Image deblurring remains a central research area within image processing, critical for its role in enhancing image quality and facilitating clearer visual representations across diverse applications. This paper tackles the optimization problem of image deblurring, assuming a known blurring kernel. We introduce an improved optimal proximal gradient algorithm (IOptISTA), which builds upon the optimal gradient method and a weighting matrix, to efficiently address the non-blind image deblurring problem. Based on two regularization cases, namely the $l_1$ norm and total variation norm, we perform numerical experiments to assess the performance of our proposed algorithm. The results indicate that our algorithm yields enhanced PSNR and SSIM values, as well as a reduced tolerance, compared to existing methods.
This is a review on blind image deblurring. First, we formulate the blind image deblurring problem and explain why it is challenging. Next, we bring some psychological and cognitive studies on the way our human vision system deblurs. Then, relying on several previous reviews, we discuss the topic of metrics and datasets, which is non-trivial to blind deblurring. Finally, we introduce some typical optimization-based methods and learning-based methods.
Faculty of Computer science, University of New Brunswick（新 Brunswick大学计算机科学学院）；Department of Computer Science, Brandon University（布兰顿大学计算机科学系）；Department of Applied Computer Science, University of Winnipeg（温尼伯大学应用计算机科学系）
Ecole Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
The goal of this project is to build a deep-learning solution that deblurs cornea scans, used for medical examination. The spherical shape of the eye prevents ophtamologist from having completely sharp image. Provided with a stack of corneas from confocal images, our approach is to build a model that performs an upscaling of the images using an SR (Super Resolution) Network.
Image blurring refers to the degradation of an image wherein the image's overall sharpness decreases. Image blurring is caused by several factors. Additionally, during the image acquisition process, noise may get added to the image. Such a noisy and blurred image can be represented as the image resulting from the convolution of the original image with the associated point spread function, along with additive noise. However, the blurred image often contains inadequate information to uniquely determine the plausible original image. Based on the availability of blurring information, image deblurr…
Huazhong University of Science and Technology（华中科技大学）；University of California, Merced（加州大学默塞德分校）；Nanjing University of Science and Technology（南京理工大学）
We present an effective blind image deblurring method based on a data-driven discriminative prior.Our work is motivated by the fact that a good image prior should favor clear images over blurred images.In this work, we formulate the image prior as a binary classifier which can be achieved by a deep convolutional neural network (CNN).The learned prior is able to distinguish whether an input image is clear or not.Embedded into the maximum a posterior (MAP) framework, it helps blind deblurring in various scenarios, including natural, face, text, and low-illumination images.However, it is difficul…
Blind image deblurring is a particularly challenging inverse problem where the blur kernel is unknown and must be estimated en route to recover the deblurred image. The problem is of strong practical relevance since many imaging devices such as cellphone cameras, must rely on deblurring algorithms to yield satisfactory image quality. Despite significant research effort, handling large motions remains an open problem. In this paper, we develop a new method called Blind Image Deblurring using Row-Column Sparsity (BD-RCS) to address this issue. Specifically, we model the outer product of kernel a…
Peking University（北京大学）；National Institute of Informatics（国立信息学研究所）；Harbin Institute of Technology（哈尔滨工业大学）
Blind image deblurring, i.e., deblurring without knowledge of the blur kernel, is a highly ill-posed problem. The problem can be solved in two parts: i) estimate a blur kernel from the blurry image, and ii) given estimated blur kernel, de-convolve blurry input to restore the target image. In this paper, we propose a graph-based blind image deblurring algorithm by interpreting an image patch as a signal on a weighted graph. Specifically, we first argue that a skeleton image---a proxy that retains the strong gradients of the target but smooths out the details---can be used to accurately estimate…
Indian Institute of Technology Gandhinagar（甘地纳加尔印度理工学院）
Blind motion deblurring involves reconstructing a sharp image from an observation that is blurry. It is a problem that is ill-posed and lies in the categories of image restoration problems. The training data-based methods for image deblurring mostly involve training models that take a lot of time. These models are data-hungry i.e., they require a lot of training data to generate satisfactory results. Recently, there are various image feature learning methods developed which relieve us of the need for training data and perform image restoration and image synthesis, e.g., DIP, InGAN, and SinGAN.…
