Thomas Maugey
email: thomas.maugey@inria.fr

Towards digital sobriety: why improving the energy efficiency of video streaming is not enough

Keywords: video streaming, global warming, energy reduction, rebound effect

IPCC conclusions are unequivocal: we must divide our greenhouse gas emissions by two before 2030 if we want to maintain the global warming below 1.5°C in 2100. Hence, it becomes urgent to aim sobriety. Contrary to what is often claimed, digital technologies must also target global emission reduction, as their impact on the climate is huge and exploding every year. Among the digital world's emissions, those related to video processing and streaming are significant. At the same time, a lot of research efforts are currently done to reduce the energy consumed by video transmission algorithms or infrastructures. In this paper, we demonstrate that, even though such research works are crucial, they are not sufficient to enable global video streaming emissions reductions. The conclusion is that we must collectively think of other complementary solutions.

I. INTRODUCTION

Contrary to popular belief, the digital world is not intangible nor dematerialized. According to [START_REF] Ferreboeuf | Lean ICT-towards digital sobriety[END_REF], its emission of greenhouse gas (GHG) corresponded to 4% of the global world emissions in 2019 (equivalent to the world air traffic), and this percentage could be doubled in 2025. Concretely, this GHG emission is due to (i) the energy consumed by the data centers (19%), the network (16%) and the terminals (20%) but also due to (ii) the production of computers, TVs, smartphones (45%). Online video streaming takes a significant part among these emissions. In 2018, online video streaming constituted 60% of the global data flow, and, by itself, generated 1% of the global emissions [START_REF] Efoui-Hess | Climate crisis: The unsustainable use of online video[END_REF] (as much as a country like Spain emits). And this quantity explodes: video traffic is estimated to grow by 79% between 2021 and 2027 [START_REF] Jonsson | Ericsson mobility report[END_REF].

At the same time, the sixth report of the Intergovernmental Panel on Climate Change (IPCC) [START_REF] Zhongming | Ar6 synthesis report: Climate change 2022[END_REF] states that if we want to keep the global warming under 1.5°C (Paris agreement), one should target a global emissions decrease of 50% when compared to those of 2019. This corresponds to a decrease of 7.6% per year [START_REF] Masson-Delmotte | Global warming of 1.5 c[END_REF]. They also state that this is not the path that is currently taken. Hence, every part of our society must urgently aim sobriety. This is for example the case of online video streaming. So the question is simple: what are the solutions that should be envisaged to halve by 2030 the GHG emissions due to video streaming ?

As these emissions are correlated (not necessarily proportional) to the video data flow, one could rely on the progress of video compression algorithms to decrease the size of every individual video. For example, the last H.266/VVC standard reaches 50% of gain when compared with the previous codec H.265/HEVC [START_REF] Bross | Overview of the versatile video coding (vvc) standard and its applications[END_REF] published 7 years before. Even though similar gains can be expected for the future codec, this cannot, by itself, enables a drastic emissions reduction, because of at least three reasons. First, the size reduction is achieved on individual videos and not on the global flow. In other words, video compression does not fight against the number of videos that is created or consumed. Yet, this amount of video streamed will explode in the coming years [START_REF] Bross | Overview of the versatile video coding (vvc) standard and its applications[END_REF]. Second, compression gains are usually not exploited to decrease the required bandwidth but rather to increase the video resolution or to create new video usages (IoT, Virtual Reality, etc.) [START_REF] Bross | Overview of the versatile video coding (vvc) standard and its applications[END_REF]. Last, compression gains are achieved only at the expense of huge algorithm complexity. Encoding a video becomes more and more complex and thus requires, for each codec, more energy [START_REF] Pakdaman | Complexity analysis of next-generation vvc encoding and decoding[END_REF], [START_REF] Bossen | Vvc complexity and software implementation analysis[END_REF].

Another cause of video streaming gas emissions is precisely the complexity of the operations all along the transmission chain (including the video coding algorithms as mentioned just above). Intensive research efforts have been made to decrease this complexity and thus the consumed energy. Without sake of completeness, works have been conducted to improve energy efficiency of 5G/6G [START_REF] Usama | A survey on recent trends and open issues in energy efficiency of 5g[END_REF]- [START_REF] Andersson | Improving energy performance in 5g networks and beyond[END_REF], rooting protocole [START_REF] Hamidouche | Low energy-efficient clustering and routing based on genetic algorithm in wsns[END_REF], [START_REF] Dogra | Energy-efficient routing protocol for next-generation application in the internet of things and wireless sensor networks[END_REF], video encoding/decoding [START_REF] Amestoy | Tunable vvc frame partitioning based on lightweight machine learning[END_REF]- [START_REF] Hamidouche | Hardware-friendly multiple transform selection module for the vvc standard[END_REF], display [START_REF] Le Meur | Energy-aware images: Quality of experience vs energy reduction[END_REF].

