I Introduction
Models of signals and images based on sparsity, lowrank, and other properties are useful in image and video processing. In illposed or illconditioned inverse problems, it is often useful to employ signal models that reflect known or assumed properties of the latent images. Such models are often used to construct appropriate regularization. For example, the sparsity of images in wavelet or discrete cosine transform (DCT) domains has been exploited for image and video reconstruction tasks [1, 2, 3]. In particular, the learning of such models has been explored in various settings [4, 5, 6, 7, 8, 9, 10], where they may potentially outperform fixed models since they adapt to signals and signal classes.
There has been growing interest in such dictionary learningbased image restoration or reconstruction methods [11, 12, 13]. For example, in blind compressed sensing [10, 12, 13], a dictionary for the underlying image or video is estimated together with the image from undersampled measurements. This allows the dictionary to adapt to the current data, which may enable learning novel features, providing improved reconstructions.
For inverse problems involving largescale or streaming data, e.g., in interventional imaging or restoring (e.g, denoising, inpainting) large or streaming videos, etc., it is often critical to obtain reconstructions in an online or timesequential (or datasequential) manner to limit latency. Batch methods that process all the data at once are often prohibitively expensive in terms of time and memory usage, and infeasible for streaming data. Methods for online learning of dictionary and sparsifying transform models from streaming signals including noisy data have been recently proposed [14, 15, 16, 17, 18] and shown to outperform stateoftheart methods [19, 20, 21]
including deep learning methods for video denoising.
This paper focuses on methods for dynamic image reconstruction from limited measurements such as video or medical imaging data. One such important class of dynamic image reconstruction problems arises in dynamic magnetic resonance imaging (dMRI), where the data are inherently or naturally undersampled because the object is changing as the data (samples in kspace or Fourier space of the object acquired sequentially over time) is collected. Various techniques have been proposed for reconstructing such dynamic image sequences from limited measurements [22, 23, 24]. Such methods may achieve improved spatial or temporal resolution by using more explicit signal models compared to conventional approaches (such as kspace data sharing in dMRI, where data is pooled in time to make sets of kspace data with sufficient samples, such as in the form of a Casorati matrix [25]); these methods typically achieve increased accuracy at the price of increased computation.
While some reconstruction techniques are driven by sparsity models and assume that the image sequence is sparse in some transform domain or dictionary [22], other methods exploit lowrank or other kinds of sophisticated models [25, 26, 27, 28, 29, 24, 30]. For example, L+S methods [24, 30] assume that the image sequence can be decomposed as the sum of lowrank and sparse (either directly sparse or sparse in some known transform domain) components that are estimated from measurements. Dictionary learningbased approaches including with lowrank models provide promising performance for dynamic image reconstruction [12, 31, 32]. Although these methods allow adaptivity to data and provide improved reconstructions, they involve batch processing and typically expensive computation and memory use. Next, we outline our contributions, particularly a new online framework that address these issues of stateoftheart batch reconstruction methods.
Ia Contributions
This paper investigates a framework for Online Adaptive Image Reconstruction, dubbed OnAIR, exploiting learned dictionaries. In particular, we model spatiotemporal patches of the underlying dynamic image sequence as sparse in an (unknown) dictionary, and propose a method to jointly and sequentially (over time) estimate the dictionary, sparse codes, and images from streaming measurements. Various constraints are considered for the dictionary model such as a unitary matrix, or a matrix whose atoms/columns are lowrank when reshaped into spatiotemporal matrices. The proposed OnAIR algorithms involve simple and efficient updates and require a small, fixed amount of data to be stored in memory at a time, greatly reducing the computation and memory demands during processing compared to conventional batch methods that process all of the data simultaneously.
Numerical experiments demonstrate the effectiveness of the proposed OnAIR methods for performing video inpainting from subsampled and noisy pixels, and dynamic MRI reconstruction from very limited kt space measurements. The experiments show that our proposed methods are able to learn dictionaries adaptively from corrupted measurements with important representational features that improve the quality of the reconstructions compared to nonadaptive schemes. Moreover, the OnAIR methods provide better image quality than batch schemes with a single adapted dictionary since they allow the dictionary to change over time for changing (dynamic) videos. At the same time, they require much lower runtimes than adaptive batch (or offline) reconstructions.
Short versions of this work appeared recently in [33] and [34]. This paper extends those initial works by exploring OnAIR methods with multiple constraints on the learned models including a novel Unitary Dictionary (OnAIRUD) constraint, and a Lowrank Dictionary atom (OnAIRLD) constraint. Such constraints may offer a degree of algorithmic efficiency and robustness to artifacts in limited data settings. For example, the updates per time instant in OnAIRUD are simpler and nonsequential, enabling it to be faster in practice than the other OnAIR schemes. Importantly, this paper reports extensive numerical experiments investigating and evaluating the performance of the proposed online methods in multiple inverse problems, and comparing their performance against recent related online and batch methods in the literature. Finally, we also compare the performance of the OnAIR methods to an oracle online scheme, where the dictionary is learned offline from the ground truth data, and show that the proposed online learning from highly limited and corrupted streaming measurements performs comparably to the oracle scheme.
