Achieving Rotational Invariance with Bessel-Convolutional Neural Networks
Authors: Valentin Delchevalerie
Adrien Bibal
Benoit Frenay
Alexandre Mayer
Venue: NeurIPS 2021 Poster (2021)
Keywords: Rotational invariance
Convolutional neural networks
Bessel functions
Bessel-Convolutional neural networks

Abstract: For many applications in image analysis, learning models that are invariant to translations and rotations is paramount. This is the case, for example, in medical imaging where the objects of interest can appear at arbitrary positions, with arbitrary orientations. As of today, Convolutional Neural Networks (CNN) are one of the most powerful tools for image analysis. They achieve, thanks to convolutions, an invariance with respect to translations. In this work, we present a new type of convolutional layer that takes advantage of Bessel functions, well known in physics, to build Bessel-CNNs (B-CNNs) that are invariant to all the continuous set of possible rotation angles by design.

Achieving Rotational Invariance with Bessel-Convolutiona Neural Networks
Valentin Delchevalerie, 1 Adrien Bibal,² 2 Benoît Frénay, 2- Alexandre Mayer PReCISE, NADI & naXys institutes, University of Namur, Belgium 2 PReCISE, NADI institute, University of Namur, Belgium Department of Physies, naXys institute, University of Namur, Belgium [valentin, deeeeaaede,, adrien.bibal, benoit.frenay. alexandre.maye mayer )
@ unaauurbb
Abstract
For many applications in image analysis, learning models that invariant translations and rotations is paramount. This is the case, for example, in medical imaging where the objects of interest can appear at arbitrary positions, with arbitrary orientations. As of today, Convolutional Neural Networks (CNN) are one of the most powerful tools for image analysis. They achieve, thanks convolutions, an invariance with respect to translations. In this work, we present a new type of convolutional layer that takes advantage of Bessel functions, well known in physics, to build Messel-CNNG (B-CNNs) that are invariant to all the continuous set of possible rotation angles by design.
Introduction
Deep learning models, and more particularly Convolutional Neural Networks (CNNs), are known as being among the most powerful tools for image analysis. For this reason, they still constantly upgraded in order to achieve better performance M. One of the main reasons why CNNs are so much used in computer vision lies in the fact that they achieve translation invariance thanks to convolutions. Filters sweep the image locally and patterns can be recognized regardless of their absolute position in the image. However, some other important types of invariance are more difficult to obtain. It is for example the case for the rotational invariance, which is relevant for many applications. One could for example consider medical imaging where tissues, cells, tumors or other objects of interest have local, arbitrary, orientation in the images (2]. Another example is satellite imaging of ships, where both the global orientation of the satellite and the local orientation of ship are arbitrary [.
Multiple works proposed solutions in order to bring rotational invariance in CNNs. However, many of them (i) only make the model more robust to rotations without providing guarantees regarding the rotational invariance 4 5]
7 8. (ii) only provide guarantees for a finite set of rotation 15 or (iii) only handle global rotational invariance, while local one is 14
angles 10 12
13
sometimes more relevant (4,
5 6. In addition to that, and more similarly to the method proposed in this paper, other works present solutions to build CNNs that are rotational invariant for all rotation
angles, while providing mathematical guarantees. One can for example cite [6 17 18 Our work in the continuity of those. We integrate new kind of convolutional layer in CNNs to build Neural Networks (B-CNNs). In B-CNNs, Bessel functions from physics are used to build a representation of the images that is more adapted to deal with rotations. propose new way to use this representation in order to compute feature maps in rotational equivariant way (see Figure [ for an example). This rotational equivariance can then lead to rotational invariance with proper choice of architecture.
"B. Frénay and A. Mayer are co-last authors.
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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Figure 1: This figure illustrates the lack of rotational equivariance in standard CNNs (left), as opposed to the rotational equivariance of B-CNNs (right). For each triplet of images, the first image is the input image, the second is the feature map obtained, and the last one is the feature map reoriented to make the comparison easier. It can be seen that the last images in the two left triplets for the standard CNN are different. Hence, different orientation of the input image produces a different feature map. However, feature maps are identical except for a rotation for the B-CNN on the right.
2 formally defines the problem of rotational invariance in CNNs and provides the necessary notations. Next, Section presents the related works bringing rotational invariance into CNNs. Our method, B-CNN, then introduced in Section 4 along with some background on Bessel functions. Experiments are described and discussed in Section [ Finally, more general discussion about some aspects of B-CNNs, as well as some future works, are presented in Section before concluding the paper in Section 7
Section
Background and problem definition
For several applications in computer vision, patterns of interest may appear in arbitrary orientations. This typically happens in medical imaging (2), satellite imaging (3) astronomical imaging [I9], texture recognition (20), etc. In such cases, the fact that the orientation is meaningless be used as a prior during learning through feature extraction that is orientation-agnostic. However, building deep learning models that present guarantees with respect to the rotational invariance Indeed, in the particular case of Convolutional Neural Networks (CNNs), the computation of a feature map is clearly not rotation invariant since, general, for a rotation of an angle a, we have
trivial.
l 4-1ks
l 1 1-1kg-
Σ 0
Σ m=
m,y m K (m, n) # Σ TV
(x -
1 n.) K (m,
(1)
==
where z' = cos a y sin a, 1 = x sin a + y cos T is the input of the convolutional layer and K represents a particular filter of size k x l22 Figure Iillstrrats this lack of invariance.
