in probabilistic graphical models (PGMs) refers to the set of the techniques that carry out inference over groups of random variables (or states) that behave similarly[Jha et al.2010, Kimmig et al.2015]. A vast body of theoretical work develops a variety of lifted inference techniques, both exact (e.g., [Poole2003, Braz et al.2005, Singla and Domingos2008, Kersting2012]) and approximate (e.g., [Singla et al.2014, Van den Broeck and Niepert2015]). Most of these works develop technical ideas applicable to generic subclasses of PGMs, and the accompanying experiments are aimed at providing first proofs of concepts. However, little work exists on transferring these ideas to the top domain-specific algorithms for real-world applications.
Algorithms for NLP, computational biology, and computer vision (CV) problems make heavy use of PGM machinery (e.g., [Blei et al.2003, Friedman2004, Szeliski et al.2008]). But, they also include significant problem-specific insights to get high performance. Barring a handful of exceptions [Jernite et al.2015, Nath and Domingos2016], lifted inference hasn’t been applied directly to such algorithms.
We study the potential value of lifting to CV problems such as image denoising, stereo vision, and image segmentation. Most CV problems are structured output prediction tasks, typically assigning a label to each pixel. A large class of solutions are PGM-based: they define a Markov Random Field (MRF) that has each pixel as a node, with unary potential that depends on pixel value, and pairwise neighborhood potentials that favor similar labels to neighboring pixels.
We see three main challenges in applying existing lifted inference literature to these problems. First, most existing algorithms focus on computing marginals [Singla and Domingos2008, Kersting et al.2009, Gogate and Domingos2011, Niepert2012, Anand et al.2016, Anand et al.2017] instead of MAP inference. Second, among the algorithms performing lifted MAP [Noessner et al.2013, Mladenov et al.2014, Sarkhel et al.2014, Mittal et al.2014], many of the algorithms focus on exact lifting. This breaks the kind of symmetries we need to compute since different pixels may not have exact same neighborhood. Third, the few algorithms that perform approximate lifting for MAP, e.g. [Sarkhel et al.2015], can’t handle a distinct unary potential on every node. This is essential for our application since image pixels take ordinal values in three channels.
In response, we develop an approximate lifted MAP inference algorithm which can effectively handle unary potentials. We initialize our algorithm by merging together pixels having the same order of top- labels based on the unary potential values. We then adapt an existing symmetry finding algorithm [Kersting et al.2009] to discover groupings which also have similar neighborhoods. We refer to our groupings as lifted pixels. We impose the constraint that all pixels in a lifted pixel must be assigned the same label. Our approximate lifting reduces the model size drastically leading to significant time savings. Unfortunately, such approximate lifting could adversely impact solution quality. However, we vary the degree of approximation in symmetry finding to output a sequence of coarse-to-fine models with varying quality-time trade-offs. By switching between such models, we develop a coarse-to-fine (C2F) inference procedure applicable to many CV problems.
We formalize these ideas in a novel template for using lifted inference in CV. We test C2F lifted inference on two problems: stereo matching and image segmentation. We start with one of the best MRF-based solvers each for both problems – neither of these are vanilla MRF solvers. Mozerov & Weijer TSGO use a two-way energy minimization to effectively handle occluded regions in stereo matching. Co-operative cuts [Kohli et al.2013] for image segmentation use concave functions over a predefined set of pixel pairs to correctly segment images with sharp edges. We implement C2F inference on top of both these algorithms and find that C2F versions have a strong anytime behavior – given any amount of inference time, they output a much higher quality (and are never worse) than their unlifted counterparts, and don’t suffer any loss in the final quality. Overall, our contributions are:
We present an approximate lifted MAP algorithm that can efficiently handle a large number of distinct unary potentials.
We develop a novel template for applying lifted inference in structured prediction tasks in CV. We provide methods that output progressively finer approx. symmetries, leading to a C2F lifted inference procedure.
We implement C2F inference over a near state-of-the-art stereo matching algorithm, and one of the best MRF-based image segmentation algorithms. We release our implementation for wider use by the community.111https://github.com/dair-iitd/c2fi4cv/
We find that C2F has a much superior anytime behavior. For stereo matching it achieves 60% better quality on average in time-constrained settings. For image segmentation C2F reaches convergence in 33% less time.
