Fourier Analysis of the 2D Screened Poisson Equation for Gradient Domain Problems

Bhat, P. N.; Curless, Brian; Cohen, Michael F.; Zitnick, C. Lawrence

doi:10.1007/978-3-540-88688-4_9

Cited by 100 publications

(93 citation statements)

References 27 publications

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“…Gradient-domain methods directly estimate image-space gradients using coherent paths, and then reconstruct a final image with Poisson reconstruction [Bhat et al 2008]. The seminal work of gradient-domain Metropolis light transport [Lehtinen et al 2013] inspired several relevant follow-up works.…”

Section: Related Workmentioning

confidence: 99%

Gradient-domain volumetric photon density estimation

et al. 2018

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Fig. 1. Equal-time (5 minutes) renderings of a smoky Kitchen scene. Gradient-domain volumetric rendering techniques with L1 reconstruction converge faster than primal-domain volumetric rendering technique. The relMSE error metric has a unitless scale of 10 −2 .Gradient-domain rendering can improve the convergence of surface-based light transport by exploiting smoothness in image space. Scenes with participating media exhibit similar smoothness and could potentially benefit from gradient-domain techniques. We introduce the first gradient-domain formulation of image synthesis with homogeneous participating media, including four novel and efficient gradient-domain volumetric density estimation algorithms. We show that naïve extensions of gradient domain path-space and density estimation methods to volumetric media, while functional, can result in inefficient estimators. Focussing on point-, beam-and plane-based gradient-domain estimators, we introduce a novel shift mapping that eliminates redundancies in the naïve formulations using spatial relaxation within the volume. We show that gradient-domain volumetric rendering improve convergence compared to primal domain state-of-the-art, across a suite of scenes. Our formulation and algorithms support progressive estimation and are easy to incorporate atop existing renderers.Authors' addresses: Adrien Gruson, adrien

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Section: Related Workmentioning

confidence: 99%

Gradient-domain volumetric photon density estimation

et al. 2018

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“…The existence and uniqueness of solutions to Equa-tions (1) - (2), provided that the data terms are regular enough, is a standard result in linear PDE theory [5].…”

Section: Poisson Equation On a Continuous Domainmentioning

confidence: 99%

“…(10) Equation (10) illustrates a Poisson-like problem in which pixels (2,3) and (3,2) are set to the values g 23 and g 32 respectively, and a condition on the Laplacian of the remaining pixels is imposed. The…”

Section: Discrete Modelmentioning

confidence: 99%

“…To implement the previous simple two grid correction scheme, we must define the interpolation/restriction operators I 2). Before proceeding with these definitions, it is worth commenting why the seven steps described above are important for a successful definition of multigrid schemes.…”

Section: Correct the Fine Grid Approximation Vmentioning

confidence: 99%

“…It is at the core of many powerful image processing methods [2,9,11,13,14,17] and has been extensively studied for its practical applications and its interesting mathematical properties. In the past decade, the volume of digital data has grown exponentially, and large amounts of high resolution digital images are uploaded and processed everyday.…”

Section: Introductionmentioning

confidence: 99%

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An Analysis and Implementation of Multigrid Poisson Solvers With Verified Linear

Martino¹,

Facciolo²

2018

Image Processing on Line

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The Poisson equation is the most studied partial differential equation, and it allows to formulate many useful image processing methods in an elegant and efficient mathematical framework. Using different variations of data terms and boundary conditions, Poisson-like problems can be developed, e.g. for local contrast enhancement, inpainting, or image seamless cloning among many other applications. Multigrid solvers are among the most efficient numerical solvers for discrete Poisson-like equations. However, their correct implementation relies on: (i) the proper definition of the discrete problem, (ii) the right choice of interpolation and restriction operators, and (iii) the adequate formulation of the problem across different scales and layers. In the present work we address these aspects, and we provide a mathematical and practical description of multigrid methods. In addition, we present an alternative to the extended formulation of Poisson equation proposed in 2011 by Mainberger et al. The proposed formulation of the problem suits better multigrid methods, in particular, because it has important mathematical properties that can be exploited to define the problem at different scales in a intuitive and natural way. In addition, common iterative solvers and Poisson-like problems are empirically analyzed and compared. For example, the complexity of problems is compared when the topology of Dirichlet boundary conditions changes in the interior of the regular domain of the image. The main contribution of this work is the development and detailed description of an implementation of a multigrid numerical solver which converges in linear time. Source CodeThe reviewed source code and documentation of a Matlab implementation for Multigrid Poisson solvers and the applications described in this work are available from the web page of this article 1 . Usage instructions are included in the README.txt file of the archive.

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The role of superior image composition in children's analogical reasoning

Chen

Guo

et al. 2019

Softw Pract Exp

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Analogical reasoning, as a higher cognitive ability, can help children make inferences about a novel situation. It is vital to help children's analogical reasoning development. However, the traditional intervention methods are simple and the effects cannot maintain. Aiming at this problem, the present study was the first to use computer technology especially image composition technique to promote children's analogical reasoning from an interdisciplinary perspective. Specifically, one minimum region entropy based composition model was proposed. On the one hand, sparse coding model and spatial pyramid matching model were used for searching semantically matching images. On the other hand, minimum region entropy model could contribute to composite the candidate region into an ideal position. Furthermore, we set up a database using massive images and adequate experiments based on it to verify the model's effectiveness and robustness. What's more important, we applied the improved image composition to analogical reasoning task. The results showed that the performance of intervention group was obviously better than control group during intervention stages and posttest stage. In general, the present study not only demonstrated the advantages of the improved image composition but also revealed composition's remarkable contribution for getting analogical relationship by children.

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Fourier Analysis of the 2D Screened Poisson Equation for Gradient Domain Problems

Cited by 100 publications

References 27 publications

Gradient-domain volumetric photon density estimation

Gradient-domain volumetric photon density estimation

An Analysis and Implementation of Multigrid Poisson Solvers With Verified Linear

The role of superior image composition in children's analogical reasoning

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