We introduce a novel aggregation method to efficiently perform image denoising. Preliminary filters are aggregated in a non-linear fashion, using a new metric of pixel proximity based on how the pool of filters reaches a consensus. We provide a theoretical bound to support our aggregation scheme, its numerical performance is illustrated and we show that the aggregate significantly outperforms each of the preliminary filters.
Carnegie Mellon University（卡内基梅隆大学）；University of British Columbia（不列颠哥伦比亚大学）；SenseTime Research（商汤科技研究院）；University of California, Merced（加州大学默塞德分校）
Existing denoising methods typically restore clear results by aggregating pixels from the noisy input. Instead of relying on hand-crafted aggregation schemes, we propose to explicitly learn this process with deep neural networks. We present a spatial pixel aggregation network and learn the pixel sampling and averaging strategies for image denoising. The proposed model naturally adapts to image structures and can effectively improve the denoised results. Furthermore, we develop a spatio-temporal pixel aggregation network for video denoising to efficiently sample pixels across the spatio-tempora…
King Fahd University of Petroleum and Minerals（法赫德国王石油与矿业大学）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
In spite of the improvements achieved by the several denoising algorithms over the years, many of them still fail at preserving the fine details of the image after denoising. This is as a result of the smooth-out effect they have on the images. Most neural network-based algorithms have achieved better quantitative performance than the classical denoising algorithms. However, they also suffer from qualitative (visual) performance as a result of the smooth-out effect. In this paper, we propose an algorithm to address this shortcoming. We propose a deep collaborative filtering-based (Deep-CoFiB) …
The bilateral filter is known to be quite effective in denoising images corrupted with small dosages of additive Gaussian noise. The denoising performance of the filter, however, is known to degrade quickly with the increase in noise level. Several adaptations of the filter have been proposed in the literature to address this shortcoming, but often at a substantial computational overhead. In this paper, we report a simple pre-processing step that can substantially improve the denoising performance of the bilateral filter, at almost no additional cost. The modified filter is designed to be robu…
The bilateral filter is a useful nonlinear filter which without smoothing edges, it does spatial averaging. In the literature, the effectiveness of this method for image denoising is shown. In this paper, an extension of this method is proposed which is based on complex wavelet transform. In fact, the bilateral filtering is applied to the low-frequency (approximation) subbands of the decomposed image using complex wavelet transform, while the thresholding approach is applied to the high frequency subbands. Using the bilateral filter in the complex wavelet domain forms a new image denoising fra…
Indian Institute of Technology, Hyderabad（印度理工学院海得拉巴分校）；Indian Institute of Science（印度科学学院）
The bilateral filter has diverse applications in image processing, computer vision, and computational photography. In particular, this non-linear filter is quite effective in denoising images corrupted with additive Gaussian noise. The filter, however, is known to perform poorly at large noise levels. Several adaptations of the filter have been proposed in the literature to address this shortcoming, but often at an added computational cost. In this paper, we report a simple yet effective modification that improves the denoising performance of the bilateral filter at almost no additional cost. …
This paper introduces a new approach to non-local means image denoising. Instead of using all pixels located in the search window for estimating the value of a pixel, we identify the highly corrupted pixels and assign less weight to these pixels. This method is called robust non-local means. Numerical and subjective evaluations using ultrasound images show good performances of the proposed denoising method in recovering the shape of edges and important detailed components, in comparison to traditional ultrasound image denoising methods
Jiangxi University of Finance and Economics（江西财经大学）；Chinese Academy of Sciences（中国科学院）
A mainstream type of the state of the arts (SOTAs) based on convolutional neural network (CNN) for real image denoising contains two sub-problems, i.e., noise estimation and non-blind denoising. This paper considers real noise approximated by heteroscedastic Gaussian/Poisson Gaussian distributions with in-camera signal processing pipelines. The related works always exploit the estimated noise prior via channel-wise concatenation followed by a convolutional layer with spatially sharing kernels. Due to the variable modes of noise strength and frequency details of all feature positions, this desi…
York University（约克大学）；Weizmann Institute of Science（魏茨曼科学研究所）