2.1 Computer Vision Problems as MRFs
Most computer vision problems are structured output prediction problems and their PGM-based solutions often follow similar formulations. They cast the tasks into the problem of finding the lowest energy assignment over grid-structured MRFs (denoted by ). The random variables in these MRFs are the set of pixels in the input image. Given a set of labels , the task of structured output prediction is to label each pixel with a label from . The MRFs have two kinds of potentials () – unary and higher-order. Unary potentials are defined over each individual pixel, and usually incorporate pixel intensity, color, and other pixel features. Higher order potentials operate over cliques (pairs or more) of neighboring pixels and typically express some form of spatial homophily – “neighboring pixels are more likely to have similar labels.” While the general PGM structure of various tasks are similar, the specific potential tables and label spaces are task-dependent.
The goal is to find the MAP assignment over this MRF, which is equivalent to energy minimization (by defining energy as negative log of potentials). We denote the negative log of unary potentials by , and that of higher-order potentials by .222In the interest of readability, we say ‘potential’ to mean ‘negative log of potential’ in the rest of the paper. Thus, energy of a complete assignment can be defined as:
Here denotes the assignment restricted to the set of variables in the potential . And the output of the algorithm is the assignment :
2.2 Symmetries in Graphical Models
Lifting an algorithm often requires computing a set of symmetries that can be exploited by that algorithm. For PGMs, two popular methods for symmetry computation are color passing for computing symmetries of variables [Kersting et al.2009], and graph isomorphism for symmetries of states [Niepert2012, Bui et al.2013]. Since our work is based on color passing, we explain it in more detail.
Color passing for an MRF operates over a colored bipartite graph containing nodes for all variables and potentials, and each node is assigned a color. The graph is initialized as follows: all variables nodes get a common color; all potential nodes with exactly same potential tables are assigned a unique color. Now, in an iterative color passing scheme, in each iteration, each variable node passes its color to all neighboring potential nodes. The potential nodes store incoming color signatures in a vector, append their own color to it, and send the vector back to variable nodes. The variable nodes stack these incoming vectors. New colors are assigned to each node based on the set of incoming messages such that two nodes with same messages are assigned the same unique color. This process is repeated until convergence, i.e., no further change in colors.
A coloring of the bipartite graph defines a partition of variable nodes such that all nodes of the same color form a partition element. Each iteration of color passing creates successively finer partitions, since two variable nodes, once assigned different colors, can never get the same color.
3 Lifted Computer Vision Framework
In this section, we will describe our generic template which can be used to lift a large class of vision applications including those in stereo, segmentation etc. Our template can be seen as transforming the original problem space to a reduced problem space over which the original inference algorithm can now be applied much more efficiently. Specifically, our description in this section is entirely algorithm independent.
We will focus on MAP inference which is the inference task of choice for most vision applications (refer Section 2).
The key insight in our formulation is based on the realization that pixels which are involved in the same (or similar) kinds of unary and higher order potentials, and have the same (or similar) neighborhoods, are likely to have the same MAP value. Therefore, if somehow we could discover such sets of pixels a priori, we could explicitly enforce these pixels to have the same value while searching for the solution, substantially reducing the problem size and still preserving the optimal MAP assignment(s). Since in general doing this exactly may lead to a degenerate network, we do it approximately. Hence trading-off speed for marginal loss in solution quality. The loss in solution quality is offset by resorting to coarse-to-fine inference where we start with a crude approximation, and gradually make it finer, to guarantee optimality at the end while still obtaining significant gains. We next describe the details of our approach.
3.1 Obtaining a Reduced Problem
Consider an energy minimization problem over a PGM . Let denote the set of labels over which variables in the set can vary. Let denote a partition of into disjoint subsets, i.e., , , when , and . We refer to each as a partition element. Correspondingly, let us define as a set of partition variables, where there is a one to one correspondence between partition elements and the partition variables and each partition variable takes values in the set . Let denote the partition element to which belongs. Let denote a subset of variables. We say that a partition element is represented in the set if s.t. .
Given a subset of variables , let be a potential defined over . Let denote an assignment to variables in the set . Let denote the value taken by a variable in . We say that an assignment respects a partition if the variables in belonging to the same partition element have the same label in , i.e., . Next, we introduce the notion of a reduced potential.
