2.1.

Description

The algorithm of Van Dam and Lane describes a technique for deriving the wavefront aberration from two intensity measurements ($I_1$ and $I_2$), acquired at two different defocused planes by using the first derivative of the wavefront rather than contrast proportional to the Laplacian of the phase.¹

This is a new scheme to detect the wavefront directly using the linear relationship obtained from geometrical optics between the slope of the wavefront and the displacement of the photon.

The intensity defines a probability density function of the photon arrival and the method is based on the evolution of the cumulative density function of the intensity with geometric propagation.

In one dimension, the problem is easily solved with a histogram specification procedure with a linear relationship between the slope of the wavefront and the difference in the abscissas of the histograms. In two dimensions, the method requires the use of the Radon transform. In fact, as the authors mention in Ref. 1: “The algorithm consists only of matrix multiplications including the Radon transform and the least-squares reconstruction and one-dimensional interpolations. Consequently, it is suitable for real time implementation in an adaptive optics system.” Furthermore, the method is insensitive to scintillation at the aperture.

2.2.

CUDA Implementation

The Van Dam and Lane algorithm, applied to the entire pupil of the telescope, can recover the atmospherical wavefront phase. The implementation of the algorithm of Van Dam and Lane on specialized hardware, GPU or Field Programmable Gates Arrays type, is needed to achieve real time (the characteristic time of the atmosphere). In our case, we have decided to implement it first in CUDA (GPU). As there is no data dependency in the calculations, it was decided to assign a thread to each pixel of the image. These threads are grouped into blocks of $256\times 256$. Note, that all calculations are performed on GPU (not CPU) whether it be calculation of derivatives, diffractions, interpolations, or matrix products. The derivatives of the Zernike polynomials have been implemented using the definition by Noll for both derivatives $\partial x$ and $\partial y$.³

After the atmospheric contribution to the wavefront is extracted, the algorithm of Van Dam and Lane is, again, applied in parallel over the 36 telescope segments in order to obtain the local tip-tilt. Table 1 summarizes the results of such an implementation on several GPU card models for various final resolutions of the wavefront phase map. As can be seen, it is possible to work within the time stability of the atmosphere, below the 10 ms, up to resolutions of about $1024\times 1024$ pixels.

Table 1

Computation times (ms).

2.3.

Zernike Polynomials Restoration

As an initial test of the algorithm, several recoveries of the wavefront phases have been done on circular pupils. The algorithm recovers the Zernike polynomials of the wavefront phase. The polynomials can be retrieved either individually or as a combination. In the combined case, the algorithm maintains the ratio between the coefficients.

As seen in Fig. 1, the algorithm is able to retrieve the Zernike polynomials that comprise the original phase. For this test, we used the first 50 Zernike polynomials on a $512\times 512$ pixel image. The proportionality is maintained as can be seen by observing the similarity coefficients: mean squared error (MSE), peak signal-to-noise ratio (PSNR)⁴ and mean structural similarity (MSSIM).⁵

Fig. 1

(a) Left: original wavefront phase. Right: recovered wavefront phase. (b) Retrieved coefficients versus Zernike polynomials. With mean squared error $(\mathrm{MSE})=22.0772$, peak signal-to-noise ratio $(\mathrm{PSNR})=34.6914$, and mean structural similarity $(\mathrm{MSSIM})=0.992188$.

The MSSIM index is a method for measuring the similarity between two images. The MSSIM index is the measuring of image quality based on an initial uncompressed or distortion-free image as reference. MSSIM is designed to improve on traditional methods, such as PSNR and MSE, which have proven to be inconsistent with human eye perception. The range of positive values in the MSSIM index is between zero and one. A value of zero means no correlation between images or vectors. A value of one means the two compared images are equal.

Figure 2 shows several recoveries comparing the original and the restored wavefront maps. The original phases are atmospherical wavefronts simulated using a 496 Zernike polynomial expansion.⁶ An 8 m diameter telescope and a 20 cm Fried diameter were considered. The algorithm is able to retrieve the first 120 Zernike polynomials that comprise the simulated atmospherical wavefront phases.

