The Scaled Gradient Projection Method: An Application to Nonconvex Optimization

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1 2332 PIERS Proceedings, Prague, Czech Republic, July 6 9, 2015 The Scaled Gradient Projection Method: An Application to Nonconvex Optimization M. Prato 1, A. La Camera 2, S. Bonettini 3, and M. Bertero 2 1 Dipartimento di Scienze Fisiche, Inormatiche e Matematiche Università di Modena e Reggio Emilia, Via Campi 213/b, Modena 41125, Italy 2 Dipartimento di Inormatica, Bioingegneria, Robotica e Ingegneria dei Sistemi Università di Genova, Via Dodecaneso 35, Genova 16145, Italy 3 Dipartimento di Matematica e Inormatica Università di Ferrara, Via Saragat 1, Ferrara 44122, Italy Abstract The scaled gradient projection (SGP) method is a variable metric orward-bacward algorithm designed or constrained dierentiable optimization problems, as those obtained by reormulating several signal and image processing problems according to standard statistical approaches. The main SGP eatures are a variable scaling matrix multiplying the gradient direction at each iteration and an adaptive steplength parameter chosen by generalizing the well-nown Barzilai-Borwein rules. An interesting result is that SGP can be exploited within an alternating minimization approach in order to address optimization problems in which the unnown can be splitted in several blocs, each with a given convex and closed easible set. Classical examples o applications belonging to this class are the non-negative matrix actorization and the blind deconvolution problems. In this wor we applied this method to the blind deconvolution o multiple images o the same target obtained with dierent PSFs. In particular, or our experiments we considered the NASA unded Fizeau intererometer LBTI o the Large Binocular Telescope, which is already operating on Mount Graham and has provided the irst Fizeau images, demonstrating the possibility o reaching the resolution o a 22.8 m telescope. Due to the Poisson nature o the noise aecting the measured images, the resulting optimization problem consists in the minimization o the sum o several Kullbac-Leibler divergences, constrained in suitable easible sets accounting or the dierent eatures to be preserved in the object and the PSFs. 1. INTRODUCTION Intererometry is a standard way exploited in astronomical imaging to obtain high angular resolution starting rom two or more telescopes with smaller diameters. Famous examples o astronomical intererometers are the Very Large Telescope Intererometer (VLTI), the Navy Prototype Optical Intererometer (NPOI) and the Center or High Angular Resolution Astronomy (CHARA) array. A urther Fizeau intererometer which recently provided its irst images is that located at the Large Binocular Telescope (LBT, Mount Graham, Arizona) and called Large Binocular Telescope Intererometer (LBTI [1]). With its two 8.4-m primary mirrors, separated by a 14.4-m center-tocenter distance, and mounted on a common alt-az mount, LBT can be considered the very irst Extremely Large Telescope (ELT). The adaptive secondary mirrors provide high-order adaptive optics corrections which produce images with Strehl Ratio (SR), i.e., the ratio o pea diraction intensity o an aberrated versus perect waveorm (see, e.g., [9]), greater than 0.9. LBTI collects the light rom both primary mirrors and combines the two beams on the image plane (Fizeau imaging). In order to obtain a uniorm coverage o the requency plane, several LBTI images at dierent rotation angles are needed. Each o these measured images will be aected by several sources o noise, whose statistical nature includes both Poisson (due to, e.g., radiation rom the object, bacground, dar current) and Gaussian (read-out-noise (RON) due to the ampliier) components. It has been shown [14] that the latter one can be handled by adding the variance o the RON to the measured images and the bacground and considering the resulting data as purely Poissonian. As a result o this, a statistical approach to the image reconstruction problem leads to the minimization o the sum o a given number p (equal to the number o images acquired by LBTI) o Kullbac- Leibler divergences min KL(, ω 1,..., ω p ) Ω ω 1,...,ωp Ωω p n 2 ( (g ) i (g ) i ln (ω + b ) i =1 ) + (ω + b ) i (g ) i, (1)

