We propose a novel deep network architecture for image denoising based on a Gaussian Conditional Random Field (GCRF) model. In contrast to the existing discriminative denoising methods that train a separate model for each noise level, the proposed deep network explicitly models the input noise variance and hence is capable of handling a range of noise levels. Our deep network, which we refer to as deep GCRF network, consists of two sub-networks: (i) a parameter generation network that generates the pairwise potential parameters based on the noisy input image, and (ii) an inference network whose layers perform the computations involved in an iterative GCRF inference procedure. We train the entire deep GCRF network (both parameter generation and inference networks) discriminatively in an end-to-end fashion by maximizing the peak signal-to-noise ratio measure. Experiments on Berkeley segmentation and PASCALVOC datasets show that the proposed deep GCRF network outperforms state-of-the-art image denoising approaches for several noise levels.
GCRFs were first introduced in @cite_12 by modeling the parameters of the conditional distribution of output given input as a function of the input image. The precision matrix associated with each image patch was modeled as a linear combination of twelve derivative filter-based matrices. The combination weights were chosen as a parametric function of the responses of the input image to a set of oriented edge and bar filters, and the parameters were learned using discriminative training. This GCRF model was extended to Regression Tree Fields (RTFs) in @cite_27 , where regression trees were used for selecting the parameters of Gaussians defined over image patches. These regression trees used responses of the input image to various hand-chosen filters for selecting an appropriate leaf node for each image patch. This RTF-based model was trained by iteratively growing the regression trees and optimizing the Gaussian parameters at leaf nodes. Recently, a cascade of RTFs @cite_4 has also been used for image restoration tasks. In contrast to the RTF-based approaches, all the components of our network are differentiable, and hence it can be trained end-­to-­end using standard gradient-based techniques.
@cite_27 @cite_4 @cite_12
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Conditional random fields (CRFs) are popular discriminative models for computer vision and have been successfully applied in the domain of image restoration, especially to image denoising. For image deblurring, however, discriminative approaches have been mostly lacking. We posit two reasons for this: First, the blur kernel is often only known at test time, requiring any discriminative approach to cope with considerable variability. Second, given this variability it is quite difficult to construct suitable features for discriminative prediction. To address these challenges we first show a connection between common half-quadratic inference for generative image priors and Gaussian CRFs. Based on this analysis, we then propose a cascade model for image restoration that consists of a Gaussian CRF at each stage. Each stage of our cascade is semi-parametric, i.e., it depends on the instance-specific parameters of the restoration problem, such as the blur kernel. We train our model by loss minimization with synthetically generated training data. Our experiments show that when applied to non-blind image deblurring, the proposed approach is efficient and yields state-of-the-art restoration quality on images corrupted with synthetic and real blur. Moreover, we demonstrate its suitability for image denoising, where we achieve competitive results for grayscale and color images. Markov random field (MRF) models are a popular tool for vision and image processing. Gaussian MRF models are particularly convenient to work with because they can be implemented using matrix and linear algebra routines. However, recent research has focused on on discrete-valued and non-convex MRF models because Gaussian models tend to over-smooth images and blur edges. In this paper, we show how to train a Gaussian conditional random field (GCRF) model that overcomes this weakness and can outperform the non-convex field of experts model on the task of denoising images. A key advantage of the GCRF model is that the parameters of the model can be optimized efficiently on relatively large images. The competitive performance of the GCRF model and the ease of optimizing its parameters make the GCRF model an attractive option for vision and image processing applications. After a decade of rapid progress in image denoising, recent methods seem to have reached a performance limit. Nonetheless, we find that state-of-the-art denoising methods are visually clearly distinguishable and possess complementary strengths and failure modes. Motivated by this observation, we introduce a powerful non-parametric image restoration framework based on Regression Tree Fields (RTF). Our restoration model is a densely-connected tractable conditional random field that leverages existing methods to produce an image-dependent, globally consistent prediction. We estimate the conditional structure and parameters of our model from training data so as to directly optimize for popular performance measures. In terms of peak