IB Organization
The rest of this paper is organized as follows. Section II reviews the dictionary learning framework that forms the basis of our online schemes. Section III describes our formulations for online adaptive image reconstruction and Section IV presents algorithms for solving the problems. Section V presents extensive numerical experiments that demonstrate the promising performance of our proposed methods in inverse problem settings such as video inpainting and dynamic MRI reconstruction from highly limited data. Finally, Section VI concludes with proposals for future work.
Ii Dictionary Learning Models
Here, we briefly review some prior works on dictionary learning that helps build our OnAIR framework. Given a set of signals (or vectorized image patches) that are represented as columns of a matrix
, the goal of dictionary learning (DL) is to learn a dictionary and a matrix of sparse codes such that . Traditionally, the DL problem is often formulated [4] as follows:(1)  
s.t. 
where and denote the th column of and the th column of , respectively, and denotes a target sparsity level for each signal. Here, the “norm” counts the number of nonzero entries of its argument, and the columns of are set to unit norm to avoid scaling ambiguity between and [35]. Alternatives to (1) exist that replace the “norm” constraint with other sparsitypromoting constraints, or enforce additional properties on the dictionary [36, 37, 38], or enable online dictionary learning [14].
Dictionary learning algorithms [39, 4, 40, 41, 14, 42] typically attempt to solve (1) or its variants in an alternating manner by performing a sparse coding step (updating ) followed by a dictionary update step (updating ). Methods such as KSVD [4] also partially update the coefficients in in the dictionary update step, while some recent methods update the variables jointly and iteratively [43]. Algorithms for (1) typically repeatedly update (involves an NPhard sparse coding problem) and tend to be computationally expensive.
The DINOKAT learning problem [13] is an alternative dictionary learning framework that imposes a lowrank constraint on reshaped dictionary atoms (columns). The corresponding problem formulation is
(2)  
s.t. 
where the “norm” counts the total number of nonzeros in . The operator reshapes dictionary atoms into matrices of size for some and such that , and is the maximum allowed rank for each reshaped atom. The dimensions of the reshaped atoms can be chosen on an applicationspecific basis. For example, when learning from 2D image patches, the reshaped atoms can have the dimensions of the 2D patch. In the case where spatiotemporal (3D) patches are vectorized and extracted from dynamic data, the atoms could be reshaped into spacetime (2D) matrices. Spatiotemporal patches of videos often have high temporal correlation, so they may be well represented by a dictionary with lowrank spacetime (reshaped) atoms [13]. The parameter in (2) controls the overall sparsity of . Penalizing the overall or aggregate sparsity of enables variable sparsity levels across training signals (a more flexible model than (1)). The constraints for prevent pathologies (e.g., unbounded algorithm iterates) due to the noncoercive objective [44]. In practice, we set very large, and the constraint is typically inactive. We proposed an efficient algorithm in [13] for Problem (2) that requires less computation than algorithms such as KSVD.
Another alternative to the DL problems in (1) and (2) involves replacing the constraints in (2) with the unitary constraint , where is the identity matrix. Learned unitary operators work well in image denoising and more general reconstruction problems [45, 46]. Moreover, algorithms for learning unitary dictionaries tend to be computationally cheap [45]. Our OnAIR framework exploits some of the aforementioned dictionary structures.
Iii Problem Formulations
This section formulates datadriven online image reconstruction. First, we propose an online image reconstruction framework based on an adaptive dictionary regularizer as in (2). Let denote the sequence of dynamic image frames to be reconstructed. We assume that noisy, undersampled linear measurements of these frames are observed. We process the streaming measurements in minibatches, with each minibatch containing measurements of consecutive frames. Let
denote the vectorized version of the 3D tensor obtained by (temporally) concatenating
consecutive frames of the unknown dynamic images. In practice, we construct the sequence using a sliding window (over time) strategy, which may involve overlapping or nonoverlapping minibatches. We model the spatiotemporal (3D) patches of each as sparse with respect to a latent dictionary . Under this model, we propose to solve the following online dictionary learningdriven image reconstruction problem for each times.t. 
Here indexes time, and denotes the (typically undersampled) measurements that are related to the underlying frames (that we would like to reconstruct) through the linear sensing operator . For example, in video inpainting, samples a subset of pixels in , or in dynamic MRI, corresponds to an undersampled Fourier encoding. The operator is a patch extraction matrix that extracts the th spatiotemporal patch from as a vector. A total of (possibly) overlapping 3D patches are assumed. Matrix with is the synthesis dictionary to be learned and is the unknown sparse code for the th patch of , with . Matrix has as its columns, and its aggregate (total) sparsity is penalized. The weights are regularization parameters that control the relative adaptive dictionary regularization and sparsity of , respectively, in the model.