Two important notions are central in this work: rotational invariance and rotational equivariance, On the one hand, if f (I) represents the computation of feature maps in a particular layer of CNN, f (T) is rotational invariant if f (R (a) 1) = (I), Va [0, 2T), where R (a: is an operator that rotates by an angle a. However,
(R (a) I) R (a) f (1). Va E [0, 2x) ). then f (I) rotational cquivariant since feature maps are identical except for rotation (see Figure D]. For particular architectures where the feature maps the final layer have a size of 1 1, this rotational equivariance leads to rotational invariance. Indeed, if is one single value, R (a) I = I. Such a strong rotational invariance something usually not guaranteed, or sometimes only for a finite subset of [0, 2x).
3 Related work on rotational invariance for CNNs
The use of Bessel functions in the Machine Learning community has already been proposed for image denoising, classification or recognition purposes. One can for example cite [2] 22 23 24), which already use a decomposition using Bessel functions similar to the one used in this paper. However, these works differ from ours since their objective is to build rotational invariant covariance matrix, and to obtain rotational invariant models on top of this, while the objective of B-CNNs is to perform rotational invariant convolutional operation between filters and images.

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Methods for introducing rotational invariance in CNNs can be categorized in different groups: those (i) that just make CNNs more robust to rotations, without providing any mathematical guarantees. those (ii) that provide mathematical guarantees but only for finite set of rotations (typically obtained by discrete group of symmetries), and finally those (iii) that introduce continuous rotational invariance while providing mathematical guarantees.
Among the first category of methods, one can cite data augmentation 4 spatial transformer net- works (STN) [5). rotation-invariant and fisher discriminative CNNs (RFID-CNNs) [6], deformable CNNs 77, and SIFT-CNNS [8] For the second category, one of the best-known method is Group CNNs (G-CNNs) 191, which define the transformations for which an invariance is desired using the theory of groups and symmetries. However, one can also cite maassformaiion-invaiian pooling (TI-POOLING) [TO]. deep symmetry networks (DSN) [[T] stecrable CNNs [12]. steerable filter
CNNs (SFCNNs) 13,, dense steerable filter CNNs 14, and spherical CNNs [5] Finally, there are already few methods in the third category. i.e. methods that are invariant to the continuous set of rotations. General eq))-vauivariant steerable CNNs 17 present a general solution to obtain E(2)- equivariant CNNs by constraining the kernels using a group representation, LieConv 3 generalizes equivariance to arbitrary transformations from Lie groups, and Harmonic networks [6 use circular harmonics to achieve a rotational invariance by maintaining disentanglement of rotation orders in the network. Our work is in the continuity of these techniques. It proposes a new framework to achieve the invariance. For example, compared to harmonic networks, B B-CNNs present different advantages: (i) it involves real-valued feature maps, which make B-CNNs casier to use with already existing techniques (it can use classic non-linear activation functions, classic batch- norichoommaization. etc.), (ii) it uses an arbitrary number of rotation orders, and (iii) it can also be made invariant to other types of invariance in future works.
In the next section, we introduce Bessel-CNNs, based on the use of the Bessel functions well known in physics, to make CNNs invariant by design to all rotation angles in the continuous set [0, 2x).
4 Exploiting Bessel functions for CNNs
The mathematical background on Bessel functions and the motivation of their use for rotational invariant image recognition tasks are presented in this section. Some of these developments are inspired from the use of Bessel functions in physics (25). These mathematical developments lead to new type of Convolutional Neural Networks (CNN) that we call ConsollCannllltionnal Neural Network (B-CNN). Mathematical evidence of a rigorous rotational invariance is pointed out.
4.1 Bessel functions
Bessel functions are partieularly well known in physics because they arise when solving some important problems in polar or cylindrical coordinates. For example, they appear when dealing with wave or heat propagation. Bessel functions may be of different order v and are defined as a particular solution of the Bessel's differential equation
d'y dx
dy dx
+ (~2- 0.
x²²
This equation involves two particular kinds of Bessel functions Ju (x) and Y, (z), called the Bessel function of the first and of the second kind, respectively. In this work, v must be an integer and we only consider Ju (x) since Y, (z) diverges for x O. For E N. J/ (x) and J_ (x) are not independent and
(x) (-))= (x).
(3)
Bessel functions also satisfy
J. (-1
(4)
which means that J, is an even function if v/ is even, and an odd function otherwise. Bessel functions of the first kind can be used to build particular basis for the representation in polar coordinates of images defined in a circular domain of radius R, as
R PJB (kusp) dp,
Ju (Kusp) ii0 Vv,j E J where Nos = 1/1 2T
(5)
3

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forms an orthonormal basis for all squared- -interbbbee functions f such that f D2 C R - R (where the domain Da is a disk in R²². Indeed, we can show that
J
R 2m I."
P [NugJo (Kens)) e eivoy N J. (kvi ewoo) d0dp = ke
(6)
the k are solutions of either Ju (kv = 0 or J (Mv = 0 126]. We use J (kv. = 0 as condition for the ky because it makes it more convenient to represent arbitrary functions (by the fact that for = 0 there is a ku.j = 0 with this condition, we can use (0.0p))e = to describe an arbitrary constant intensity) [25]. The key advantage of using this basis is that can deal easily with rotations. Later, this basis will be used to build new representation for images.
4.2 Bessel coefficients
An arbitrary function V/ (p, 0) E E. where p and are polar coordinates, can be expressed with the basis in Equation (5) as
00
V (p.