Let be a set of variables and let denote its partition. Given the potential , the reduced potential is defined to be the restriction of to those labeling assignments of which respect the partition . Equivalently, we can define the reduced potential over the set of partition variables which are represented in the set .
For example, consider a potential defined over three Boolean variables. The table for would have entries. Consider the partition where and . Then, the reduced potential is the restriction of to those rows in the table where . Hence has four rows in its table and equivalently can be thought of defining a potential over the possible combinations of and variables. We are now ready to define a reduced graphical model.
Let represent a PGM. Given a partition of , the reduced graphical model is the graphical model defined over the set of partition variables such that every potential in is replaced by the corresponding reduced potential in .
Let and denote the energies of the states and in and , respectively. The following theorem relates the energies of the states in the two graphical models.
For every assignment of in , there is a corresponding assignment of such that .
The theorem can be proved by noting that each potential in was obtained by restricting the original potential to those assignments where variables in belonging to the same partition took the same label. Since this correspondence is true for every potential in the reduced set, to obtain the desired state , for every variable we simply assign it the label of its partition in .
Let and be the MAP states (i.e. having the minimum energy) for and , respectively. Then, .
The process of reduction can be seen as curtailing the entire search space to those assignments where variables in the same partition take the same label. A reduction in the problem space will lead to computational gains but might result in loss of solution quality, where the solution quality can be captured by the difference between and . Therefore, we need to trade-off the balance between the two.
Intuitively, a good problem reduction will keep those variables in the same partition which are likely to have the same value in the optimal assignment for the original problem. How do we find such variables without actually solving the inference task? We will describe one such technique in Section 3.3.
There is another perspective. Instead of solving one reduced problem, we can instead work with a series of reduced problems which successively get closer to the optimal solution. The initial reductions are coarser and far from optimal, but can be solved efficiently to quickly reach in the region where the solution lies. The successive iterations can then refine the solution iteratively getting closer to the optimal. This leads us to the coarse-to-fine inference described next.
3.2 Coarse to Fine Inference
We will define a framework for C2F (coarse-to-fine) inference so that we maintain the computational advantage while still preserving optimality. In the following, for ease of notation, we will drop the superscript in to denote the partition of . Therefore, will refer to both the partition as well as the set of partition variables. Before we describe our algorithm, let us start with some definitions.
Let and be two partitions of . We say that is coarser than , denoted as , if such that . We equivalently say that is finer than .
It is easy to see that defines a partition of itself which is the finest among all partitions, i.e., such that is a partition of , . We also refer it to as the degenerate partition. For ease of notation, we will denote the finest partition by (same as ). We will refer to the corresponding PGM as (same as ). Next, we state a theorem which relates two partitions with each other.
Let and be two partitions of such that . Then can be seen as a partition of the set .
The proof of this lemma is straightforward and is omitted due to lack of space. Consider a set of coarse to fine partitions given as . Let respectively denote the reduced problem, energy function and MAP assignment for the partition . Using Lemma 1, is a partition of . Then, using Theorem 3.1, we have for every assignment to variables in , there is an assignment to variables in such that . Also, using Corollary 3.1, we have . Together, these two statements imply that starting from the coarsest partition, we can gradually keep on improving the solution as we move to finer partitions.
Our C2F set-up assumes an iterative MAP inference algorithm which has the anytime property i.e., can produce solutions of increasing quality with time. C2F Function (see Algorithm 1) takes 3 inputs: a set of C2F partitions , inference algorithm , and a stopping criteria . The algorithm in turn takes three inputs: PGM , starting_assignment , stopping criteria . outputs an approximation to the MAP solution once the stopping criteria is met. Starting with the coarsest partition (
in line 2), a start state is picked for the coarsest problem to be solved (line 3). In each iteration (line 4), C2F finds the MAP estimate for the current problem () using algorithm (line 5). This solution is then mapped to a same energy solution of the next finer partition (line 6) which becomes the starting state for the next run of A. The solution is thus successively refined in each iteration. The process is repeated until we reach the finest level of partition. In the end, is run on the finest partition and the resultant solution is output (lines 9,10). Since the last partition in the set is the original problem , optimality with respect to is guaranteed.
Next, we describe how to use the color passing algorithm (Section 2) to get a series of partitions which get successively finer. Our C2F algorithm can then be applied on this set of partitions to get anytime solutions of high quality while being computationally efficient.
