# A Survey of Orthogonal Moments for Image Representation: Theory, Implementation, and Evaluation

arXiv: https://arxiv.org/abs/2103.14799 · HTML: https://mlsanity.com/paper/2103.14799/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 255 references that are in the map.
- First posted on arXiv 27 Mar 2021.

## Builds on (most cited first, 22 of 22)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Spatial Pyramid Pooling in Deep Convolutional Networks for Visual Recognition](https://mlsanity.com/paper/1406.4729/) (2014)
- [Recent Advances in Convolutional Neural Networks](https://mlsanity.com/paper/1512.07108/) (2015)
- [HPatches: A benchmark and evaluation of handcrafted and learned local descriptors](https://mlsanity.com/paper/1704.05939/) (2017)
- [Quaternion Knowledge Graph Embeddings](https://mlsanity.com/paper/1904.10281/) (2019)
- [PCANet: A Simple Deep Learning Baseline for Image Classification?](https://mlsanity.com/paper/1404.3606/) (2014)
- [SIFT Meets CNN: A Decade Survey of Instance Retrieval](https://mlsanity.com/paper/1608.01807/) (2016)
- [Quaternion Convolutional Neural Networks](https://mlsanity.com/paper/1903.00658/) (2019)
- [Towards Robust Pattern Recognition: A Review](https://mlsanity.com/paper/2006.06976/) (2020)
- [Image Projective Invariants](https://mlsanity.com/paper/1707.05950/) (2017)
- [A comparison of dense region detectors for image search and fine-grained classification](https://mlsanity.com/paper/1410.8151/) (2014)
- [Fast and Efficient Calculations of Structural Invariants of Chirality](https://mlsanity.com/paper/1711.05866/) (2017)
- [MomentsNet: a simple learning-free method for binary image recognition](https://mlsanity.com/paper/1702.06767/) (2017)
- [Shape DNA: Basic Generating Functions for Geometric Moment Invariants](https://mlsanity.com/paper/1703.02242/) (2017)
- [Fast and accurate computation of orthogonal moments for texture analysis](https://mlsanity.com/paper/1803.00638/) (2018)
- [Differential and integral invariants under Mobius transformation](https://mlsanity.com/paper/1808.10083/) (2018)
- [Dual affine moment invariants](https://mlsanity.com/paper/1911.08233/) (2019)
- [Isomorphism between Differential and Moment Invariants under Affine Transform](https://mlsanity.com/paper/1705.08264/) (2017)
- [Reflection Invariant and Symmetry Detection](https://mlsanity.com/paper/1705.10768/) (2017)

## Built on by (most cited first, 8 of 8)

- [DeepWSD: Projecting Degradations in Perceptual Space to Wasserstein Distance in Deep Feature Space](https://mlsanity.com/paper/2208.03323/) (2022)
- [A Principled Design of Image Representation: Towards Forensic Tasks](https://mlsanity.com/paper/2203.00913/) (2022)
- [Representing Noisy Image Without Denoising](https://mlsanity.com/paper/2301.07409/) (2023)
- [MomentsNeRF: Leveraging Orthogonal Moments for Few-Shot Neural Rendering](https://mlsanity.com/paper/2407.02668/) (2024)
- [Spatial-Frequency Discriminability for Revealing Adversarial Perturbations](https://mlsanity.com/paper/2305.10856/) (2023)
- [Hierarchical Invariance for Robust and Interpretable Vision Tasks at Larger Scales](https://mlsanity.com/paper/2402.15430/) (2024)
- [Polar Separable Transform for Efficient Orthogonal Rotation-Invariant Image Representation](https://mlsanity.com/paper/2510.09125/) (2025)
- [Varifold Moment Invariants for Sustainable and Explainable Contour Feature Extraction](https://mlsanity.com/paper/2606.07333/) (2026)

## Abstract (arXiv)

Image representation is an important topic in computer vision and pattern recognition. It plays a fundamental role in a range of applications towards understanding visual contents. Moment-based image representation has been reported to be effective in satisfying the core conditions of semantic description due to its beneficial mathematical properties, especially geometric invariance and independence. This paper presents a comprehensive survey of the orthogonal moments for image representation, covering recent advances in fast/accurate calculation, robustness/invariance optimization, definition extension, and application. We also create a software package for a variety of widely-used orthogonal moments and evaluate such methods in a same base. The presented theory analysis, software implementation, and evaluation results can support the community, particularly in developing novel techniques and promoting real-world applications.