Σ
Vns NogJy (kuu e000 e
==0
where the Bessel coefficients L Mnj are obtained by projecting vi (p, 0) on this particular basis
2nt /,²² R
V Ves -
P [NNusJu (krss) e'voj" (p. 0) d0dp.
In order be mathematically exact when describing (p, 0) using its Bessel coefficients, one should consider all v E {-x..... ) and all j E {0.... x) to represent V (p, faithfully. Nevertheless, as only a finite to be chosen. By taking into account Properties (3) and (4) when looking at Equation (8) and as (p,0) I. we end up with the relations
of coefficients can be considered in practice, values of Vmaa and Jmax need
(V
v-) = (-1) (W
(3 = (-1) 10/t1310 (V2,)),
where R and 3 stands for the real and imaginary part, respectively. One can then only compute the Bessel coefficients for v (resp j) in {0...., Vmar (resp. ] since & and Yes are not independent. Also, note that, when or j increases, kv.j also increases. The values of Vmax and Jmaz can then be determined through a mathematical insight about the maximal value of kes to use in given problem (for example, using the Nyquist frequency, in (2)).
To understand now why working with Bessel coefficients is particularly useful, one can see how an arbitrary rotation of & (p. 0) by angle a modifies its Bessel coefficients V Vuj. To do so, let us
consider sprot (p, V (p, a)) for an angle a E [0, 2m). Its Bessel coefficients are given by
rot / 2n de f. R
dpp [NujJu (ku e eiro grot (p,
(10)
By defining 0' = = a, this leads to
2m d0'
[. R
dpp Ju (kvsp) " v (p. (p, eiva Vuyee
(11)
This motivates our work, as a rotation of an arbitrary function by an angle a only modifies its Bessel coefficients by multiplying factor e iia which makes rotations conveniently expressed in the Fourier-Bessee transform domain (analogously to how the Fourier transform maps translations to multiplications by complex exponentials). Therefore, if V (p, 8) represents an image, its Bessel coefficients constitute a more adapted representation to build rotational invariant operations.
4.3 Defining a rotational invariant operation
In order to build rotational invariant operation between the Bessel coefficients Vny and filter made of the same number of complex numbers Ky.j. one can consider
1
2n
K V
Ky Vuise Vense-ivape -ira ( da.
(12)
2T
Jn

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This operation is, by design, necessarily invariant to rotations as Equation [II shows that multiplying each Ve. by eiva equivalent to applying rotation the image. By performing the integration for a going from 0 to 2m, the image achieves a complete rotation around itself. Therefore, K y does not depend anymore on the particular initial orientation of the image. Note that, without the squared modulus, we can show that only the coefficients Vo. contribute to the final results. The operation would therefore rely on an incomplete representation of the image.
Using Equation as such may look like a computationally expensive brute force strategy, since requires to evaluate the numerical integration over a. However, given that
2
R (am) (ax)) |mm 25] cos (0m - -0,) Σ S (am) ) (az) |mm cos (0m - 0z) Σ
0:2
m.j
m.j
Σ
R (ay) 3 (am) I2m sin (fm 0g)
(13)
m.j
where 0g and 2;
|2/l are complex numbers, the squared modulus in Equation
becomes
Σ R (K2r) R (K |Vn Werg cos (Ous Ouis - (v v'))
+
Σ
3 (KWr) 3 (K)rg) WV Vey cos (fv.) -a (v-vv vv))
v'
-2 Σ
3 (Kts) R (K)r) Wes Varg sin (Ous - Our - -
(14)
v'
if wr 1 - The three terms should be integrated over thanks to Equation the trigonometrie functions are a-dependent. A simple integration leads to
12 but only
I. sc (0p.) - o - a (v v') da = 2T se (Ou.j Ou) if v'
(15)
otherwise,
where se can represent the cosine or the sine function. Therefore,
K
Σ R
R (Kt Mous Mese cos (Ou.) Ow (Kt,)
Σ
(K))) 3 (Kig.) Mi Mndl cos (0nj 0v(
"
- 3 (Kz)) R (Kzjy) | Vay sin (0u.) 02 Ow'r)),
Σ K.J,S'
(16)
and by using once again the result presented in Equation 13 we have
K. V
Σ Σ K KK Mpsl?
(17)
which is therefore equivalent to Equation 12 but much more convenient. By construction, the results obtained by Equation (17) do not depend on the particular orientation the image. Furthermore, K & E E (with K E C and L Pr C). which is something convenient in order to use, for example, classic activation functions. Using Equation|] [2 (or Equation [7p to achieve the rotational invariance one of the main contribution of this paper.
4.4 Bessel-Convolutional Neural Networks (B-CNNs)
In standard CNNs, a feature map is the result of the convolution between the image and a filter of 1 x li weights, as described in Equation (I). The weights of this filter are real numbers that are tuned during the learning process of the network. In Bessel-CCnvvlutional Neural Networks (B-CNNs), the process to build feature maps is slightly different as the representation of the image based on Bessel
5

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Input
Feature map
kn.
1
Vug (m.
(m.))