Even though these research results are crucial and make the energy expenditure more optimized, we demonstrate in this paper that this is insufficient to achieve the goal set by the IPCC. We build a simple model inspired by [START_REF] Bol | Moore's law and ict innovation in the anthropocene[END_REF] and derived from the Kaya identity. This model enables to compare the order of magnitude of the energy reduction on one side and the digital affluence on the other side. Results show that even drastic energy reduction cannot cope with the forecasted video data flow explosion.

II. METHODOLOGY

In [START_REF] Kaya | Environment, energy, and economy: strategies for sustainability[END_REF], Kaya et al. proposed a model stating that the global carbon footprint can be expressed as the product of four factors: population, affluence, expressed in Gross Domestic Product (GDP), energy intensity (per unit of GDP), and carbon intensity (GHG emissions per unit of energy consumed). Even though the limitations of such simple model have been largely discussed, this formulation enables to express the carbon footprint with readily available data, highlighting at the same time the elements on which one could act to reduce emissions. This model as been employed by the IPCC for its report in 2000 [START_REF] Nakicenovic | Special report on emissions scenarios[END_REF].

Recently, Bol et al. has adapted this model to evaluate the Information and Communication Technology (ICT) carbon footprint evolution [START_REF] Bol | Moore's law and ict innovation in the anthropocene[END_REF]. In their model, the economic factors have been avoided, and the four above factors have been reformulated such that they focus on the ICT sector. In our paper, we go one step further by restricting the model to the video streaming emissions evaluation. The proposed formula is illustrated in Fig. 1. In this formula, the GHG emissions of video streaming η is expressed as:

η = N • A • E • C (1)
The population is denoted by N and corresponds to the amount of people who are equipped with a device able to create or watch a video online (e.g., mobile, computer, tablet).

The "video affluence" per user is denoted by A. It corresponds to the amount of video data streamed by user, expressed in GB.

The video streaming energy intensity E corresponds to the energy needed to stream 1 GB of video data, expressed in kWh. This is the factor on which video streaming energy optimization can operate.

The carbon intensity C indicates the amount of GHG emitted for one kWh of consumed energy. This simple model is not intended to be used to estimate the magnitude of the global emissions, but rather their evolution over time. Concretely, we define α N , α A , α E and α C the evolution coefficients for each of the factors between time t and t -1, i.e., the Compound Annual Growth Rate (CAGR). The global evolution can be now expressed recursively as:

η(t) = N (t) • A(t) • E(t) • C(t), (2)
where

N (t) = α N N (t -1) (3) A(t) = α A A(t -1) (4) E(t) = α E E(t -1) (5) C(t) = α C C(t -1). (6)

III. SCENARIOS AND RESULTS

We now simulate the emissions based on the model in [START_REF] Efoui-Hess | Climate crisis: The unsustainable use of online video[END_REF]. For the number of equipped users N , we take the evolution of the number of internet users. In [START_REF] Cisco | Cisco annual internet report[END_REF], the CAGR of internet users is estimated at 6% between 2018 and 2023. For the video affluence A, we take the forecasted CAGR of the Mobile data traffic per smartphone between 2021 and 2027 estimated at 24% [START_REF] Jonsson | Ericsson mobility report[END_REF]. For the carbon intensity factor C, we take an optimistic CAGR of -1% [START_REF]Co2 emissions in 2022[END_REF].

For the energy intensity E, we consider different scenarios, from the most pessimistic to the most optimistic, as seen in Fig. 2.

S1 (no energy saving): α E = 1. This scenario is pessimistic in the sense that it does not consider the huge research effort dedicated to energy efficency optimization in video streaming. Another way to see it, is that each gain of energy is annihilated by another increase in energy. This is for example what may happen if video compression gains are achieved with extremely high encoding complexity. We can see in Fig. 2(a) that this worst case scenario leads to an explosion of the GHG emissions (multiplied by ten between 2019 and 2030).

S2 (realistic energy saving): α E = 0.9. This take into account the energy optimization initiatives, by considering a realistic energy decrease. Even though it is hard to consider what is realistic or not, this CAGR enables in 10 years to decrease the energy by more than 50%. This is typically the kind of gains that are achieved for video compression in a same period of time [START_REF] Zhang | An overview of emerging video coding standards[END_REF]. Unfortunately, this scenario depicted in Fig. 2(b) also leads to a GHG emissions explosion.