We present a novel, blind, single image deblurring method that utilizes information regarding blur kernels. Our model solves the deblurring problem by dividing it into two successive tasks: (1) blur kernel estimation and (2) sharp image restoration. We first introduce a kernel estimation network that produces adaptive blur kernels based on the analysis of the blurred image. The network learns the blur pattern of the input image and trains to generate the estimation of image-specific blur kernels. Subsequently, we propose a deblurring network that restores sharp images using the estimated blur …
Peking University（北京大学）；Harbin Institute of Technology（哈尔滨工业大学）；National Institute of Informatics（国立信息学研究所）
Blind image deblurring, i.e., deblurring without knowledge of the blur kernel, is a highly ill-posed problem. The problem can be solved in two parts: i) estimate a blur kernel from the blurry image, and ii) given estimated blur kernel, de-convolve blurry input to restore the target image. In this paper, by interpreting an image patch as a signal on a weighted graph, we first argue that a skeleton image---a proxy that retains the strong gradients of the target but smooths out the details---can be used to accurately estimate the blur kernel and has a unique bi-modal edge weight distribution. We …
In recent years, deep neural network-based restoration methods have achieved state-of-the-art results in various image deblurring tasks. However, one major drawback of deep learning-based deblurring networks is that large amounts of blurry-clean image pairs are required for training to achieve good performance. Moreover, deep networks often fail to perform well when the blurry images and the blur kernels during testing are very different from the ones used during training. This happens mainly because of the overfitting of the network parameters on the training data. In this work, we present a …
Centre for Quantum Computation & Intelligent Systems（量子计算与智能系统中心）；Faculty of Engineering & Information Technology（工程与信息技术学院）；University of Technology Sydney（悉尼科技大学）
This paper comprehensively reviews the recent development of image deblurring, including non-blind/blind, spatially invariant/variant deblurring techniques. Indeed, these techniques share the same objective of inferring a latent sharp image from one or several corresponding blurry images, while the blind deblurring techniques are also required to derive an accurate blur kernel. Considering the critical role of image restoration in modern imaging systems to provide high-quality images under complex environments such as motion, undesirable lighting conditions, and imperfect system components, im…
INRIA（法国国家信息与自动化研究所）；Département d’informatique de l’ENS（巴黎高等师范学院计算机科学系）；ENS（巴黎高等师范学院）；CNRS（法国国家科学研究中心）；PSL University（巴黎文理研究大学）；Xi’an Jiaotong University（西安交通大学）
Non-blind image deblurring is typically formulated as a linear least-squares problem regularized by natural priors on the corresponding sharp picture's gradients, which can be solved, for example, using a half-quadratic splitting method with Richardson fixed-point iterations for its least-squares updates and a proximal operator for the auxiliary variable updates. We propose to precondition the Richardson solver using approximate inverse filters of the (known) blur and natural image prior kernels. Using convolutions instead of a generic linear preconditioner allows extremely efficient parameter…
In this paper, we study the problem of recovering a sharp version of a given blurry image when the blur kernel is unknown. Previous methods often introduce an image-independent regularizer (such as Gaussian or sparse priors) on the desired blur kernel. We shall show that the blurry image itself encodes rich information about the blur kernel. Such information can be found through analyzing and comparing how the spectrum of an image as a convolution operator changes before and after blurring. Our analysis leads to an effective convex regularizer on the blur kernel which depends only on the given…
Multi-scale Frequency Enhancement Network for Blind Image Deblurring
The Institute of Systems Engineering, Academy of Military Science（军事科学院系统工程研究所）；School of Artificial Intelligence and Computer Science, Jiangnan University（江南大学人工智能与计算机学院）；Jiangsu Provincial Engineering Laboratory of Pattern Recognition and Computational Intelligence, Jiangnan University（江南大学江苏省模式识别与计算智能工程实验室）；College of Artificial Intelligence, China University of Petroleum(beijing)（中国石油大学（北京）人工智能学院）；Beijing Key Laboratory of Petroleum Data Mining, China University of Petroleum(beijing)（中国石油大学（北京）北京市石油数据挖掘重点实验室）