Figure 2: Illustration of how the feature maps are computed in B-CNNs.
coefficients are used. A filter K made of (Umaz + 1) x (Jmas + 1) complex weights K2.j is used to sweep the image sub-regions of size k x k. Note that Pmaz and Jmaz are generally larger than 11 and k2, which results in an increase of the number of trainable parameters in cach layer. However, as it easier for B-CNNs deal with rotational invariance, we show in Section [5 architectures with less layers than for other methods still achieve similar or even better performances. For each sub-region of the image centered position n1), its Bessel coefficients V (//m..))
computed, and
a (m.n)
- K V (m. Mg
(18)
computed, where a (m,.) is real value that invariant rotation, i.e., does depend on the particular orientation of the sub-region. All the values obtained for the different sub-regions
constitute the feature map, which is then rotation equivariant. This is schematized in Figure
4.5 An efficient implementation of B-CNNs
Implementing B-CNNs as described in Section can for each sub-region of the image. However, developing Equation (88 with Equation
can be slow, as it requires to evaluate Equation (8)
gives
I, NugJJ (ku,))) wew.wrggmmm. (m..)) (p. 0) pdpd0)
2x
R
a (m.
Σ
K
2m (
R wimm) (0,0) E KigNNojd (kuzp) e ivo pdpd0
Σ1
(19)
where (mm.n) (p,0) represents a particular sub-region of the image centered at (m, n) in polar coordinates. When going back to a Cartesian coordinates system, one can get
a (m.n) Σ 1
1mm I,n v) Σ Kin EKig (HegV x²) (a drdyy drdyl 2
(20)
~u Ju (ku,p)) is defined as where 7 (Ix, y) is the input of the layer, 0 (x, = x aretan and
Jn (kvsp) Jn (ku.jp) if S R otherwise.
(21)
Finally, by defining Trg (I, y) = NusJu (Birs v x²
y2 e (1.()))) Equation 20 leads to
= (z, Σ Kig (I, 12
Σ 11 (x, (z. 12
(22)
where denotes convolutional product and F, (x. y) corresponds to a filter modified by the Bessel functions. Compared to a standard CNN, the only difference is that, actually, a layer in B-CNN performs Umex + convolutions instead of one only, and it also needs to update the filters with the Bessel functions. Nevertheless, Tug (ar, y) part of the model and only needs to be computed once, as it does not depend on the input of the layer, but only the choice of Poaw and Jmaz.

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5 Experiments
This section presents the details of the experiments used to test the rotational invariance of B-CNNs, including the datasets (Section[5.1) the baseline architectures (Section 5.2) and the experimental setup (Section 5.3). Experimental results are presented in Section and a discussion on the rotational equivariance of feature maps proposed in Section 55,
5.1 Datasets
In this comparative study, three datasets are used in an increasing complexity. In all of them, orientations present in the test set are not present in the training set. The first dataset is MNIST (27). a classic baseline of 28 x 28 images of handwritten digits, where images in the test set are randomly rotated by an angle in [0, 2T). For all runs, the training set contains 60, 000 images and the test set 10, 000 images. The second dataset is Outex-TC-00010
dataset (28), a classic dataset for a more complex classification involving both global and local rotations. contains 128 x 128 grayscale images of 24 particular textures. The training set contains 480 images with 20 orientations, and the test set is composed of 3840 images with 160 orientations. The third dataset made of 128 x 128 x brain MRI images [29] that are either cancerous or non-cancerous. This dataset is used as real-world, more complex dataset, which contains local entities (the brain tumors) that can occur at arbitrary orientations. Each image is cropped to make the brain fill the entire image. The training and test sets contain 190 and 63 images, respectively.
5.2 Baseline architectures
Experiments aim to evaluate how good B-CNNs are for problems where rotational invariance is desired. We decided to use G-CNNs, as well as standard CNNs in our comparative study. For G-CNNs, the symmetry group C4 is used (rotations of angle 3. n. 2 and 2T). For each dataset, one architecture based on CNNs, G-CNNs and B-CNNs has been chosen by picking the one that performs the best on a validation set. Importance is given to the fact that the three models contain roughly the same number of trainable parameters, in order to perform fair comparisons. Attention is also paid to the fact that the feature map in the last convolution layer should be of size 1 1, in order to achieve rotational invariance for G-CNNs and B-CNNs.
3r
The standard CNN architecture for MNIST is made of 7 conv. layers with 32 filters for the first two layers, and 64 filters of size 3 3 for the other layers; one max- max-pooling layer that operates over x regions after the second and last layers; and one dense layer with 10 units. The G-CNN architecture similar, except that the number of filters is divided by 2 in order to keep the same number of parameters. The B-CNN for MNIST is made of 5 Bessel-conyolutional (B-Conv) layers (one of 16 filters with Umar = 11, Jmas = and k = 3: three of 16 filters with Umer = 9, Jmaz = 7 and k = 5; and one of 32 filters with Vmaz = 7, Jmox = 5 and k- = 7), a max-pooling layer over 2 2 regions after the fourth layer and a dense layer with 10 units. Activation functions are relu, except for the B-CNN where tanh leads to better performances on the validation set. Adam optimization with learning rate of 0.0001 and an exponential decay rate of 0.9 is used for the three models.