S3 (optimistic energy saving): α E = 0.8. In order to go further, we now consider a scenario that bet on (super) optimistic energy savings: every year the video streaming energy decreases by 20%. This means that in 2023, the energy necessary to stream one GB corresponds ton only 10% of the one in 2019. We can see in Fig. 2(c) that even with such gigantic gain, the GHG emissions is not stabilized and still slightly grows.

S3 (unrealistic energy saving): α E = 0.7. We have finally considered a scenario with an energy intensity suffiently decreasing to fulfill the Paris agreement (see Fig. 2(d)). It corresponds to the unrealistic decrease of 30% per year, which means that in 2023, the level of energy necessary to stream one GB corresponds to only 1% of the one in 2019. At this stage, we must emphasize the simplicity of the model. With the aforementioned curve, we do not intend to propose an accurate GHG emissions evaluation, as the complexity of the chain requires more evolved models [START_REF] Guennebaud | Assessing VoD pressure on network power consumption[END_REF]. Another very important limitation is that this model neglects the rebound effect (the fact that energy saving may accelerate the video data consumption explosion). However, this model still enables to put the energy efficiency optimization efforts in perspective with the growth of video usage per user. And the conclusion is clear: the video affluence (the amount of video data streamed per user) grows much too fast. This is the reason why, we explore in the next section, the case where video affluence is limited or decreases.

IV. LIMIT THE VIDEO AFFLUENCE

As seen in the previous section, optimizing the energy intensity E is not enough to achieve the GHG emissions decrease of Paris agreement, mostly because a too big increase of video affluence A. Let us now study scenarios in which the video affluence is limited, or even decreased. This cannot be achieved with technology improvement, and is only possible thanks to laws or regulations (as it is claimed by some actors [START_REF] Efoui-Hess | Climate crisis: The unsustainable use of online video[END_REF]).

In Fig. 3(a), we simulate the case where a decision have been taken to limit the video affluence at its nowadays level. In parallel, the realistic energy saving of scenario S2 are pursued.

In that case, the GHG emissions decreases, but not enough. In 2030, we don't even achieve the level of 2019's GHG emissions.

In Fig. 3(b), we go further, and we consider a regulation rule making the video affluence decrease by 10% every year. We can see that, with again the energy saving of scenario S2, the Paris agreement objectives is achieved in 2030. In this scenario, the amount of allowed video data streamed per user is decreased when compared to the one in 2019.

Once again, the simplicity of the model prevents us to use it as an accurate forecast. However, it shows that, if energy optimization is not sufficient by itself to decrease GHG emissions, it becomes interesting and crucial when completed by, for example, laws and regulations restraining the allowed amount of data per user.

V. CONCLUSION

In this paper, we have studied the effect of the numerous methods performing energy optimization over the video transmission pipeline. We have shown that the forecasted explosion of the amount of data per user annihilate such efforts. To achieve Paris agreement objectives (half emissions in 2030 when compared to 2019), the academic, industrial and politic actors must also consider laws and regulation to limit the video data consumption. (b) αA = 0.9 after 2023 Fig. 3. GHG emissions for the case when video affluence A is limited or is decreased. This simulation is run for the realistic scenario S2 of energy saving.

Fig. 1 .

 1 Fig. 1. Rough video streaming carbon footprint modeling inspired by the Kaya identity.

1 2019 9 2019 8 2019 7 Fig. 2 .

 19872 Fig. 2. Evolution of GHG emissions due to video streaming for different scenario of yearly energy saving α E .

  of users A: video usage (GB/user) E: energy intensity (kWh/GB) C: carbon intensity (GHG emission/kW) η: video streaming carbon footprint (GhG emissions) Paris agreement of users A: video usage (GB/user) E: energy intensity (kWh/GB) C: carbon intensity (GHG emission/kW) η: video streaming carbon footprint (GhG emissions) Paris agreement


Amine Bohi
email: amine.bohi@lsis.org

Dario Prandi
email: prandi@ceremade.dauphine.fr

Vincente Guis
email: guis@univ-tln.fr

Frédéric Bouchara
email: bouchara@univ-tln.fr

Jean-Paul P Gauthier
email: gauthier@univ-tln.fr

Fourier descriptors based on the structure of the human primary visual cortex with applications to object recognition

Keywords: Descriptor, Fourier transform, hexagonal grid, geometric transformations, support vector machine, object recognition

In this paper we propose a supervised object recognition method using new global features and inspired by the model of the human primary visual cortex V1 as the semidiscrete roto-translation group SE(2, N ) = Z N ⋊ R 2 . The proposed technique is based on generalized Fourier descriptors on the latter group, which are invariant to natural geometric transformations (rotations, translations). These descriptors are then used to feed an SVM classifier. We have tested our method against the COIL-100 image database and the ORL face database, and compared it with other techniques based on traditional descriptors, global and local. The obtained results have shown that our approach looks extremely efficient and stable to noise, in presence of which it outperforms the other techniques analyzed in the paper.

