Paper:
ADVERSARIAL IMAGINATION PRIORS
Given an image, humans effortlessly run the image formation process backwards in their minds: they can tell albedo from shading, foreground from background, and imagine the occluded parts of the scene behind foreground objects. In this work, we propose a weakly supervised inversion machine trained to generate similar imaginations that when rendered using differentiable, graphics-like decoders, produce the original visual input. We constrain the imagination spaces by providing exemplar memory repositories in the form of foreground segmented objects, albedo, shading, background scenes and imposing adversarial losses on the imagination spaces. Our model learns to perform such inversion with weak supervision, without ever having seen paired annotated data, that is, without having seen the image paired with the corresponding ground-truth imaginations. We demonstrate our method by applying it to three Computer Vision tasks: image in-painting, intrinsic decomposition and object segmentation, each task having its own differentiable renderer. Data driven adversarial imagination priors effectively guide inversion, minimize the need for hand designed priors of smoothness or good continuation, or the need for paired annotated data. Consider Figure 1. We imagine a missing triangle occluding three small black circles rather than three carefully arranged pacman shapes – which is what the pixels depict. In (b), we do not perceive two parts of the sea separated by a standing person, rather a continuous sea landscape. In (c), we explain the input as a ”masked 8” rather than two semicircles. Consistent explanations of visual observations in terms of familiar concepts and memories we call “imaginations”. Imaginations invert the image formation process and propose 3D shape, camera pose, scene layering, spatial layout, albedo, shading, inpainted, un-occluded perceptions of the world, necessary for the understanding of the visual scene and interaction with it. Gestalt philosophers (Smith (1988)) proposed a set or principles to explain formation of such percepts, such as, closure, center surround pop-out, good continuity, smoothness etc, which many works attempt to hand design principles to incorporate those into computational frameworks of e.g., perceptual grouping (Yu (2003)). In this work, we present a learning-based inversion model that uses data-driven priors instead. We propose a computational model that addresses inverse problems in Computer Vision using adversarial imagination priors. Figure 2 illustrates our model. It is comprised of a generator neural network that given a visual input predicts visual imaginations, such as, in-painted image, un-occluded background scene, object segmentation, albedo and shading etc. Relevant memories, assumed to +
Consider Figure 1. We imagine a missing triangle occluding three small black circles rather than three carefully arranged pacman shapes – which is what the pixels depict. In (b), we do not perceive two parts of the sea separated by a standing person, rather a continuous sea landscape. In (c), we explain the input as a ”masked 8” rather than two semicircles. Consistent explanations of visual observations in terms of familiar concepts and memories we call “imaginations”. Imaginations invert the image formation process and propose 3D shape, camera pose, scene layering, spatial layout, albedo, shading, inpainted, un-occluded perceptions of the world, necessary for the understanding of the visual scene and interaction with it. Gestalt philosophers (Smith (1988)) proposed a set or principles to explain formation of such percepts, such as, closure, center surround pop-out, good continuity, smoothness etc, which many works attempt to hand design principles to incorporate those into computational frameworks of e.g., perceptual grouping (Yu (2003)). In this work, we present a learning-based inversion model that uses data-driven priors instead.
We propose a computational model that addresses inverse problems in Computer Vision using adversarial imagination priors. Figure 2 illustrates our model. It is comprised of a generator neural network that given a visual input predicts visual imaginations, such as, in-painted image, un-occluded background scene, object segmentation, albedo and shading etc. Relevant memories, assumed to
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How much is it?
Under review as a conference paper at ICLR 2017
Distribution Matching
have been acquired from past experience, are retrieved based on coarse attribute matching. Fullyconvolutional discriminator networks match statistics of the generated imaginations with retrieved relevant memories. A non-parametric graphics-like differentiable renderer projects such imaginations accordingly and reconstructs the original image. Our model is trained using a combination of adversarial and reconstruction losses.
Architectures we explore ensure the original images can be reconstructed from the inferred imaginations using basic, parameter-free differentiable renderers. This particular choice of decoder function further enforces the imagination spaces to take the particular desired forms, along with the adversarial priors. We are inspired by work of capsules (Tieleman (2014)) that first introduced such domain specific, graphics-like decoders for image generation. We empirically validate the choice of such decoders against standard parametric deconvolutional networks employed by previous works, e.g., inverse graphics network of Kulkarni et al. (2015b).