Problem (P1) has a form often used for online optimization [14, 47]. The objective is a weighted average of the function , with the constraints defining the feasible sets for and the sparse codes. Problem (P1) jointly estimates the adaptive dictionary model for the patches of together with the underlying image frames. Note that, for each time index , we solve (P1) only for the latest group of frames and the latest sparse coefficients , while the previous images and sparse coefficients are set to their estimates from previous minibatches (i.e., and for ). However, the dictionary is adapted to all the spatiotemporal patches observed up to time .^{1}^{1}1We emphasize the global dependence of on all previous data by using the optimization variable within the objective rather than a timeindexed variable as done for the variables and . Assuming , the objective in (P1) with respect to acts as a surrogate (upper bound) for the usual empirical (batch) risk function [14, 47] that uses the optimal reconstructions and sparse codes (i.e., those that minimize the cost) for all the data. An exponential forgetting factor with is used for the terms in (P1), and is a normalization constant for the objective. The forgetting factor diminishes the influence of “old” data on the dictionary adaptation process. When the dynamic object or scene changes slowly over time, a large (close to 1) is preferable so that past information has more influence and vice versa.
As written in (P1), the dictionary is updated based on patches from all previous times; however, the proposed algorithm does not need to store this information during optimization. Indeed, our algorithm in Section IV computes only a few constantsized matrices that contain the necessary cumulative (over time) information to solve (P1).
When minibatches and do not overlap (i.e., no common frames), each frame is reconstructed exactly once in its corresponding window in (P1). However, it is often beneficial to construct the ’s using an overlapping sliding window strategy [17], in which case a frame may be reconstructed in multiple windows (minibatches of frames). In this case, we independently produce estimates for each time index as indicated in (P1), and then we produce a final estimate of the underlying frame by computing a weighted average of the reconstructions of that frame from each window in which it appeared. We found empirically that an exponentially weighted average (similar to that in (P1)) performed better than alternatives such as an unweighted average of the estimates from each window or using only the most recent reconstruction from the latest window.
In (P1), we imposed a lowrank constraint on the dictionary atoms. As an alternative, we consider constraining the dictionary to be a unitary matrix. The online optimization problem in this case is as follows:
s.t. 
where all terms in (P2) are defined as in (P1). Note that (P2) does not require the norm constraints on the sparse coefficients because the unitary constraint on the dictionary precludes the possibility of repeated dictionary atoms, which was the motivation for including these constraints in (P1) [44].
Iv Algorithms and Properties
This section presents the algorithms for Problems (P1) and (P2) and their properties. We propose an alternating minimizationtype scheme for (P1) and (P2) and exploit the online nature of the optimization to minimize the costs efficiently. At each time index , we alternate a few times between updating while holding fixed (the dictionary learning step) and then updating with held fixed (the image update step). For each , we use a warm start for (initializing) the alternating approach. We initialize the dictionary with the most recent dictionary (). Frames of that were estimated in the previous (temporal) windows are initialized with the most recent
weighted reconstructions, and new frames are initialized using simple approaches (e.g., interpolation in the case of inpainting). Initializing the sparse coefficients
with the codes estimated in the preceding window () worked well in practice. All updates are performed efficiently and with modest memory usage as will be shown next. Figure 1 provides a graphical flowchart depicting our proposed online scheme. The next subsections present the alternating scheme for each time in more detail.Iva Dictionary Learning Step for (P1)
Let . Minimizing (P1) with respect to yields the following optimization problem:
(3)  
s.t. 
where is the matrix whose columns contain the patches for , and is the th column of . We use a block coordinate descent approach [44] (with few iterations) to update the sparse coefficients and atoms (columns of ) in (3) sequentially. For each , we first minimize (3) with respect to keeping the other variables fixed (sparse coding step), and then we update keeping the other variables fixed (dictionary atom update step). These updates are performed in an efficient online manner as detailed next.
IvA1 Sparse Coding Step
Minimizing (3) with respect to leads to the following subproblem:
(4) 
where the matrix
(5) 
is defined based on the most recent estimates of the other atoms and sparse coefficients. The solution to (4), assuming , is given by [44]
(6) 
where is the elementwise hard thresholding operator that sets entries with (complex) magnitude less than to zero and leaves other entries unaffected, is a length vector of ones, and denote elementwise multiplication and elementwise minimum respectively, and is computed elementwise, with denoting the phase. We do not construct in (6) explicitly; rather we efficiently compute the matrixvector product based on the most recent estimates of each quantity using sparse matrixvector operations [44].
IvA2 Dictionary Atom Update Step
Here, we minimize (3) with respect to . This update uses past information via the forgetting factor . Let and denote the weighted patches and sparse coefficients, respectively, and let and denote the matrices formed by stacking the ’s horizontally and ’s vertically, respectively, for times 1 to . Finally, define using the most recent estimates of all variables, with denoting the th column of . Using this notation, the minimization of (3) with respect to becomes
(7) 
Let be the rank
truncated singular value decomposition (SVD) of the matrix
that is obtained by computing the leading singular vectors and singular values of the full SVD . Then a solution to (7) is given by [13](8) 
where is any matrix^{2}^{2}2In our experiments, we set to be the reshaped first column of the identity matrix, which worked well. of appropriate dimension with rank at most such that .