signal-to-noise-ratio (PSNR), our model improves on the best published denoising method by at least 0.26dB across a range of noise levels. Our most practical variant still yields statistically significant improvements, yet is over 20× faster than the strongest competitor. Our approach is well-suited for many more image restoration and low-level vision problems, as evidenced by substantial gains in tasks such as removal of JPEG blocking artefacts. We propose a novel end-to-end trainable deep network architecture for image denoising based on a Gaussian Conditional Random Field (GCRF) model. In contrast to the existing discriminative denoising methods that train a separate model for each individual noise level, the proposed deep network explicitly models the input noise variance and hence is capable of handling a range of noise levels. Our deep network, which we refer to as deep GCRF network, consists of two sub-networks: (i) a parameter generation network that generates the pairwise potential parameters based on the noisy input image, and (ii) an inference network whose layers perform the computations involved in an iterative GCRF inference procedure. We train two deep GCRF networks (each network operates over a range of noise levels: one for low input noise levels and one for high input noise levels) discriminatively by maximizing the peak signal-to-noise ratio measure. Experiments on Berkeley segmentation and PASCALVOC datasets show that the proposed approach produces results on par with the state of-the-art without training a separate network for each individual noise level. This paper presents a novel deep architecture, called neural regression forest (NRF), for depth estimation from a single image. NRF combines random forests and convolutional neural networks (CNNs). Scanning windows extracted from the image represent samples which are passed down the trees of NRF for predicting their depth. At every tree node, the sample is filtered with a CNN associated with that node. Results of the convolutional filtering are passed to left and right children nodes, i.e., corresponding CNNs, with a Bernoulli probability, until the leaves, where depth estimations are made. CNNs at every node are designed to have fewer parameters than seen in recent work, but their stacked processing along a path in the tree effectively amounts to a deeper CNN. NRF allows for parallelizable training of all "shallow" CNNs, and efficient enforcing of smoothness in depth estimation results. Our evaluation on the benchmark Make3D and NYUv2 datasets demonstrates that NRF outperforms the state of the art, and gracefully handles gradually decreasing training datasets. Many computer vision problems can be formulated in a Bayesian framework with Markov Random Field (MRF) or Conditional Random Field (CRF) priors. Usually, the model assumes that a full Maximum A Posteriori (MAP) estimation will be performed for inference, which can be really slow in practice. In this paper, we argue that through appropriate training, a MRF CRF model can be trained to perform very well on a suboptimal inference algorithm. The model is trained together with a fast inference algorithm through an optimization of a loss function on a training set containing pairs of input images and desired outputs. A validation set can be used in this approach to estimate the generalization performance of the trained system. We apply the proposed method to an image denoising application, training a Fields of Experts MRF together with a 1-4 iteration gradient descent inference algorithm. Experimental validation on unseen data shows that the proposed training approach obtains an improved benchmark performance as well as a 1000-3000 times speedup compared to the Fields of Experts MRF trained with contrastive divergence. Using the new approach, image denoising can be performed in real-time, at 8 fps on a single CPU for a 256 × 256 image sequence, with close to state-of-the-art accuracy. This article presents a statistical theory for texture modeling. This theory combines filtering theory and Markov random field modeling through the maximum entropy principle, and interprets and clarifies many previous concepts and methods for texture analysis and synthesis from a unified point of view. Our theory characterizes the ensemble of images I with the same texture appearance by a probability distribution f(I) on a random field, and the objective of texture modeling is to make inference about f(I), given a set of observed texture examples.In our theory, texture modeling consists of two steps. (1) A set of filters is selected from a general filter bank to capture features of the texture, these filters are applied to observed texture images, and the histograms of the filtered images are extracted. These histograms are estimates of the marginal distributions of f( I). This step is called feature extraction. (2) The maximum entropy principle is employed to derive a distribution p(I), which is restricted to have the same marginal distributions as those in (1). This p(I) is considered as an estimate of f( I). This step is called feature fusion. A stepwise algorithm is proposed to choose filters from a general filter bank. The resulting model, called FRAME (Filters, Random fields And Maximum Entropy), is a Markov random field (MRF) model, but with a much enriched