2 Progress In Electromagnetics Research Symposium Proceedings 2333 where: is the n n image to be recovered; g is the -th n n measured image, and b the corresponding n n bacground radiation; ω is the unnown n n point spread unction (PSF) related to g, which has to be estimated during the reconstruction procedure; Ω and Ω ω are the easible sets o and ω, accounting or physical constraints that object and PSFs have to satisy. Since we are assuming that the PSFs are unnown, problem (1) is a so-called blind deconvolution problem [7], which is highly ill-posed due to the ininite solutions available. Moreover, rom the optimization point o view the objective unction in (1) is nonconvex, thus leading to the presence o multiple stationary points. A possible way to address this ind o problems is to introduce suitable constraints on the unnown to reduce the set o easible solutions. In particular, in [11, 12] the ollowing easible sets have been considered { } { } n 2 Ω = R n2 0, i = 1 p n 2 n 2 (g b ) i, Ω ω = ω R n2 0 ω s, ω i = 1, p =1 where the upper bound s is estimated rom the nowledge o the SR. Moreover, in [12] an alternating minimization approach or the solution o (1) has been proposed based on the scaled gradient projection (SGP) method [3]. The aim o this paper is to exploit the same optimization approach to clash with a urther deect which characterizes intererometric images. As nown, the PSF can be described as the product o the Airy unction (representing the PSF o a single mirror) by the intererometric term (1 + cos θ), where θ depends on the current pixel and the orientation o the baseline (besides the other parameters o the acquisition system) 1. I we consider a (small) dierence in the two optical paths, i.e., there is a residual error when the system corrects or the so-called piston error, then the intererometric term becomes (1 + cos(θ + φ)). This phase error φ moves the position o the ringes in the direction o the baseline (and thereore in the direction orthogonal to the ringes themselves) with the result that the PSF is not symmetric with respect to the original PSF. The corresponding image is aected by the same eect. 2. THE CYCLIC BLOCK COORDINATE GRADIENT PROJECTION METHOD The cyclic bloc coordinate gradient projection (CBCGP) method is an alternating minimization algorithm originally proposed in [2] in the context o non-negative matrix actorization [6] and adapted in ollowing wors to astronomical blind deconvolution problems [4, 11, 12]. The main idea at the basis o this approach is a partial minimization over each bloc o variables in a Gauss-Seidel style perormed inexactly by means o the scaled gradient projection (SGP) method [3]. The SGP algorithm is a irst-order method which applies to a general constrained optimization problem min ϕ(x). (2) x Ω The main steps o SGP are reported in Algorithm 1, where D is the set o diagonal positive deinite matrices with diagonal elements bounded in a positive interval. I ϕ is equal to the Kullbac-Leibler divergence and the PSFs are normalized to unit volume, a requently used scaling matrix is the one borrowed rom the Richardson-Lucy method [8, 13] and deined as { 1 { [D ] ii = max µ, min µ, x ()}}, with µ > 1 (see, e.g., [3, 5, 10]). As concerns the choice o the steplength parameter α, we adopted a suitable alternation o the Barzilai and Borwein rules (see [3, 10] or urther details). The CBCGP scheme applied to problem (1) is reported in Algorithm 2. 1 For more details: θ(n 1, n 2 ) = A(n 1 cos α + n 2 sin α), where α is the hour angle o the baseline and A = 2π B/λ, being the pixel size o the image in rad/px, B the distance between the two mirrors and λ the observed wavelength.