3.3 C2F Partitioning for Computer Vision
We now adapt the general color passing algorithm to MRFs for CV problems. Unfortunately, unary potentials make color passing highly ineffective. Different pixels have different RGB values and intensities, leading to almost every pixel getting a different unary potential. Naive application of color passing splits almost all variables into their own partitions, and lifting offers little value.
A natural approximation is to define a threshold, such that two unary potentials within that threshold be initialized with the same color. Our experiments show limited success with this scheme because because two pixels may have the same label even when their actual unary potentials are very different. What is more important is relative importance given to each label than the actual potential value.
In response, we adapt color passing for CV by initializing it as before, but with one key change: we initialize two unary potential nodes with the same color if their lowest energy labels have the same order for the top labels (we call this unary split threshold). Experiments reveal that this approximation leads to effective partitions for lifted inference.
Finally, we can easily construct a sequence of coarse-to-fine partitions in the natural course of color passing’s execution – every iteration of color passing creates a finer partition. Moreover, as an alternative approach, we may also increase . In our implementations, we intersperse the two, i.e., before every next step we pick one of two choices: either, we run another iteration of color passing; or, we increase by one, and split each variable partition based on the lowest energy labels of its constituent variables.
We parameterize to denote the partition from the current state of color passing, which has been run till iterations and unary split threshold is . It is easy to prove that another iteration of color passing or splitting by increasing as above leads to a finer partition. I.e., and . We refer to each element of a partition of variables as a lifted pixel, since it is a subset of pixels.
4 Lifted Inference for Stereo Matching
We first demonstrate the value of lifted inference in the context of stereo matching [Scharstein and Szeliski2002]. It aims to find pixel correspondences in a set of images of the same scene, which can be used to further estimate the 3D scene. Formally, two images and corresponding to images of the scene from a left camera and a right camera are taken such that both cameras are at same horizontal level. The goal is to compute a disparity labeling for every pixel such that corresponds to . We build a lifted version of TSGO [Mozerov and van de Weijer2015], as it is MRF-based and ranks on the Middlebury Stereo Evaluation Version 2 leaderboard.333http://vision.middlebury.edu/stereo/eval/
Background on TSGO: TSGO treats stereo matching as a two-step energy minimization, where the first step is on a fully connected MRF with pairwise potentials and the second is on a conventional locally connected MRF. Lack of space precludes a detailed description of the first step. At the high level, TSGO runs one iteration of message passing on fully connected MRF, computes marginals of each pixel , which act as unary potentials for the MRF of second step.
The pairwise potential used in step two is , where is a truncated linear function of , and takes one of three distinct values depending on color difference between pixels. The MAP assignment computes the lowest energy assignment of disparities for every pixel for this MRF.
Lifted TSGO: Since step two is costlier, we build its lifted version as discussed in previous section. For color passing, two unary potential nodes are initialized with the same color if their lowest energy labels exactly match (). Other initializations are consistent with original color passing for general MRFs. A sequence of coarse-to-fine models is outputted as per Section 3.3. C2F TSGO uses outputs from the sequence , , and then refines to the original MRF. Model refinement is triggered whenever energy hasn’t decreased in the last four iterations of alpha expansion (this becomes the stopping criteria in Algorithm 1).
Experiments: Our experiments build on top of the existing TSGO implementation444http://www.cvc.uab.es/mozerov/Stereo/, but we change the minimization algorithm in step two to alpha expansion fusion [Lempitsky et al.2010] from OpenGM2 library [Andres et al.2010, Kappes et al.2015], as it improves the speed of the base implementation. We use the benchmark Middlebury Stereo datasets of 2003, 2005 and 2006 [Scharstein and Szeliski2003, Hirschmuller and Scharstein2007]. For the 2003 dataset, quarter-size images are used and for others, third-size images are used. The label space is of size 85 (85 distinct disparity labels).