While deep learning (DL) architectures like convolutional neural networks (CNNs) have enabled effective solutions in image denoising, in general their implementations overly rely on training data, lack interpretability, and require tuning of a large parameter set. In this paper, we combine classical graph signal filtering with deep feature learning into a competitive hybrid design -- one that utilizes interpretable analytical low-pass graph filters and employs 80% fewer network parameters than state-of-the-art DL denoising scheme DnCNN. Specifically, to construct a suitable similarity graph fo…
Image denoising is one of the fundamental problems in image processing. In this paper, a novel approach to suppress noise from the image is conducted by applying the interquartile range (IQR) which is one of the statistical methods used to detect outlier effect from a dataset. A window of size kXk was implemented to support IQR filter. Each pixel outside the IQR range of the kXk window is treated as noisy pixel. The estimation of the noisy pixels was obtained by local averaging. The essential advantage of applying IQR filter is to preserve edge sharpness better of the original image. A variety…
Filter2Noise: A Framework for Interpretable and Zero-Shot Low-Dose CT Image Denoising
Pattern Recognition Lab, Friedrich-Alexander-University Erlangen-Nuremberg（模式识别实验室，弗里德里希-亚历山大大学埃尔朗根-纽伦堡）；Siemens Healthineers AG（西门子医疗影像有限公司）；Department of Radiology, State Key Laboratory of Complex Severe and Rare Diseases, Peking Union Medical College Hospital, Chinese Academy of Medical Sciences and Peking Union Medical College（放射科，复杂严重罕见疾病国家重点实验室，北京协和医院，中国医学科学院及北京协和医学院）；Theranostics and Translational Research Center, National Infrastructures for Translational Medicine, Institute of Clinical Medicine, Peking Union Medical College Hospital, Chinese Academy of Medical Sciences and Peking Union Medical College（诊疗与转化研究中心，国家转化医学基础设施，临床医学研究院，北京协和医院，中国医学科学院及北京协和医学院）
Indian Institute of Technology Guwahati（印度理工学院古瓦哈蒂分校）；UiT The Arctic University of Norway（挪威特罗姆瑟大学（UiT 挪威北极大学））；Indian Institute of Technology Dhanbad（印度理工学院丹巴德分校）
Scanning acoustic microscopy (SAM) has been employed since microscopic images are widely used for biomedical or materials research. Acoustic imaging is an important and well-established method used in nondestructive testing (NDT), bio-medical imaging, and structural health monitoring.The imaging is frequently carried out with signals of low amplitude, which might result in leading that are noisy and lacking in details of image information. In this work, we attempted to analyze SAM images acquired from low amplitude signals and employed a block matching filter over time domain signals to obtain…
Low-resolution and signal-dependent noise distribution in positron emission tomography (PET) images makes denoising process an inevitable step prior to qualitative and quantitative image analysis tasks. Conventional PET denoising methods either over-smooth small-sized structures due to resolution limitation or make incorrect assumptions about the noise characteristics. Therefore, clinically important quantitative information may be corrupted. To address these challenges, we introduced a novel approach to remove signal-dependent noise in the PET images where the noise distribution was considere…
In this new computing paradigm, named quantum computing, researchers from all over the world are taking their first steps in designing quantum circuits for image processing, through a difficult process of knowledge transfer. This effort is named Quantum Image Processing, an emerging research field pushed by powerful parallel computing capabilities of quantum computers. This work goes in this direction and proposes the challenging development of a powerful method of image denoising, such as the Total Variation (TV) model, in a quantum environment. The proposed Quantum TV is described and its su…
University of Massachusetts Medical School（马萨诸塞大学医学院）；The University of Akron（阿克伦大学）
Noise is a major issue while transferring images through all kinds of electronic communication. One of the most common noise in electronic communication is an impulse noise which is caused by unstable voltage. In this paper, the comparison of known image denoising techniques is discussed and a new technique using the decision based approach has been used for the removal of impulse noise. All these methods can primarily preserve image details while suppressing impulsive noise. The principle of these techniques is at first introduced and then analysed with various simulation results using MATLAB…
Mitsubishi Electric Research Labs (MERL)（三菱电机研究实验室）
In 3D image/video acquisition, different views are often captured with varying noise levels across the views. In this paper, we propose a graph-based image enhancement technique that uses a higher quality view to enhance a degraded view. A depth map is utilized as auxiliary information to match the perspectives of the two views. Our method performs graph-based filtering of the noisy image by directly computing a projection of the image to be filtered onto a lower dimensional Krylov subspace of the graph Laplacian. We discuss two graph spectral denoising methods: first using Chebyshev polynomia…