The standard CNN for Outex is made of 5 conv. layers (one with 48 filters of size 5 x 5, one with 64 filters of size 5 x 5, two with 64 filters of size 3 3 and one with 128 filters of size 3), one max-pooling layer over 3 x 3 regions after the first layer, one over 2 x 2 regions after the second and third layers, and one dense layer with 24 units. The G-CNN architecture for Outex similar, but the number of filters is divided by 2 again to have the same number of parameters. The model based on B-CNN is made of 3 B-Conv layers (one with 16 filters with Pmaoz = 15, Jmaz = 11 and k = 15; one with 32 filters with Mmaz 10, Jmaz 9 and k = 9; and one with 32 filters with Vmaa = 7. Imes and k = 7), a max-pooling layer over x 3 regions after the first layer, one over 2 x 2 regions after the two others, and dense layer with 24 units. Again, activation functions are relu, except for B-CNN where tanh are used. Adam optimization with a learning rate of 0.0005 and an exponential decay rate of 0.9 is used for the three models.
The standard CNN for brain MRI is made of 5 conv. layers (one with 32 filters of size x 7, one with 32 filters of size 5 5. two with 64 filters of size 5 x 5 and one with 64 filters of size 3 x 3), one max- max-pooling layer over 3 x regions after the first layer, one over x 2 regions after the second and third layers, and one dense layer with 2 units. The G-CNN architecture is similar but the number
7

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160
50-
G-CXN B-CNN
Acurcey (8)
Accurcty
G-Chh
G-5NN BKNN
21-
CNN
50-
130
Looch
Acsency 150
G-CNN
C-CMN
G-CNN
21-
CVN
CMN
CNN
:00
300
130
151
301
Epoch
(a) Performance MNIST
(b) Performance Outex
(c) Performance brain MRI
Figure 3: Mean test accuracy over 40 runs with 99% confidence intervals for B-CNN, G-CNN and standard CNN on MNIST, Outex and brain MRL the first row, the training sets contain input images as they are originally in the dataset, whereas data augmentation is used in the second row.
of filters is once again divided by 2. The model based on B-CNN is made B-Conv layers (one with 32 filters with Vinar 11, Jmax = 11 and & = 9; one with 16 filters with Vmaz 9, Jmaa 9 and & 7: one with 16 filters with Vnne 7, Jmas and k 5). max-pooling layer over 3 x 3 regions after the first layer, one over x regions after the three other layers, and a dense layer with 2 units. Again, activation functions are relu, except for B B-CNN where tanh are used. Adam optimization with learning rate of 0.001 and a exponential decay rate of 0.9 is used for the three models.
and & 5; and one with 32 filters with
Umaz Jmaz
5.3 Experimental setup
Each experiment consists of 40 independent runs and mean accuracy scores with 99% confidence intervals are reported. In order to isolate intrinsic rotational invariance, each model is trained for each dataset in two different settings. In the first setting, images are rotated all during training. In the second setting, input images are rotated by random angle during training (a form of data augmentation). In order to avoid border effects due to the appearance black corners when images are rotated for the Outex dataset, images are cropped from an initial size of 128 x 128 to 88 x 88. This is not performed for MNIST and brain MRI, since image corners are already black.
In order to highlight the ability of B-CNNs to use the invariance for the continuous set angles as prior knowledge, another experiment in low data setting is also performed. This experiment is exactly the same as the previous one on the Outex dataset (with rotation of the input images), except that 75% of the training is removed, and placed in the test set.
5.4 Experimental results
Figure 3 shows the mean test accuracy of B B-CNNs, G-CNNs and standard CNNs on MNIST, Outex and brain MRI. The first observation is that standard CNNs have poor performance in almost all cases, which justifies the need for better methods. A second observation is that using data augmentation (second row), as opposed to using the input images as they originally are in the dataset (first row), increases the performance of all methods in all cases. Furthermore, in the setup without data augmentation, can be observed that B-C B-CNNs largely outperform G-CNNs and standard CNNs on MNIST, This indicates that G-CNN need data augmentation in addition to the groups of symmetry to learn rotational invariance, while Bessel convolutions intrinsically impose the invariance.
The results are interestingly reversed for Outex, where B-CNNs perform better than G-CNNs with data augmentation, while being roughly equivalent without data augmentation. The good performance of G- G-CNNs without data augmentation on Outex may be explained by the fact that the texture images

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contain an intrinsic data augmentation (a rotation of an image can be present in the original dataset). This is not the case of MNIST, where all images the original dataset have the same orientation.
It can also be observed that the variance of G-CNNs is generally higher than B-CNNs, due to the fact that discrete of rotations is encoded into the groups of the G-CNNs. Because of that, mismatch can happen between the rotations learned with the randomly rotated input images and the groups on the one hand, and the rotation of the images in the test on the other hand. This mismatch is minimized for B-CNNs, as the continuous set of rotation angles is imposed by design. Thanks to that, the predictions of B- B-CNNs are more stable than the one of G-CNNs.
The generalized instability in performance for all methods on brain MRI (see Figure 3c) can be explained by the small number of instances used at each epoch (190 for training and 63 for testing). This characteristic of the dataset may also explain that B-CNN achieves better performance more quickly, as the rotational invariance is present by design (i.e., invariance is by opposition to G-CNN that more heavily relies on the training instances to learn this invariance.
prior knowledge),
Figure 4 presents the results obtained on the Outex dataset in low data setting, where 75% of the training data is removed. When, in the initial experiment presented in Figure 3b) we slightly outperformed G-CNNs by less than 5% accuracy, in this experiment where less data is provided during training. we outperform G-CNNs by a little bit less than 15%. Furthermore, the learning curves again much more stable with B-CNNs.