Our model can infers visual imaginations without having seen paired annotations, that is, each input image paired with the corresponding ground-truth. Instead, repositories of relevant memories suffice, in the form of collections of albedo, shading, segmented objects, complete background scenes etc. This distinguishes it from previous works that rely on supervision for decomposing an image into imaginations (Kulkarni et al. (2015b)) or that train image conditioned generator networks using a combination of adversarial and L2 reconstruction loss on imaginations -such as works of Pathak et al. (2016) for in-painting, work of Jiajun Wu (2016) for 3D object reconstruction, work of Dong et al. (2015) for super-resolution. In these works, L2 loss is used to condition the imagination on the input image, e.g., in Jiajun Wu (2016), the adversarial loss ensures the generated voxel grid looks like a 3D object and the L2 loss ensures it is the 3D object corresponding to that particular input image instead of an arbitrary one. Instead, we employ a different method of conditioning: we add a reconstruction loss after our graphics-like decoder, when imaginations are projected (rendered back) to image pixels, reconstructing the original image; our model is like an autoencoder in that sense. In this way, we do not need paired supervision, we can take advantage of unlabelled data and we do not discriminate between training and test phases: adversarial priors useful for inversion can be employed at any time, and the relevant memory repositories may be updated. However, any available annotated pairs can always be used to pretrain the generator network. We did consider such small amount of pretraining in one of the considered tasks and this in this paper to emphasize the power of adversarial priors.
Our model enables feedback from the input image directly to its memory priors: the relevant memory engine retrieves memories based on matching of attribute/ feature descriptors. In this way, priors are tailored to the visual input which alleviates the rare sample problem of traditional training methods, which suffer from the imbalance of training data samples: some examples are way more typical than others, and thus more represented on the neural network weights. Hard negative mining has been used to fight such skewness of training distributions. We empirically show that fixing the prior
distribution instead of adapting it results in undesirable, wrong imaginations, and makes the balance of adversarial and reconstruction losses dependent on example by example basis.
In our model, distribution matching of predicted imaginations and retrieved memories concerns local image statistics. In contrast to most previous use cases of adversarial networks, our work (1) conditions imaginations to the input image -does not generate from random noise- and (2) has a feedback loop by projecting imaginations back to the image through rendering and L2 reconstruction loss. Both (1) and (2) constrain the imagination space and thus our adversarial distribution matching cares mostly about local statistics, rather than global structure, texture matching rather than semantic content. For example, for grass in-painting, we do not care whether the imagination looks similar or exactly the same as a retrieve grass image, we only want to make sure each part in the imagination follows a grass-like texture. We propose fully-convolutional discriminator networks, that predict real versus fake binary tests densely across the feature grid, rather than once for the whole image. This accelerates training, makes our model robust to the size of the network input size and across different field-of-views.
In summary, our contributions are as follows:
• A weakly supervised model for inverse problems given visual input based on adversarial imagination priors and graphics-like decoders.
• Relevant memory retrieval for informative adversarial priors. • Fully-convolutional discriminator networks for matching local image statistic distributions
robust to network input size and image field-of-view.
We demonstrate our model in the tasks of image in-painting, figure-ground layer extraction and intrinsic image decomposition. We show successful imagination prediction without using paired ground-truth annotations. We are working towards updating the draft with inverse problems in videos to convey the generality of the proposed model.
1 RELATED WORK
Vision as an inference problem Both Computer and Human Vision fields have worked towards models that given visual observations attempt to infer hidden properties about the visual scene by inverting the image formation process, ”un”-doing camera projection, occlusions, motion blur, downsampling, image masking. Examples are inferring 3D shape and camera pose in videos or images in Tomasi & Kanade (1992), decoupling 3D shape, lighting and albedo interactions in Barron & Malik (2013); Kong et al. (2014), inferring scene depth segmentation layering in Yang et al. (2012), super-resolving low resolution input in Yang et al. (2010), filling in pixels in masked (”hole”) images (in-painting) Efros & Leung (1999) etc.