In this paper we propose a measure of anisotropy as a quality parameter to estimate the amount of noise in noisy images. The anisotropy of an image can be determined through a directional measure, using an appropriate statistical distribution of the information contained in the image. This new measure is achieved through a stack filtering paradigm. First, we define a local directional entropy, based on the distribution of 0's and 1's in the neigborhood of every pixel location of each stack level. Then the entropy variation of this directional entropy is used to define an anisotropic measure. T…
Sparse representation of images under certain transform domain has been playing a fundamental role in image restoration tasks. One such representative method is the widely used wavelet tight frame systems. Instead of adopting fixed filters for constructing a tight frame to sparsely model any input image, a data-driven tight frame was proposed for the sparse representation of images, and shown to be very efficient for image denoising very recently. However, in this method the number of framelet filters used for constructing a tight frame is the same as the length of filters. In fact, through fu…
Nonlinear bilateral filters (BF) deliver a fine blend of computational simplicity and blur-free denoising. However, little is known about their nature, noise-suppressing properties, and optimal choices of filter parameters. Our study is meant to fill this gap-explaining the underlying mechanism of bilateral filtering and providing the methodology for optimal filter selection. Practical application to CT image denoising is discussed to illustrate our results.
The Chinese University of Hong Kong（香港中文大学）；SmartMore
Despite the quality improvement brought by the recent methods, video super-resolution (SR) is still very challenging, especially for videos that are low-light and noisy. The current best solution is to subsequently employ best models of video SR, denoising, and illumination enhancement, but doing so often lowers the image quality, due to the inconsistency between the models. This paper presents a new parametric representation called the Deep Parametric 3D Filters (DP3DF), which incorporates local spatiotemporal information to enable simultaneous denoising, illumination enhancement, and SR effi…
In this paper, we propose a novel image denoising algorithm exploiting features from both spatial as well as transformed domain. We implement intensity-invariance based improved grouping for collaborative support-agnostic sparse reconstruction. For collaboration firstly, we stack similar-structured patches via intensity-invariant correlation measure. The grouped patches collaborate to yield desirable sparse estimates for noise filtering. This is because similar patches share the same support in the transformed domain, such similar supports can be used as probabilities of active taps to refine …
Tampere University of Technology（坦佩雷理工大学）；Noiseless Imaging Ltd（Noiseless Imaging有限公司）
We introduce a paradigm for nonlocal sparsity reinforced deep convolutional neural network denoising. It is a combination of a local multiscale denoising by a convolutional neural network (CNN) based denoiser and a nonlocal denoising based on a nonlocal filter (NLF) exploiting the mutual similarities between groups of patches. CNN models are leveraged with noise levels that progressively decrease at every iteration of our framework, while their output is regularized by a nonlocal prior implicit within the NLF. Unlike complicated neural networks that embed the nonlocality prior within the layer…
IDF: Iterative Dynamic Filtering Networks for Generalizable Image Denoising
Department of Computer Science, Hanyang University（翰阳大学计算机科学系）；School of Computing and Artificial Intelligence, Southwest Jiaotong University（西南交通大学计算机与人工智能学院）
Image textures, as a kind of local variations, provide important information for human visual system. Many image textures, especially the small-scale or stochastic textures are rich in high-frequency variations, and are difficult to be preserved. Current state-of-the-art denoising algorithms typically adopt a nonlocal approach consisting of image patch grouping and group-wise denoising filtering. To achieve a better image denoising while preserving the variations in texture, we first adaptively group high correlated image patches with the same kinds of texture elements (texels) via an adaptive…
Sri Venkateswara College of Engineering, Anna University（安娜大学斯里文卡特斯瓦拉工程学院）
This paper proposes a new technique based on nonlinear Adaptive Median filter (AMF) for image restoration. Image denoising is a common procedure in digital image processing aiming at the removal of noise, which may corrupt an image during its acquisition or transmission, while retaining its quality. This procedure is traditionally performed in the spatial or frequency domain by filtering. The aim of image enhancement is to reconstruct the true image from the corrupted image. The process of image acquisition frequently leads to degradation and the quality of the digitized image becomes inferior…
University of California at San Diego（加州大学圣地亚哥分校）；Purdue University（普渡大学）