5.5 On the equivariance of feature maps
In addition their performance stability, B-CNNs also offer certain stability from the equivariance of their feature maps. Unlike standard CNNs in Figure the feature maps of B-CNNs extract the exact same latent features, matter the orientation of the input image. This characteristic of B- B-CNNs can help in providing trust to its users. For instance, medical experts would see that the model extract the same latent features for tumorous brain images, and this no matter the orientation of the tumor.
Figure 5 shows the equivariance B-CNNs and G-CNNs. The feature maps in the figure are randomly taken from the second convolution layer of the B-CNN and G-CNN. It can be observed that the feature maps in the first row (for a non-rotated image from Outex) for the B-CNN are generally the same as those of the second row (corresponding to the same image, but rotated of 60%, see Figure 5a). However, when training a G-CNN with the same number of layers and the same filter sizes, its feature maps change significantly after rotating the input image (see Figure
In addition to Figure 5 we also performed a quantitative study order to assess the rotational equivariance of B- B-CNNs, G-CNNs and CNNs. To do so, we performed the same experiment as the previous one, but for the whole test set and for random rotational angles. For each sample, the 4 feature maps that are the most active are min-max scaled between () and 1 (this step is needed since feature maps of different architectures may have absolute values at a different scale). Then, the difference between the non-rotated and the rotated feature maps is computed (the difference between the first and second row in Figure 5). The bigger this value, the less equivariant the method is. We obtained mean value of 1.77% 2.78% for the B-CNN, 10.64% + 5.64% for the G-CNN and 5.73% + 5.64% for the CNN. B- B-CNNs therefore achieve a better rotational equivariance.
100
G-CNN
80
B-CNN
Accuracy (%)
CNN
60-
40-
0-
50
100
150
200
Epoch
Figure 4: Performance on the Outex dataset in low data setting (120 images in the training set).

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(a) Some feature maps of a B-CNN
(b) Some feature maps of a G-CNN
The first row corresponds to random feature maps for a non-rotated image from the Outex dataset. The second row corresponds to the same feature maps, but for 60 rotated version of the image. To ease the comparison with the first row, these feature maps have been reoriented. The corner pixels of the images of the second row are hidden they correspond to pixels that are out of the bound. For the sake of comparison, the same pixels have been hidden in the images of the first row.
Figure 5:
6 Discussion and future work
In practice, we are not always able to obtain a perfect rotational invariance. The main reason is that the Bessel functions are discretely sampled filters. It directly follows that, if too small filters are used, perfect rotational invariance may not be achieved due to discretization errors. This is the reason why data augmentation helps, even when using B-CNNs. Therefore, using larger filters helps to achieve a better rotational invariance. It would be interesting to consider other interpolation methods improve the sampling of input images and reduce this effect (similarly to what is already done by [30] to ensure translation invariance in CNNs).
grid, with resolution fixed by the size of the
In future works, one could consider bringing several updates to B-CNNs. Firstly, the actual V'aax + 1 convolutions can be converted to only one convolution. Secondly, performances of B-CNNs are deeply dependent in this work on the choice of Vmar and Jmaz- However, it is possible consider the Nyquist frequency in order to rather impose an upper limit to the ku.j as done in 21. This will both minimize the aliasing effect and maximize the amount of information preserved by the Bessel coefficients. Thirdly, it is also possible to bring other types of invariance using the same framework, like for example invariance under reflections (in order to make B-CNNs B(2)-invariant).
Before concluding, we would like to point out again that B-CNNs achieve a global rotational invariance by a succession of local invariance, provided by the multiplication between the image and the filters (it is the case for all similar techniques). This property can probably make B-CNNs and similar models more easily fooled by adversarial techniques. In the future, it could be interesting to evaluate the robustness of such rotation invariant CNNs, as adversarial techniques is an important concern for many applications.
7 Conclusion
This paper proposes new kind of CNNs, called Bessel CNNs (or B-CNNs), that are intrinsically invariant to the rotation of input images. In order to achieve this, new convolutional layer has been designed, based on Bessel functions from physics. Experiments show that B-CNNs do not need any preprocessing of the input images to be rotational invariant, and that the full rotational invariance brought by the Bessel convolutions lead to more stable results. We conclude that Bessel convolutions can be used in any application involving global and/or local rotations of images.
Acknowledgments and Disclosure of Funding
The authors want to thank José Oramas M. from the University of Antwerp for his insights on the paper. A.M. is funded by the Fund for Scientific Research (F.R.S.-FNRS)) of Belgium. V.D. benefits from the support of the Walloon region with Ph.D. grant from FRIA (F.RSS.-FRRS. This research used resources of PTCI at UNamur, supported by the F.R.S.-FNRS under the convention 2.5020. 11.
