Equivariant and SE(2)-Invariant Neural Network Leveraging

Fourier-Based Descriptors for 2D Image Classiﬁcation

Emna Ghorbel

1,2 a

, Achraf Ghorbel

and Faouzi Ghorbel

CRISTAL Laboratory, GRIFT Research Group ENSI, La Manouba University, 2010, La Manouba, Tunisia

Medtech, South Mediterranean University, Tunis, Tunisia

Keywords:

Equivariance, Invariance, Fourier based-Descriptors, Neural Networks, CNNs.

Abstract:

This paper introduces a novel deep learning framework for 2D shape classiﬁcation that emphasizes equivari-

ance and invariance through Generalized Finite Fourier-based Descriptors (GFID). Instead of relying on raw

images, we extract contours from 2D shapes and compute equivariant, invariant, and stable descriptors, which

represent shapes as column vectors in complex space. This approach achieves invariance to parameteriza-

tion and rigid transformations, while reducing the number of network parameters. We evaluate the proposed

lightweight neural network framework by testing it against a simple CNN and a pre-trained InceptionV3, ﬁrst

using the original test set and then with rotated and translated images from well-known benchmarks. Ex-

perimental results demonstrate the effectiveness of our method under rigid transformations, showcasing the

beneﬁts of Fourier-based invariants for robust classiﬁcation.

1 INTRODUCTION

Deep Learning (DL) has recently gained widespread

popularity in the ﬁelds of computer vision and ma-

chine learning due to its remarkable performance in

a variety of tasks including image classiﬁcation and

object detection (Guo et al., 2016; Li et al., 2015).

Despite these advancements, challenges remain, par-

ticularly in managing variability introduced by trans-

formations such as rotation, shifting, and noise, which

can signiﬁcantly affect model accuracy (Lyle et al.,

2020; Quiroga et al., 2023; Ruderman et al., 2018).

Many existing deep learning models, especially

convolutional neural networks (CNNs), typically rely

heavily on raw image data, rendering them vulnerable

to these transformations. While these models aim to

create effective representations, they often fall short

in achieving the necessary invariance and stability,

which are crucial for robust performance across di-

verse scenarios. As a result, their effectiveness can be

compromised when faced with even minor alterations

in input data.

To address these limitations, recent research has

shifted its focus toward the use of descriptors that

inherently offer invariance to transformations (Mau-

rya et al., 2024; Wang et al., 2024; Shi et al., 2024;

https://orcid.org/0000-0002-6179-1358

Quiroga et al., 2023; Li et al., 2024; Delchevalerie

et al., 2021). However, many existing approaches still

rely on raw pixel data, which can undermine the po-

tential of descriptors. Furthermore, these methods of-

ten fail to ensure both equivariance and invariance,

resulting in decreased stability when faced with trans-

formed data. The descriptors used in such approaches

typically lack the ability to fully verify both proper-

ties.

In that context, we propose a novel deep learning

framework that leverages Generalized Finite Fourier-

based Invariant Descriptors (GFID) (Ghorbel et al.,

2022) for image classiﬁcation. Our approach involves

extracting shapes from images, applying arc-length

parameterization on the resulting contours, and com-

puting invariant descriptors represented as column

vectors in complex space. This representation ensures

the model’s equivariance and invariance to rigid trans-

formations (rotation and translation), while also en-

abling a lightweight architecture that allows for faster

computations and seamless integration into existing

systems. We rigorously evaluate the performance of

our framework against a traditional CNN and the pre-

trained InceptionV3 on well-established datasets, in-

cluding MNIST (LeCun et al., 1998), Fashion MNIST

(Research, 2017), and Hand Gesture Recognition

(rishabh arya, 2021). The results highlight the effec-

tiveness of GFID-based Neural Network, offering a

210

Ghorbel, E., Ghorbel, A. and Ghorbel, F.

Equivariant and SE(2)-Invariant Neural Network Leveraging Fourier-Based Descriptors for 2D Image Classiﬁcation.