We propose a data-dependent denoising procedure to restore noisy images. Different from existing denoising algorithms which search for patches from either the noisy image or a generic database, the new algorithm finds patches from a database that contains only relevant patches. We formulate the denoising problem as an optimal filter design problem and make two contributions. First, we determine the basis function of the denoising filter by solving a group sparsity minimization problem. The optimization formulation generalizes existing denoising algorithms and offers systematic analysis of the …
School of Computer and Communication Sciences, EPFL（洛桑联邦理工学院计算机与通信科学学院）
Deep image denoisers achieve state-of-the-art results but with a hidden cost. As witnessed in recent literature, these deep networks are capable of overfitting their training distributions, causing inaccurate hallucinations to be added to the output and generalizing poorly to varying data. For better control and interpretability over a deep denoiser, we propose a novel framework exploiting a denoising network. We call it controllable confidence-based image denoising (CCID). In this framework, we exploit the outputs of a deep denoising network alongside an image convolved with a reliable filter…
We propose an adaptive approach for non local means (NLM) image filtering termed as non local adaptive clipped means (NLACM), which reduces the effect of outliers and improves the denoising quality as compared to traditional NLM. Common method to neglect outliers from a data population is computation of mean in a range defined by mean and standard deviation. In NLACM we perform the median within the defined range based on statistical estimation of the neighbourhood region of a pixel to be denoised. As parameters of the range are independent of any additional input and is based on local intensi…
Multispectral image denoising with optimized vector non-local mean filter
CSE Department, College of Engineering, Qatar University, P.O. Box 2713, Doha-Qatar（卡塔尔大学工程学院计算机科学与工程系）；LE2i Lab, UMR CNRS 6306, University of Burgundy, BP 47870, 21078, Dijon-France（布尔格斯大学）；Department of Computer Science, University of Biskra, 07000, Algeria（阿尔及利亚比斯克拉大学计算机科学系）
Video Denoising is one of the fundamental tasks of any videoprocessing pipeline. It is different from image denoising due to the tem-poral aspects of video frames, and any image denoising approach appliedto videos will result in flickering. The proposed method makes use oftemporal as well as spatial dimensions of video frames as part of a two-stage pipeline. Each stage in the architecture named as Spatio-TemporalNetwork uses a channel-wise attention mechanism to forward the encodersignal to the decoder side. The Attention Block used in this paper usessoft attention to ranks the filters for bet…
Princeton University（普林斯顿大学）；University of Central Florida（中佛罗里达大学）
The problem of image restoration in cryo-EM entails correcting for the effects of the Contrast Transfer Function (CTF) and noise. Popular methods for image restoration include `phase flipping', which corrects only for the Fourier phases but not amplitudes, and Wiener filtering, which requires the spectral signal to noise ratio. We propose a new image restoration method which we call `Covariance Wiener Filtering' (CWF). In CWF, the covariance matrix of the projection images is used within the classical Wiener filtering framework for solving the image restoration deconvolution problem. Our estim…
Lehigh University（理海大学）；South China University of Technology（华南理工大学）
Filtering multi-dimensional images such as color images, color videos, multispectral images and magnetic resonance images is challenging in terms of both effectiveness and efficiency. Leveraging the nonlocal self-similarity (NLSS) characteristic of images and sparse representation in the transform domain, the block-matching and 3D filtering (BM3D) based methods show powerful denoising performance. Recently, numerous new approaches with different regularization terms, transforms and advanced deep neural network (DNN) architectures are proposed to improve denoising quality. In this paper, we ext…
Northeastern University at Qinhuangdao（东北大学秦皇岛分校）；Univ Bretagne-Sud（南布列塔尼大学）；Northeastern University（东北大学）；Tianjin University（天津大学）
Following the pioneering works of Rudin, Osher and Fatemi on total variation (TV) and of Buades, Coll and Morel on non-local means (NL-means), the last decade has seen a large number of denoising methods mixing these two approaches, starting with the nonlocal total variation (NLTV) model. The present article proposes an analysis of the NLTV model for image denoising as well as a number of improvements, the most important of which being to apply the denoising both in the space domain and in the Fourier domain, in order to exploit the complementarity of the representation of image data in both d…
National University of Defense Technology（国防科技大学）；University of Science and Technology Beijing（北京科技大学）
