Abstract
Neural fields are evolving towards a general-purpose continuous representation for visual computing. Yet, despite their numerous appealing properties, they are hardly amenable to signal processing. As a remedy, we present a method to perform general continuous convolutions with general continuous signals such as neural fields. Observing that piecewise polynomial kernels reduce to a sparse set of Dirac deltas after repeated differentiation, we leverage convolution identities and train a repeated integral field to efficiently execute large-scale convolutions. We demonstrate our approach on a variety of data modalities and spatially-varying kernels.
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- J Harold Ahlberg, Edwin Norman Nilson, and Joseph Leonard Walsh. 2016. The Theory of Splines and Their Applications. Vol. 38. Elsevier.Google Scholar
- Jonathan T. Barron, Ben Mildenhall, Matthew Tancik, Peter Hedman, Ricardo Martin-Brualla, and Pratul P. Srinivasan. 2021. Mip-NeRF: A Multiscale Representation for Anti-Aliasing Neural Radiance Fields. ICCV (2021).Google Scholar
- Jonathan T. Barron, Ben Mildenhall, Dor Verbin, Pratul P. Srinivasan, and Peter Hedman. 2022. Mip-NeRF 360: Unbounded Anti-Aliased Neural Radiance Fields. CVPR (2022).Google Scholar
- E Oran Brigham. 1988. The fast Fourier transform and its applications. Prentice-Hall, Inc.Google ScholarDigital Library
- Jiawen Chen, Sylvain Paris, and Frédo Durand. 2007. Real-time edge-aware image processing with the bilateral grid. ACM Trans. Graph. 26, 3 (2007), 1--9.Google ScholarDigital Library
- Franklin C Crow. 1984. Summed-area tables for texture mapping. In SIGGRAPH. 207--212.Google Scholar
- Yilun Du, M. Katherine Collins, B. Joshua Tenenbaum, and Vincent Sitzmann. 2021. Learning Signal-Agnostic Manifolds of Neural Fields. In NeurIPS.Google Scholar
- Emilien Dupont, Adam Goliński, Milad Alizadeh, Yee Whye Teh, and Arnaud Doucet. 2021. Coin: Compression with implicit neural representations. In ICLR (Neural Compression Workshop).Google Scholar
- Emilien Dupont, Hyunjik Kim, S. M. Ali Eslami, Danilo Jimenez Rezende, and Dan Rosenbaum. 2022a. From data to functa: Your data point is a function and you can treat it like one, Vol. 162. PMLR, 5694--5725.Google Scholar
- Emilien Dupont, Yee Whye Teh, and Arnaud Doucet. 2022b. Generative Models as Distributions of Functions, Vol. 151. PMLR, 2989--3015.Google Scholar
- Ziya Erkoç, Fangchang Ma, Qi Shan, Matthias Nießner, and Angela Dai. 2023. Hyperdiffusion: Generating implicit neural fields with weight-space diffusion. In ICCV.Google Scholar
- Zeev Farbman, Raanan Fattal, and Dani Lischinski. 2011. Convolution pyramids. ACM Trans. Graph. 30, 6 (2011), 1--8.Google ScholarDigital Library
- Rizal Fathony, Anit Kumar Sahu, Devin Willmott, and J Zico Kolter. 2020. Multiplicative filter networks. In ICLR.Google Scholar
- Alain Fournier and Eugene Fiume. 1988. Constant-time filtering with space-variant kernels. ACM Trans. Graph. 22, 4 (1988), 229--238.Google ScholarDigital Library
- William T Freeman, Edward H Adelson, et al. 1991. The design and use of steerable filters. IEEE TPAMI 13, 9 (1991), 891--906.Google ScholarDigital Library
- Paul S Heckbert. 1986. Filtering by repeated integration. ACM Trans. Graph. 20, 4 (1986), 315--321.Google ScholarDigital Library
- Pedro Hermosilla, Tobias Ritschel, Pere-Pau Vázquez, Àlvar Vinacua, and Timo Ropinski. 2018. Monte carlo convolution for learning on non-uniformly sampled point clouds. ACM Trans. Graph. 37, 6 (2018), 1--12.Google ScholarDigital Library
- Amir Hertz, Or Perel, Raja Giryes, Olga Sorkine-Hornung, and Daniel Cohen-Or. 2021. Sape: Spatially-adaptive progressive encoding for neural optimization. NeurIPS 34 (2021), 8820--8832.Google Scholar
- Brian K. S. Isaac-Medina, Chris G. Willcocks, and Toby P. Breckon. 2023. Exact-NeRF: An Exploration of a Precise Volumetric Parameterization for Neural Radiance Fields. CVPR (2023).Google Scholar
