordingly. Figure 2 shows this variation in a representative sample of SEEDS images. The top-left PSF is unsaturated, while the other panels each show a typical PSF of a different ADI sequence. The PSF varies strongly with atmospheric conditions and the performance of AO188, HiCIAO's 188-actuator AO system (Hayano et al. 2008), which itself depends on stellar brightness. To capture this variation, we proceed empirically, ultimately building a set of three PSF templates from nearly 3000 individual exposures of saturated stars observed as part of the SEEDS survey.

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We extract three templates from 3000 images by iteratively registering the frames and performing PCA. We initially centroid the frames by fitting Moffat profiles. We manually remove outliers—some of which result from bad data, others from the failure of the algorithm—and use a scaled, unsaturated PSF to flag saturated pixels. We then rescale the data to a common flux and perform PCA on the sequence of images. We use weighted PCA, estimating the noise at each unsaturated pixel as the sum of read noise and photon noise, and ignore saturated pixels.

We now use the mean PSF and the first two principal components from this first pass to refine our centroids. We first re-estimate the centroid of each image by flagging saturated pixels (those with at least 70% of the maximum intensity on the detector) and centroiding the greatest concentration of such pixels. We then compute the rms distance r_rms of the saturated pixels from the provisional center, and mask all pixels within 1.5r_rms. We model the variance of the intensity at each remaining pixel as shot noise plus read noise, and fit the frame with a linear combination of the mean PSF and the first two principal components by minimizing

$$\begin{equation} \chi ^2 = \sum _{{\rm pix}\, i} \frac{1}{\sigma _i^2} \left(I_i - \sum _{{\rm tmpl}\, j} \alpha _j T_{ij} \right)^2, \end{equation} \tag{ 2 }$$

where j indexes the templates T (T₀ represents the mean PSF), i indexes the pixels, and α_j are free parameters. We then shift the template PSF by an integer number of pixels relative to the image (thereby avoiding any artifacts from interpolation), and compute the best-fit χ² as a function of positional offset. Finally, we centroid the χ² map by fitting a 2D quadratic, and take the location of this peak to be the centroid of the image. We then register all of the frames, scale them to a common flux, mask saturated pixels, weight the other pixels, and perform PCA. We repeat this entire sequence of steps one more time to build our final set of PSF templates, which we take to be the mean PSF and the first two principal components.

We register each sequence of saturated frames using the same method described in the previous section. The algorithm initially proceeds in four steps.

• 1.  
  Flag saturated pixels, centroid the greatest concentration of such pixels to compute a provisional center.
• 2.  
  Mask pixels near the provisional center, weight all other pixels by photon and read noise.
• 3.  
  Fit the PSF templates using χ² at an integer-pixel grid of offsets.
• 4.  
  Centroid the map of χ² merit statistics.

We then recenter each frame and compute the average PSF of the ADI sequence. As Figure 2 shows, the average PSF in a given sequence of images can be significantly asymmetric. We therefore compute relative centroids according to the procedure just described, and separately determine the centroid of the average PSF. We do not re-compute a separate PSF model for each ADI sequence. Doing so offers no improvement in performance, and appears to introduce slight interpolation artifacts.

A peculiar feature of HiCIAO makes it possible to visually estimate the centroid of a saturated frame. The HgCdTe H2RG chip is reset pixel-by-pixel and then read out twice; the interface computer records the difference between the two readings. As a result, pixels that saturate rapidly tend to show no difference between the two readings, and hence, are recorded as having zero intensity. We therefore allow the user to interactively select the absolute centroid of the average of an ADI sequence. Figure 3 shows a sample centroid verification image; the innermost circle has a radius of 4 HiCIAO pixels, or about 38 mas. As Figure 3 indicates, it is usually possible to absolutely centroid a sequence of saturated images to an accuracy of at least ∼0.5 pixels, around 5 mas. We then add this user-determined offset to each frame's centroid. In this way, ACORNS-ADI determines the relative centroids automatically, while the user selects the absolute centroid, in the form of a single offset for the sequence of images.