vocabulary and hence much stronger descriptive ability than the previous MRF models used for texture modeling. Gibbs sampler is adopted to synthesize texture images by drawing typical samples from p(I), thus the model is verified by seeing whether the synthesized texture images have similar visual appearances to the texture images being modeled. Experiments on a variety of 1D and 2D textures are described to illustrate our theory and to show the performance of our algorithms. These experiments demonstrate that many textures which are previously considered as from different categories can be modeled and synthesized in a common framework. We propose a new semisupervised learning algorithm, referred to as patch distribution compatible semisupervised dimension reduction, for face and human gait recognition. Each image (a face image or an average human silhouette image) is first represented as a set of local patch features and it is further characterized as the corresponding patch distribution feature, which can be expressed as an image-specific Gaussian mixture model (GMM) adapted from the universal background model. Assuming that the individual components of the image-specific GMMs from all the training images reside on a submanifold, we assign a component-level prediction label matrix to each individual GMM component and introduce a new regularizer based on a set of local submanifold smoothness assumptions in our objective function. We also constrain each component-level prediction label matrix to be consistent with the image-level prediction label matrix , as well as enforce to be close to the given labels for the labeled samples. We further use a linear regression function to provide embeddings for the training data and the unseen test data. Inspired by the recent work flexible manifold embedding, we additionally integrate the regression residue in our objective function to measure the mismatch between and , such that our method can better cope with the data sampled from a nonlinear manifold. Finally, the optimal solutions of the component-level prediction label matrix , the image-level prediction label matrix , the projection matrix , and the bias term b can be simultaneously obtained. Comprehensive experiments on three benchmark face databases CMU PIE, FERET, and AR as well as the USF HumanID gait database clearly demonstrate the effectiveness of our algorithm over other state-of-the-art semisupervised dimension reduction methods. We show that a generative random field model, which we call generative ConvNet, can be derived from the commonly used discriminative ConvNet, by assuming a ConvNet for multicategory classification and assuming one of the categories is a base category generated by a reference distribution. If we further assume that the non-linearity in the ConvNet is Rectified Linear Unit (ReLU) and the reference distribution is Gaussian white noise, then we obtain a generative ConvNet model that is unique among energy-based models: The model is piecewise Gaussian, and the means of the Gaussian pieces are defined by an auto-encoder, where the filters in the bottom-up encoding become the basis functions in the top-down decoding, and the binary activation variables detected by the filters in the bottom-up convolution process become the coefficients of the basis functions in the top-down deconvolution process. The Langevin dynamics for sampling the generative ConvNet is driven by the reconstruction error of this autoencoder. The contrastive divergence learning of the generative ConvNet reconstructs the training images by the auto-encoder. The maximum likelihood learning algorithm can synthesize realistic natural image patterns. In this work we introduce a structured prediction model that endows the Deep Gaussian Conditional Random Field (G-CRF) with a densely connected graph structure. We keep memory and computational complexity under control by expressing the pairwise interactions as inner products of low-dimensional, learnable embeddings. The G-CRF system matrix is therefore low-rank, allowing us to solve the resulting system in a few milliseconds on the GPU by using conjugate gradient. As in G-CRF, inference is exact, the unary and pairwise terms are jointly trained end-to-end by using analytic expressions for the gradients, while we also develop even faster, Potts-type variants of our embeddings. We show that the learned embeddings capture pixel-to-pixel affinities in a task-specific manner, while our approach achieves state of the art results on three challenging benchmarks, namely semantic segmentation, human part segmentation, and saliency estimation. Our implementation is fully GPU based, built on top of the Caffe library, and is available at https: github.com siddharthachandra gcrf-v2.0. A novel algorithm based on random forests with surrogate splits is proposed to address the classification problem of incomplete data without imputation.The algorithm allows each tree to cast a vote even the voting process is interrupted by missing attributes.Experimental results on various acknowledged datasets show that the proposed method is robust and efficient. Random forests (RF) is known as an efficient algorithm in classification, however it depends on the integrity of datasets. Conventional methods in dealing with missing values usually employ estimation