Introduction

Object recognition is a fundamental problem in computer vision and keeps attracting more and more at-tention nowadays. Its concepts have been applied in multiple fields, as manufacturing, surveillance system, optical character recognition, face recognition, etc.

Almost every object recognition algorithm proposed in the literature is based on the computation of certain features of the image, which allow to characterize the object depicted and to discriminate it from others. In particular, since objects can appear at different locations and with different sizes, it is desirable for such features to be invariant by translation, rotation and scale. These invariant features can be global, i.e. computed taking into account the whole image, or local, i.e. computed considering only neighborhoods of key-points in the image.

In this paper we focus on Fourier descriptors, an important class of global invariant features used since the seventies [START_REF] Granlund | Fourier Preprocessing for Hand Print Character Recognition[END_REF][START_REF] Zahn | Fourier descriptors for plane closed curves[END_REF] based on algebraic properties of the Fourier transform. In particular, inspired by some neurophysiological facts on the structure of the human primary visual cortex, we extend this theory to define features invariant to translation and rotation and we apply them for invariant object recognition in SVM context. These results are then compared with those obtained with another important class of global invariant features, the moment invariants (see Appendix A), used e.g. in [START_REF] Chou | Pavement distress classification using neural networks[END_REF][START_REF] Raja | Artificial neural networks based war scene classification using invariant moments and glcm features: A comparative study[END_REF][START_REF] Rajasekaran | Image recognition using simplified fuzzy artmap augmented with a moment based feature extractor[END_REF], and with two different local invariants. For more information on object recognition via local features we refer to [START_REF] Bay | Surf: Speeded up robust features[END_REF][START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF][START_REF] Ke | Pca-sift: A more distinctive representation for local image descriptors[END_REF][START_REF] Lowe | Distinctive image features from scaleinvariant keypoints[END_REF][START_REF] Mikolajczyk | A performance evaluation of local descriptors[END_REF][START_REF] Morel | Asift: A new framework for fully affine invariant image comparison[END_REF].

Our choice of a global approach is motivated by the better results obtained by these methods in presence of noise, luminance changes and other different alterations, with respect to algorithms based on local invariant features [START_REF] Choksuriwong | Comparative study of global invariant descriptors for object recognition[END_REF]. Indeed, under these conditions, key-points detectors used in the local approach produce key-points that are not relevant for the object recognition.

In the following we briefly introduce the theory of Fourier descriptors, before discussing the framework used in this paper and our contributions.

Fourier descriptors

The basic idea behind Fourier descriptors is that the action of an abelian locally compact group G on functions in L 2 (G) is much easier to treat at the level of their Fourier transforms. In the specific case of images, f, g ∈ L 2 (R 2 ), this is expressed by the well-known equivalence for the translation of a ∈ R 2 :

f (x) = g(x -a) ∀x ∈ R 2 ⇐⇒ f (λ) = e i a,λ ĝ(λ) ∀λ ∈ R 2 , (1)
where the Fourier transform is defined 1

by f (λ) = R 2 f (x) e -i λ,x dx, ∀λ ∈ R 2 .
In this setting, Fourier descriptors are quantities associated with functions of L 2 (R 2 ) that can be easily computed starting from their Fourier representation and that are invariant under the action of translations. Ideally, a Fourier descriptor should be complete, meaning that for any couple of images f, g ∈ L 2 (R 2 ) the equality of the Fourier descriptor is equivalent to the equality of f and g up to translations. Indeed, the lack of completeness could yield to problems in applications, notably to false positives in the classification.

However, a result as strong as completeness is usually out of reach and unnecessary for practical applications. In this case, one looks for Fourier descriptors that are at least weakly complete, meaning that they are complete on a sufficiently big subset of L 2 (R 2 ), usually either open and dense or at least residual, i.e. the intersection of countably many open and dense sets. This guarantees that the Fourier descriptor will correctly classify a sufficiently large class of images.

