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...inal images obtained using a face detector; and (b) average of the reconstructed low-rank images. We obtain an initial estimate of the transformation in each image using the Viola-Jones face detector =-=[39]-=-. We again align the images to an 80 ×60 canonical frame. For this experiment, we use affine transformations G = Aff(2) in RASL, to cope with the large pose variability in LFW. Since there is no groun...

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... are essential for its practical use. Fortunately, a recent flurry of work on high-dimensional nuclear norm minimization has shown that such problems are well within the capabilities of a standard PC =-=[19, 20, 2, 22, 17]-=-. In this section, we show how one such fast first-order method, the Accelerated Proximal Gradient (APG) algorithm [2, 22, 17], can be adapted to efficiently solve (6). The APG approach replaces the e...

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... classification. Intelligently harnessing the information encoded in these large sets of images seems to require more efficient and effective solutions to the long-standing batch image alignment task =-=[18, 3]-=-: given many images of an object or objects of interest, align them to a fixed canonical template. To a large extent, progress in batch image alignment has ∗ This work was supported by grants NSF IIS ...

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... small fraction of all pixels in an image, we can model them as sparse errors whose nonzero entries can have arbitrarily large magnitude. This model has been successfully employed in face recognition =-=[19]-=-. In addition to occlusions, real images typically contain some noise of small magnitude in each pixel. To keep our discussion simple, we assume here that such noise is negligible in magnitude as comp...

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...ch as zooming in on a single dark pixel or a dark region in the images. i=1 3 This kind of iterative linearization has a long history in gradient algorithms for batch image alignment (see, e.g., [9], =-=[20]-=- and references therein). More recently a similar iterative convex programming approach was proposed for single-to-batch image alignment in face recognition [21]. April 14, 2011 DRAFTREVISED MANUSCRI...

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...rg, and reference IEEECS Log Number TPAMI-2010-07-0582. Digital Object Identifier no. 10.1109/TPAMI.2011.282. 0162-8828/12/$31.00 2012 IEEE Published by the IEEE Computer Society minimization [15], =-=[16]-=- have shown that it is indeed possible to efficiently and exactly recover low-rank matrices despite significant corruption, using tools from convex programming. These developments prompt us to revisit...

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...assification. Intelligently harnessing the information encoded in these large sets of images seems to require more efficient and effective solutions to the long-standing batch image alignment problem =-=[2]-=-, [3]: Given many images of an object or objects of interest, align them to a fixed canonical template. (a) Original images (b) Aligned images (c) Low-rank component (d) Recovered errors (e) Average o...

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...ely low-rank. This assumption holds ,i = 1,...,n are quite generally. For example, if the I0 i images of some convex Lambertian object under varying illumination, then a rank-9 approximation suffices =-=[1]-=-. Being able to correctly identify this low-dimensional structure is crucial for many vision tasks such as face recognition. Modeling corruption. In practice, however, this lowrank structure can be ea...

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..., and YouTube has led to a dramatic increase in the amount of visual data available online. Within the computer vision community, this has inspired a renewed interest in large, unconstrained datasets =-=[1]-=-. Such data pose steep challenges to existing vision algorithms: significant illumination variation, partial occlusion, as well as poor or even no alignment (see Figure 1(a) for example). This last di...

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...en two terms has been studied extensively as the alternating direction method of multipliers in the optimization literature and its convergence has been well established for various cases [32], [33], =-=[34]-=-. In particular, the convergence for the Principal Component Pursuit problem – essentially problem (7) without the term associated with ∆τ – has been established in [29]. Recently, [35] obtained a con...

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...al for its practical use. Fortunately, a recent flurry of work on high-dimensional nuclear norm minimization has shown that such problems are well within the capabilities of a standard PC [27], [28], =-=[29]-=-. In this section, we show how one such fast first-order method, the Augmented Lagrange Multiplier (ALM) algorithm [29], [30], [16], can be adapted to efficiently solve (7). The basic idea of the ALM ...