The goal of blind image deblurring is to recover a sharp image from a motion blurred one without knowing the camera motion. Current state-of-the-art methods have a remarkably good performance on images with no noise or very low noise levels. However, the noiseless assumption is not realistic considering that low light conditions are the main reason for the presence of motion blur due to requiring longer exposure times. In fact, motion blur and high to moderate noise often appear together. Most works approach this problem by first estimating the blur kernel $k$ and then deconvolving the noisy b…
Blind image deblurring (BID) is an ill-posed inverse problem, usually addressed by imposing prior knowledge on the (unknown) image and on the blurring filter. Most of the work on BID has focused on natural images, using image priors based on statistical properties of generic natural images. However, in many applications, it is known that the image being recovered belongs to some specific class (e.g., text, face, fingerprints), and exploiting this knowledge allows obtaining more accurate priors. In this work, we propose a method where a Gaussian mixture model (GMM) is used to learn a class-adap…
Blind Image deblurring tries to estimate blurriness and a latent image out of a blurred image. This estimation, as being an ill-posed problem, requires imposing restrictions on the latent image or a blur kernel that represents blurriness. Different from recent studies that impose some priors on the latent image, this paper regulates the structure of the blur kernel. We propose a kernel mixture structure while using the Gaussian kernel as a base kernel. By combining multiple Gaussian kernels structurally enhanced in terms of scales and centers, the kernel mixture becomes capable of modeling nea…
Dalian University of Technology（大连理工大学）；Guilin University of Electronic Technology（桂林电子科技大学）
Blind image deblurring plays a very important role in many vision and multimedia applications. Most existing works tend to introduce complex priors to estimate the sharp image structures for blur kernel estimation. However, it has been verified that directly optimizing these models is challenging and easy to fall into degenerate solutions. Although several experience-based heuristic inference strategies, including trained networks and designed iterations, have been developed, it is still hard to obtain theoretically guaranteed accurate solutions. In this work, a collaborative learning framewor…
Hebei University of Technology（河北工业大学）；Australian National University（澳大利亚国立大学）；Tencent（腾讯）
Single-image super-resolution (SR) and multi-frame SR are two ways to super resolve low-resolution images. Single-Image SR generally handles each image independently, but ignores the temporal information implied in continuing frames. Multi-frame SR is able to model the temporal dependency via capturing motion information. However, it relies on neighbouring frames which are not always available in the real world. Meanwhile, slight camera shake easily causes heavy motion blur on long-distance-shot low-resolution images. To address these problems, a Blind Motion Deblurring Super-Reslution Network…
The Pennsylvania State University（宾夕法尼亚州立大学）；Technion, Israel Institute of Technology（以色列理工学院）
Blind image deblurring remains a topic of enduring interest. Learning based approaches, especially those that employ neural networks have emerged to complement traditional model based methods and in many cases achieve vastly enhanced performance. That said, neural network approaches are generally empirically designed and the underlying structures are difficult to interpret. In recent years, a promising technique called algorithm unrolling has been developed that has helped connect iterative algorithms such as those for sparse coding to neural network architectures. However, such connections ha…
Australian National University（澳大利亚国立大学）；Australian Centre for Robotic Vision（澳大利亚机器人视觉中心）；Northwestern Polytechnical University（西北工业大学）
The image blurring process is generally modelled as the convolution of a blur kernel with a latent image. Therefore, the estimation of the blur kernel is essentially important for blind image deblurring. Unlike existing approaches which focus on approaching the problem by enforcing various priors on the blur kernel and the latent image, we are aiming at obtaining a high quality blur kernel directly by studying the problem in the frequency domain. We show that the auto-correlation of the absolute phase-only image can provide faithful information about the motion (e.g. the motion direction and m…
Blind image deblurring is an important yet very challenging problem in low-level vision. Traditional optimization based methods generally formulate this task as a maximum-a-posteriori estimation or variational inference problem, whose performance highly relies on the handcraft priors for both the latent image and the blur kernel. In contrast, recent deep learning methods generally learn, from a large collection of training images, deep neural networks (DNNs) directly mapping the blurry image to the clean one or to the blur kernel, paying less attention to the physical degradation process of th…