Various Fourier descriptors have been defined in the literature [START_REF] Granlund | Fourier Preprocessing for Hand Print Character Recognition[END_REF][START_REF] Kakarala | The bispectrum as a source of phasesensitive invariants for fourier descriptors: A grouptheoretic approach[END_REF][START_REF] Kuhl | Elliptic Fourier features of a closed contour[END_REF][START_REF] Smach | Generalized Fourier descriptors with applications to objects recognition in SVM context[END_REF][START_REF] Zahn | Fourier descriptors for plane closed curves[END_REF]. In this work we are mainly interested in the following two, whose invariance w.r.t. translations can be checked via [START_REF] Barut | Theory of group representations and applications[END_REF].

-Power-spectrum: the quantity PS f (λ) := | f (λ)| 2 for λ ∈ R 2 , which is the Fourier transform of the auto-correlation function

a f (x) := R 2
f (y) f (y + x) dy. 1 Here we use a non-unitary definition of the Fourier transform for future convenience in computations.

It is easy to show that the power-spectrum is not weakly complete, and indeed it is used in texture synthesis to identify the translation invariant Gaussian distribution of textures [START_REF] Galerne | Random phase textures: Theory and synthesis[END_REF].

-Bispectrum: an extension of the power-spectrum, it is the quantity BS f (λ 1 , λ 2 ) := f (λ 1 ) f (λ 2 ) f (λ 1 + λ 2 ), or equivalently the Fourier transform of the triple correlation, defined as

a 3,f (x 1 , x 2 ) := R 2
f (y) f (y + x 1 ) f (y + x 2 ) dy.

These descriptors are complete on compactly supported functions of L 2 (R 2 ) and are well established in statistical signal processing. See e.g. [START_REF] Dubnov | Polyspectra as measures of sound texture and timbre[END_REF], where they are applied to sound texture recognition.

These two Fourier descriptors can be easily generalized to functions on L 2 (G) of a locally compact abelian group G to obtain invariants under the action of G. This can be applied, for example, to 2D shape recognition. However, when working with images, these descriptors are unsatisfying. Indeed, they are invariant only under translations, and so cannot be used to classify images under the action of rotations.

Framework of the paper

In this paper, following a line of research started in [START_REF] Smach | Generalized Fourier descriptors with applications to objects recognition in SVM context[END_REF], we present a theoretical framework that allows us to build generalized Fourier descriptors which are invariant w.r.t. (semidiscrete) roto-translations of images. We exploit the following two facts:

-It is possible to define a natural generalization of the power spectrum and the bispectrum on noncommutative groups, as it has been done in [START_REF] Kakarala | The bispectrum as a source of phasesensitive invariants for fourier descriptors: A grouptheoretic approach[END_REF][START_REF] Smach | Generalized Fourier descriptors with applications to objects recognition in SVM context[END_REF]. -Contributions of some of the authors to a fairly recent model of the human primary visual cortex V1 [START_REF] Boscain | Hypoelliptic diffusion and human vision: a semidiscrete new twist on the petitot theory[END_REF][START_REF] Boscain | Image reconstruction via non-isotropic diffusion in Dubins / Reed -Shepp-like control systems[END_REF] have shown that the latter can be modeled as the semidiscrete group of roto-translations From these facts, a natural pipeline for invariant object recognition is the following:

SE(2, N ) = Z N ⋊ R
1. Given an image f ∈ L 2 (R 2 ) lift it to a cortical stim- ulus Lf ∈ L 2 (SE(2, N )).

Compute the generalized Fourier descriptors of Lf

on the non-commutative group SE(2, N ).

3. If the lift of another image g ∈ L2 (R 2 ) have the same Fourier descriptors as Lf , deduce that Lf ≈ Lg up to the action of SE(2, N ). 4. Thanks to the left-invariance and injectivity of the lift L, obtain that also f ≈ g up to the action of SE(2, N ).

This pipeline was already investigated in [START_REF] Smach | Generalized Fourier descriptors with applications to objects recognition in SVM context[END_REF], where the authors considered a non-left-invariant lift, the cyclic lift. For this lift they then proved a weak completeness result of the generalized bispectrum for images, represented as L 2 (R 2 ) functions with support inside a fixed compact set.

In this paper we consider the same question for leftinvariant lifts, where the situation turns out to be more complicated. In particular, as explained in the following section, to ensure the weak completeness we are led to consider "stronger" invariants than the generalized bispectrum. However, as observed in Remark 4, the actual computation of these stronger invariants on lifted images requires N times less computational time and space w.r.t. the computation of the generalized bispectrum of cyclically lifted images.