The main computation in (8) is computing , since the SVD of the small (i.e., spacetime matrix with typically) matrix has negligible computational cost. In principle, the matrixvector multiplication depends on all past information processed by the streaming algorithm; however, it can be recursively computed using constant time and memory. Indeed, observe that
(9)  
where denotes the th element of a vector . The vectors and depend on all previous data, but they can be recursively computed over time as
(10) 
where for each column index , the matrix is understood to contain the latest versions of the sparse codes already updated (sequentially) during the dictionary learning step. Using these recursive formulas, the product can be readily computed each time in our algorithm. Thus, the update in (8) can be performed in a fully online manner (i.e., without storing all the past data).
Alternatively, we can collect the vectors and as columns of matrices and , and perform the following recursive updates once at the end of of the overall algorithm for (P1) (i.e., once per minibatch of frames):
(11) 
Here denotes the final sparse codes estimates from the algorithm for (P1). In this case, when performing the inner update (9), the contributions of the two terms on the right hand side of (10) are incorporated separately. The matrices and are small, constantsized matrices whose dimensions are independent of the time index and the dimensions of the frame sequence, so they are stored for efficient use in the next minibatch. Moreover, the matrix is sparse, so all the matrixmatrix multiplications in (11) (or the matrixvector multiplications in (10)) are computed using efficient sparse operations.
IvB The Unitary Dictionary Variation
In the case of (P2), unlike for (P1), we do not perform block coordinate descent over the columns of and . Instead we minimize (P2) with respect to each of the matrices and directly and exploit simple closedform solutions for the matrixvalued updates. The following subsections describe the solutions to the and update subproblems.
IvB1 Sparse Coding Step
Since is a unitary matrix, minimizing (P2) with respect to yields the following subproblem:
(12) 
where is again the matrix whose th column is . The solution to (12) is given by the simple elementwise hardthresholding operation
(13) 
IvB2 Dictionary Update Step
Minimizing (P2) with respect to results in the following optimization:
(14) 
where is used for notational convenience. Using the definitions of the matrices and from the dictionary atom updates in Section IVA, we can equivalently write (14) as
(15) 
Problem (15) is a wellknown orthogonal Procrustes problem [48]. The solution is given by
(16) 
where is a full SVD of . Similarly as in (11), let . The matrix can be recursively updated over time according to (11) (or (10)), so the dictionary update step can be performed efficiently and fully online.
IvC Image Update Step
Minimizing (P1) or (P2) with respect to the minibatch yields the following quadratic subproblem:
(17) 
Problem (17) is a least squares problem with normal equation
(18) 
In applications such as video denoising or inpainting, the matrix premultiplying in (18) is a diagonal matrix that can be efficiently precomputed and inverted. More generally, in inverse problems where the matrix premultiplying (i.e., in particular) in (18) is not diagonal or readily diagonalizable (e.g., in dynamic MRI with multiple coils), we minimize (17) by applying an iterative optimization method. One can use any classical algorithm in this case, such as the conjugate gradient (CG) method. For the experiments shown in Section V, we used a few iterations (indexed by ) of the simple proximal gradient method [49, 50] with updates of the form
(19) 
where and the proximal operator of a function is . The proximal operation in (19) corresponds to a simple least squares problem that is solved efficiently by inverting a diagonal matrix (arising from the normal equation of the proximal operation), which is precomputed. A constant stepsize suffices for convergence [49]. Moreover, the iterations (19) monotonically decrease the objective in (17) when a constant step size is used [50].
Fig. 2 summarizes the overall OnAIR algorithms for the online optimization Problems (P1) and (P2). In practice, we typically use more iterations for reconstructing the first minibatch of frames, to create a good warm start for further efficient online optimization. The initial in the algorithm can be set to an analytical dictionary (e.g., based on the DCT or wavelets) and we set the initial
to a zero matrix.
OnAIR Algorithms for (P1) and (P2) 

Inputs : measurement sequence , weights and , rank , upper bound , forgetting factor , number of dictionary learning iterations , number of proximal gradient iterations , and the number of outer iterations per minibatch . 
Outputs : reconstructed dynamic image sequence . 
Initialization: Initial (first) estimates of new incoming (measured) frames, initial and at , and . 
For = , do 
Warm Start: Set , , and is set as follows: frames estimated in previous minibatches are set to a weighted average of those estimates and new frames are set as in . 
For = repeat 

End 
Recursive Updates: Update and using (11). 
Output Updates: Set , and . For frames not occurring in future windows, is set to a weighted average of estimates over minibatches. 
End 
IvD Computational Cost and Convergence
The computational cost for each time index of the proposed algorithms for solving the online image reconstruction problems (P1) and (P2) scales as , where with assumed, and is the number of (overlapping) patches in each temporal window. The cost is dominated by various matrixvector multiplications. Assuming each window’s length , the memory (storage) requirement for the proposed algorithm scales as , which is the space required to store the image patches of when performing the updates for (P1) or (P2). Typically the minibatch size is small (to allow better tracking of temporal dynamics), so the number and maximum temporal width of 3D patches in each window are also small, ensuring modest memory usage for the proposed online methods.