This paper proposes a novel approach to regularize the \textit{ill-posed} and \textit{non-linear} blind image deconvolution (blind deblurring) using deep generative networks as priors. We employ two separate generative models --- one trained to produce sharp images while the other trained to generate blur kernels from lower-dimensional parameters. To deblur, we propose an alternating gradient descent scheme operating in the latent lower-dimensional space of each of the pretrained generative models. Our experiments show promising deblurring results on images even under large blurs, and heavy no…
One of the most important issues in the image processing is the approximation of the image that has been lost due to the blurring process. These types of matters are divided into non-blind and blind problems. The second type of problem is more complex in terms of calculations than the first problems due to the unknown of original image and point spread function estimation. In the present paper, an algorithm based on coarse-to-fine iterative by $l_0-αl_1$ regularization and framelet transform is introduced to approximate the spread function estimation. Framelet transfer improves the restored ke…
Peking University（北京大学）；Cooperative Media-net Innovation Center（协同媒体网创新中心）；Beida (Binhai) Information Research（北大（滨海）信息研究院）；Harbin Institute of Technology（哈尔滨工业大学）
Blind image deblurring is a challenging problem in computer vision, which aims to restore both the blur kernel and the latent sharp image from only a blurry observation. Inspired by the prevalent self-example prior in image super-resolution, in this paper, we observe that a coarse enough image down-sampled from a blurry observation is approximately a low-resolution version of the latent sharp image. We prove this phenomenon theoretically and define the coarse enough image as a latent structure prior of the unknown sharp image. Starting from this prior, we propose to restore sharp images from t…
DWDN: Deep Wiener Deconvolution Network for Non-Blind Image Deblurring
School of Computer Science and Engineering, Nanjing University of Science and Technology（南京理工大学计算机科学与工程学院）；Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）；Department of Computer Science, TU Darmstadt（达姆施塔特工业大学计算机科学系）
Centro de Informática, Universidade Federal de Pernambuco（伯南布哥联邦大学信息中心）；Motorola Mobility LLC (a Lenovo Company)（摩托罗拉移动有限责任公司（联想旗下公司））
We propose a new incremental aggregation algorithm for multi-image deblurring with automatic image selection. The primary motivation is that current bursts deblurring methods do not handle well situations in which misalignment or out-of-context frames are present in the burst. These real-life situations result in poor reconstructions or manual selection of the images that will be used to deblur. Automatically selecting best frames within the burst to improve the base reconstruction is challenging because the amount of possible images fusions is equal to the power set cardinal. Here, we approac…
This paper proposes a novel approach to regularize the ill-posed blind image deconvolution (blind image deblurring) problem using deep generative networks. We employ two separate deep generative models - one trained to produce sharp images while the other trained to generate blur kernels from lower dimensional parameters. To deblur, we propose an alternating gradient descent scheme operating in the latent lower-dimensional space of each of the pretrained generative models. Our experiments show excellent deblurring results even under large blurs and heavy noise. To improve the performance on ri…
Chongqing University of Posts and Telecommunications（重庆邮电大学）；Chongqing University（重庆大学）
Most existing deep-learning-based single image dynamic scene blind deblurring (SIDSBD) methods usually design deep networks to directly remove the spatially-variant motion blurs from one inputted motion blurred image, without blur kernels estimation. In this paper, inspired by the Projective Motion Path Blur (PMPB) model and deformable convolution, we propose a novel constrained deformable convolutional network (CDCN) for efficient single image dynamic scene blind deblurring, which simultaneously achieves accurate spatially-variant motion blur kernels estimation and the high-quality image rest…
Tampere Research Center（坦佩雷研究中心）；Huawei Technologies（华为技术有限公司）
Blind single image deblurring has been a challenge over many decades due to the ill-posed nature of the problem. In this paper, we propose a single-frame blind deblurring solution with the aid of Laplacian filters. Utilized Residual Dense Network has proven its strengths in superresolution task, thus we selected it as a baseline architecture. We evaluated the proposed solution with state-of-art DNN methods on a benchmark dataset. The proposed method shows significant improvement in image quality measured objectively and subjectively.
