# Zero Pixel Directional Boundary by Vector Transform

arXiv: https://arxiv.org/abs/2203.08795 · HTML: https://mlsanity.com/paper/2203.08795/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 74 references that are in the map.
- First posted on arXiv 16 Mar 2022.

## Builds on (most cited first, 30 of 36)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [You Only Look Once: Unified, Real-Time Object Detection](https://mlsanity.com/paper/1506.02640/) (2015)
- [The Cityscapes Dataset for Semantic Urban Scene Understanding](https://mlsanity.com/paper/1604.01685/) (2016)
- [DeepLab: Semantic Image Segmentation with Deep Convolutional Nets, Atrous Convolution, and Fully Connected CRFs](https://mlsanity.com/paper/1606.00915/) (2016)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Aggregated Residual Transformations for Deep Neural Networks](https://mlsanity.com/paper/1611.05431/) (2016)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Deep High-Resolution Representation Learning for Visual Recognition](https://mlsanity.com/paper/1908.07919/) (2019)
- [Per-Pixel Classification is Not All You Need for Semantic Segmentation](https://mlsanity.com/paper/2107.06278/) (2021)
- [Panoptic Segmentation](https://mlsanity.com/paper/1801.00868/) (2018)
- [Holistically-Nested Edge Detection](https://mlsanity.com/paper/1504.06375/) (2015)
- [Panoptic-DeepLab](https://mlsanity.com/paper/1910.04751/) (2019)
- [MaX-DeepLab: End-to-End Panoptic Segmentation with Mask Transformers](https://mlsanity.com/paper/2012.00759/) (2020)
- [clDice -- A Novel Topology-Preserving Loss Function for Tubular Structure Segmentation](https://mlsanity.com/paper/2003.07311/) (2020)
- [Boundary loss for highly unbalanced segmentation](https://mlsanity.com/paper/1812.07032/) (2018)
- [ParseNet: Looking Wider to See Better](https://mlsanity.com/paper/1506.04579/) (2015)
- [Richer Convolutional Features for Edge Detection](https://mlsanity.com/paper/1612.02103/) (2016)
- [Learning to predict crisp boundaries](https://mlsanity.com/paper/1807.10097/) (2018)
- [EpicFlow: Edge-Preserving Interpolation of Correspondences for Optical Flow](https://mlsanity.com/paper/1501.02565/) (2015)
- [Bi-Directional Cascade Network for Perceptual Edge Detection](https://mlsanity.com/paper/1902.10903/) (2019)
- [The Edge of Depth: Explicit Constraints between Segmentation and Depth](https://mlsanity.com/paper/2004.00171/) (2020)
- [DeepEdge: A Multi-Scale Bifurcated Deep Network for Top-Down Contour Detection](https://mlsanity.com/paper/1412.1123/) (2014)
- [Active Boundary Loss for Semantic Segmentation](https://mlsanity.com/paper/2102.02696/) (2021)
- [CASENet: Deep Category-Aware Semantic Edge Detection](https://mlsanity.com/paper/1705.09759/) (2017)
- [Pushing the Boundaries of Boundary Detection using Deep Learning](https://mlsanity.com/paper/1511.07386/) (2015)
- [Object Contour Detection with a Fully Convolutional Encoder-Decoder Network](https://mlsanity.com/paper/1603.04530/) (2016)
- [Distance Map Loss Penalty Term for Semantic Segmentation](https://mlsanity.com/paper/1908.03679/) (2019)

## Built on by (most cited first, 2 of 2)

- [Neural Vector Fields for Implicit Surface Representation and Inference](https://mlsanity.com/paper/2204.06552/) (2022)
- [MatchED: Crisp Edge Detection Using End-to-End, Matching-based Supervision](https://mlsanity.com/paper/2602.20689/) (2026)

## Abstract (arXiv)

Boundaries are among the primary visual cues used by human and computer vision systems. One of the key problems in boundary detection is the label representation, which typically leads to class imbalance and, as a consequence, to thick boundaries that require non-differential post-processing steps to be thinned. In this paper, we re-interpret boundaries as 1-D surfaces and formulate a one-to-one vector transform function that allows for training of boundary prediction completely avoiding the class imbalance issue. Specifically, we define the boundary representation at any point as the unit vector pointing to the closest boundary surface. Our problem formulation leads to the estimation of direction as well as richer contextual information of the boundary, and, if desired, the availability of zero-pixel thin boundaries also at training time. Our method uses no hyper-parameter in the training loss and a fixed stable hyper-parameter at inference. We provide theoretical justification/discussions of the vector transform representation. We evaluate the proposed loss method using a standard architecture and show the excellent performance over other losses and representations on several datasets. Code is available at https://github.com/edomel/BoundaryVT.