Contributions of the paper

Let K ⊂ R 2 be any compact set, representing the size of the images under consideration. According to the pipeline for object recognition introduced above, the weak completeness of the generalized Fourier descriptors on images can be proved in two steps:

1. Prove the completeness of the generalized Fourier descriptors on some residual set

G ⊂ L 2 (Z N × K) of cortical stimuli; 2. Prove that for some residual set R ⊂ L 2 (R 2 ) of images with support in K we have L(R) ⊂ G.
The first point is addressed in Theorem 1, where is identified an open and dense set G ⊂ L 2 (Z N × K) on which the combination of the generalized powerspectrum and bispectrum holds. This generalizes the result in [START_REF] Smach | Generalized Fourier descriptors with applications to objects recognition in SVM context[END_REF], where the same result was proved for a residual subset of the range of the cyclic lift.

Unfortunately, it turns out that for this set G and a left-invariant lift L there is no hope of finding a set R ⊂ L 2 (R 2 ) satisfying the second point above. We are then led to introduce stronger Fourier descriptors, the rotational power-spectrum and bispectrum, which are invariant only w.r.t. rotations. To solve this problem we preprocess images by centering them at their barycenter, a procedure that is essential also in [START_REF] Smach | Generalized Fourier descriptors with applications to objects recognition in SVM context[END_REF]. Theorem 2 then shows that the resulting invariants are complete for an open and dense set of functions in L 2 (K), for any compact K ⊂ R 2 . The proof of this completeness requires fine technical tools from harmonic analysis and the theory of circulant operators, and for this reason we only present a sketch of it, evidencing the technical difficulties. A complete proof will be presented in a forthcoming paper by the second and last authors.

Finally, in Theorem 3 we show that, under mild assumptions on the mother wavelet Ψ , to check the equality of all these Fourier descriptors it is sufficient to compute simple quantities computed from the 2D Fourier transform of the image. This allow for an efficient implementation on regular hexagonal grids. After using these descriptors to feed a SVM based classifier, we compare their performances with those of Hu and Zernike moments, the Fourier-Mellin transform and some wellknown local descriptors. To this purpose, we test them on two large databases: the COIL-100 2 object recognition database, composed of 7200 objects presenting rotation and scale changes [START_REF] Nene | Columbia object image library (coil-20)[END_REF], and the ORL3 face database, on which different human faces are subjects to several kind of variations.

Structure of the paper

The remainder of the paper is organized as follows. In Section 2, we present the features of a mathematical model of the primary visual cortex V 1 that are essential to our approach. In Section 3, we introduce some generalities on the Fourier Transform on the semidiscrete group of roto-translation SE(2, N ). In Section 4, we describe the natural generalization of the power-spectrum and the bispectrum on R 2 to SE(2, N ). We then prove the weak completeness result (Theorem 1) and show that under the chosen lift operator this does not imply weak completeness for images. Finally, we introduce the rotational power-spectrum and bispectrum and sketch the proof of the corresponding weak completeness result (Theorem 2) for images. We end this section with some result on the practical computation of these descriptors. In Section 5 we illustrate some numerical results where these descriptors are compared with those obtained via global descriptors such as Zernike moments, Hu moments, Fourier-Mellin transform, and local ones like the SIFT and HoG descriptors. Finally, we conclude with some practical suggestions in Section 6.

A mathematical model of the primary visual cortex V1

As mentioned in the introduction, the main novelty of our approach is its connection with a fairly recent model of the human primary visual cortex V1 due to Petitot and Citti-Sarti [START_REF] Citti | A cortical based model of perceptual completion in the roto-translation space[END_REF][START_REF] Petitot | Neurogéométrie de la vision -Modèles mathématiques et physiques des architectures fonctionnelles[END_REF] and our recent contributions [START_REF] Boscain | Anthropomorphic image reconstruction via hypoelliptic diffusion[END_REF][START_REF] Boscain | Hypoelliptic diffusion and human vision: a semidiscrete new twist on the petitot theory[END_REF][START_REF] Boscain | Image reconstruction via non-isotropic diffusion in Dubins / Reed -Shepp-like control systems[END_REF][START_REF] Prandi | Image processing in the semidiscrete group of rototranslations[END_REF]. The theory of orientation scores introduced in [START_REF] Duits | Left-invariant parabolic Evolutions on SE(2) and Contour Enhancement via Invertible Orientation Scores. Part I: linear left-invariant diffusion equations on SE(2)[END_REF]16] is also strongly connected with this work, in particular for its exploitation of left-invariant lift operators. We also mention [START_REF] Sifre | Rotation, scaling and deformation invariant scattering for texture discrimination[END_REF], where image invariants based on the structure of the roto-translation group SE(2) are introduced for textures. In this section we present the features of this model that are essential to our approach.