Tsinghua University（清华大学）；Beijing Institute of Technology（北京理工大学）
Under low-light environment, handheld photography suffers from severe camera shake under long exposure settings. Although existing deblurring algorithms have shown promising performance on well-exposed blurry images, they still cannot cope with low-light snapshots. Sophisticated noise and saturation regions are two dominating challenges in practical low-light deblurring. In this work, we propose a novel non-blind deblurring method dubbed image and feature space Wiener deconvolution network (INFWIDE) to tackle these problems systematically. In terms of algorithm design, INFWIDE proposes a two-b…
Istituto Italiano di Tecnologia（意大利技术研究院）；Ca’Foscari University of Venice（威尼斯大学）
Most computer vision and machine learning-based approaches for historical document analysis are tailored to grayscale or RGB images and thus, mostly exploit their spatial information. Multispectral (MS) and hyperspectral (HS) images contain, next to the spatial information, much richer spectral information than RGB images (usually spreading beyond the visible spectral range) that can facilitate more effective feature extraction, more accurate classification and recognition, and thus, improved analysis. Although utilization of rich spectral information can improve historical document analysis t…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；University of Tübingen（图宾根大学）
We present a neural network model approach for multi-frame blind deconvolution. The discriminative approach adopts and combines two recent techniques for image deblurring into a single neural network architecture. Our proposed hybrid-architecture combines the explicit prediction of a deconvolution filter and non-trivial averaging of Fourier coefficients in the frequency domain. In order to make full use of the information contained in all images in one burst, the proposed network embeds smaller networks, which explicitly allow the model to transfer information between images in early layers. O…
Motion blur is a known issue in photography, as it limits the exposure time while capturing moving objects. Extensive research has been carried to compensate for it. In this work, a computational imaging approach for motion deblurring is proposed and demonstrated. Using dynamic phase-coding in the lens aperture during the image acquisition, the trajectory of the motion is encoded in an intermediate optical image. This encoding embeds both the motion direction and extent by coloring the spatial blur of each object. The color cues serve as prior information for a blind deblurring process, implem…
University of Electronic Science and Technology of China（电子科技大学）；The Chinese University of Hong Kong（香港中文大学）；Northeast Normal University（东北师范大学）
Non-blind deblurring methods achieve decent performance under the accurate blur kernel assumption. Since the kernel uncertainty (i.e. kernel error) is inevitable in practice, semi-blind deblurring is suggested to handle it by introducing the prior of the kernel (or induced) error. However, how to design a suitable prior for the kernel (or induced) error remains challenging. Hand-crafted prior, incorporating domain knowledge, generally performs well but may lead to poor performance when kernel (or induced) error is complex. Data-driven prior, which excessively depends on the diversity and abund…
Hefei University of Technology（合肥工业大学）；Zhengzhou University（郑州大学）；University of Technology Sydney（悉尼科技大学）
Blind image deblurring (BID) remains a challenging and significant task. Benefiting from the strong fitting ability of deep learning, paired data-driven supervised BID method has obtained great progress. However, paired data are usually synthesized by hand, and the realistic blurs are more complex than synthetic ones, which makes the supervised methods inept at modeling realistic blurs and hinders their real-world applications. As such, unsupervised deep BID method without paired data offers certain advantages, but current methods still suffer from some drawbacks, e.g., bulky model size, long …
A Multi-scale Generalized Shrinkage Threshold Network for Image Blind Deblurring in Remote Sensing
School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）；School of Automation and Electronic Information, Xiangtan University（湘潭大学自动化与电子信息学院）
Shanghai Advanced Research Institute, Chinese Academy of Sciences（中国科学院上海高等研究院）
Currently, many blind deblurring methods assume blurred images are noise-free and perform unsatisfactorily on the blurry images with noise. Unfortunately, noise is quite common in real scenes. A straightforward solution is to denoise images before deblurring them. However, even state-of-the-art denoisers cannot guarantee to remove noise entirely. Slight residual noise in the denoised images could cause significant artifacts in the deblurring stage. To tackle this problem, we propose a cascaded framework consisting of a denoiser subnetwork and a deblurring subnetwork. In contrast to previous me…
Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Science（中国科学院长春光学精密机械与物理研究所）；Changchun University of Science and Technology（长春理工大学）