3 2334 PIERS Proceedings, Prague, Czech Republic, July 6 9, 2015 Algorithm 2: Scaled gradient projection (SGP) method Choose the starting point x (0) Ω, set the parameters β, δ (0, 1), 0 < α min < α max. For = 0, 1, 2,... do the ollowing steps: Step 1. Choose the steplength parameter α [α min, α max ] and the scaling matrix D D; Step 2. Projection: y () = P Ω,D 1(x () α D ϕ(x () )); Step 3. Descent direction: d () = y () x () ; Step 4. Set λ = 1; Step 5. Bactracing loop: I ϕ(x () + λ d () ) ϕ(x () ) + βλ ϕ(x () ) T d () then go to Step 6; Else set λ = δλ and go to Step 5. i Step 6. Set x (+1) = x () + λ d (). Algorithm 3: Cyclic bloc coordinate gradient projection (CBCGP) method Choose the starting points (0) Ω, ω (0) 1,..., ω(0) p Ω ω, and two integers N, N ω 1. For h = 0, 1, 2,... do the ollowing steps: Step. Choose an integer 1 N (h) Algorithm 2 to problem starting rom the point (h). Step ω. For = 1,..., p do the ollowing steps: Choose an integer 1 N ω (h) Algorithm 2 to problem starting rom the point ω (h). N and compute (h+1) by applying N (h) iterations o min KL(, ω (h) 1,..., Ω ω(h) p ) (3) N ω and compute ω (h+1) by applying N (h) ω iterations o min KL( (h+1), ω (h+1) 1,..., ω (h+1) ω Ω 1, ω, ω(h) +1,..., ω(h) p ) (4) ω 3. NUMERICAL EXPERIMENTS We simulated several diraction limited, SR = 1, LBTI PSFs changing the values o the mentioned phase error. Each set o PSFs is a cube o three PSFs with baseline angles at 0, 60, and 120 and corresponds to a dierent value o φ (ranging rom 0, 0.2,..., 1.0). An horizontal cut o the 0 PSFs is shown in the let panel o Fig. 1. By choosing a model o star cluster (based on the nine brightest stars o the Pleiades, as described in [10, 11]), we used these PSFs or generating several multiple images. We adopted a pixel-size o 11 mas and we supposed to acquire images in M-band (λ = 4.8 µm). We applied our blind algorithm by using as initialization the autocorrelation o the ideal diraction limited PSF (i.e., the PSF with no phase error). We remar that this choice has a SR o about 0.35, as described in our previous papers, with positions o the ringes that do not correspond with those in the images. Thereore our goal is to veriy whether the algorithm is able to reconstruct both the object and the PSFs and what is the maximum phase error which gives satisying results. Following [11, 12], we chose N (h) = 50 and N (h) ω = 1 or all h,, and we perormed 1000 outer

4 Progress In Electromagnetics Research Symposium Proceedings 2335 (a) (b) (c) Figure 1: (a) The horizontal cut o the simulated PSFs or each o the phase error φ and or the baseline angle o 0. A positive value moves the position o the ringes towards let. (b) The input image corresponding to φ = 1.0 or the baseline angle o 0. (c) The 0 PSF used or the initialization. The images are shown in log scale with a zoom actor o 2. Table 1: Astrometric error (AE), magnitude average relative error (MARE) and root mean square error (RMSE) or dierent phase errors φ. For the three PSFs, the values in bracets denote the errors ater re-centering. The normalized value o the objective unction is also shown. φ AE (pixels) MARE RMSE 0 RMSE 60 RMSE 120 2/(pn 2 )KL 0.0 < % 0.7% 0.7% < % 0.6% 0.6% % (4.5%) 28.0% (16.5%) 26.6% (16.2%) % (15.9% ) 45.2% (18.3%) 20.0% (11.8%) % (20.0%) 64.1% (9.3%) 39.1% (9.2%) % (18.8%) 64.5% (9.1%) 39.0% (9.1%) (a) (b) Figure 2: (a) The horizontal cut o the reconstructed PSFs or each o the phase error φ. (b) The six reconstructed PSFs corresponding to φ = 0, 0.2,..., 1.0 (in order rom let to right, rom top to bottom) and or the baseline angle o 0. The images are shown in log scale with a zoom actor o 4. iterations. In order to evaluate the quality o the reconstructions we computed three errors: the astrometric error (AE), the magnitude average relative error (MARE), and the root mean square