and imputation approaches whose efficiency is tied to the assumptions of data features. Recently, algorithm of surrogate decisions in RF was developed and this paper proposes a random forests algorithm with modified surrogate splits (Adjusted Weight Voting Random Forest, AWVRF) which is able to address the incomplete data without imputation.Differing from the present surrogate method, in AWVRF algorithm, when the primary splitting attribute and the surrogate attributes of an internal node are all missing, the undergoing instance is allowed to exit at the current node with a vote. Then the weight of the vote is adjusted by the strength of the involved attributes and the final decision is made by weighted voting. AWVRF does not comprise imputation step, thus it is independent of data features.AWVRF is compared with the methods of mean imputation, LeoFill, knnimpute, BPCAfill and conventional RF with surrogate decisions (surrRF) using 50 times repeated 5-fold cross validation on 10 acknowledged datasets. In a total of 22 experiment settings, the method of AWVRF harvests the highest accuracy in 14 settings and the largest AUC in 7 settings, exhibiting its superiority over other methods. Compared with surrRF, AWVRF is significantly more efficient and remain good discrimination of prediction. Experimental results show that the present AWVRF algorithm can successfully handle the classification task for incomplete data. Abstract Visual tracking algorithms based on structured output support vector machine (SOSVM) have demonstrated excellent performance. However, sampling methods and optimization strategies of SOSVM undesirably increase the computational overloads, which hinder real-time application of these algorithms. Moreover, due to the lack of high-dimensional features and dense training samples, SOSVM-based algorithms are unstable to deal with various challenging scenarios, such as occlusions and scale variations. Recently, visual tracking algorithms based on discriminative correlation filters (DCF), especially the combination of DCF and features from deep convolutional neural networks (CNN), have been successfully applied to visual tracking, and attains surprisingly good performance on recent benchmarks. The success is mainly attributed to two aspects: the circular correlation properties of DCF and the powerful representation capabilities of CNN features. Nevertheless, compared with SOSVM, DCF-based algorithms are restricted to simple ridge regression which has a weaker discriminative ability. In this paper, a novel circular and structural operator tracker (CSOT) is proposed for high performance visual tracking, it not only possesses the powerful discriminative capability of SOSVM but also efficiently inherits the superior computational efficiency of DCF. Based on the proposed circular and structural operators, a set of primal confidence score maps can be obtained by circular correlating feature maps with their corresponding structural correlation filters. Furthermore, an implicit interpolation is applied to convert the multi-resolution feature maps to the continuous domain and make all primal confidence score maps have the same spatial resolution. Then, we exploit an efficient ensemble post-processor based on relative entropy, which can coalesce primal confidence score maps and create an optimal confidence score map for more accurate localization. The target is localized on the peak of the optimal confidence score map. Besides, we introduce a collaborative optimization strategy to update circular and structural operators by iteratively training structural correlation filters, which significantly reduces computational complexity and improves robustness. Experimental results demonstrate that our approach achieves state-of-the-art performance in mean AUC scores of 71.5 and 69.4 on the OTB2013 and OTB2015 benchmarks respectively, and obtains a third-best expected average overlap (EAO) score of 29.8 on the VOT2017 benchmark. We show that a generative random field model, which we call generative ConvNet, can be derived from the commonly used discriminative ConvNet, by assuming a ConvNet for multi-category classification and assuming one of the categories is a base category generated by a reference distribution. If we further assume that the non-linearity in the ConvNet is Rectified Linear Unit (ReLU) and the reference distribution is Gaussian white noise, then we obtain a generative ConvNet model that is unique among energy-based models: The model is piecewise Gaussian, and the means of the Gaussian pieces are defined by an auto-encoder, where the filters in the bottom-up encoding become the basis functions in the top-down decoding, and the binary activation variables detected by the filters in the bottom-up convolution process become the coefficients of the basis functions in the top-down deconvolution process. The Langevin dynamics for sampling the generative ConvNet is driven by the reconstruction error of this auto-encoder. The contrastive divergence learning of the generative ConvNet reconstructs the training images by the auto-encoder. The maximum likelihood learning algorithm can synthesize realistic natural image patterns. Until recently, the lack of ground truth data has hindered the application of discriminative structured prediction techniques to the stereo