We compare our coarse-to-fine TSGO (using partitions) against vanilla TSGO. Figures 1(a,b) show the aggregate plots of energy (and error) vs. time. We observe that C2F TSGO reaches the same optima as TSGO, but in less than half the time. It has a much superior anytime performance – if inference time is given as a deadline, C2F TSGO obtains 59.59% less error on average over randomly sampled deadlines. We also eyeball the outputs of C2F TSGO and TSGO and find them to be visually similar. Figure 2 shows a sample qualitative comparison. Figure 2 shows five of the ten largest partition elements in the partition from . Clearly, the partition elements formed are not contiguous, and seem to capture variables that are likely to get the same assignment. This underscores the value of our lifting framework for CV problems.
We also compare our partitioning strategy with threshold partitioning discussed in Section 3.3. We merge two pixels in thresholding scheme if the L1-norm distance of their unary potentials is less than a threshold. For each partition induced by our approach, we find a value of threshold that has roughly the same number of lifted pixels. Figure 1(c) shows that partitions based on and converges to a much lower energy quickly compared to the corresponding threshold values ( and respectively). For , convergence is slower compared to corresponding threshold () but eventually has significantly better quality.
5 Lifted Inference for Image Segmentation
We now demonstrate the general nature of our lifted CV framework by applying it to a second task. We choose multi-label interactive image segmentation, where the goal is to segment an image based on a seed labeling (true labels for a few pixels) provided as input. Like many other CV problems, this also has an MRF-based solution, with the best label-assignment generally obtained by MAP inference using graph cuts or loopy belief propagation [Boykov et al.2001, Szeliski et al.2008].
However, MRFs with only pairwise potentials are known to suffer from short-boundary bias – they prefer segmentations with shorter boundaries, because pairwise potentials penalize every pair of boundary pixels. This leads to incorrect labeling for sharp edge objects. Kohli et al. kohli2013principled use CoGC, cooperative graph cuts [Jegelka and Bilmes2011], to develop one of the best MRF-based solvers that overcome this bias.
Background on CoGC: Traditional MRFs linearly penalize the number of label discontinuities at edges (boundary pixel pairs), but CoGC penalizes the number of types of label discontinuities through the use of a concave energy function over groups of ordered edges. It first clusters all edges on the basis of color differences, and later applies a concave function separately over the number of times a specific discontinuity type is present in each edge group . Their carefully engineered CoGC energy function is as follows:
where unary potentials depend on colors of seed pixels, is a concave function, the indicator function, and depends on the color difference between . Intuitively, collects all edges with similar discontinuities and penalizes them sub-linearly, thus reducing the short-boundary bias in the model.
The usage of a concave function makes the MRF higher order with cliques over edge groups. However, the model is shown to reduce to a pairwise hierarchical MRF through the addition of auxiliary variables.
Lifted CoGC: CoGC is lifted using the framework of Section 3, with one additional change. We cluster edge groups using color difference and the position of the edge. Edge groups that are formed only on the basis of color difference make the error of grouping different segment’s boundaries into a single group. For e.g., it erroneously cluster boundaries between white cow and grass, and sky and grass together in the top image in Figure 3.
Coarse-to-fine partitions are obtained by the method described in Section 3.3. C2F CoGC uses outputs from the sequence before refining to the original MRF. Model refinement is triggered if energy has not reduced over the last iterations.
Experiments: Our experiments use the implementation of Cooperative Graph Cuts as provided by [Kohli et al.2013].555Available at https://github.com/aosokin/coopCuts_CVPR2013 Energy minimization is performed using alpha expansion [Boykov et al.2001]. The implementation of CoGC performs a greedy descent on auxiliary variables while performing alpha expansion on the remaining variables, as described in Kohli et. al. kohli2013principled. The dataset used is provided with the implementation. It is a part of the MSRC V2 dataset.666Available at https://www.microsoft.com/en-us/research/ project/image-understanding/?from=http%3A%2F%2Fresearch. microsoft.com%2Fvision%2Fcambridge%2Frecognition%2F.
Figure 3 shows three individual energy vs. time plots. Results on other images are similar. We find that C2F CoGC algorithm converges to the same energy as CoGC in about two-thirds the time on average. Overall, C2F CoGC achieves a much better anytime performance than other lifted and unlifted CoGC.
Similar to Section 4, refined partitions attain better quality than coarser ones at the expense of time. Since the implementation performs a greedy descent over auxiliary variables, refinement of current partition also resets the auxiliary variables to the last value that produced a change. Notice that energy minimization on output of attains a lower energy than on . This observation drives our decision to refine by increasing . Qualitatively, C2F CoGC produces the same labeling as CoGC. Finally, similar to stereo matching, partitions based on thresholding scheme perform significantly worse compared to for image segmentation as well.