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...SCRIPT SUBMITTED TO IEEE TRANS. PAMI, APRIL 2011. 3 To a large extent, progress in batch image alignment has been driven by the introduction of increasingly sophisticated measures of image similarity =-=[4]-=-. Learned-Miller’s influential congealing algorithm seeks an alignment that minimizes the sum of entropies of pixel values at each pixel location in the batch of aligned images [5], [6]. If we stack t...

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... between two terms has been studied extensively as the alternating direction method of multipliers in the optimization literature and its convergence has been well established for various cases [32], =-=[33]-=-, [34]. In particular, the convergence for the Principal Component Pursuit problem – essentially problem (7) without the term associated with ∆τ – has been established in [29]. Recently, [35] obtained...

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...em of fitting a low-rank model to highly corrupted data [14], a problem that until recently lacked a polynomial-time algorithm with strong performance guarantees. Recent advances in rank minimization =-=[15]-=-, [16] have shown that it is indeed possible to efficiently and exactly recover low-rank matrices despite significant corruption, using tools from convex programming. These developments prompt us to r...

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... are essential for its practical use. Fortunately, a recent flurry of work on high-dimensional nuclear norm minimization has shown that such problems are well within the capabilities of a standard PC =-=[27]-=-, [28], [29]. In this section, we show how one such fast first-order method, the Augmented Lagrange Multiplier (ALM) algorithm [29], [30], [16], can be adapted to efficiently solve (7). The basic idea...

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...transformations from a finitedimensional group G that has a parametric representation, such as the similarity group SE(2) × R+, the 2-D affine group Aff(2), and the planar homography group GL(3) (see =-=[18]-=- for more details on transformation groups). Consolidating the above two models, we formulate the image alignment problem as follows. Suppose that I1,I2,...,In represent n input images of the same obj...

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...ees of robustness or convergence rate. This somewhat unsatisfactory status quo is mainly due to the extremely difficult nature of the core problem of fitting a low-rank model to highly corrupted data =-=[14]-=-, a problem that until recently lacked a polynomial-time algorithm with strong performance guarantees. Recent advances in rank IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 34, ...

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...nating between two terms has been studied extensively as the alternating direction method of multipliers in the optimization literature and its convergence has been well established for various cases =-=[32]-=-, [33], [34]. In particular, the convergence for the Principal Component Pursuit problem – essentially problem (7) without the term associated with ∆τ – has been established in [29]. Recently, [35] ob...

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...batch image alignment (see, e.g., [9], [20] and references therein). More recently a similar iterative convex programming approach was proposed for single-to-batch image alignment in face recognition =-=[21]-=-. April 14, 2011 DRAFTREVISED MANUSCRIPT SUBMITTED TO IEEE TRANS. PAMI, APRIL 2011. 9 Algorithm 1 (Outer loop of RASL) INPUT: Images I1,...,In ∈ R w×h , initial transformations τ1,...,τn in a certain...

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...age similarity [4]. Learned-Miller’s influential congealing algorithm seeks an alignment that minimizes the sum of entropies of pixel values at each pixel location in the batch of aligned images [5], =-=[6]-=-. If we stack the aligned images as the columns of a large matrix, this criterion demands that each row of this matrix be nearly constant. Conversely, the least squares congealing procedure of [7], [8...

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...ontain some noise of small magnitude in each pixel. This can be easily augmented into our model by adding a noise matrix Z of bounded magnitude to the equality constraint in (7). It has been shown in =-=[22]-=- that sparse and low-rank matrix decomposition (without transformations) by convex optimization is stable to additive Gaussian noise of small magnitude, in addition to sparse errors. It may be possibl...

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...of image similarity [4]. Learned-Miller’s influential congealing algorithm seeks an alignment that minimizes the sum of entropies of pixel values at each pixel location in the batch of aligned images =-=[5]-=-, [6]. If we stack the aligned images as the columns of a large matrix, this criterion demands that each row of this matrix be nearly constant. Conversely, the least squares congealing procedure of [7...