While the overall computational cost and memory requirements for each time index are similar for the algorithms for (P1) and (P2), the simple matrixvalued forms of the dictionary and sparse code updates for (P2) result in practice, in a severalfold decrease in actual runtimes due to optimizations inherent to matrixvalued computations in modern linear algebra libraries.
The proposed algorithms involve either exact block coordinate descent updates or for example, proximal gradient iterations (with appropriate step size) when the matrix premultiplying in (18) is not readily diagonalizable. These updates are guaranteed to monotonically decrease the objectives in (P1) and (P2) for each time index . Whether the overall iterate sequence produced by the algorithms also converges over time (see e.g., [14]) is an interesting open question that we leave for future work.
% Missing Pixels  Coastguard  Bus  Flower Garden  

50%  60%  70%  80%  90%  50%  60%  70%  80%  90%  50%  60%  70%  80%  90%  
OnAIRFD  33.1  31.4  29.6  27.3  22.5  28.7  27.1  25.5  23.7  21.5  24.4  22.8  21.0  18.8  15.8 
OnAIRUD  33.8  31.3  28.1  24.8  21.9  29.7  27.6  25.5  23.4  21.1  24.4  22.1  19.6  17.1  15.6 
Online (DCT)  32.7  30.3  27.8  25.3  22.6  28.4  26.7  25.0  23.1  20.8  23.3  21.6  19.9  18.1  16.3 
Batch Learning  33.1  31.2  29.1  26.3  22.8  27.8  26.3  24.7  22.9  20.9  23.5  21.8  20.1  18.2  16.1 
Interpolation (3D)  29.8  28.5  27.3  25.9  24.1  27.3  25.7  24.0  22.1  20.0  20.6  19.6  18.5  17.5  16.4 
Interpolation (2D)  28.2  26.5  24.9  23.1  21.1  26.0  24.8  23.7  22.5  21.1  20.1  18.8  17.5  16.2  14.8 
V Numerical Experiments
This section presents extensive numerical experiments illustrating the usefulness of the proposed OnAIR methods. We consider two inverse problem applications in this work: video reconstruction (inpainting) from noisy and subsampled pixels, and dynamic MRI reconstruction from highly undersampled kt space data. The algorithm for (P1) is dubbed OnAIRLD when the parameter is lower than its maximum setting (i.e., lowrank dictionary atoms) and is dubbed OnAIRFD (i.e., fullrank dictionary atoms) otherwise. The algorithm for (P2) is dubbed OnAIRUD (unitary dictionary). The following subsections discuss the applications and results. A link to software to reproduce our results will be provided at http://web.eecs.umich.edu/~fessler/.
Va Video Inpainting
VA1 Framework
First, we consider video inpainting or reconstruction from subsampled and potentially noisy pixels. We work with the publicly available videos^{3}^{3}3The data is available at http://www.cs.tut.fi/~foi/GCFBM3D/. provided by the authors of the BM4D method [51]. We process the first frames of each video at native resolution. We measure a (uniform) random subset of the pixels in each frame of the videos, and also simulate additive (zero mean) Gaussian noise for the measured pixels in some experiments. The proposed OnAIR methods are then used to reconstruct the videos from their corrupted (noisy and/or subsampled) measurements.
The parameters for the proposed OnAIR methods are chosen as follows. We used a sliding (temporal) window of length
frames with a temporal stride of 1 frame, to reconstruct (maximally overlapping) minibatches of frames. In each window, we extracted
overlapping spatiotemporal patches with a spatial stride of 2 pixels. We learned a square dictionary, and the operator reshaped dictionary atoms into spacetime matrices. We ran the OnAIR algorithms for (P1) and (P2) for iterations in each temporal window (minibatch), with inner iteration (of block coordinate descent or alternation) for updating , and used a direct reconstruction as per (18) in the image update step. A forgetting factor was observed to work well. We ran the algorithms for more () iterations for the first minibatch of data, for which the initial was the discrete cosine transform (DCT) matrix and the initial sparse codes were zero. Newly arrived frames were first initialized (i.e., in the first minibatch they occur) using 2D cubic interpolation. We simulated various levels of subsampling of the videos (with and without noise), and we chose or fullrank atoms in (P1), which outperformed lowrank atoms in the experiments here. The videos in this section have substantial temporal motion, so allowing fullrank atoms enabled the algorithm to better learn the dynamic temporal features of the data. Section VB demonstrates the usefulness of lowrank atoms in (P1). We tuned the weights and for (P1) and (P2) to achieve good reconstruction quality at an intermediate undersampling factor (70% missing pixels) for each video, and used the same parameters for other factors.We measure the performance of our methods using the (3D) peak signaltonoise ratio (PSNR) metric that is computed as the ratio of the peak pixel intensity in the video to the root mean square reconstruction error relative to the ground truth video. All PSNR values are expressed in decibels (dB).