- Markus Kettunen, Marco Manzi, Miika Aittala, Jaakko Lehtinen, Frédo Durand, and Matthias Zwicker. 2015. Gradient-domain path tracing. ACM Trans. Graph. 34, 4 (2015), 1--13.Google ScholarDigital Library
- Diederik P. Kingma and Jimmy Ba. 2015. Adam: A Method for Stochastic Optimization. In ICLR.Google Scholar
- Georgios Kopanas, Thomas Leimkühler, Gilles Rainer, Clément Jambon, and George Drettakis. 2022. Neural Point Catacaustics for Novel-View Synthesis of Reflections. ACM Trans. Graph. 41, 6 (2022), 1--15.Google ScholarDigital Library
- Jaakko Lehtinen, Jacob Munkberg, Jon Hasselgren, Samuli Laine, Tero Karras, Miika Aittala, and Timo Aila. 2018. Noise2Noise: Learning Image Restoration without Clean Data. In ICML. 2965--2974.Google Scholar
- Thomas Leimkühler, Hans-Peter Seidel, and Tobias Ritschel. 2018. Laplacian Kernel Splatting for Efficient Depth-of-field and Motion Blur Synthesis or Reconstruction. ACM Trans. Graph. 37, 4 (2018), 1--11.Google ScholarDigital Library
- Tony Lindeberg. 2013. Scale-space theory in computer vision. Vol. 256. Springer Science & Business Media.Google Scholar
- David B Lindell, Julien NP Martel, and Gordon Wetzstein. 2021. Autoint: Automatic integration for fast neural volume rendering. In CVPR. 14556--14565.Google Scholar
- David B Lindell, Dave Van Veen, Jeong Joon Park, and Gordon Wetzstein. 2022. Bacon: Band-limited coordinate networks for multiscale scene representation. In CVPR. 16252--16262.Google Scholar
- Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoorthi, and Ren Ng. 2020. Nerf: Representing scenes as neural radiance fields for view synthesis. In ECCV. 405--421.Google Scholar
- Thomas W. Mitchel, Benedict Brown, David Koller, Tim Weyrich, Szymon Rusinkiewicz, and Michael Kazhdan. 2020. Efficient Spatially Adaptive Convolution and Correlation. Technical Report 2006.13188. arXiv preprint.Google Scholar
- Thomas Müller, Alex Evans, Christoph Schied, and Alexander Keller. 2022. Instant Neural Graphics Primitives with a Multiresolution Hash Encoding. ACM Trans. Graph. 41, 4 (2022), 1--15.Google ScholarDigital Library
- Harald Niederreiter. 1992. Low-discrepancy point sets obtained by digital constructions over finite fields. Czechoslovak Mathematical Journal 42, 1 (1992), 143--166.Google ScholarCross Ref
- Jeong Joon Park, Peter Florence, Julian Straub, Richard Newcombe, and Steven Love-grove. 2019. Deepsdf: Learning continuous signed distance functions for shape representation. In CVPR. 165--174.Google Scholar
- Keunhong Park, Utkarsh Sinha, Jonathan T Barron, Sofien Bouaziz, Dan B Goldman, Steven M Seitz, and Ricardo Martin-Brualla. 2021. Nerfies: Deformable neural radiance fields. In ICCV. 5865--5874.Google Scholar
- Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. 2017. Automatic differentiation in pytorch. (2017).Google Scholar
- Kenneth Perlin. 1984. Personal communication with Paul Heckbert, mentioned in Heckbert [1986].Google Scholar
- Nicholas Sharp and Alec Jacobson. 2022. Spelunking the Deep: Guaranteed Queries on General Neural Implicit Surfaces via Range Analysis. ACM Trans. Graph. 41, 4 (2022), 1--16.Google ScholarDigital Library
- Assaf Shocher, Ben Feinstein, Niv Haim, and Michal Irani. 2020. From discrete to continuous convolution layers. arXiv preprint arXiv:2006.11120 (2020).Google Scholar
- Patrice Simard, Léon Bottou, Patrick Haffner, and Yann LeCun. 1998. Boxlets: a fast convolution algorithm for signal processing and neural networks. NeurIPS 11 (1998).Google Scholar
- Gurprit Singh, Cengiz Öztireli, Abdalla GM Ahmed, David Coeurjolly, Kartic Subr, Oliver Deussen, Victor Ostromoukhov, Ravi Ramamoorthi, and Wojciech Jarosz. 2019. Analysis of sample correlations for Monte Carlo rendering. In Comp. Graph. Forum, Vol. 38. Wiley Online Library, 473--491.Google Scholar