DOI: 10.5220/0013143300003890

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 17th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2025) - Volume 2, pages 210-215

ISBN: 978-989-758-737-5; ISSN: 2184-433X

robust and efﬁcient solution for image classiﬁcation

when dealing with transformed data.

This paper is organized as follows: In Section

2, we present our proposed approach, including the

GFID neural network framework. Section 3 describes

our experimental setup and results, while Section 4

concludes the paper and discusses avenues for future

research.

2 PROPOSED APPROACH

In this section, a novel equivariant, invariant and sta-

ble neural network framework designed for image

classiﬁcation leveraging Generalized Finite Fourier-

based Invariant Descriptors (GFID) (Ghorbel et al.,

2022; Ghorbel and Ghorbel, 2024) is proposed.

2.1 Generalized Finite Fourier-Based

Invariant Descriptor

Here, we recall the the Generalized Finite Fourier In-

variant Descriptor (GFID) methematical formulation

and its inverse function.

From (Ghorbel and Ghorbel, 2024), F

(γ) is cal-

culated as the Fast Fourier Transform (FFT) of N

samples extracted uniformly from a normalized arc

length parameterization (n.a.l.p.) of a given curve γ.

We select a positive integer n

such that 1 < n

< N,

along with two strictly positive real numbers p and

q. Therefore, the GFID descriptor corresponding to

the complex vector (I

) residing in the ﬁnite com-

plex vector space C

N−1

is computed as follows for

all 1 ≤ n ≤ N − 1,







−n−1

n−n

−1

−n−p−1

−1

n−n

−q

if F

and F

−1

̸= 0

0 if F

= 0

Where |·| denotes the modulus operator. The GFID

exhibits crucial invariance properties with respect

to curve parametrization, Euclidean transformations,

and the choice of starting point on the curve. It has

been demonstrated that these descriptors are stable

against subtle curve deformations and are invertible,

allowing for the unique reconstruction of the original

curve from its GFID up to a Euclidean transformation.

Importantly, small modiﬁcations to the GFID result in

reconstructed curves that closely resemble the origi-

nal shape, which enhances the model’s robustness.

The analytical inverse formula for the GFID can

be expressed as,

= I

−p

∆

−q

∆

i(nθ

+θ

)

Figure 1: Modulus of GFID descriptors demonstrating in-

variance to rigid transformations and robustness to minor

shape changes. (a) Original shape, (b) Transformed shape,

where ∆ = p + q + 1. The Inverse Fast Fourier Trans-

form (IFFT) of (F

) enables reconstruction of the

original curve up to a translation deﬁned by F

, a rota-

tion determined by θ

, and a starting point represented

by θ

where the variables θ

and θ

correspond to the

arguments of F

and F

−1

, respectively.

Figure 1 illustrates the invariance and stability

properties of the GFID descriptors. The ﬁrst row dis-

plays the GFID modulus of the original shapes. The

second row illustrates the transformed (rotation+shift)

shapes along with their corresponding GFID modu-

lus, demonstrating invariance under euclidean trans-

formations. The third row presents a different shape

belonging to the original’s class and its GFID modu-

lus, highlighting the stability of the descriptors under

shape variations. Overall, these observations conﬁrm

that the GFID descriptors maintain robustness against

both rigid transformations and subtle shape changes.

Given these properties, the GFID is integrated into our

neural network framework, making the model equiv-

ariant, invariant, and stable.

2.2 Neural Network Framework Using

GFID Descriptors

Here, the neural network architecture using GFID de-

scriptors for shape classiﬁcation is presented.

At the ﬁrst stage, the image dataset is converted

into binary images, from which contours are ex-

tracted. These contours are then resampled using arc-

length parameterization. GFID descriptors are sub-

sequently computed on the resampled contours, pre-

serving essential shape information while ensuring

equivariance and robustness to geometric transforma-

tions. After that, the GFID vectors are divided into

Equivariant and SE(2)-Invariant Neural Network Leveraging Fourier-Based Descriptors for 2D Image Classiﬁcation

211

(a) (b) (c) (d)

Figure 2: The GFID computation pipeline: (a) Original image, (b) Contour detection and extraction (the red line), (c) Contour

reparameterization based on arc-length, (d) GFID modulus.