5 2336 PIERS Proceedings, Prague, Czech Republic, July 6 9, 2015 error (RMSE). The three errors are deined as: AE = 1 q q (xi x i ) 2 + (y i ỹ i ) 2 ; MARE = 1 q q m i m i m i ; RMSE = ω ω 2 ω 2 where q is the number o stars, (x i, y i ) and ( x i, ỹ i ) are the reconstructed centroid and the original position o each star, m i and m i are the reconstructed and the true magnitudes, ω and ω are the reconstructed and true PSFs and 2 denotes the Euclidean norm. All the errors are reported in Table 1. As concerns RMSE, it is interesting to note that the values are quite high or all cases with φ > 0.2. Indeed, since each star is reconstructed with a small, but signiicant, astrometric error, this causes a small displacement o the reconstructed PSF with respect to the original one used in the computation o the RMSE. Thereore we re-centered each PSF by means o a sub-pixel-precision procedure and we computed again the RMSE (the new values are shown in parenthesis). O course, in case o real images this operation is not necessary. In conclusion, the proposed strategy seems to provide good reconstructions o the PSFs only i the measured images are aected by a low phase error, although the magnitude o the stars are in general well recovered. Future wor will include the addition o explicit regularization terms in the objective unction speciically aimed at preserving the eatures which characterize the LBTI PSFs. ACKNOWLEDGMENT The present paper has been supported by the Italian Ministry or University and Research (projects FIRB Futuro in Ricerca 2012, contract RBFR12M3AC, and PRIN 2012, contract 2012MTE38N) and the Italian Gruppo Nazionale per il Calcolo Scientiico (GNCS). REFERENCES 1. Bailey, V. P., P. M. Hinz, PA. T. uglisi, et al., Large binocular telescope intererometer adaptive optics: On-sy perormance and lessons learned, Proc. SPIE, Vol. 9148, , Bonettini, S., Inexact bloc coordinate descent methods with application to the nonnegative matrix actorization, IMA J. Numer. Anal., Vol. 31, No. 4, , Bonettini, S., R. Zanella, and L. Zanni, A scaled gradient projection method or constrained image deblurring, Inverse Probl., Vol. 25, No. 1, , Bonettini, S., A. Cornelio, and M. Prato, A new semiblind deconvolution approach or Fourierbased image restoration: An application in astronomy, SIAM J. Imaging Sci., Vol. 6, No. 3, , Bonettini, S. and M. Prato, Accelerated gradient methods or the X-ray imaging o solar lares, Inverse Probl., Vol. 30, No. 5, , Lee, D. D. and H. S. Seung, Learning the part o objects rom non-negative matrix actorization, Nature, Vol. 401, No. 6755, , Levin, A., Y. Weiss, F. Durand, and W. T. Freeman, Understanding and evaluating blind deconvolution algorithms, IEEE Conerence on Computer Vision and Pattern Recognition, 2009, CVPR 2009, , Paris, France, Jun Lucy, L. B., An iterative technique or the rectiication o observed distributions, Astronom. J., Vol. 79, No. 6, , Mahajan, V. N., Strehl ratio or primary aberrations in terms o their aberration variance, J. Opt. Soc. Am., Vol. 73, No. 6, , Prato, M., R. Cavicchioli, L. Zanni, P. Boccacci, and M. Bertero, Eicient deconvolution methods or astronomical imaging: Algorithms and IDL-GPU codes, Astron. Astrophys., Vol. 539, A133, Prato, M., A. La Camera, S. Bonettini, and M. Bertero, A convergent blind deconvolution method or post-adaptive-optics astronomical imaging, Inverse Probl., Vol. 29, No. 6, , Prato, M., A. La Camera, S. Bonettini, S. Rebegoldi, M. Bertero, and P. Boccacci, A blind deconvolution method or ground-based telescopes, New Astron., Vol. 40, 1 13, Richardson, W. H., Bayesian based iterative method o image restoration, J. Opt. Soc. Amer., Vol. 62, No. 1, 55 59, Snyder, D. L., A. M. Hammoud, and R. L. White, Image recovery rom data acquired with a charge-coupled-device camera, J. Opt. Soc. Am. A, Vol. 10, No. 5, , 1993.