problem. In this paper we use ground truth data sets that we have recently constructed to explore different model structures and parameter learning techniques. To estimate parameters in Markov random fields (MRFs) via maximum likelihood one usually needs to perform approximate probabilistic inference. Conditional random fields (CRFs) are discriminative versions of traditional MRFs. We explore a number of novel CRF model structures including a CRF for stereo matching with an explicit occlusion model. CRFs require expensive inference steps for each iteration of optimization and inference is particularly slow when there are many discrete states. We explore belief propagation, variational message passing and graph cuts as inference methods during learning and compare with learning via pseudolikelihood. To accelerate approximate inference we have developed a new method called sparse variational message passing which can reduce inference time by an order of magnitude with negligible loss in quality. Learning using sparse variational message passing improves upon previous approaches using graph cuts and allows efficient learning over large data sets when energy functions violate the constraints imposed by graph cuts. In this work we propose a structured prediction technique that combines the virtues of Gaussian Conditional Random Fields (G-CRF) with Deep Learning: (a) our structured prediction task has a unique global optimum that is obtained exactly from the solution of a linear system (b) the gradients of our model parameters are analytically computed using closed form expressions, in contrast to the memory-demanding contemporary deep structured prediction approaches [1, 2] that rely on back-propagation-through-time, (c) our pairwise terms do not have to be simple hand-crafted expressions, as in the line of works building on the DenseCRF [1, 3], but can rather be ‘discovered’ from data through deep architectures, and (d) out system can trained in an end-to-end manner. Building on standard tools from numerical analysis we develop very efficient algorithms for inference and learning, as well as a customized technique adapted to the semantic segmentation task. This efficiency allows us to explore more sophisticated architectures for structured prediction in deep learning: we introduce multi-resolution architectures to couple information across scales in a joint optimization framework, yielding systematic improvements. We demonstrate the utility of our approach on the challenging VOC PASCAL 2012 image segmentation benchmark, showing substantial improvements over strong baselines. We make all of our code and experiments available at https: github.com siddharthachandra gcrf. The task of modeling the distribution of a large number of tree species under future climate scenarios presents unique challenges. First, the model must be robust enough to handle climate data outside the current range without producing unacceptable instability in the output. In addition, the technique should have automatic search mechanisms built in to select the most appropriate values for input model parameters for each species so that minimal effort is required when these parameters are fine-tuned for individual tree species. We evaluated four statistical models—Regression Tree Analysis (RTA), Bagging Trees (BT), Random Forests (RF), and Multivariate Adaptive Regression Splines (MARS)—for predictive vegetation mapping under current and future climate scenarios according to the Canadian Climate Centre global circulation model. To test, we applied these techniques to four tree species common in the eastern United States: loblolly pine (Pinus taeda), sugar maple (Acer saccharum), American beech (Fagus grandifolia), and white oak (Quercus alba). When the four techniques were assessed with Kappa and fuzzy Kappa statistics, RF and BT were superior in reproducing current importance value (a measure of basal area in addition to abundance) distributions for the four tree species, as derived from approximately 100,000 USDA Forest Service’s Forest Inventory and Analysis plots. Future estimates of suitable habitat after climate change were visually more reasonable with BT and RF, with slightly better performance by RF as assessed by Kappa statistics, correlation estimates, and spatial distribution of importance values. Although RTA did not perform as well as BT and RF, it provided interpretive models for species whose distributions were captured well by our current set of predictors. MARS was adequate for predicting current distributions but unacceptable for future climate. We consider RTA, BT, and RF modeling approaches, especially when used together to take advantage of their individual strengths, to be robust for predictive mapping and recommend their inclusion in the ecological toolbox. Recently, models based on conditional random fields (CRF) have produced promising results on labeling sequential data in several scientific fields. However, in the vision task of continuous action recognition, the observations of visual features have dimensions as high as hundreds or even thousands. This might pose severe difficulties on parameter estimation and even degrade the performance. To bridge the gap between the high dimensional observations and the random fields, we propose a novel model that replace the