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Unfortunately, it is difficult to independently verify the accuracy of the absolute centroid. The SEEDS survey does have a few image sequences with a bright, unsaturated star, and those taken under favorable conditions confirm an accuracy of ≲0.5 pixels. Image sequences with poor AO performance appear to have larger errors of ∼1–2 pixels. These errors remain impossible to verify on most sequences. We recommend that the user choose a conservative error σ_cen, and scale the sensitivity by the factor

$$\begin{equation} \left(1 + \frac{\sigma _\phi \sigma _{\rm cen}}{R_{\rm PSF}} \right)^{-1}, \end{equation} \tag{ 3 }$$

where σ_ϕ is the standard deviation in parallactic angle and R_PSF is the diameter of the PSF core. Any error in the absolute centroid will smear out a companion by a fractional amount roughly proportional to the field rotation, and inversely proportional to the size of the PSF.

We measure the performance of our new centroiding algorithm using ADI image sequences of single stars. The performance consists of the centroids' relative accuracy, which is completely determined by ACORNS-ADI, and their absolute accuracy, which is determined by the user and is much more difficult to assess. The accuracy of the relative centroids affects the quality of the data reduction, but not its astrometry. Poor image registration will smear out speckles and point sources but without introducing systematic offsets. The astrometric error in the reduced data is thus equal to the error in the absolute centroid, and must be estimated by the user. This varies from data set to data set but for data with good AO performance, is generally ≲0.5 pixels, ∼5 mas.

We can assign upper limits to the errors in image registration using the scatter in fitted centroid positions. This scatter will be due both to tracking errors and PSF fitting errors, which we assume to be uncorrelated. We further assume that tracking errors dominate the slow drift in the PSF position. Before the installation of an atmospheric dispersion corrector (ADC; Egner et al. 2010) in AO188, the PSF would drift over the course of an observation due to differential atmospheric refraction between the visible, in which the AO system guides, and the near-infrared, in which HiCIAO observes. This effect was mostly, though not entirely, eliminated by the installation of the ADC. More recent observations at high airmass (Thalmann et al. 2011) indicate that these slow drifts occur at the level of at most a few pixels (tens of mas) over the course of an ADI sequence.

As we wish to measure only the frame-to-frame fluctuations in the fitted centroids, we fit and subtract a low-order polynomial from the centroid positions in each ADI sequence. An alternative approach, measuring the positional difference between successive frames and dividing by $\sqrt{2}$, gives nearly identical results. We then compute the rms scatter of the residual centroid positions, excluding the most discrepant 1% of points to remove outliers. For a true Gaussian distribution, this outlier exclusion reduces the variance by 10%; we correct our measured variances for this effect. The worst data sets, those in which the PSF varies most and positional jitter is most likely to be real, show an rms scatter of ∼0.3 pixels (3 mas) in both the horizontal and vertical directions, or ∼0.4 pixels overall. Most of the data are much better; 12 of the 21 ADI sequences we used to build the template PSFs showed overall residual rms scatters of less than 0.2 pixels (2 mas), and 17 of 21 had rms scatters of less than 0.3 pixels (3 mas).

Given the variation and asymmetries in HiCIAO's PSF, we allow the user to interactively determine the absolute centroid of an image sequence. However, the algorithm presented above can register a series of images to a typical precision of ∼0.2 pixels, or 2 mas. By centroiding a map of χ², which itself is computed only at integer pixel offsets, our new method avoids interpolating images or PSF templates. This makes it relatively fast and free of systematics.

A simple way to model the PSF is to use the median of all frames in an ADI sequence (see Marois et al. 2006). The model PSF will then include all structures and companions in the FOV averaged over all position angles in the image sequence. Azimuthally extended sources will appear in the model PSF much as they do in individual exposures, while point sources will be smeared out by field rotation.

In this simple technique, the same model PSF is subtracted from each frame. The frames are then de-rotated to a common reference position and co-added. Variations in the PSF, such as a changing Strehl ratio, will strongly degrade the sensitivity of the final, processed image to point sources. A point source itself will suffer from a fractional flux loss roughly proportional to the ratio of the size of the PSF core to the amount of field rotation at its location,

$$\begin{equation} f_{\rm loss} \sim \frac{\lambda }{D} \frac{1}{r_{\rm sep} \Delta \phi }, \end{equation} \tag{ 4 }$$

where Δϕ is the total field rotation and r_sep is the angular separation between the point source and the central star. For a companion at r_sep ∼ 1'' with an H-band PSF core λ/D ∼ 005 and a total field rotation of 60°, f_loss ∼ 5%.