Multimodality Inverse problems are ill-posed: There are many imagination solutions whose projection or rendering would result in the same visual image. Multi-modality of the desired hidden representation causes methods that rely on maximum likelihood to suffer from regression-to-themean problem. Despite this fact, direct feed-forward neural networks regressors or classifiers have been trained in a supervised way to achieve such inversion, e.g., depth estimation in Eigen et al. (2014), albedo estimation in Narihira et al. (2015b), volumetric inference in Firman et al. (2016), super-resolution in Dong et al. (2015) etc. The argument is that with large enough receptive fields, ambiguities of inversion are diminished. However, such approaches require human supervision, may not generalize well enough to handle different inputs, adapt to the example at hand effectively, or achieve global consistency of the solution (Narihira et al. (2015a)).
Generative adversarial networks (Goodfellow et al. (2014); Sebastian Nowozin (2016); Radford et al. (2015)) instead have shown to minimize Jensen-Shannon divergence between the matched distributions and exhibit a mode seeking behaviour (Theis et al. (2016)) desirable for inversion. Our adversarial priors can be thought as a surrogate to true perceptual losses, which would involve humans in the loop and would be very expensive to obtain in practise.
Concurrent work of Sønderby et al. (2016) proposes a model for super-resolution that does not required paired supervision, similar to our model. They have an adversarial loss in the high resolution
image and their decoder is a downsampler. Their model is a special case of our model, in which we consider general graphics-like decoders, tailored to each task. Further, they do not consider relevant memory retrieval and do not consider fully-convolutional decoders.
Priors Other research approaches on inverse problems do not employ learning but rather rely on hand designed priors, such as sparsity in Yang et al. (2010), spatial smoothness (for optical flow, depth, albedo etc), temporal smoothness (for shading in Kong et al. (2014)), low-rank 3D shape or trajectory priors in Akhter et al. (2008); Wu et al. (2016), deformable 3D scene models in Kulkarni et al. (2015a). Such hand designed priors, though do not suffer from generalization issues, cannot exploit data available effectively.
Our work proposes data driven priors implemented through adversarial distribution matching between inferred imaginations and retrieval memories. Such priors exploit unlabelled data available in the form of imagination repositories, do not suffer from training and test discrepancies, do not need paired supervision and alleviate the engineering burden of designing good prior models.
Feedback Feedback is visual processing has been incorporated in recent computational models through iterative processing, where each step produces a better estimate of the relevant memory, let is be image reconstruction Raiko et al. (2014), body pose estimation Carreira et al. (2015) etc. Such feedback is incorporated in our adversarial prior model through a memory retrieval mechanism which uses coarse feature extraction and attributes on unoccluded parts of the visual input to retrieve relevant examples, and thus influence the reconstruction in an example by example case, alleviating the problem of data imbalance and finetuning, catastrophic forgetting, hard negative mining of traditional training paradigms.
Domain specific non-parametric decoders Model architectures we explore are based on the fact that the inferred imaginations are such that the original image can be reconstructed using basic, parameter-free operations, such as, camera projection, that project inferred 3D and camera pose to 2D scene Handa et al. (2016), pointwise multiplication for image decomposition, layering that assembles different imaginations based on their depth and segmentation masks. Our work is inspired by work of Tieleman (2014) which proposes capsules, a model for image generation by assembling 2D image pieces and their poses predicted from the encoder into one canvas.
2 MODEL
Our model is illustrated in Figure 2. Given a set of images X = {xi, x2, · · · , xn}, and a memory database M , a generator network inverts each image x into a set of imaginations z1,z2, · · ·zK , which, (1) when rendered back to pixels, the projection should match the corresponding input image; and (2) the imagination statistics should match the distribution of relevant memories retrieved from M through a memory retrieval engine. Our model is trained to minimize the combination of (1) an image reconstruction loss and (2) an adversarial imagination loss that constraints the imagination space(s). The imagination spaces and renderer architecture depend on the inversion task. We consider thress tasks in this work: 1) image in-painting, 2) intrinsic image decomposition, 3) figure-ground layer extraction.
We denote the generator as a mapping function from input to imaginations G(x), the renderer as a mapping function from the imagination to the input image P (z), the image retrieval engine as a mapping function from memories M and input image x to relevant memories R(M,x), and the discriminator for imposing distribution matching between imaginations and retrieved memories as D. In case of multiple imagination spaces (e.g., shading and albedo, in-painted background and foreground object mask etc.) we will use Gi(x) to denote the i-th imagination proposed by the generator. Suppose there are K imagination spaces, the memory retrieval engine will need to retrieved K relevant memory that corresponds to each of the imaginations. Besides, we also need K discriminators to look after each generated imagination. Here we use Ri(M,x) and Di to denote
the corresponding retrieved relevant memory and discriminator for the i-th imagination space. Our loss reads as follows:
min D max G Ex∈X ||P (G(x))− x||2︸ ︷︷ ︸ reconstruction loss +β n∑ i=1 logDi(Ri(M,x)) + log(1−Di(Gi(x)))︸ ︷︷ ︸ adversarial loss , (1)
where β the relative weight of reconstruction and adversarial losses.