- Vincent Sitzmann, Julien Martel, Alexander Bergman, David Lindell, and Gordon Wetzstein. 2020. Implicit neural representations with periodic activation functions. NeurIPS 33 (2020), 7462--7473.Google Scholar
- Vincent Sitzmann, Michael Zollhöfer, and Gordon Wetzstein. 2019. Scene Representation Networks: Continuous 3D-Structure-Aware Neural Scene Representations. In NeurIPS.Google Scholar
- Ilya Meerovich Sobol. 1967. On the distribution of points in a cube and the approximate evaluation of integrals. Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki 7, 4 (1967), 784--802.Google Scholar
- Kenneth O Stanley. 2007. Compositional pattern producing networks: A novel abstraction of development. Genetic programming and evolvable machines 8 (2007), 131--162.Google Scholar
- Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan Barron, and Ren Ng. 2020. Fourier features let networks learn high frequency functions in low dimensional domains. NeurIPS 33 (2020), 7537--7547.Google Scholar
- Ayush Tewari, Justus Thies, Ben Mildenhall, Pratul Srinivasan, Edgar Tretschk, W Yifan, Christoph Lassner, Vincent Sitzmann, Ricardo Martin-Brualla, Stephen Lombardi, et al. 2022. Advances in neural rendering. Comp. Graph. Forum 41, 2 (2022), 703--735.Google ScholarCross Ref
- Carlo Tomasi and Roberto Manduchi. 1998. Bilateral filtering for gray and color images. In ICCV. 839--846.Google Scholar
- Edgar Tretschk, Ayush Tewari, Vladislav Golyanik, Michael Zollhöfer, Christoph Lassner, and Christian Theobalt. 2021. Non-rigid neural radiance fields: Reconstruction and novel view synthesis of a dynamic scene from monocular video. In ICCV. 12959--12970.Google Scholar
- Cristina Vasconcelos, Kevin Swersky, Mark Matthews, Milad Hashemi, Cengiz Oztireli, and Andrea Tagliasacchi. 2023. CUF: Continuous Upsampling Filters. In CVPR.Google Scholar
- Delio Vicini, Sébastien Speierer, and Wenzel Jakob. 2022. Differentiable signed distance function rendering. ACM Trans. Graph. 41, 4 (2022), 1--18.Google ScholarDigital Library
- Paul Viola and Michael Jones. 2001. Rapid object detection using a boosted cascade of simple features. In CVPR, Vol. 1. I-I.Google ScholarCross Ref
- Shenlong Wang, Simon Suo, Wei-Chiu Ma, Andrei Pokrovsky, and Raquel Urtasun. 2018. Deep parametric continuous convolutional neural networks. In CVPR. 2589--2597.Google Scholar
- Yinhuai Wang, Shuzhou Yang, Yujie Hu, and Jian Zhang. 2022. NeRFocus: Neural Radiance Field for 3D Synthetic Defocus. arXiv preprint arXiv:2203.05189 (2022).Google Scholar
- Lance Williams. 1983. Pyramidal parametrics. In SIGGRAPH, Vol. 17. 1--11.Google ScholarDigital Library
- Andrew P Witkin. 1987. Scale-space filtering. In Readings in Computer Vision. 329--332.Google Scholar
- Yiheng Xie, Towaki Takikawa, Shunsuke Saito, Or Litany, Shiqin Yan, Numair Khan, Federico Tombari, James Tompkin, Vincent Sitzmann, and Srinath Sridhar. 2022. Neural fields in visual computing and beyond. Comp. Graph. Forum 41, 2 (2022), 641--676.Google ScholarCross Ref
- Dejia Xu, Peihao Wang, Yifan Jiang, Zhiwen Fan, and Zhangyang Wang. 2022. Signal Processing for Implicit Neural Representations. In NeurIPS.Google Scholar
- Guandao Yang, Serge Belongie, Bharath Hariharan, and Vladlen Koltun. 2021. Geometry processing with neural fields. NeurIPS 34 (2021), 22483--22497.Google Scholar
- Guandao Yang, Sagie Benaim, Varun Jampani, Kyle Genova, Jonathan Barron, Thomas Funkhouser, Bharath Hariharan, and Serge Belongie. 2022. Polynomial neural fields for subband decomposition and manipulation. NeuRIPS 35 (2022), 4401--4415.Google Scholar
- Yu-Jie Yuan, Yang-Tian Sun, Yu-Kun Lai, Yuewen Ma, Rongfei Jia, and Lin Gao. 2022. NeRF-editing: geometry editing of neural radiance fields. In ICCV. 18353--18364.Google Scholar
- Xian-Da Zhang. 2022. Modern signal processing. In Modern Signal Processing. De Gruyter.Google Scholar
Index Terms
- Neural Field Convolutions by Repeated Differentiation
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