We compare the performance of the proposed OnAIRFD () and OnAIRUD methods for (P1) and (P2) with that of an identical online algorithm but which uses a fixed DCT dictionary. We also produce reconstructions using a “batch” version of the method for (P1) [13] (with ) that processes all video frames jointly.^{4}^{4}4This batch learningbased reconstruction method is equivalent to the proposed online method for (P1) with set to the total number of frames in the video. For each comparison method, we used the same patch dimensions, initializations, etc., and tuned the parameters and of each method individually. The one exception is that we used a spatial patch stride of 4 pixels for the Batch Learning method rather than the 2 pixel spatial stride used for the online methods. This change was made because the batch method is memory intensive—it extracts and processes image patches from the entire video concurrently—so it was necessary to process fewer patches to make the computation feasible. We ran the batch learningbased reconstruction for 20 iterations, which worked well. Finally, we also compare the OnAIR reconstructions to baseline reconstructions produced by 2D (framebyframe cubic) interpolation and 3D interpolation (using the natural neighbor method in MATLAB).
% Missing Pixels  Coastguard (25 dB PSNR)  

50%  60%  70%  80%  90%  
OnAIRFD  28.6  27.9  27.2  26.1  23.9 
OnAIRUD  29.4  28.6  26.6  24.9  22.0 
Online (DCT)  28.6  27.8  26.7  25.1  22.9 
Batch Learning  28.6  27.8  26.6  25.1  22.5 
Interpolation (3D)  26.2  25.9  25.4  24.6  23.4 
Interpolation (2D)  24.7  24.0  23.0  21.9  20.2 
VA2 Results
Table I lists the PSNR values (in dB) for the various reconstruction methods for different levels of subsampling (from 50% to 90% missing pixels) of three videos (without added noise) from the BM4D dataset. Table II shows analogous results for the Coastguard video when i.i.d. zeromean Gaussian noise with 25 dB PSNR was added before sampling the pixels. The proposed OnAIRFD method typically provides the best PSNRs at higher undersampling factors, and the proposed OnAIRUD method (with a more structured/unitary^{5}^{5}5This also corresponds to a unitary sparsifying transform model [6], with denoting the sparsifying transform or operator. ) performs better at lower undersampling factors.
The best PSNRs achieved by the OnAIR methods are typically better (by up to 1.4 dB) than for the Batch Learning scheme. Both OnAIR variations also typically outperform (by up to 2 dB) the online method with a fixed DCT dictionary (the initial in our methods), suggesting that the dictionaries learned by the proposed methods are successfully adapting to and reconstructing underlying features of the videos.
Figs. 3 and 4 show the original and reconstructed frames for a few representative frames of the Coastguard and Flower Garden videos. Results for multiple methods are shown. Fig. 3 shows that the proposed OnAIRFD method produces visually more accurate reconstructions of the texture in the waves and also produces fewer artifacts near the boats in the water and the rocks on the shore. Fig. 4 illustrates that the proposed OnAIR method produces a sharper reconstruction with less smoothing artifacts than the online method with a fixed DCT dictionary and the batch learningbased reconstruction method, and it is less noisy than the interpolationbased reconstruction.
VA3 Properties
Fig. 5 shows the framebyframe (2D) PSNRs for the Coastguard video inpainted from 70% missing pixels, using various methods. Clearly the proposed OnAIRFD method achieves generally higher PSNRs across frames of the video. The overall trends in PSNR over frames are similar across the methods and are due to motion in the original videos, with more motion generally resulting in lower PSNRs.
Fig. 6 shows a representative example of a (final) learned dictionary produced by the proposed OnAIRFD method for the Bus video, along with the initial (at ) DCT dictionary. The dictionaries each contain 320 atoms, each of which is an spacetime tensor. We visualize each atom by plotting the first slice of the atom and also plotting the profile from a vertical slice through the middle of each atom tensor. The (first slice) images show that the learned dictionary has adapted to both smooth and sharp gradients in the image, and the dynamic (evolved) nature of the profiles shows that the dictionary atoms have adapted to temporal trends in the data.
To assess the relative efficiency of the OnAIR methods compared to the Batch Learning method, we measured runtimes on the Coastguard video with the same fixed patch sizes, patch strides, and numbers of iterations for each method.^{6}^{6}6Experiments were performed in MATLAB R2017b on a 2016 MacBook Pro with a 2.7 GHz Intel Core i7 processor and 16 GB RAM. In this case, the OnAIRFD and OnAIRUD methods were 1.8x and 4.0x faster, respectively, than the Batch Learning method. In addition, the OnAIR methods typically require much fewer iterations (per minibatch) compared to the batch method to achieve their best reconstruction accuracies, so, in practice, when utilizing half as many outer iterations, the OnAIRFD and OnAIRUD methods were approximately 3.6x and 8.0x faster, respectively, than the batch scheme.