10

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12


Reviews:
[{"review_id": "t0WYFplcUjJ", "reviewer": "Reviewer_Vncf", "summary": "The paper shows how Bessel functions may be used in the context of a convolutional network to obtain better recognition performance for rotated images.  Conceptually a Bessel function expansion is applied locally at each point in an image, which is then filtered with a standard convolutional filter bank (eq. 18).  The authors describe a more efficient implementation that folds the Bessel function expansion into the kernel convolution (eq. 22).  Performance is described on MNIST, Outex, and MRI images of tumors, showing improvement for the Bessel function approach.", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "This seems like a reasonable approach.  However I think the authors should be more explicit about the fact that they are only addressing *local* rotational invariance of feature descriptors, as opposed to global rotational invariance to shapes in an image.  It would appear that the global rotational invariance is achieved simply by training on objects in all different rotations, which seems pretty brute force.  But the other methods being compared to appear to do the same thing.  Perhaps this is state of the art, but in my view rather primitive.  There doesn't appear to be any attempt to reason about the geometry of an object, just a bunch of filters and averaging.  \n\nAlso the Bessel function descriptor simply averages over all orientations, so it loses all information about the local configuration of shape (with respect to orientation).  It would seem that such an approach could be easily fooled by adversarial examples or nonsensical images that provide the same signature in terms of these descriptors but without a consistent global shape, something to think about.\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes", "review_text": "", "extra_scores": {"ethical_concerns": "None", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "1 hour"}, "extra_text": {}}, {"review_id": "MEKQQxXJdP5", "reviewer": "Reviewer_iKDt", "summary": "The authors propose a representation based on Bessel functions as a replacement for standard convolutional layers which guarantees rotation invariance. They discuss the properties of Bessel functions and motivate the new rotation-invariant layers. Experiments show that the proposed layers indeed improve rotation invariance; in fact, the proposed scheme seems to outperform the existing schemes based on group representations which only implement invariance for a discrete set of angles.\n", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Overall this is a solid paper about a creative idea. It would be great to see a bit more discussion about the peculiarities of the proposed layers and to tighten the presentation. Another welcome addition would be a precise statement about when perfect rotation invariance can be achieved, and a discussion about why we don't see it in experiments. Perhaps a reference to similar issues that plague ordinary convolutional layers (see below) and a comparison with the proposed Bessel representations. Below I list a bag of comments not ordered by any particular criterion:\n\n- Please state for which space the used Bessel functions form a basis\n\n- To streamline notation (e.g. in (7), (8)) I suggest to define normalized versions of Bessel functions\n\n- When you write \"In order to be mathematically exact\", what are you referring to? \n\n- I am a bit puzzled by \"as \\nu or j increases, k_{\\nu, j} also increasse. As J_\\nu(x) tends to zero when x increases, a large value for \\nu and j means that \\Psi_{\\nu, j} will only slightly influence the reconstruction of \\Psi\". This is true for images that are low-pass in the Bessel basis. If the image contains high-frequencies then it is not true. In particular, how about an image $u(x) = J_\\nu(k_{\\nu,j}\\rho)$ for large \\nu? \n- Certainly the above makes sense for images with finite pixels, but then the requisite value of \\nu_max can be determined exactly from the sampling theorem. (NB: I strongly suggest to replace \"The values of \\nu_max, j_max can be determined through a mathematical insights about...\" by something like \"The values... can be determined by considering ... \").\n\n- One thing that I find at the same time interesting and confusing is that you use the Fourier-Bessel power spectrum. This clearly immediately gives invariance, but a \"canonical\" way to get invariance (as in some of the cited prior work) is to compute a convolution over rotations. What is the technical difficulty in doing that for all angles? And what are the implications or drawbacks of the fact that your representations are non-negative? \n\n- Fourier-Bessel representations are commonly used in signal processing to obtain rotation-invariant representations. One good example is https://arxiv.org/abs/1211.1968 or fast steerable PCA (https://arxiv.org/abs/1412.0781). It would be good to compare your work to these representations. There are many other examples which would be acknowledging and commenting under prior work.\n\n- Some of your experimental results where you improve over C5 convolutions in G-CNNs are not surprising since 4 rotation angles is a coarse discretization. It would be great to compare to finer discretizations of 8 or 16 angles.\n\n- Section 5.2 gives a great amount of detail about the architectures which is a priori good, but it throws a brick in the flow of the paper. Perhaps some of it could be put in the appendix? (Not all of it is really important to appreciate the experiments)\n\n- Could you elaborate why you don't get perfect invariance in experiments? Is there a setting / dataset where you could expect perfect invariance? Could you state a theorem? In the context of translations, perfect {equi, in}variance is precluded by nonlinearities and downsampling, but it can be restored by modifying the pooling strategy (https://openaccess.thecvf.com/content/CVPR2021/papers/Chaman_Truly_Shift-Invariant_Convolutional_Neural_Networks_CVPR_2021_paper.pdf, https://arxiv.org/pdf/2105.04040.pdf). Could a similar strategy help achieve perfect rotation invariance in your Bessel-Fourier framework?\n\n- How does the training time / complexity of your neural networks compare to that of standard CNNs?\n\n\nSome misprints / suggestions:\n\n- mathematical basis -> basis\n- which really conveniently expresses rotation -> which conveniently expresses rotations in the Fourier-Bessel transform domain (perhaps: analogously to how the Fourier transform maps translations to multiplications by complex exponentials)\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3-4"}, "extra_text": {}}, {"review_id": "DLMpmClQvHU", "reviewer": "Reviewer_nHDN", "summary": "This paper proposes a method to achieve rotational equivariance (and to some extent, invariance) in convolutional neural networks. The method works by representing the convolution kernel in a Bessel Function basis (which is naturally rotation equivariant). This results in a convolutional operator which, though requires more convolution operators (one per Bessel coefficient used compared to one in regular CNNs), outputs a rotation equivariant feature map, unlike normal CNNs.\nThe method is demonstrated to work on three simple datasets - MNIST (with rotations) a Texture dataset and a small MRI dataset. ", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Originality:\nThe proposed method is different from prior work and quite an elegant solution to the problem.