Figure 3: GFID-NN : The Neural Network Framework us-

ing GFID descriptors.

real and imaginary components and normalized for

consistent scaling. We implement a dual-input model

architecture: one input layer processes the real part,

while the other handles the imaginary part. Each path-

way includes dense layers with ReLU activation func-

tions. The outputs from the real and imaginary path-

ways are concatenated to create a uniﬁed representa-

tion, which undergoes additional processing through

dense layers before being classiﬁed via a softmax out-

put layer. Figure 2 illustrates the GFID computation

pipeline going from the original image to the GFID

description.

Figure 3 presents the Neural Network Framework

using GFID descriptors namely GFID-NN. Note that

the inverse function of the GFID enables reconstruc-

tion at each layer of the proposed neural network.

However, this aspect will be addressed in future work.

Table 1: Comparison of parameters Models Across

Datasets.

Dataset Model Param.

MNIST

GFID-NN 15,648

Simple-CNN 53,322

InceptionV3 22,020,490

FashionMNIST

GFID-NN 15,648

Simple-CNN 53,322

InceptionV3 22,020,490

HandGesture

GFID-NN 16,608

Simple-CNN 84,712

InceptionV3 22,040,980

2.3 Algorithm for GFID Computation

The detailed steps for computing GFID descriptors

are outlined in Algorithm 1. These steps ensure ro-

bust extraction of invariant features.

Input: Curve γ, number of resampling points

N, parameters n

, p, q

Output: GFID vector I

∈ C

N−1

Step 1: Curve reparameterization

Resample uniformly γ to N points:

{γ

, γ

, . . . , γ

Step 2: Compute Fast Fourier Transform

Compute {F

}

N−1

n=1

, the FFT coefﬁcients of the

resampled curve.

Step 3: Compute GFID Invariants

for n = 1 to N − 1 do

if F

̸= 0 and F

−1

̸= 0 then

←

·F

−n−1

·F

n−n

−1

−n−p−1

·|F

−1

n−n

−q

;

else

← 0;

end

Step 4: Return GFID Vector

return I

= {I

}

N−1

n=1

Algorithm 1: Generalized Finite Fourier-based Invariant

Descriptor (GFID) Algorithm.

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

212

Figure 4: Samples from each of the three datasets: (a)

MNIST (b) Hand Gesture Recognition (c) Fashion MNIST.

3 EXPERIMENTS

In this part, we present results to validate the proposed

method for allowing invariance in 2D image classiﬁ-

cation.

3.1 Datasets

The MNIST dataset (LeCun et al., 1998) is a well-

known benchmark in the ﬁeld of machine learning

and computer vision, consisting of 70,000 grayscale

images of handwritten digits from 0 to 9. Each im-

age is 28x28 pixels, providing a standardized format

for training and testing classiﬁcation algorithms. The

dataset is divided into 60,000 training samples and

10,000 test samples, enabling robust evaluation of

model performance. MNIST serves as a foundational

dataset for assessing the effectiveness of various clas-

siﬁcation techniques, making it a popular choice for

initial experiments in digit recognition tasks.

The Fashion MNIST dataset (Research, 2017)

serves as a more challenging alternative to the orig-

inal MNIST, comprising 70,000 grayscale images of

clothing items from 10 different categories, including

T-shirts, trousers, dresses, and shoes. Like MNIST,

each image in Fashion MNIST is also 28x28 pix-

els, allowing for direct comparisons between models

trained on both datasets. The dataset is structured into

60,000 training images and 10,000 test images.

Hand Gesture Recognition Dataset (rishabh arya,

2021) contains total 24000 images of 20 different

gestures. This dataset primarily use for hand ges-

ture recognition task. Figure 4 displays representative

samples from each of the three datasets.