Image diffusion plays a fundamental role for the task of image denoising. Recently proposed trainable nonlinear reaction diffusion (TNRD) model defines a simple but very effective framework for image denoising. However, as the TNRD model is a local model, the diffusion behavior of which is purely controlled by information of local patches, it is prone to create artifacts in the homogenous regions and over-smooth highly textured regions, especially in the case of strong noise levels. Meanwhile, it is widely known that the non-local self-similarity (NSS) prior stands as an effective image prior …
The Chinese University of Hong Kong（香港中文大学）；Shanghai AI Laboratory（上海人工智能实验室）；SenseBrain（思脑）
Controllable image denoising aims to generate clean samples with human perceptual priors and balance sharpness and smoothness. In traditional filter-based denoising methods, this can be easily achieved by adjusting the filtering strength. However, for NN (Neural Network)-based models, adjusting the final denoising strength requires performing network inference each time, making it almost impossible for real-time user interaction. In this paper, we introduce Real-time Controllable Denoising (RCD), the first deep image and video denoising pipeline that provides a fully controllable user interfac…
GIK Institute of Engineering Sciences and Technology（GIK工程科学与技术学院）
Among the patch-based image denoising processing methods, smooth ordering of local patches (patch ordering) has been shown to give state-of-art results. For image denoising the patch ordering method forms two large TSPs (Traveling Salesman Problem) comprised of nodes in N-dimensional space. Ten approximate solutions of the two large TSPs are then used in a filtering process to form the reconstructed image. Use of large TSPs makes patch ordering a computationally intensive method. A modified patch ordering method for image denoising is proposed. In the proposed method, several smaller-sized TSP…
This paper proposes a new procedure in order to improve the performance of block matching and 3-D filtering (BM3D) image denoising algorithm. It is demonstrated that it is possible to achieve a better performance than that of BM3D algorithm in a variety of noise levels. This method changes BM3D algorithm parameter values according to noise level, removes prefiltering, which is used in high noise level; therefore Peak Signal-to-Noise Ratio (PSNR) and visual quality get improved, and BM3D complexities and processing time are reduced. This improved BM3D algorithm is extended and used to denoise s…
Many patch-based image denoising algorithms can be formulated as applying a smoothing filter to the noisy image. Expressed as matrices, the smoothing filters must be row normalized so that each row sums to unity. Surprisingly, if we apply a column normalization before the row normalization, the performance of the smoothing filter can often be significantly improved. Prior works showed that such performance gain is related to the Sinkhorn-Knopp balancing algorithm, an iterative procedure that symmetrizes a row-stochastic matrix to a doubly-stochastic matrix. However, a complete understanding of…
Adaptive Dynamic Filtering Network for Image Denoising
School of Computer Science and Information Engineering, Hefei University of Technology (HFUT)（合肥工业大学计算机与信息工程学院）；Intelligent Interconnected Systems Laboratory of Anhui Province (HFUT)（安徽省智能互联系统实验室（合肥工业大学））；Guangxi Academy of Sciences（广西科学院）；Intelligent Manufacturing Institute of HFUT（合肥工业大学智能制造研究院）
Gaussian noise removal is an interesting area in digital image processing not only to improve the visual quality, but for its impact on other post-processing algorithms like image registration or segmentation. Many presented state-of-the-art denoising methods are based on the self-similarity or patch-based image processing. Specifically, Non-Local Means (NLM) as a patch-based filter has gained increasing attention in recent years. Essentially, this filter tends to obtain the noise-less signal value by computing the Gaussian-weighted Euclidean distance between the patch under-processing and oth…
Image denoising is a critical issue in the field of digital image processing. This paper proposes a novel Salt & Pepper noise suppression by developing a Kriging Interpolation Filter (KIF) for image denoising. Gray-level images degraded with Salt & Pepper noise have been considered. A sequential search for noise detection was made using kXk window size to determine non-noisy pixels only. The non-noisy pixels are passed into Kriging interpolation method to predict their absent neighbor pixels that were noisy pixels at the first phase. The utilization of Kriging interpolation filter proves that …
Nanjing University（南京大学）；Southeast University（东南大学）；University of Oulu（奥卢大学）
Group-based sparse representation has shown great potential in image denoising. However, most existing methods only consider the nonlocal self-similarity (NSS) prior of noisy input image. That is, the similar patches are collected only from degraded input, which makes the quality of image denoising largely depend on the input itself. However, such methods often suffer from a common drawback that the denoising performance may degrade quickly with increasing noise levels. In this paper we propose a new prior model, called group sparsity residual constraint (GSRC). Unlike the conventional group-b…