2.1 IMAGINATION GENERATOR G
Given an image, the generator outputs one or more imaginations. In the tasks we consider, imaginations have a retinotopic representation, that is, they have the same size as the input image. Our generators are convolutional/deconvolutional neural networks with skip-layer connections from the encoding to the decoding layers. Skip-layer connections much improve the precision of the produced imaginations. We share weights of the first convolutional layers across multiple imagination spaces. Figure 3 shows the generator architectures we used for the different inversion tasks.
2.2 IMAGINATION RENDERER P
Below we present our domain-specific renderers for three Computer Vision tasks: image in-painting, intrinsic image decomposition and figure-ground layer extraction.
Image in-painting The input is a masked image x, an image whose content is covered by a black contiguent mask m. The task is to invert such masking and produce an imagination that corresponds to the complete (in-painted) image before the masking operation, as shown in Figure 3 (a). The rendering function P in this case is defined as P (z) = m z,where denotes pointwise multiplication. Intrinsic image decomposition Given an image x, the generator generates albedo z1 and shading z2, as shown in Figure 3 (b). For Lambertian surfaces that the product of albedo and shading should recover the original image, we thus define our renderer to be: P (z1,z2) =z1 z2. Note that we need two discriminator networks, one that controls the statistics distribution of generated albedos and one that controls the statistics distribution of generated shading imaginations. In practise, instead of pointwise multiplication, we used addition in the log space.
Figure-ground layer extraction In this task, given an image, we want to invert the layering superimposition caused by the objects against their background and produce imaginations of the segmented objects and in-painted background scene. Given an image x, the generator outputs a foreground segmentation mask zm, corresponding image foreground z1 = x zm and an in-painted background z2 such that the in-painted background matches the relevant background memories and the image foreground matches memories of segmented relevant objects with clean (black) background, as shown in Figure 3 (c). Our renderer in this case is defined to be: P (z1,z2,zm) = (1−zm) z2+z1 x, that is, it overlays the object on the in-painted background.
2.3 FULLY-CONVOLUTIONAL DISCRIMINATOR D
We propose fully-convolutional discriminator architectures for matching local image statistics between inferred imaginations and retrieved relevant memories. Fully-convolutional discriminators employ many -instead of one- classifiers centered at grid points of the feature maps, that calculate the confidence scores of being real of fake pattern for each of the local receptive fields, in different layers of the network. Fully-convolutional discriminators allow better generalization from relevant memories to generated imaginations as they match only local statistics and not global patterns. Further, they are much faster and more stable to train as the number of examples fed into the discriminator increases. In our experimental section, we show empirically that fully-convolutional adversarial loss accelerates and stabilizes training.
2.4 MEMORY RETRIEVAL ENGINE R
Given an input image and a memory database of the same imagination types we want to generate, e.g., albedos of natural images, the memory retrieval engine retrieves the most relevant memories. The details of memory retrieval depend on the inversion task.
Image in-painting In this case, we measure the L2 pixel distance between the visible part of the input image and images in our memory database, and retrieve the top nearest neighbors.
Intrinsic image decomposition We retrieve relevant shadings by L2 pixel matching between the grayscale version of the input image and albedo memories. We retrieve relevant albedos by computing pixel matching between the input image and albedo memories.
Figure-ground layer extraction Our foreground object memories are segmented objects, as shown in Figure 3 (c) z1. We retrieve relevant segmented objects according to an object detector output, that makes sure our segmented object imaginations agree on the object category with retrieved memories (here the object category of interest is “chair”). For the in-painted background memories, we use L2 pixel distance between the current image and images from the SUN scene dataset.
In any of the aforementioned inversion tasks, after the model starts to generate reasonable initial imaginations, we can use those to retrieve more relevant memories. Such iterative feedback between memory and visual processing though very reasonable to do, we did not consider it in this work to keep the framework simple.