3.2 Implementation Settings

The GFID-NN model is developed using the Tensor-

Flow and Keras frameworks. For the implementation

of the GFID module, we set the hyperparameters (

= 2, p = 1, q = 1, N = 100) following the param-

eter studies conducted in (Ghorbel et al., 2022; Ghor-

bel and Ghorbel, 2024). The ﬁrst layer of the neu-

ral network processes 50 × 2 features derived from

the GFID descriptors. Besides, we implemented a

simpliﬁed Convolutional Neural Network (CNN) ar-

chitecture for image classiﬁcation tasks, speciﬁcally

designed for comparison with the GFID-NN model.

The CNN processes grayscale images of size 28×28

with a single channel and consists of two convolu-

tional layers, featuring 32 and 64 ﬁlters, respectively.

The output from the last pooling layer is ﬂattened and

passed through a fully connected output layer with

softmax activation. In the same way, we employed the

pre-trained InceptionV3 architecture (Szegedy et al.,

2015) imported from Keras where input images are

resized to 299×299 pixels. Also, we used the Data

Augmentation framework from keras of rotation and

translation that we called Rigid-aug. All training

is conducted over 30 epochs with a batch size of

32 on a single T4 GPU. During the training phase,

we conﬁgure the following settings: (1) Loss Func-

tion: Sparse Categorical Cross-Entropy, (2) Opti-

mizer: Adam with a learning rate of 10

−3

, and (3)

Metric: Accuracy.

3.3 Model Complexities

In terms of computational complexity, the Simple-

CNN model has a complexity of O(n

), where n is

the input size (28×28 for grayscale images). This is

mainly due to the convolution operations and fully

connected layers. The complexity increases quadrat-

ically with the size of the image as the model applies

convolutions and then processes the features through

dense layers. The InceptionV3 model is more com-

plex, with convolutional layers contributing a com-

plexity of O(n

), but the fully connected layers result

in a higher complexity of O(n

). This is due to the

deeper architecture and larger number of parameters

in the model, especially when working with larger

input images (224×224). The additional layers and

computations required for the inception modules lead

to signiﬁcantly higher computational costs. In com-

parison, the GFID-NN model beneﬁts from a more

efﬁcient feature extraction process. By converting im-

ages into contours and applying Fast Fourier Trans-

form (FFT) to extract GFID descriptors, the feature

extraction step has a complexity of O(n logn), where

n represents the number of contour points (typically

50 or fewer for smaller images). This efﬁcient pro-

cess reduces the dimensionality of the input, making

it computationally lighter. After the feature extrac-

tion, the model processes the data through dense lay-

ers, where the complexity scales with the number of

neurons. However, the overall complexity remains

lower compared to other models due to the smaller

input size and compact feature representation.

Equivariant and SE(2)-Invariant Neural Network Leveraging Fourier-Based Descriptors for 2D Image Classiﬁcation

213

Table 2: Performance Metrics for Different Models on Original and Transformed Tests for MNIST, Fashion MNIST, and

Hand Gesture Recognition.

Dataset Model Original Test Transformed Test

Acc. Pr. Rec. F1Sc. Acc. Pr. Rec. F1Sc.

MNIST

(LeCun et al., 1998)

InceptionV3 0.9926 0.9927 0.9927 0.9927 0.3289 0.4068 0.3289 0.3404

InceptionV3

(+Rigid-aug)

0.9934 0.9935 0.9935 0.9935 0.3806 0.4475 0.3806 0.3715

CNN (Simple) 0.8395 0.8426 0.8395 0.8346 0.7886 0.8167 0.7886 0.7907

GFID-NN 0.8561 0.8553 0.8546 0.8543 0.8431 0.8435 0.8422 0.8434

Fashion

MNIST

(Research, 2017)

InceptionV3 0.9151 0.9149 0.9152 0.9149 0.1641 0.3064 0.1642 0.1521

InceptionV3

(+Rigid-aug)

0.8853 0.8877 0.8854 0.8859 0.2784 0.3490 0.2785 0.2470

CNN (Simple) 0.8597 0.7212 0.7125 0.7106 0.5213 0.5864 0.5214 0.4905

GFID-NN 0.6947 0.6924 0.6945 0.6928 0.6893 0.6824 0.6893 0.6828

Hand Gesture

Recognition

(rishabh arya, 2021)

InceptionV3 1 1 1 1 0.0766 0.2212 0.0766 0.0473

InceptionV3

(+Rigid-aug)

1 1 1 1 0.2722 0.3799 0.2722 0.2159

CNN (Simple) 1 1 1 1 0.1155 0.2306 0.1155 0.1078

GFID-NN 0.9863 0.9863 0.9862 0.9862 0.9712 0.9712 0.9711 0.9711

3.4 Model Parameters Across Datasets

In this section, we analyze the number of trainable pa-

rameters across various architectures to gain insights

into their structure and impact on performance.