Azimuthally extended sources, like disks, will be suppressed by a factor

$$\begin{equation} f_{\rm loss}(\phi) \sim \int _{\phi - \Delta \phi /2}^{\phi + \Delta \phi /2} \frac{I(\theta)\,d\theta }{I(\phi)}. \end{equation} \tag{ 5 }$$

If we expand I as a Taylor series about ϕ, all of the odd terms vanish due to the symmetry of the integral. In other words, azimuthal gradients, as well as azimuthally symmetric sources, are completely suppressed; only higher-order features survive a median PSF subtraction.

The left two panels of Figure 4 show the original PSFs in annuli, smeared out by the field rotation of the data set, and the effective PSFs after a mean PSF subtraction. The latter are, on average, identical to the effective PSFs produced by a median PSF subtraction. The data set shown had 155 exposures, with a total field rotation of ∼30°, and is typical of a SEEDS observation. The integrated flux in the mean PSF-subtracted image is 0 to within 0.1% of the flux in the raw PSF image.

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Median PSF subtraction is simple both conceptually and computationally, and has been successfully used to measure the geometry of circumstellar disks (Thalmann et al. 2011). However, as discussed above, this technique preserves only high-azimuthal-order disk features; as a result, it is only effective on a small subset of the SEEDS disk sample. Furthermore, other techniques such as LOCI, described in the following section, are much more sensitive to point sources. We include median subtraction as an option in ACORNS-ADI but generally recommend against its use. We use it here mainly as a baseline against which to measure the performance of other algorithms.

LOCI, a technique for empirical PSF modeling, was introduced by Lafrenière et al. (2007b). The LOCI algorithm models the PSF in an ADI frame as a local linear combination of other frames in the sequence, with the coefficients calculated using simple least-squares. In each region of frame i, LOCI takes the other frames {j} and fits coefficients {α_ij}, eventually producing an image of residual intensity,

$$\begin{equation} {\cal R}_i = I_i - \sum _j \alpha _{ij} I_j. \end{equation} \tag{ 6 }$$

LOCI fits for the {α_ij} over optimization regions that are typically several hundred PSF footprints—several thousand pixels—in size. The subtraction regions are generally at least a factor of 10 smaller. We give more details about the calculation of the LOCI coefficients in Appendix A. Because LOCI uses least-squares fitting, the solution for the {α_ij} is a linear problem, with the size of the resulting linear system set by the number of frames in an ADI sequence.

To interpret a LOCI-processed ADI sequence, artificial point sources are added and the data are re-reduced (Lafrenière et al. 2007b). Here, we define the LOCI effective PSF to be the difference between the final, reduced image with and without a faint point source. The right panels of Figure 4 show the effective PSF after reducing a sample SEEDS data set with LOCI. The data set, with 155 frames and a total field rotation of ∼30°, represents a typical SEEDS observation. To ensure that each effective PSF is independent from the others, we add and reduce the faint companions one at a time. We use optimization regions 200 PSF footprints in size and a minimum field rotation of 70% of the PSF full width at half-maximum (FWHM) as our fiducial LOCI parameters.

As with median PSF subtraction (Section 5.1), LOCI subtracts azimuthally displaced copies of a faint source, producing negative "wings" in the effective PSF with an integrated flux approximately equal to the flux in the core. Indeed, the integrated flux in each LOCI panel is zero to within 0.5% of the flux in the original PSFs (top-left panel). The bottom-right panel shows the effect of a random positional jitter on the effective PSFs; such a jitter could be due to unmodeled instabilities in the PSF or image registration errors. These jitters, even if only 1/6 of the PSF FWHM in each coordinate, smear out the PSF cores and significantly degrade the sensitivity of observations (see Figure 4), emphasizing the need for reliable, sub-pixel image registration.

Especially at small separations from the central star, LOCI suppresses companion flux more than does median PSF subtraction. This is partly because the best reference frames tend to be nearby in time (and hence have little relative field rotation). Panels (a)–(c) of Figure 5 demonstrate this effect. Panel (b) shows azimuthally displaced PSF copies weighted by the LOCI coefficients obtained without adding a faint companion; panel (c), which closely resembles the mean-subtracted (lower left) panel of Figure 4, shows the effective PSFs after this step. Because of LOCI's angular protection criterion, there is little intensity suppression in the PSF cores in panel (b). However, this does not account for the full amount of flux loss (panel (e)).