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...es [32], [33], [34]. In particular, the convergence for the Principal Component Pursuit problem – essentially problem (7) without the term associated with ∆τ – has been established in [29]. Recently, =-=[35]-=- obtained a convergence result for certain three-term alternation applied to the noisy principal component pursuit problem (see also [36]). However, [35] reflects a very similar theory-practice gap – ...

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...e a log-determinant measure that can be viewed as a smooth surrogate for the rank function [10]. The low-rank objective can also be directly enforced, as in Transformed Component Analysis (TCA) [11], =-=[12]-=-, which uses an EM algorithm to fit a low-dimensional linear model, subject to domain transformations drawn from a known group. A major drawback of the above approaches is that they do not simultaneou...

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...inimize a log-determinant measure that can be viewed as a smooth surrogate for the rank function [10]. The low-rank objective can also be directly enforced, as in Transformed Component Analysis (TCA) =-=[11]-=-, [12], which uses an EM algorithm to fit a low-dimensional linear model, subject to domain transformations drawn from a known group. A major drawback of the above approaches is that they do not simul...

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..., are required for the algorithm to converge. It is important to realize that, in general, manifolds formed by transformed images may not be C 2 (or not even C 1 ), due to the presence of sharp edges =-=[26]-=-. However, in our case, we can view the digital images Ii ◦τi as resampling transformations of an ideal bandlimited reconstruction Îi obtained from the digital image Ii, in which case the mapping x ↦→...

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...ariations and gross pixel corruptions or partial occlusions that often occur in real images (e.g., shadows, hats, glasses in Figure 1). The Robust Parameterized Component Analysis (RPCA) algorithm of =-=[13]-=- also fits a low-rank model, and uses a robust fitting function to reduce the influence of corruption and occlusion. Unfortunately, this leads to a difficult, nonconvex optimization problem, with no t...

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...ly 20% of the pixels in each image (i.e., ρ = 0.2). The results are shown in Figure 7. We observe that the output images are well-aligned with respect to each other and free of corruptions. Recently, =-=[38]-=- proposed an image denoising algorithm based on low-rank matrix completion. Our method differs from that work in three main aspects. Firstly, we denoise the images globally instead of in a patch-based...

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...5], [6]. If we stack the aligned images as the columns of a large matrix, this criterion demands that each row of this matrix be nearly constant. Conversely, the least squares congealing procedure of =-=[7]-=-, [8] seeks an alignment that minimizes the sum of squared distances between pairs of images, and hence demands that the columns be nearly constant. In both cases, if the criterion is satisfied exactl...

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...lem of fitting a low-rank model to highly corrupted data [8], a problem that until recently lacked a polynomial-time algorithm with strong performance guarantees. Recent advances in rank minimization =-=[5, 4]-=- have shown that it is indeed possible to efficiently and exactly recover low-rank matrices despite significant corruption, using tools from convex programming. These developments prompt us to revisit...

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...ssential for its practical use. Fortunately, a recent flurry of work on high-dimensional nuclear norm minimization has shown that such problems are well within the capabilities of a standard PC [27], =-=[28]-=-, [29]. In this section, we show how one such fast first-order method, the Augmented Lagrange Multiplier (ALM) algorithm [29], [30], [16], can be adapted to efficiently solve (7). The basic idea of th...

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...lgorithms was extensively studied in the late 1970’s and early 1980’s, and they continue to draw attention today [23]. We draw upon this body of work, in particular results of Jittorntrum and Osborne =-=[24]-=- (building on work of Cromme [25]) to understand the local convergence of RASL. The result of [24] concerns the problem of minimizing the composition of a norm ‖ · ‖⋄ : R n → R with a C 2 mapping f : ...