\n\nQuality:\nI think this is quite a solid paper in terms of quality all in all, though there are a few shortcomings. The motivation is clear, and the proposed solution indeed addresses the problem directly. The mathematical analysis is, though I didn't check in detail, extensive and well presented and the bottom line (i.e convolving with Kernels obtained with the projection on the Bessel function) is clear and actionable. I think the main weak point is the experimental evaluation - though I am convinced that the method is able to produce rotationally equivariant feature maps I am not sure it has been demonstrated to be useful, or, in fact, significantly better than competing methods or even a regular CNN (certainly with augmentations). The datasets used are too simple and the results, even in the ideal case (other than MNIST without augmentation) are not convincing.\nAnother point I wish the authors would address is what happens in cases where one actually wants orientation information to be preserved and processed by the network? would a combination of the proposed method and regular CNNs work? how? \n\nClarity:\nThough the writing the is rough at times I had no problem following the paper. It is nicely presented and well structured.\n\nSignificance:\nI think this is another weak point of the paper - I was not convinced this is an actual significant problem - not because learning rotation equivariance is not important, but because it seems current methods solve it quite well, certainly in more complex settings (where the data is rich). Nevertheless the analysis is interesting and may be useful in some specific contexts.\n\nPost discussion update: Following the author rebuttal and discussion with reviewer I have updated my score and I am now leaning for acceptance of this paper.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Discussed above, no societal impact I am concerned about.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3"}, "extra_text": {}}, {"review_id": "CNivyMAyTo-", "reviewer": "Reviewer_9Dgi", "summary": "This paper proposes a new type of convolutional layer that takes advantage of Bessel functions, which are very well known in physics, to achieve rotational equivariance in convolutional neural networks. The main idea of the approach is as follows: in a general convolutional layer, a convolution is performed with a sub region of the image and the filter. But here, the sub region is first transformed to a polar coordinate system using a Bessel function as the basis, and the properties of the Bessel function are utilized here to define a filter operation that integrates over all possible angles of the sub region, thus making the operation rotation invariant, while ensuring its scalable to deep networks. The main contribution of the paper, therefore, is to propose a technique that achieves both local and global rotational equivariance in convolutional neural networks over the continuous set of all possible angles, while making it scalable and mathematically rigorous. ", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Originality: \n- The proposed method is definitely very novel, and the authors claim that to the best of their knowledge, this is the first method to introduce Bessel functions in machine learning, as well as to propose a rigorous rotational invariance over the continuous set of all possible angles in CNNs. \n- The related work is comprehensive and very well compiled. Also, the authors make it clear how their approach stands out, with the main drawbacks of the other approaches being not providing rigorous rotational invariance or not covering the entire spectrum of angles. \n\nQuality: \n- The main idea of the paper, using Bessel functions in polar coordinate system to define a filter that is rotationally invariant, is supported by rigorous mathematical evidence. All derivations involved in obtaining the final form of the proposed filter operation are very clearly provided.\n- Comprehensive experiments are conducted, while comparing with the state of the art techniques, to provide empirical evidence to support their claims.\n- This is a complete piece of work with no unexplained or incomplete sections.  \n\nClarity: \n- The paper overall is very well written. Its clear, understandable, and very well organized. The code is also provided in the supplementary material. \n- It would be really helpful if the authors can expand a bit more, and add some derivations, in section 4.1, especially for equations 5 and 6, if required in an appendix. \n- It would be helpful if the authors can further explain the symbols in equation 9. \n- In Figure 3, one observation is that data augmentation (which includes only random rotations) provided considerable improvements to BCNN. GCNN with data augmentation gave better results than BCNN without data augmentation is several scenarios. Further, in section 5.4, the authors commented that \"GCNNs need data augmentation in addition to the groups of symmetry to learn rotational invariance, while Bessel convolutions intrinsically impose the invariance\". But the observations clearly show that data augmentation helps a lot with improving even the BCNN model. The authors too acknowledge this in section 5.4. Can the authors provide their opinion on why data augmentation might be providing those improvements to BCNN, when it should be intrinsically invariant to rotations? \n\nSignificance: \n- I think the introduction of Bessel functions is a very important contribution of the paper. It has a lot of scope for further work from the community. Further, with the strong guarantees provided by the paper, unlike any other preceding work, this paper definitely advances the state of the art in the area of rotational invariance in CNNs. \n\nFurther Comments: \n- It's very interesting to note that this approach seemed to provide an advantage in low data scenarios, and provide more stability due to equivariance of the feature maps. It would be interesting to see further exploration into the implications in low data settings and robustness in future work. \n- No large datasets were used in the experiments, which can be used to test the scalability of the proposed approach. It would be very helpful if the authors provide results (if required in an appendix) on some large dataset like ImageNet, with a well-known deep network like a ResNet, etc. This is to analyze how the proposed approach will scale with large models and datasets, and if similar gains can be noticed in such a scenario too. ", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "I think the main limitation of this work is the proposed Bessel convolutional layer is computationally more expensive than a normal convolutional layer. The authors did acknowledge this and provided an explanation on how the prior knowledge helps in achieving similar results with smaller networks that can compensate for the computational differences. In addition to that, I think it will be very helpful if the authors provided some results with a large dataset and a deep network too.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}]

Author rebuttal:

Meta-review: After lively discussion, all reviewers found this paper interesting and thought the paper ought to be accepted. Everyone liked the idea of using Bessel functions to achieve local rotational invariance clever and found the technique to be potentially quite impactful -- especially in the low data regime. Reviewers also found the experimental results compelling.
Decision: Accept (Poster)