Sri Jayachamrajendra College of Engineering（斯里贾亚查姆拉金德拉工程学院）
Due to the factors like processing power limitations and channel capabilities images are often down sampled and transmitted at low bit rates resulting in a low resolution compressed image. High resolution images can be reconstructed from several blurred, noisy and down sampled low resolution images using a computational process know as super resolution reconstruction. Super-resolution is the process of combining multiple aliased low-quality images to produce a high resolution, high-quality image. The problem of recovering a high resolution image progressively from a sequence of low resolution …
Tampere University of Technology（坦佩雷理工大学）；National Aerospace University（国立航空航天大学）
This paper studies the problem of full reference visual quality assessment of denoised images with a special emphasis on images with low contrast and noise-like texture. Denoising of such images together with noise removal often results in image details loss or smoothing. A new test image database, FLT, containing 75 noise-free "reference" images and 300 filtered ("distorted") images is developed. Each reference image, corrupted by an additive white Gaussian noise, is denoised by the BM3D filter with four different values of threshold parameter (four levels of noise suppression). After carryin…
We propose a novel direction to improve the denoising quality of filtering-based denoising algorithms in real time by predicting the best filter parameter value using a Convolutional Neural Network (CNN). We take the use case of BM3D, the state-of-the-art filtering-based denoising algorithm, to demonstrate and validate our approach. We propose and train a simple, shallow CNN to predict in real time, the optimum filter parameter value, given the input noisy image. Each training example consists of a noisy input image (training data) and the filter parameter value that produces the best output (…
New Jersey Institute of Technology（新泽西理工学院）；Michigan State University（密歇根州立大学）；University of Notre Dame（圣母大学）；Snap Inc.（Snap公司）
Graph Neural Networks (GNNs) have risen to prominence in learning representations for graph structured data. A single GNN layer typically consists of a feature transformation and a feature aggregation operation. The former normally uses feed-forward networks to transform features, while the latter aggregates the transformed features over the graph. Numerous recent works have proposed GNN models with different designs in the aggregation operation. In this work, we establish mathematically that the aggregation processes in a group of representative GNN models including GCN, GAT, PPNP, and APPNP …
The core operation of current Graph Neural Networks (GNNs) is the aggregation enabled by the graph Laplacian or message passing, which filters the neighborhood node information. Though effective for various tasks, in this paper, we show that they are potentially a problematic factor underlying all GNN methods for learning on certain datasets, as they force the node representations similar, making the nodes gradually lose their identity and become indistinguishable. Hence, we augment the aggregation operations with their dual, i.e. diversification operators that make the node more distinct and …
One of the classical problems in machine learning and data mining is feature selection. A feature selection algorithm is expected to be quick, and at the same time it should show high performance. MeLiF algorithm effectively solves this problem using ensembles of ranking filters. This article describes two different ways to improve MeLiF algorithm performance with parallelization. Experiments show that proposed schemes significantly improves algorithm performance and increase feature selection quality.
Carnegie Mellon University（卡内基梅隆大学）；Tandon School of Engineering, New York University（纽约大学坦登工程学院）
This work studies the denoising of piecewise smooth graph signals that exhibit inhomogeneous levels of smoothness over a graph, where the value at each node can be vector-valued. We extend the graph trend filtering framework to denoising vector-valued graph signals with a family of non-convex regularizers, which exhibit superior recovery performance over existing convex regularizers. Using an oracle inequality, we establish the statistical error rates of first-order stationary points of the proposed non-convex method for generic graphs. Furthermore, we present an ADMM-based algorithm to solve …
We discuss the problem of adaptive discrete-time signal denoising in the situation where the signal to be recovered admits a "linear oracle" -- an unknown linear estimate that takes the form of convolution of observations with a time-invariant filter. It was shown by Juditsky and Nemirovski (2009) that when the $\ell_2$-norm of the oracle filter is small enough, such oracle can be "mimicked" by an efficiently computable adaptive estimate of the same structure with an observation-driven filter. The filter in question was obtained as a solution to the optimization problem in which the $\ell_\inf…