3 EXPERIMENTS
We show results of our model for (1) image in-painting, (2) intrinsic image decomposition and (3) figure-ground layer extraction. The corresponding model architectures are shown in Figure 3 and further training details are provided in the Appendix.
3.1 IMAGE IN-PAINTING
We used the MNIST dataset and masked parts of its digit images. Specifically, we randomly selected 2500 samples of digits 0, 1, 2, 3 from the dataset and overlayed a squared mask at the center over them to create our input images. Our memory databaseM contains 1000 samples for each of the ten digits. We purposefully designed such distribution mismatch between the input image dataset and memory database to study the usefulness of retrieved memories under a controlled setup. The set of digit images contained in M does not intersect the set of images we used to create our input images. In other words, the groundtruth imaginations for our input images are not contained in our memory database.
Figure 4 shows the results of four in-painting models: (1) a baseline with L2 pixel loss between imaginations and retrieved relevant memories (BmemL2), (2) our model with memories retrieved uniformly at random from M rather than conditioned on input images (we supress our memory retrieval engine R) (Bmemrand), (3) our model with memories retrieved uniformly at random from M and with larger weight on the reconstruction loss (BmemrandHR), (4) our model.
Treating retrieved relevant memories as the golden ground-truth produces blurry images, as shown in Figure 4 Row 2. L2 matching optimizes the wrong objective, aside of the fact that it suffers from regression-to-the-mean error even with perfectly correct paired ground-truth, as noted in previous works (e.g., Sønderby et al. (2016)). Bmemrand produces imaginations that look like reasonable digits but do not match the corresponding input image, as shown in Figure 4 Row 3, rightmost column. Such discrepancy between memories and desired imagination distributions cannot be corrected by increasing the reconstruction loss over the adversarial loss (BmemrandHR), shown in Figure 4 Row 4. Then, the resulting imaginations do not look like correct digits anymore. Our model correctly in-paints the masked digits, as shown in Figure 4 Row 5.
For each input digit image we show in Figure 4 Row 6 the closest retrieved memory from our engine R. By comparing the output of our model with the closest memory, we see that we learn to
Inputs
Bmemrand BmemrandHR our model
ima ination BmemL2
Retrieved Memory With R Ra dom draw
interpolate on the memory space and form an imagination that fits the current input image, without copy pasting, as a nearest neighbor memory engine alone would do.
3.2 INTRINSIC IMAGE DECOMPOSITION
We use the MIT intrinsic image dataset of Grosse et al. (2009). We use ten objects for training and ten objects for testing. During training, our inputs are images of the training objects and our memory database contains albedos and shadings for the training objects. At test time, we just evaluate our generator on images of the test objects, without finetuning our model. We used random image cropping for data augmentation as described in the Appendix.
Figure 5 Left shows results of our model which never uses paired annotations, that is, does not have access to pairing of each RGB image with its ground-truth albedo and shading. The results are comparable to an oracle model that has access to such paired supervision and optimizes a regression loss, similar to previous work of Narihira et al. (2015b), shown in Figure 5 Right. Our model effectively generalizes to unseen objects (Figure 5 Bottom Right). Figure 6 shows how using fullyconvolutional discriminators on albedo and shading stabilize training and converge faster.
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regression with full supervisionOur approach with unpaired groundtruth
Testing on unseen objects
Figure 5: MIT intrinsic decomposition with unpaired shading and albedo. I: Input Image, A: Inferred albedo, S: Inferred shading, R: Reconstructed Image using A and S. Left: inferred albedo and shading using our weakly supervised method. Right: inferred albedo and shading using a fully supervised model that minimizes regression loss. The bottom part shows the results of decompositions on unseen objects.
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Fully-Conn D Fully-Conv D + Fully-Conn D Fully-Conv D
Figure 6: L2 distance between inferred albedo imagination and the ground truth. Fully-convolutional discriminators (purple and green lines) converge faster than fully connected ones, that employ only one fake/real classifier per image.
3.3 FIGURE-GROUND LAYER EXTRACTION
We use the seeing 3D chairs dataset of Aubry et al. (2014) as object memory database which contains 1200 different chairs and the SUN scene dataset Xiao et al. (2010) as the background memory database which contains 131000 images. Our input images are generated by randomly selecting an SUN image and cropping it to 64 × 64. After that, we overlaid an chair image on top of it as our input image.