Since it is well-known [START_REF] Hubel | Receptive fields of single neurones in the cat's striate cortex[END_REF] that neurons in V1 are sensitive not only to positions in the visual field but also to local orientations and that it is reasonable to assume these orientations to be finite, in [START_REF] Boscain | Hypoelliptic diffusion and human vision: a semidiscrete new twist on the petitot theory[END_REF] V1 has been modeled as the semidiscrete group of roto-translations

SE(2, N ) = Z N ⋊ R 2 for some even N ∈ N. Letting R k be the rotation of 2π/k, the (non-commutative) group operation of SE(2, N ) is (x, k)(y, r) = (x + R k y, k + r).
Here, we are implicitly identifying k + r with k + r mod N .

Visual stimuli f ∈ L 2 (R 2 ) are assumed to be lifted to activation patterns in L 2 (SE(2, N )) by a lift operator L : L 2 (R 2 ) → L 2 (SE(2, N )). Motivated by neurophysiological evidence, we then assume that (H) The lift operator L is linear and is defined as

Lf (x, k) := R 2 f (y) Ψ (R -k (y -x)) dy, (2)
for a given mother wavelet Ψ ∈ L 2 (R 2 ) such that L is injective and bounded.

Remark 1

This assumption means that the lift operator under consideration is the wavelet transform w.r.t. Ψ (See, e.g., [START_REF] Führ | Continuous wavelet transforms from semidirect products : Cyclic representations and Plancherel measure[END_REF]). The fact that L be injective and bounded is then equivalent to the fact that the mother wavelet Ψ is weakly admissible, i.e., is such that the map

λ ∈ R 2 → k∈ZN | Ψ (R -k λ)| 2
is strictly positive and essentially bounded.

As a consequence of the above assumption, the lift operation L is left-invariant w.r.t. to the action of SE(2, N ). Namely,

Λ(x, k) • L = L • π(x, k). (3)
Here Λ and π are the actions of SE(2, N ) on L 2 (SE(2, N )) and L 2 (R 2 ) respectively. That is,

[Λ(x, k)ϕ](y, r) = ϕ (x, k) -1 (y, r) = ϕ(R -k (y -x), k + r), [π(x, k)f ](y) = f (x, k) -1 y = f (R -k (y -x)).
Formula (3) can be seen as a semidiscrete version of the shift-twist symmetry [START_REF] Bressloff | Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex[END_REF].

The main observation for our purposes is that (3) means that two images f and g ∈ L 2 (R 2 ) can be deduced via roto-translation (i.e., f = π(x, k)g for some (x, k) ∈ SE(2, N )) if and only if their lifts can be deduced via Λ(x, k).

Preliminaries on non-commutative harmonic analysis

In this section we introduce some generalities on the (non-commutative) Fourier transform on SE(2, N ), an essential tool to define and compute the Fourier descriptors we are interested in. We refer to [START_REF] Barut | Theory of group representations and applications[END_REF][START_REF] Hewitt | Abstract harmonic analysis[END_REF] for a general introduction to the topic.

Since SE(2, N ) is a non-commutative unimodular group, the Fourier transform of ϕ ∈ L 2 (SE(2, N )) is an operator associating to each (continuous) irreducible unitary representations T λ of SE(2, N ) some Hilbert-Schmidt operator on the Hilbert space where T λ acts. Here, λ is an index taking values in the dual object of SE(2, N ), which is denoted by SE(2, N ) and is the set of equivalence classes of irreducible unitary representations.

The set of irreducible representations of a semi-direct product group can be obtained via Mackey's machinery (see, e.g., [1, Ch. 17.1, Theorems 4 and 5]). Accordingly, SE(2, N ) is parametrized by the orbits of the (contragredient) action of rotations {R k } k∈ZN on R 2 , i.e., by the slice S ⊂ R 2 which in polar coordinates is (0, +∞)× [0, 2π/N ). Additionally, corresponding to the origin, there are the characters of Z N . Namely, to each λ ∈ S corresponds the representation T λ acting on C N via

T λ (x, k)v = diag h (e i λ,R h x ) • S k v = e i λ,R h x v h+k N -1 h=0 , (4)
where we denoted by diag h v h the diagonal matrix of diagonal v ∈ C N and by S the shift operator (Sv) j = v j+1 , so that (S k v) j = v j+k . On the other hand, to each k ∈ Z N corresponds the representation on C given by z → e i 2πk N z. Since it is possible to show that to invert the Fourier transform it is enough to consider only the representations parametrized by S, we will henceforth ignore the Z N part of the dual.