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As we show in Appendix A, the addition of a faint source perturbs the LOCI coefficients themselves. Rather than minimizing the least-squares equation for the companion-free PSF (Equation (6) squared and summed over pixels), we instead minimize

$$\begin{equation} \sum _{\rm pixels} \left((I_i + I^{\prime }_i) - \sum _j \alpha ^{\prime }_{ij} (I_j + I^{\prime }_j ) \right)^2, \end{equation} \tag{ 7 }$$

where I'_i is the perturbing (companion) intensity in frame i from a faint companion, and {α'_ij} are the perturbed LOCI coefficients. We use an approximation for I'_i, minimize Equation (7), and linearize it about the unperturbed coefficients {α_ij} to derive the fractional flux suppression. Appendix A gives the full derivation. This additional effect is the reason why adding too many reference frames can actually degrade LOCI's performance. To a certain degree, which varies according to AO performance and radial separation (and the number of reference frames), LOCI can fit anything. Panel (d) shows this effect in the same SEEDS data sequence; at small separations, it can suppress companion flux by an additional factor of ∼2.

The product of these two effects, the subtraction of azimuthally displaced PSF copies and the perturbation of the LOCI equations, accounts for the suppression of companion flux in a LOCI reduction. Both vary as a function of position but are nearly independent of companion flux. Figure 6 shows the results of aperture photometry on actual LOCI PSFs like those in the right panels of Figure 4. The error bars on individual points indicate the scatter in relative photometry as a function of source flux; that these are zero for the blue points (those with no positional jitter) indicates that the LOCI effective PSFs are linear in source flux. However, positional jitter, whether from AO tracking errors or from poor image registration, can introduce significant systematic errors into the recovered sensitivities.

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To avoid adding test sources everywhere on the FOV, we produce a map of flux suppression using the method derived in Appendix A. As an alternative, we could add faint sources to densely populate the FOV. However, these sources would have to be reduced independently of one another. If there is more than one source in an optimization region, the effect of the sources on the LOCI subtraction coefficients will change (see Equation (A6)). In general, because the linear system will be more heavily constrained, the residual intensity will be larger, and the user will overestimate his or her sensitivity.

The orange curve in Figure 6 indicates the radial profile of our map of simulated flux loss, with the hatched region covering a spread of ±2σ. Because we do not compute the perturbation of the LOCI coefficients self-consistently, and because we neglect asymmetries in the companion PSF, we do not capture the full range of positional variability in relative photometry. For this reason, we recommend using the mean flux suppression at a companion's separation to calibrate its flux. We also recommend adding the azimuthal variance in the partial flux subtraction to the usual annular variance in intensity. Our model is computationally simple and generally does an excellent job of reproducing the typical relative photometry at all angular separations; it is also free of the systematic error that we would introduce by adding and reducing many point sources simultaneously.

Several authors (e.g., Marois et al. 2010; Soummer et al. 2011; Pueyo et al. 2012; Currie et al. 2012) have recently introduced refinements to the LOCI algorithm discussed above. These include reducing the size of subtraction regions to a single pixel, masking a small area around each subtraction region, preconditioning the design matrix, and selecting relatively correlated subsets of the full data as reference frames. We find that for SEEDS data, none of these steps offer a significant improvement in sensitivity. The ineffectiveness of masking an area around each subtraction region is particularly surprising. While this refinement does reduce flux suppression, it does so at the cost of additional noise. It appears that, for SEEDS data, LOCI is often as effective at fitting and removing sources as it is at fitting and removing noise. Even the subtraction of a radial profile from each image, a component of the original LOCI algorithm (Lafrenière et al. 2007b), does not improve sensitivity in typical SEEDS data.

One refinement, suggested by Marois et al. (2010) and Pueyo et al. (2012), does improve the sensitivity of some SEEDS data. Because LOCI can fit sources and noise equally well when given enough reference frames, it is preferable to reduce groups of ∼100 frames at a time; a number somewhat smaller than the size (in PSF footprints) of the optimization regions. We implement this refinement by adding the capability to process data in an integer number of groups of frames, with a single group being equivalent to a normal LOCI reduction.

We introduce three refinements of our own in addition to those listed above.