We use 200 background images and 200 chair images to generate a small subset of ”labelled data”, where we provide the network with ground-truth of mask and background and train the network using regression loss, as described in the Appendix. Such small scale supervised pretraining suffices for stability of our model in this task; it is very realistic to assume the existence of such strong sparse supervision. Figure 7 shows the inferred mask and background we obtain.
4 CONCLUSION
We have presented a weakly supervised inverse model of images that predicts imaginations of hidden representations which then renders through image formation or layering to reconstruct the original image. It regularizes the inferred hidden representations using convolutional adversarial priors by distribution matching against retrieved relevant memories. It does not assume paired supervision and can handle multimodal imagination spaces. We have empirically validated are design choices of fully-convolutional adversarial discriminator networks and relevant memory retrieval. We believe the proposed learning paradigm better exploits unlabelled data in the form of images, depth maps, albedo, shading or segmentation maps and complements well human paired annotations.
We are working towards updating the paper with two inversion problems in videos, visual odometry and motion object segmentation. Videos allow for much stronger observation module, with imagination projections from frame to frame, as well as temporal constraining of the imaginations in time. Further, we are working towards quantifying generalization of our imaginations from training to test images, specifically measuring how well our model can do with increasing dissimilarity between memories in the database and input images.

Review:
The paper describes a network architecture for inverse problems in computer vision. Example inverse problems considered are image inpainting, computing intrinsic image decomposition and foreground/background separation.
The architecture is composed of (i) a generator that produces target (latent) output (such as foreground / background regions), 
(ii) renderer that composes that latent output back to the image that can be compared with the input to measure reconstruction error, 
and (iii) adversarial prior that ensures the target output (latent) image respects a certain image statistics.
Strong  points.
- The proposed architecture with memory database is interesting and appears to be novel. 
Weak points:
- Experimental results are only proof-of-concept in toy set-ups and do not clearly demonstrate benefits of the proposed architecture.
- It is unclear whether the memory retrieval engine that retrieves images based on L2 distance on pixel values is going generalize to other more realistic scenarios. 
- Clarity. The clarity of explanation can be also improved (see below).
Detailed evaluation.
Originality:
- The novelty of this work lies in the (iii) adversarial prior that places an adversarial loss between the generated latent output and a single image retrieved from a large unlabelled database of target output examples (called memory). The adversarial prior has a convolutional form matching local image statistics, rather than the entire image.  The particular form of network architecture with the memory-based fully convolutional adversarial loss appears to be novel and potentially interesting.
- Motivation for the Architecture. The weakest point of the proposed architecture is the "Memory retrieval engine" R (section 2.4),
where images are retrieved from the memory by measuring L2 distance on pixel intensities. While this maybe ok for simple problems considered in this work, it is unclear how this can generalize to other more complicated datasets and problems.  
This should be better discussed, better justified and ideally results in some more realistic set-up shown (see below).
Quality:
- Experiments. Results are shown for inpainting of MNIST digits, intrinsic image decomposition on the MIT intrinsic image database, and figure/ground layer extraction on the synthesized dataset of 3D chairs rendered onto background from real photographs.  
 The experimental validation of the model is not very strong and proof-of-concept only. All the experiments are performed in simplified toy set-ups. The MNIST digit inpainting is far from current state-of-the-art on image inpainting in real photographs (see e.g. Pathak et al., 2016). The foreground background separation is done on  only synthetically generated test data. Even for intrinsic image demposition problem there is now relatively large-scale dataset of (Bell et al., 2014), see the citation below.  
While this is probably ok for the ICLR paper, it diminishes the significance of the work. Is this model going to be useful in a real settings? One possibility to address this would be to focus on one of the problems and show results on a challenging state-of-the-art data. It would be great to see the benefits of the memory database. 
S. Bell, K. Bala, and N. Snavely. Intrinsic images in the wild.
ACM Transactions on Graphics, 33(4):159, 2014.
Clarity:
- The clarity of the writing can be improved. I found some of the terminology of the paper, specially the “imagination” and “memory” confusing. From figure 2, it is not clear how the “memories” for the given input image are obtained, which also took me some time to understand.
- To help understand the proposed architecture, it would be useful to draw an illustration of what is happening in the "feature space”, similar in spirit e.g. to figure 2 in