6 Related Work
There is a large body of work on exact lifting, both marginal [Kersting et al.2009, Gogate and Domingos2011, Niepert2012, Mittal et al.2015] and MAP [Kersting et al.2009, Gogate and Domingos2011, Niepert2012, Sarkhel et al.2014, Mittal et al.2014], which is not directly applicable to our setting. There is some recent work on approximate lifting [Van den Broeck and Darwiche2013, Venugopal and Gogate2014, Singla et al.2014, Sarkhel et al.2015, Van den Broeck and Niepert2015] but it’s focus is on marginal inference whereas we are interested in lifted MAP. Further, this work can’t handle a distinct unary potential on every node. An exception is work by Bui et al. bui&al12 which explicitly deals with lifting in presence of distinct unary potentials. Unfortunately, they make a very strong assumption of exchangeability in the absence of unaries which does not hold true in our setting since each pixel has its own unique neighborhood.
Work by Sarkhel et al. sarkhel&al15 is probably the closest to our work. They design a C2F hierarchy to cluster constants for approximate lifted MAP inference in Markov logic. In contrast, we partition ground atoms in a PGM. Like other work on approximate lifting, they can’t handle distinct unary potentials. Furthermore, they assume that their theory is provided in a normal form, i.e., without evidence, which can be a severe restriction for most practical applications. Kiddon & Domingos kiddon&domingos11 also propose C2F inference for an underlying Markov logic theory. They use a hierarchy of partitions based on a pre-specified ontology. CV does not have any such ontology available, and needs to discover partitions using the PGM directly.
Nath & Domingos nath&domingos10 exploit (approximate) lifted inference for video segmentation. They experiment on a specific video problem (different from ours), and they only compare against vanilla BP. Their initial partitioning scheme is similar to our thresholding approach, which does not work well in our experiments.
In computer vision, a popular approach to reduce the complexity of inference is to use superpixels [Achanta et al.2012, Van den Bergh et al.2012]. Superpixels are obtained by merging neighboring nodes that have similar characteristics. All pixel nodes in the same superpixel are assigned the same value during MAP inference. SLIC [Achanta et al.2012] is one of the most popular algorithms for discovering superpixels. Our approach differs from SLIC in some significant ways. First, their superpixels are local in nature whereas our algorithm can merge pixels that are far apart. This can help in merging two disconnected regions of the same object in a single lifted pixel. Second, they obtain superpixels independent of the inference algorithm, whereas we tightly integrate our lifting with the underlying inference algorithm. This can potentially lead to discovery of better partitions; indeed, this helped us tremendously in image segmentation. Third, they do not provide a C2F version of their algorithm and we did not find it straightforward to extend their approach to discover successively finer partitions. There is some recent work [Wei et al.2016] which addresses last two of these challenges by introducing a hierarchy of superpixels. In our preliminary experiments, we found that SLIC and superpixel hierarchy perform worse than our lifting approach. Performing more rigorous comparisons is a direction for future work.
7 Conclusion and Future Work
We develop a generic template for applying lifted inference to structured output prediction tasks in computer vision. We show that MRF-based CV algorithms can be lifted at different levels of abstraction, leading to methods for coarse to fine inference over a sequence of lifted models. We test our ideas on two different CV tasks of stereo matching and interactive image segmentation. We find that C2F lifting is vastly more efficient than unlifted algorithms on both tasks obtaining a superior anytime performance, and without any loss in final solution quality. To the best of our knowledge, this is the first demonstration of lifted inference in conjunction with top of the line task-specific algorithms. Although we restrict to CV in this work, we believe that our ideas are general and can be adapted to other domains such as NLP, and computational biology. We plan to explore this in the future.
We thank anonymous reviewers for their comments and suggestions. Ankit Anand is being supported by the TCS Research Scholars Program. Mausam is being supported by grants from Google and Bloomberg. Parag Singla is being supported by a DARPA grant funded under the Explainable AI (XAI) program. Both Mausam and Parag Singla are being supported by the Visvesvaraya Young Faculty Fellowships by Govt. of India. Any opinions, findings, conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views or official policies, either expressed or implied, of the funding agencies.
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