The GFID-NN model presents an innovative ap-

proach in its initial layers by transforming input im-

ages into reparameterized contours. This transforma-

tion reduces the dimensionality from a 28 × 28 ma-

trix to 100 complex elements, signiﬁcantly decreasing

the number of trainable parameters compared to tra-

ditional models. Additionally, the number of points

(N) can be reduced to 50 while preserving the overall

structure, especially for smaller images. Therefore,

by utilizing the compact and efﬁcient GFID descrip-

tors, the GFID-NN model minimizes computational

complexity while maintaining high accuracy, making

it an ideal choice for resource-constrained environ-

ments.

As illustrated in Table 1, the GFID-NN model fea-

tures a lightweight architecture with approximately

16,000 parameters. In contrast, the Simple-CNN

model, designed for the Fashion MNIST dataset, has

over 50,000 parameters due to its convolutional lay-

ers and dense output layer. The InceptionV3 model,

known for its complex architecture, signiﬁcantly sur-

passes these counts, containing over 22 million pa-

rameters.

3.5 Outcomes

In this section, we propose to analyze invariance mod-

els using performance metrics such as weighted av-

erage (w.a) Accuracy (Acc.), Precision (Pr.), Recall

(Rec.), and F1 Score (F1Sc.) across three datasets:

MNIST, Fashion MNIST, and Hand Gesture Recog-

nition. These metrics are well-suited for this multi-

ple classiﬁcation task as they provide a comprehen-

sive evaluation of model performance, particularly in

the context of imbalanced datasets.

The results are categorized into two groups: Orig-

inal Test and Transformed Test where transformations

included rotation within a range of 2π and translation

of 10 pixels. According to Table 2, InceptionV3 as

well as InceptionV3 + Rigid-aug show high accuracy

on the Original Test for MNIST (0.9926 & 0.9934)

but drop drastically (0.3289 & 0.3806) under transfor-

mations, indicating a lack of invariance. Similarly, the

models achieve respectively 0.9151 and 0.8853 accu-

racy on Fashion MNIST, with performance plummet-

ing to 0.1641 and 0.2784 post-transformation. The

simple-CNN starts with 0.8395 accuracy on MNIST

but also experiences a signiﬁcant decline to 0.7886

after transformations. For Fashion MNIST, it per-

forms slightly better with an accuracy of 0.8597 on

the Original Test, but drops to 0.5213 when trans-

formed. GFID-NN, while starting lower at 0.8561 for

MNIST, maintains better performance with 0.8431 af-

ter transformations, demonstrating its invariance. In

the Hand Gesture dataset, all models except GFID-

NN achieve 100% accuracy on the Original Test but

drop when tested with transformed data, while GFID-

NN shows 0.9863 and a robust 0.9712 under the same

conditions, highlighting its invariance effectiveness.

Therefore, results suggest that while InceptionV3 ex-

cel on unaltered data, GFID-NN offers a more reliable

performance when subjected to rigid transformations.

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

214

4 CONCLUSION

This paper introduced a novel deep learning frame-

work for 2D shape classiﬁcation based on General-

ized Finite Fourier-based Invariant Descriptors. By

extracting contours and computing invariant and sta-

ble descriptors, our model demonstrates robust per-

formance against rigid transformations, ensuring in-

variance under rotations and translations while ex-

hibiting equivariance. Experimental results on the

MNIST, Fashion MNIST, and Hand Gesture Recogni-

tion datasets show that GFID-NN outperforms tradi-

tional convolutional networks when faced with trans-

formed images. Future works will concern integrating

other invariant and stable descriptors for improving

robustness and classiﬁcation accuracy.