• 1.  
  Performing PCA on an ADI sequence and subtracting the first n components before applying LOCI, similar to the methods suggested by Soummer et al. (2012) and Amara & Quanz (2012).
• 2.  
  Including principal components as reference frames in the LOCI process.
• 3.  
  Applying LOCI twice, to overcorrect the residuals in the first application.

While we expect these refinements to be useful with a higher Strehl ratio, they seem to suppress noise and sources equally well in SEEDS data with its ∼30% Strehl ratio in the H-band. Soummer et al. (2012) and Amara & Quanz (2012) describe algorithms in which they use a library of PSF components like those we use for image registration (Section 4). Unfortunately, while these components are sufficiently good to register SEEDS images, they do not improve the sensitivity of LOCI. For SEEDS data, they are not even good enough to perform absolute centroiding to better than ∼1 pixel. Unlike a space telescope, HiCIAO's AO system must be re-tuned before each observation. As a result, the PSF variation from one observation to the next is generally much larger than the variation within a single ADI sequence. Applying an initial PCA subtraction also makes it much more difficult to understand the fractional flux loss, and hence the sensitivity to point sources. In other words, it makes the flux suppression in panel (d) of Figure 5 significantly larger and harder to estimate.

We implement all of these refinements as optional features of ACORNS-ADI. With the improved performance of SCExAO, Subaru's next-generation AO system, or when applied to data from other instruments, these refinements may become much more powerful.

The optimal method to combine a sequence of N frames (N ≫ 1) into a final, reduced image depends on the properties of the errors in each frame. For example, if the errors are independent and normally distributed, then taking the mean of all of the frames gives a combined data point with only 4N/(π(2N − 1)) ≈ 64% of the variance obtained by taking their median (Kenney & Keeping 1962). However, using the mean is not robust to outliers, and is a poor choice at small angular separations where speckle residuals may be highly non-Gaussian between frames.

The original LOCI algorithm (Lafrenière et al. 2007b) and various refinements (e.g., Soummer et al. 2011) simply use the median of their LOCI-processed frames, which may not be optimal for HiCIAO data, particularly in regions far from the central star where we expect read noise to dominate. We therefore use the trimmed mean, which is continuous between the mean and median: we sort the image sequence at each spatial location, and take the mean of the middle n points, discarding (N − n)/2 values each at the high and low end. When n = 1, this is equivalent to taking the median; when n = N, the number of images in the sequence, it is equivalent to taking the mean. We derive the efficiency of this estimator for data drawn from a normal distribution in Appendix B.

The top panel of Figure 7 shows this estimator as applied to a sample HiCIAO image sequence of 155 frames. At small angular separations, outliers are relatively common and the mean provides a poor estimator, with a standard deviation ∼20% higher than that of the median. Far from the central star, however, the picture is reversed; using the mean gives a large improvement in sensitivity. At nearly all separations, the optimal solution is somewhere in between, close to the median at small separations and close to the mean further away. We expect (and Figure 7 confirms) that the relationship between the optimal n in the trimmed mean and angular separation from the central star is essentially monotonic.

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We implement a trimmed mean estimator iteratively. We begin with the median of our image sequence and calculate its noise profile. We then calculate the noise profile for an image created by averaging more frames (or trimming fewer), and replace data points in the median image at annuli where the new estimator reduces the variance. We repeat this step, using more frames (larger n) in each successive estimator, until we use nearly all of the frames. We always trim at least 5% of the data to guard against cosmic rays and rare outliers; Figure 7 shows that this approximation incurs at most a 0.5% penalty in noise.

The bottom panel of Figure 7 shows the results of our iterative trimmed mean relative to a simple median of the frames in an image sequence. The new image represents an improvement in noise at all angular separations, with a ∼20% improvement at large separations where the frame-to-frame noise is very nearly Gaussian.

While the frame-to-frame noise at a given pixel may be significantly non-Gaussian, the distribution of trimmed means, due to the central limit theorem, is Gaussian. Our new final image thus retains the noise properties of the original LOCI algorithm; in our sample data set, it produced zero single-pixel false positives (pixels with >5σ fluctuations).

The optimal filter for detecting an object depends on both the object and the character of the noise. For noise that is Gaussian and independent at adjacent pixels, the optimal filter to search for point sources is the normalized PSF, referred to as a matched filter. In this section, we measure the performance of three filters for point sources in the HiCIAO data.