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... in the late 1970’s and early 1980’s, and they continue to draw attention today [23]. We draw upon this body of work, in particular results of Jittorntrum and Osborne [24] (building on work of Cromme =-=[25]-=-) to understand the local convergence of RASL. The result of [24] concerns the problem of minimizing the composition of a norm ‖ · ‖⋄ : R n → R with a C 2 mapping f : R p → R n : The authors of [25], ...

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...th convex function with a smooth, nonlinear mapping. The convergence behavior of such algorithms was extensively studied in the late 1970’s and early 1980’s, and they continue to draw attention today =-=[23]-=-. We draw upon this body of work, in particular results of Jittorntrum and Osborne [24] (building on work of Cromme [25]) to understand the local convergence of RASL. The result of [24] concerns the p...

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...er alternative convex optimization methods. In particular, it is about 5-10 times faster than the accelerated proximal gradient (APG) method originally proposed in the conference version of this work =-=[31]-=-. Although the convergence of the ALM method (15) has been well established in the optimization literature, we currently know of no proof that its approximation (16) converges too. The main difficulty...

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...m associated with ∆τ – has been established in [29]. Recently, [35] obtained a convergence result for certain three-term alternation applied to the noisy principal component pursuit problem (see also =-=[36]-=-). However, [35] reflects a very similar theory-practice gap – the three-term alternation for which convergence has been established is slower in practice than an alternation in the form of algorithm ...

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... Fig. 1), the matrix of aligned images might have an unknown rank higher than one. In this case, it is more appropriate to search for an alignment that minimizes the rank of the aligned images. So in =-=[9]-=-, Vedaldi et al. choose to minimize a log-determinant measure that can be viewed as a smooth surrogate for the rank function [10]. The low-rank objective can also be directly enforced, as in Transform...

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...6]. If we stack the aligned images as the columns of a large matrix, this criterion demands that each row of this matrix be nearly constant. Conversely, the least squares congealing procedure of [7], =-=[8]-=- seeks an alignment that minimizes the sum of squared distances between pairs of images, and hence demands that the columns be nearly constant. In both cases, if the criterion is satisfied exactly, th...

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...igure 1), the matrix of aligned images might have an unknown rank higher than one. In this case, it is more appropriate to search for an alignment that minimizes the rank of the aligned images. So in =-=[9]-=-, Vedaldi et. al. choose to minimize a log-determinant measure that can be viewed as a smooth surrogate for the rank function [10]. The low-rank objective can also be directly enforced, as in Transfor...

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...e a log-determinant measure that can be viewed as a smooth surrogate for the rank function [10]. The low-rank objective can also be directly enforced, as in Transformed Component Analysis (TCA) [11], =-=[12]-=-, which uses an EM algorithm to fit a low-dimensional linear model, subject to domain transformations drawn from a known group. A major drawback of the above approaches is that they do not simultaneou...

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...ees of robustness or convergence rate. This somewhat unsatisfactory status quo is mainly due to the extremely difficult nature of the core problem of fitting a low-rank model to highly corrupted data =-=[14]-=-, a problem that until recently lacked a polynomial-time algorithm with strong performance guarantees. Recent advances in rank minimization [15], [16] have shown that it is indeed possible to efficien...

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...6]. If we stack the aligned images as the columns of a large matrix, this criterion demands that each row of this matrix be nearly constant. Conversely, the least squares congealing procedure of [7], =-=[8]-=- seeks an alignment that minimizes the sum of squared distances between pairs of images, and hence demands that the columns be nearly constant. In both cases, if the criterion is satisfied exactly, th...

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...ooth convex function with a smooth, nonlinear mapping. The convergence behavior of such algorithms was extensively studied in the late 1970s and early 1980s, and they continue to draw attention today =-=[23]-=-. We draw upon this body of work, in particular results of Jittorntrum and Osborne [24] (building on the work of Cromme [25]), to understand the local convergence of RASL. The result of [24] concerns ...