The goal of blind image deblurring is to recover sharp image from one input blurred image with an unknown blur kernel. Most of image deblurring approaches focus on developing image priors, however, there is not enough attention to the influence of image details and structures on the blur kernel estimation. What is the useful image structure and how to choose a good deblurring region? In this work, we propose a deep neural network model method for selecting good regions to estimate blur kernel. First we construct image patches with labels and train a deep neural networks, then the learned model…
Department of Computing Science University of Alberta（阿尔伯塔大学计算机科学系）
Deep convolution neural networks (CNNs) play a critical role in single image super-resolution (SISR) since the amazing improvement of high performance computing. However, most of the super-resolution (SR) methods only focus on recovering bicubic degradation. Reconstructing high-resolution (HR) images from randomly blurred and noisy low-resolution (LR) images is still a challenging problem. In this paper, we propose a novel Spatial Context Hallucination Network (SCHN) for blind super-resolution without knowing the degradation kernel. We find that when the blur kernel is unknown, separate deblur…
Plug-and-Play (PnP) methods solve ill-posed inverse problems through iterative proximal algorithms by replacing a proximal operator by a denoising operation. When applied with deep neural network denoisers, these methods have shown state-of-the-art visual performance for image restoration problems. However, their theoretical convergence analysis is still incomplete. Most of the existing convergence results consider nonexpansive denoisers, which is non-realistic, or limit their analysis to strongly convex data-fidelity terms in the inverse problem to solve. Recently, it was proposed to train th…
Blind image deblurring remains a challenging problem for modern artificial neural networks. Unlike other image restoration problems, deblurring networks fail behind the performance of existing deblurring algorithms in case of uniform and 3D blur models. This follows from the diverse and profound effect that the unknown blur-kernel has on the deblurring operator. We propose a new architecture which breaks the deblurring network into an analysis network which estimates the blur, and a synthesis network that uses this kernel to deblur the image. Unlike existing deblurring networks, this design al…
In this paper, we propose a novel design of image deblurring in the form of one-shot convolution filtering that can directly convolve with naturally blurred images for restoration. The problem of optical blurring is a common disadvantage to many imaging applications that suffer from optical imperfections. Despite numerous deconvolution methods that blindly estimate blurring in either inclusive or exclusive forms, they are practically challenging due to high computational cost and low image reconstruction quality. Both conditions of high accuracy and high speed are prerequisites for high-throug…
Third Military Medical University（第三军医大学）；National University of Defense Technology（国防科技大学）
The recovery of images from the observations that are degraded by a linear operator and further corrupted by Poisson noise is an important task in modern imaging applications such as astronomical and biomedical ones. Gradient-based regularizers involve the popular total variation semi-norm have become standard techniques for Poisson image restoration due to its edge-preserving ability. Various efficient algorithms have been developed for solving the corresponding minimization problem with non-smooth regularization terms. In this paper, motivated by the idea of the alternating direction minimiz…
Real-world dynamic scene deblurring has long been a challenging task since paired blurry-sharp training data is unavailable. Conventional Maximum A Posteriori estimation and deep learning-based deblurring methods are restricted by handcrafted priors and synthetic blurry-sharp training pairs respectively, thereby failing to generalize to real dynamic blurriness. To this end, we propose a Neural Maximum A Posteriori (NeurMAP) estimation framework for training neural networks to recover blind motion information and sharp content from unpaired data. The proposed NeruMAP consists of a motion estima…
Multi-scale (MS) approaches have been widely investigated for blind single image / video deblurring that sequentially recovers deblurred images in low spatial scale first and then in high spatial scale later with the output of lower scales. MS approaches have been effective especially for severe blurs induced by large motions in high spatial scale since those can be seen as small blurs in low spatial scale. In this work, we investigate alternative approach to MS, called multi-temporal (MT) approach, for non-uniform single image deblurring. We propose incremental temporal training with construc…
INRIA（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；PSL Research University（巴黎文理研究大学）
We address the problem of non-blind deblurring and demosaicking of noisy raw images. We adapt an existing learning-based approach to RGB image deblurring to handle raw images by introducing a new interpretable module that jointly demosaicks and deblurs them. We train this model on RGB images converted into raw ones following a realistic invertible camera pipeline. We demonstrate the effectiveness of this model over two-stage approaches stacking demosaicking and deblurring modules on quantitive benchmarks. We also apply our approach to remove a camera's inherent blur (its color-dependent point-…