• 1.  
  A matched filter.
• 2.  
  A circular aperture.
• 3.  
  A "truncated" matched filter set to zero outside an aperture.
• 4.  
  A 2D median filter.

Figure 8 shows the relative performance of these filters. We truncate the matched filter at twice the radius of our fiducial 005 diameter aperture to limit the impact of the outer wings, which can depend strongly on azimuthal position (see Figure 4). We do not show the full matched filter, which fails to outperform aperture photometry even when assuming perfect knowledge of the effective PSF.

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Perhaps surprisingly, a 005 diameter (1.2λ/D at 1.6 μm) circular aperture seems to offer the best performance of all of the filters. A 2D median filter is not optimal, especially far from the central star, because the noise in the reduced frame is approximately Gaussian. The poor performance of the matched filter, on the other hand, results from the strong correlation of the residual intensity in neighboring pixels. Figure 9 demonstrates this correlation in Fourier space; it results from averaging adjacent pixels when interpolating onto a new spatial array. We interpolate each frame three times during the data reduction process: when applying the distortion correction, when recentering, and finally when de-rotating each image to a common orientation on-sky. We have verified that we can closely reproduce the power spectrum of noise at large separations by smoothing white noise.

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In addition to the suppression of noise at high spatial frequency, Figure 9 shows an increase in power at a few λ/D (a few PSF diameters), particularly in regions close to the central star. This is an artifact of the LOCI algorithm, which tends to give zero flux averaged over a spatial region larger than a PSF core. As a result, LOCI introduces an anti-correlation in intensity over scales larger than the size of the PSF. This also helps explain the poor performance of the full matched filter, which would otherwise take advantage of the negative wings in the effective PSF: large random fluctuations will also tend to be surrounded by regions of negative intensity.

To date, the vast majority of high-contrast direct imaging observations have not detected substellar companions. However, non-detections may still be used to test models of planet and brown dwarf frequency, separation, and luminosity (e.g., Lafrenière et al. 2007b; Bonavita et al. 2012). Such analyses rely on the accuracy of sensitivity maps. As we show in Figure 6, it is easy to systematically overestimate sensitivity by failing to include effects such as PSF fluctuations and image registration errors.

It has become widely accepted within the community to estimate sensitivity by computing the standard deviation in the final, combined image in annuli around the central star (e.g., Lafrenière et al. 2007b; McElwain et al. 2008; Metchev & Hillenbrand 2009; Vigan et al. 2012). At separations of more than a few λ/D, these annuli are already at least several tens of PSF footprints in size. To account for LOCI's suppression of companion flux, the data set is re-reduced after adding faint sources to compute the fractional flux loss as a function of radial separation. We begin in the same way, computing the standard deviation of the residual intensity after convolving with our chosen 005 aperture. We correct for companion flux suppression using our own method, which we describe in Section 5.3 and Appendix A.

The result of our sensitivity analysis is not simply a radial profile, but a full 2D sensitivity map, obtained with modest additional computational cost. As a final step, we multiply this map by the scaled aperture photometry of the central star in a sequence of reference images, producing a contrast map.

Most of the algorithms presented above apply to ADI data taken by any instrument. ACORNS-ADI can reduce these data with minimal modification by the user, who must supply the following.

• 1.  
  A flat-field correction and hot pixel mask.
• 2.  
  An (optional) field distortion correction.
• 3.  
  A set of PSF templates.
• 4.  
  A curve describing the integrated overlap of PSF cores.

The most difficult item to compute is the set of PSF templates. We hope to supply a set of images for each of several instruments over the coming months. The last item simply refers to the flux in an aperture displaced a certain number of pixels from the PSF centroid, and is used to compute the fractional flux loss in LOCI.

While it is easy to extend ACORNS-ADI to other instruments, we offer several notes of caution when doing so. Different instruments have different conventions for header data such as the exposure time, the number of co-adds in a frame, and the image orientation with the image rotator off. For example, in HiCIAO, the keyword "EXPTIME" refers to the total integration time of the frame, while for NIRI, "EXPTIME" refers to the integration time for each co-add. This makes it difficult to write a fully general software package. Unfortunately, ACORNS-ADI cannot detect these differing conventions automatically, and the user must be cautious.