REFERENCES

Delchevalerie, V., Bibal, A., Fr

enay, B., and Mayer, A.

(2021). Achieving rotational invariance with bessel-

convolutional neural networks. Advances in Neural

Information Processing Systems, 34:28772–28783.

Ghorbel, E. and Ghorbel, F. (2024). Data augmenta-

tion based on shape space exploration for low-size

datasets: application to 2d shape classiﬁcation. Neural

Computing and Applications, pages 1–24.

Ghorbel, E., Ghorbel, F., and M’Hiri, S. (2022). A fast

and efﬁcient shape blending by stable and analytically

invertible ﬁnite descriptors. IEEE Transactions on Im-

age Processing, 31:5788–5800.

Guo, Y., Liu, Y., Oerlemans, A., Lao, S., Wu, S., and Lew,

M. S. (2016). Deep learning for visual understanding:

A review. Neurocomputing, 187:27–48.

LeCun, Y., Bottou, L., Bengio, Y., and Haffner, P. (1998).

Gradient-based learning applied to document recogni-

tion. Proceedings of the IEEE, 86(11):2278–2324.

Li, Y., Qiu, Y., Chen, Y., He, L., and Lin, Z. (2024). Afﬁne

equivariant networks based on differential invariants.

In Proceedings of the IEEE/CVF Conference on Com-

puter Vision and Pattern Recognition, pages 5546–

5556.

Li, Y., Wang, S., Tian, Q., and Ding, X. (2015). Feature

representation for statistical-learning-based object de-

tection: A review. Pattern Recognition, 48(11):3542–

3559.

Lyle, C., van der Wilk, M., Kwiatkowska, M., Gal,

Y., and Bloem-Reddy, B. (2020). On the beneﬁts

of invariance in neural networks. arXiv preprint

arXiv:2005.00178.

Maurya, R., Pradhan, A., Thirumoorthy, G., Saravanan,

P., Sahu, G., and Karnati, M. (2024). Fouriercnn:

Skin cancer classiﬁcation using convolution neural

network fortiﬁed with fast fourier transform. In 2024

IEEE International Conference on Interdisciplinary

Approaches in Technology and Management for So-

cial Innovation (IATMSI), volume 2, pages 1–4. IEEE.

Quiroga, F. M., Torrents-Barrena, J., Lanzarini, L. C., and

Puig-Valls, D. (2023). Invariance measures for neural

networks. Applied Soft Computing, 132:109817.

Research, Z. (2017). Fashion mnist. https://github.com/

zalandoresearch/fashion-mnist. Accessed: 2024-9-

01.

rishabh arya (2021). Hand gesture. https://github.com/

rishabh-arya/Gesture-controlled-opencv-calculator.

Accessed: 2024-8-21.

Ruderman, A., Rabinowitz, N. C., Morcos, A. S., and Zo-

ran, D. (2018). Pooling is neither necessary nor suf-

ﬁcient for appropriate deformation stability in cnns.

arXiv preprint arXiv:1804.04438.

Shi, K., Zhou, X., and Gu, S. (2024). Improved im-

plicit neural representation with fourier reparameter-

ized training. In Proceedings of the IEEE/CVF Con-

ference on Computer Vision and Pattern Recognition,

pages 25985–25994.

Szegedy, C., Liu, W., Jia, Y., Sermanet, P., Reed, S.,

Anguelov, D., Erhan, D., Vanhoucke, V., and Rabi-

novich, A. (2015). Going deeper with convolutions. In

2015 IEEE Conference on Computer Vision and Pat-

tern Recognition (CVPR), pages 1–9.

Wang, J., Wu, Q., Liu, T., Wang, Y., Li, P., Yuan, T., and Ji,

Z. (2024). Fourier domain adaptation for the iden-

tiﬁcation of grape leaf diseases. Applied Sciences,

14(9):3727.

Equivariant and SE(2)-Invariant Neural Network Leveraging Fourier-Based Descriptors for 2D Image Classiﬁcation

215