observation layer of a traditional random fields model with a latent pose estimator. In training stage, the human pose is not observed in the action data, and the latent pose estimator is learned under the supervision of the labeled action data, instead of image-to-pose data. The advantage of this model is twofold. First, it learns to convert the high dimensional observations into more compact and informative representations. Second, it enables transfer learning to fully utilize the existing knowledge and data on image-to-pose relationship. The parameters of the latent pose estimator and the random fields are jointly optimized through a gradient ascent algorithm. Our approach is tested on HumanEva [1] --- a publicly available dataset. The experiments show that our approach can improve recognition accuracy over standard CRF model and its variations. The performance can be further significantly improved by using additional image-to-pose data for training. Our experiments also show that the model trained on HumanEva can generalize to different environment and human subjects. Recently there has been a lot of work on pruning filters from deep convolutional neural networks (CNNs) with the intention of reducing computations. The key idea is to rank the filters based on a certain criterion (say, l1-norm, average percentage of zeros, etc) and retain only the top ranked filters. Once the low scoring filters are pruned away the remainder of the network is fine tuned and is shown to give performance comparable to the original unpruned network. In this work, we report experiments which suggest that the comparable performance of the pruned network is not due to the specific criterion chosen but due to the inherent plasticity of deep neural networks which allows them to recover from the loss of pruned filters once the rest of the filters are fine-tuned. Specifically, we show counter-intuitive results wherein by randomly pruning 25-50 filters from deep CNNs we are able to obtain the same performance as obtained by using state of the art pruning methods. We empirically validate our claims by doing an exhaustive evaluation with VGG-16 and ResNet-50. Further, we also evaluate a real world scenario where a CNN trained on all 1000 ImageNet classes needs to be tested on only a small set of classes at test time (say, only animals). We create a new benchmark dataset from ImageNet to evaluate such class specific pruning and show that even here a random pruning strategy gives close to state of the art performance. Lastly, unlike existing approaches which mainly focus on the task of image classification, in this work we also report results on object detection. We show that using a simple random pruning strategy we can achieve significant speed up in object detection (74 improvement in fps) while retaining the same accuracy as that of the original Faster RCNN model. Recent works have shown that exploiting multi-scale representations deeply learned via convolutional neural networks (CNN) is of tremendous importance for accurate contour detection. This paper presents a novel approach for predicting contours which advances the state of the art in two fundamental aspects, i.e. multi-scale feature generation and fusion. Different from previous works directly considering multi-scale feature maps obtained from the inner layers of a primary CNN architecture, we introduce a hierarchical deep model which produces more rich and complementary representations. Furthermore, to refine and robustly fuse the representations learned at different scales, the novel Attention-Gated Conditional Random Fields (AG-CRFs) are proposed. The experiments ran on two publicly available datasets (BSDS500 and NYUDv2) demonstrate the effectiveness of the latent AG-CRF model and of the overall hierarchical framework. Statistical image reconstruction using penalized weighted least-squares (PWLS) criteria can improve image-quality in X-ray computed tomography (CT). However, the huge dynamic range of the statistical weights leads to a highly shift-variant inverse problem making it difficult to precondition and accelerate existing iterative algorithms that attack the statistical model directly. We propose to alleviate the problem by using a variable-splitting scheme that separates the shift-variant and ("nearly") invariant components of the statistical data model and also decouples the regularization term. This leads to an equivalent constrained problem that we tackle using the classical method-of-multipliers framework with alternating minimization. The specific form of our splitting yields an alternating direction method of multipliers (ADMM) algorithm with an inner-step involving a "nearly" shift-invariant linear system that is suitable for FFT-based preconditioning using cone-type filters. The proposed method can efficiently handle a variety of convex regularization criteria including smooth edge-preserving regularizers and non- smooth sparsity-promoting ones based on the l1-norm and total variation. Numerical experiments with synthetic and real in vivo human data illustrate that cone-filter preconditioners accelerate the proposed ADMM resulting in fast convergence of ADMM compared to conventional (nonlinear conjugate gradient, ordered subsets) and state-of-the-art (MFISTA, split-Bregman) algorithms that are applicable for CT.