Fig. 2

(a) Simulated atmospherical wavefront phases. (b) Restored wavefront phases. (c) Retrieved coefficients versus Zernike polynomials. The MSE, PSNR, and MSSIM coefficients are included for each map restoration.

3.1.

Simulated Data

The simulations to demonstrate the results have included the generation of blurred images $I_1$ and $I_2$ of the geometric sensor from a wavefront phase containing a random distribution of tip-tilt ($Z_2$ and $Z_3$) in the 36 segments of a large telescope.

The images $I_1$ and $I_2$ have been generated using both Fresnel propagation and the Rayleigh-Sommerfeld propagation⁷ (Fig. 3).

Fig. 3

Intra and extra defocused images obtained by Fresnel propagation.

Figure 4 shows the recovery of the tip-tilt of each of the 36 hexagonal segments in a telescope type Keck. In an extremely large telescope, as the E-ELT would be talking about more than 900 segments, extracting the local tip-tilt from only two blurred images would be extremely useful.

Fig. 4

Comparison between original segments (a) and recovered ones (b). Similarity coefficients: $\mathrm{MSE}=117.128$, $\mathrm{PSNR}=27.4442$, $\mathrm{MSSIM}=0.958495$.

Figure 5 shows a simulation in which the corresponding coefficients for $Z_2$ and $Z_3$ are retrieved. The vertical axis represents the coefficients of tip-tilt and the horizontal axis represents the corresponding segment (1 to 36). With regard to the strokes of the lines, dashed lines represent the initial coefficients and solid lines the coefficients obtained in the recovery.

Fig. 5

(a) Original and retrieved $Z_2$ coefficients for each segment ($\mathrm{MSE}=7.97959\mathrm{e}\text{-}7$, $\mathrm{PSNR}=109.111$). (b) Original and retrieved $Z_3$ coefficients for each segment ($\mathrm{MSE}=1.90616\mathrm{e}\text{-}6$, $\mathrm{PSNR}=105.329$).

3.2.

Experimental Data

The adjustment for simulated data appears to be excellent, however, it is necessary to consider the presence of atmospheric phase and detector noise. What we have done is to use real observations to test the sensitivity of the method.

Images were acquired during the observing campaigns Active Phasing Experiment (APE) at the Paranal Observatory (Chile).⁸ The aim of the experiment was to test four different techniques, Van Dam and Lane technique was not included, for cophasing the segmented mirror. As such, the instrument was installed with four sensors on the Nasmyth focus of Unit 3 of the Very Large Telescope with an APE subsystem with which the data was acquired by diffraction image phase sensing instrument (DIPSI). The segmented mirror [active segmented mirror (ASM)] was set to a known configuration and, then, the intra and extra-focal images were acquired (Fig. 6). Figure 7 shows the results of the tip-tilt recovery from the 36 central segments (three rings). The integration time was chosen to average the effects of atmospheric turbulence (10 s) and measure only the contribution to the wavefront phase of the static aberrations of the deformable mirror. The extraction of the atmospheric contribution is not the aim of this article.

Fig. 6

Intra and extra defocused images obtained by diffraction image phase sensing instrument (DIPSI).

Fig. 7

(a) Original and retrieved $Z_2$ coefficients for each segment (from top to bottom: $\mathrm{MSE}=1.8\mathrm{e}\text{-}10$, 1.5e-10, 1.6e-10 and $\mathrm{PSNR}=145.4$, 146.3, 146.2). (b) Original and retrieved $Z_3$ coefficients for each segment (from top to bottom: $\mathrm{MSE}=1.1\mathrm{e}\text{-}10$, 1.4e-10, 1.6e-10 and $\mathrm{PSNR}=147.7$, 146.5, 146.2). Note that in the DIPSI, the size of a segment is approximately 1 cm, so 1e-5 radians mean sizes of about 100 nm tip-tilt.

Note, that in the DIPSI instrument, the size of a segment is approximately 1 cm which corresponds to 1e-5 radians mean sizes of about 100 nm of tip-tilt. These values of large tip-tilt correspond to the first stages in the process of relocation of a segment recently aluminized. As can be observed, the recoveries are not completely accurate, however, they show similar behavior between original and restored local tip-tilts. New specific observations, with tip-tilt misalignments much smaller, must be done in order to assure the feasibility of this technique.