VB Dynamic MRI Reconstruction
VB1 Framework
Here, we demonstrate the usefulness of the proposed OnAIR methods for reconstructing dynamic MRI data from highly undersampled kt space measurements. We work with the multicoil (12element coil array) cardiac perfusion data [24] and the PINCAT data [12, 52] from prior works. For the cardiac perfusion data, we retrospectively undersampled the kt space using variabledensity random Cartesian undersampling with a different undersampling pattern for each time frame, and for the PINCAT data we used pseudoradial sampling with a random rotation of radial lines between frames. The undersampling factors tested are shown in Table III. For each dataset, we obtained reconstructions using the proposed OnAIR methods, the online method but with a fixed DCT dictionary, and the batch learningbased reconstruction (based on (P1)) method [13]. We also ran the recent L+S [24] and kt SLR [53]
methods, two sophisticated batch methods for dynamic MRI reconstruction that process all the frames jointly. Finally, we also computed a baseline reconstruction in each experiment by performing zeroth order interpolation across time at nonsampled kt space locations (by filling such locations with the nearest nonzero entry along time) and then backpropagating the filled kt space to image space by premultiplying with the
corresponding to fully sampled data. The first estimates () of newly arrived frames in our OnAIR methods are also computed via such zeroth order interpolation, but using only the estimated (i.e., already once processed and reconstructed) nearest (older) frames.Acceleration  Cardiac Perfusion  PINCAT  

4x  8x  12x  16x  20x  24x  5x  6x  7x  9x  14x  27x  
OnAIRLD  10.2%  12.8%  14.8%  16.7%  18.1%  18.0%  8.9%  9.7%  11.0%  12.4%  15.5%  21.8% 
Online (DCT)  10.8%  13.7%  15.8%  18.2%  20.7%  20.8%  9.5%  10.2%  11.5%  13.2%  16.4%  22.5% 
Batch Learning  10.7%  13.7%  15.9%  18.2%  22.0%  23.9%  10.0%  10.7%  11.8%  13.2%  15.9%  20.9% 
L+S  11.0%  13.8%  16.1%  18.4%  21.5%  22.5%  11.8%  12.9%  14.4%  16.6%  20.0%  25.9% 
kt SLR  11.2%  15.7%  18.4%  21.3%  24.3%  26.5%  9.8%  10.9%  12.4%  14.7%  18.2%  24.2% 
Baseline  12.8%  15.9%  18.9%  21.1%  24.5%  28.1%  22.3%  24.7%  27.5%  31.3%  36.6%  44.5% 
For the online schemes, we used spatiotemporal patches with frames per temporal window (minibatch) and a (temporal) window stride of 1 frame. We extracted overlapping patches in each minibatch using a spatial stride of 2 pixels along each dimension. We learned a square dictionary whose atoms when reshaped into spacetime matrices had rank , which worked well in our experiments. A forgetting factor of was observed to work well. We ran the online schemes for outer iterations^{7}^{7}7Prior to the first outer iteration for each minibatch in (P1), the sparse coefficients were updated using block coordinate descent iterations to allow better adaptation to new patches. per minibatch, with iteration in the dictionary learning step and proximal gradient steps in the image update step, respectively. We used outer iterations for the first minibatch to warm start the algorithm and used an initial DCT , and the initial sparse codes were zero. After one complete pass of the online schemes over all the frames, we performed another pass over the frames, using the reconstructed frames and learned dictionary from the first pass as the first initializations in the second pass. The additional pass gives a slight image quality improvement over the first (a single pass corresponds to a fully online scheme) as will be shown later.
For the batch dictionary learningbased reconstruction method, we used the same patch dimensions, strides, and initializations as for the online methods. We ran the batch method for 50 iterations. For the L+S and kt SLR methods, we used the publicly available MATLAB implementations from [54] and [55], respectively, and ran each method to convergence. The regularization parameters (weights) for all the methods here were tuned for each dataset by sweeping them over a range of values and selecting values that achieved good reconstruction quality at intermediate kt space undersampling factors. We measured the dMRI reconstruction quality using the normalized root mean square error (NRMSE) metric expressed as percentage that is computed as
(20) 
where is a candidate reconstruction and is the reference reconstruction (e.g., computed from “fully” sampled data).
Acceleration  4x  8x  12x  16x  20x  24x 

Online (oracle)  10.2%  12.4%  14.4%  16.4%  17.8%  17.9% 
OnAIRLD (2 passes)  10.2%  12.8%  14.8%  16.7%  18.1%  18.0% 
OnAIRLD (1 pass)  10.2%  12.9%  14.8%  16.6%  18.3%  18.1% 
OnAIRUD  10.5%  13.6%  15.7%  17.8%  20.4%  20.1% 
Online (DCT)  10.8%  13.7%  15.8%  18.2%  20.7%  20.8% 
VB2 Results and Comparisons
Table III shows the reconstruction NRMSE values obtained using various methods for the cardiac perfusion and PINCAT datasets at several undersampling factors. The proposed OnAIRLD method achieves lower NRMSE values in almost every case compared to the L+S and kt SLR methods, the online DCT scheme, the baseline reconstruction, and the batch learningbased reconstruction scheme. Unlike the batch schemes (i.e., L+S, kt SLR, and the batch learningbased reconstruction) that process or learn over all the data jointly, the OnAIRLD scheme only directly processes data corresponding to a minibatch of frames at any time. Yet OnAIRLD outperforms the other methods because of its memory (via the forgetting factor) and local temporal adaptivity (or tracking). These results show that the proposed OnAIR methods are wellsuited for processing streaming data.
