# Resolution learning in deep convolutional networks using scale-space theory

arXiv: https://arxiv.org/abs/2106.03412 · HTML: https://mlsanity.com/paper/2106.03412/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 57 of its 94 references that are in the map.
- First posted on arXiv 7 Jun 2021.

## Builds on (most cited first, 30 of 57)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks](https://mlsanity.com/paper/1905.11946/) (2019)
- [Feature Pyramid Networks for Object Detection](https://mlsanity.com/paper/1612.03144/) (2016)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [DeepLab: Semantic Image Segmentation with Deep Convolutional Nets, Atrous Convolution, and Fully Connected CRFs](https://mlsanity.com/paper/1606.00915/) (2016)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [Rethinking Atrous Convolution for Semantic Image Segmentation](https://mlsanity.com/paper/1706.05587/) (2017)
- [Deformable Convolutional Networks](https://mlsanity.com/paper/1703.06211/) (2017)
- [Multi-Scale Context Aggregation by Dilated Convolutions](https://mlsanity.com/paper/1511.07122/) (2015)
- [Striving for Simplicity: The All Convolutional Net](https://mlsanity.com/paper/1412.6806/) (2014)
- [Spatial Pyramid Pooling in Deep Convolutional Networks for Visual Recognition](https://mlsanity.com/paper/1406.4729/) (2014)
- [Network In Network](https://mlsanity.com/paper/1312.4400/) (2013)
- [Understanding the Effective Receptive Field in Deep Convolutional Neural Networks](https://mlsanity.com/paper/1701.04128/) (2017)
- [Making Convolutional Networks Shift-Invariant Again](https://mlsanity.com/paper/1904.11486/) (2019)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Dilated Residual Networks](https://mlsanity.com/paper/1705.09914/) (2017)
- [Auto-DeepLab: Hierarchical Neural Architecture Search for Semantic Image Segmentation](https://mlsanity.com/paper/1901.02985/) (2019)
- [Attention to Scale: Scale-aware Semantic Image Segmentation](https://mlsanity.com/paper/1511.03339/) (2015)
- [Multiscale Deep Equilibrium Models](https://mlsanity.com/paper/2006.08656/) (2020)
- [Scale-Equivariant Steerable Networks](https://mlsanity.com/paper/1910.11093/) (2019)
- [Invertible Image Rescaling](https://mlsanity.com/paper/2005.05650/) (2020)
- [Spectral Representations for Convolutional Neural Networks](https://mlsanity.com/paper/1506.03767/) (2015)
- [Learning Dynamic Routing for Semantic Segmentation](https://mlsanity.com/paper/2003.10401/) (2020)
- [Learning Filter Basis for Convolutional Neural Network Compression](https://mlsanity.com/paper/1908.08932/) (2019)
- [Gabor Convolutional Networks](https://mlsanity.com/paper/1705.01450/) (2017)

## Built on by (most cited first, 26 of 26)

- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [Generative Modelling With Inverse Heat Dissipation](https://mlsanity.com/paper/2206.13397/) (2022)
- [FlexConv: Continuous Kernel Convolutions with Differentiable Kernel Sizes](https://mlsanity.com/paper/2110.08059/) (2021)
- [Dynamic Sparse Network for Time Series Classification: Learning What to "see''](https://mlsanity.com/paper/2212.09840/) (2022)
- [Dilated convolution with learnable spacings](https://mlsanity.com/paper/2112.03740/) (2021)
- [Scale-Equivariant UNet for Histopathology Image Segmentation](https://mlsanity.com/paper/2304.04595/) (2023)
- [Covariance properties under natural image transformations for the generalized Gaussian derivative model for visual receptive fields](https://mlsanity.com/paper/2303.09803/) (2023)
- [As large as it gets: Learning infinitely large Filters via Neural Implicit Functions in the Fourier Domain](https://mlsanity.com/paper/2307.10001/) (2023)
- [Discrete approximations of Gaussian smoothing and Gaussian derivatives](https://mlsanity.com/paper/2311.11317/) (2023)
- [Frequency learning for structured CNN filters with Gaussian fractional derivatives](https://mlsanity.com/paper/2111.06660/) (2021)
- [Pooling Revisited: Your Receptive Field is Suboptimal](https://mlsanity.com/paper/2205.15254/) (2022)
- [LDConv: Linear deformable convolution for improving convolutional neural networks](https://mlsanity.com/paper/2311.11587/) (2023)
- [Receptive Field Refinement for Convolutional Neural Networks Reliably Improves Predictive Performance](https://mlsanity.com/paper/2211.14487/) (2022)
- [Dilated Convolution with Learnable Spacings: beyond bilinear interpolation](https://mlsanity.com/paper/2306.00817/) (2023)
- [Approximation properties relative to continuous scale space for hybrid discretizations of Gaussian derivative operators](https://mlsanity.com/paper/2405.05095/) (2024)
- [Rotation-Scale Equivariant Steerable Filters](https://mlsanity.com/paper/2304.04600/) (2023)
- [Scale generalisation properties of extended scale-covariant and scale-invariant Gaussian derivative networks on image datasets with spatial scaling variations](https://mlsanity.com/paper/2409.11140/) (2024)
- [Efficient Representation Learning via Adaptive Context Pooling](https://mlsanity.com/paper/2207.01844/) (2022)
- [Deep Continuous Networks](https://mlsanity.com/paper/2402.01557/) (2024)
- [PDE-CNNs: Axiomatic Derivations and Applications](https://mlsanity.com/paper/2403.15182/) (2024)
- [Scale-aware neural calibration for wide swath altimetry observations](https://mlsanity.com/paper/2302.04497/) (2023)
- [Balanced Mixture of SuperNets for Learning the CNN Pooling Architecture](https://mlsanity.com/paper/2306.11982/) (2023)
- [Scaling Continuous Kernels with Sparse Fourier Domain Learning](https://mlsanity.com/paper/2409.09875/) (2024)
- [Designing with Non-Finite Output Dimension via Fourier Coefficients of Neural Waveforms](https://mlsanity.com/paper/2212.04351/) (2022)
- [Modelling and analysis of the 8 filters from the "master key filters hypothesis" for depthwise-separable deep networks in relation to idealized receptive fields based on scale-space theory](https://mlsanity.com/paper/2509.12746/) (2025)
- [Scale-invariant Gaussian derivative residual networks](https://mlsanity.com/paper/2603.02843/) (2026)

## Abstract (arXiv)

Resolution in deep convolutional neural networks (CNNs) is typically bounded by the receptive field size through filter sizes, and subsampling layers or strided convolutions on feature maps. The optimal resolution may vary significantly depending on the dataset. Modern CNNs hard-code their resolution hyper-parameters in the network architecture which makes tuning such hyper-parameters cumbersome. We propose to do away with hard-coded resolution hyper-parameters and aim to learn the appropriate resolution from data. We use scale-space theory to obtain a self-similar parametrization of filters and make use of the N-Jet: a truncated Taylor series to approximate a filter by a learned combination of Gaussian derivative filters. The parameter sigma of the Gaussian basis controls both the amount of detail the filter encodes and the spatial extent of the filter. Since sigma is a continuous parameter, we can optimize it with respect to the loss. The proposed N-Jet layer achieves comparable performance when used in state-of-the art architectures, while learning the correct resolution in each layer automatically. We evaluate our N-Jet layer on both classification and segmentation, and we show that learning sigma is especially beneficial for inputs at multiple sizes.