Fig. 7 shows the reconstructions and reconstruction error maps (magnitudes displayed) for some representative frames from the cardiac perfusion and PINCAT datasets at 12x and 7x undersampling, respectively. The error maps indicate that the proposed OnAIRLD scheme often produces fewer artifacts compared to the existing methods.
Fig. 8 shows that the OnAIRLD scheme also typically achieves better framebyframe NRMSE compared to the other methods. Finally, Fig. 9 shows the reconstructed profiles for various methods obtained by extracting the same vertical line segment from each reconstructed frame of the PINCAT data and concatenating them. The online DCT scheme and the batch methods L+S and kt SLR show linelike or additional smoothing artifacts that are not produced by the proposed OnAIR method.
VB3 Properties
Table IV investigates the properties of the proposed OnAIR methods in more detail using the cardiac perfusion data. Specifically, it compares the NRMSE values produced by the OnAIRLD scheme with one or two passes over the data, the OnAIRUD method, and the online method with a fixed DCT dictionary. In addition, we ran the online method but with a fixed “oracle” dictionary learned from patches of the reference (true) reconstruction by solving the DINOKAT learning problem (2). The oracle dictionary was computed based on the “fully” sampled data, so it can be viewed as the “best” dictionary that one could learn from the undersampled dataset. From Table IV, we see that the NRMSE values achieved by the OnAIRLD scheme with two passes are within 0.0%  0.5% of the oracle NRMSE values, which suggests that the proposed scheme is able to learn dictionaries with good representational (recovery) qualities from highly undersampled data. Moreover, the performance of the OnAIRLD scheme with a single pass is almost identical to that with two passes, suggesting promise for fully online dMRI reconstruction. The OnAIRLD method outperformed the OnAIRUD scheme for the cardiac perfusion data indicating that temporal lowrank properties better characterize the dataset than unitary models. All the OnAIR schemes outperformed the nonadaptive yet online DCTbased scheme.
To assess the relative efficiency of the OnAIR methods compared to the Batch Learning method, we measured runtimes on the PINCAT dataset with the same parameter settings for each method used to generate the results in Table III.^{8}^{8}8Experiments were performed in MATLAB R2017b on a 2016 MacBook Pro with a 2.7 GHz Intel Core i7 processor and 16 GB RAM. With these settings, the OnAIRLD and OnAIRUD methods were 9.8x and 40.4x faster, respectively, than the Batch Learning method. One key reason for these significant performance improvements is that 50 outer iterations were required by the batch method while only 7 outer iterations were required by the online methods (after the first minibatch) to achieve the reported accuracies; this result suggests that the online dictionary adaptation performed by the OnAIR methods is both computationally efficient and allows the model to better adapt to the underlying structure of the data. The additional speedup of the OnAIRUD method with respect to OnAIRLD is attributable to the relative efficiency of the matrixvalued updates of the unitary method compared to the less optimized block coordinate descent iterations over the columns of and prescribed by OnAIRLD.
Fig. 10 shows an example of a learned dictionary produced by the proposed OnAIRLD method on the PINCAT dataset at 9x undersampling, which is compared with the initial DCT dictionary. The dictionaries have atoms, each a complexvalued spacetime tensor. Since the OnAIR LD dictionary atoms have rank when reshaped into spacetime matrices, we directly display the real and imaginary parts of the first () slice of each learned atom. The initial DCT is also displayed similarly. The (eventual) learned dictionary has clearly evolved significantly from the initial DCT atoms and has adapted to certain smooth and sharp textures at various orientations in the data.
Recall that we chose fullrank () atoms in the video inpainting experiments, while here we chose lowrank () atoms. Intuitively, lowrank atoms are a better model for the dynamic MRI data because the videos have high temporal correlation and rank atoms are necessarily constant in their temporal dimension. Conversely, the videos from Section VA contained significant camera motion and thus dictionary atoms with more temporal variation (i.e., higher rank) enabled more accurate reconstructions.
Vi Conclusions
This paper has presented a framework for online estimation of dynamic image sequences by learning dictionaries. Various properties were also studied for the learned dictionary such as a unitary property, and lowrank atoms, which offer additional efficiency or robustness to artifacts. The proposed OnAIR algorithms sequentially and efficiently update the images, dictionary, and sparse coefficients of image patches from streaming measurements. Importantly, our algorithms can process arbitrarily long video sequences with constant memory usage over time. Our numerical experiments demonstrated that the proposed methods produce accurate reconstructions for video inpainting and dynamic MRI reconstruction. The proposed methods may also be suitable for other inverse problems, including medical imaging applications such as interventional imaging, and other videoprocessing tasks from computer vision. We hope to investigate these application domains as well as demonstrate potential realtime applicability for OnAIR approaches in future work.
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