Finally, the matrix-valued Fourier coefficient of a function

ϕ ∈ L 2 (SE(2, N )) ∩ L 1 (SE(2, N )) for λ ∈ SE(2, N ) is φ(T λ ) = SE(2,N ) ϕ(a) T λ (a -1 ) da, ( 5
)
where da is the Haar measure 4 of SE(2, N ). This is essentially the same formula for the Fourier transform on R 2 , which is a scalar and is obtained using the representations λ(x) = e i x,λ acting on C.

Straightforward computations yield φ(T λ ) i,j = F (ϕ(•, i -j))(R -j λ), (6)
where we let F denote the Fourier transform on R 2 . As usual, the definition of the Fourier transform can be extended to the whole L 2 (SE(2, N )) by density arguments. Then, there exists a unique measure on SE(2, N ), the Plancherel measure, supported on S where it coincides with the restriction of the Lebesgue measure of R 2 , such that the Fourier transform is an isometry between L 2 (SE(2, N )) and L 2 ( SE(2, N )). In particular, the following inversion formula holds

ϕ(x, k) = S Tr φ(T λ ) • T λ (x, k) dλ.
The fundamental property of the non-commutative Fourier transform, generalizing [START_REF] Barut | Theory of group representations and applications[END_REF], is that for all ϕ, η ∈ L 2 (SE(2, N )) and a ∈ SE(2, N ) it holds

ϕ(x, k) = [Λ(a)η](x, k) ∀(x, k) ∈ SE(2, N ) ⇐⇒ φ(T ) = η(T ) • T -1 (a) ∀T ∈ SE(2, N ). (7)
Namely, ϕ, η can be deduced via the action of SE(2, N ) if and only if their Fourier transforms at a representation T can be deduced via multiplication by T (a).

Remark 2

The fact that the Fourier transform in (5) be matrix-valued is a direct consequence of SE(2, N ) being a Moore group, that is, that all the T λ act on finite-dimensional spaces. This is not true for the rototranslation group SE(2). As a consequence, the Fourier transform on SE(2), takes values not in the finite dimensional space of complex N × N matrices, but in the infinite dimensional space of operators over L 2 (S 1 ). This is indeed the main theoretical advantage of considering SE(2, N ). 4 That is, up to a multiplicative constant, the only left and right invariant measure on SE(2, N ). One can check that

SE(2,N) ϕ(a) da = N-1 k=0 R 2 ϕ(x, k) dx.

Decomposition of tensor product representations

Proofs of Section 4, will use a well-known fact on tensor product representations: the Induction-Reduction Theorem. (See [START_REF] Barut | Theory of group representations and applications[END_REF]). This theorem allows to decompose the tensor products of representations T λ1 ⊗ T λ2 , acting on C N ⊗C N ∼ = C N ×N , to an equivalent representation acting on k∈ZN C N , which is a block-diagonal operator whose block elements are of the form T λ1+R k λ2 . Moreover, the linear transformation realizing the equivalence is explicit.

To avoid confusion, we will henceforth denote components of vectors v ∈ C N as v(0), . . . , v(N -1), elements of k∈ZN C N as (w k ) k∈ZN where w k ∈ C N , and the components of vectors v ∈ C N ⊗ C N as v(k, h) for k, h = 0, . . . , N -1. We also remark that linear operators B on k∈ZN C N can be decomposed as B = (B k,h ) k,h∈ZN , where each B k,h is an N × N complex matrix. Namely, we have

B(w k ) k∈ZN = N -1 h=0 B k,h w h k∈ZN . (8)
Then, exploiting the commutation of the Fourier transform with equivalences of representation, the Induction-Reduction Theorem implies that for every ϕ ∈ L 2 (SE(2, N )) and any λ 1 , λ 2 ∈ S it holds

A • φ(T λ1 ⊗ T λ2 ) • A -1 = k∈ZN φ(T λ1+R k λ2 ). (9)
Here, A :

C N ⊗ C N → k∈ZN C N is given by (Av) k (h) = (A k v)(h) = v(h, h -k), ∀v ∈ C N ⊗ C N . ( 10
)
4 Fourier descriptor on SE(2, N )

In the following sections we introduce and study the Fourier descriptors on the group SE(2, N ). As already mentioned, proving a general completeness result is essentially hopeless, and we will content ourselves to prove the weak completeness. Let K ⊂ R 2 be a compact set. In the following we will be mainly concerned with functions that are compactly supported either in K or in K ×Z N ⊂ SE(2, N ).

Generalized Fourier descriptors

Following [START_REF] Smach | Generalized Fourier descriptors with applications to objects recognition in SVM context[END_REF], the power spectrum and the bispectrum on R 2 can be generalized to SE(2, N ) as follows.

Definition 1 The generalized power-spectrum and bispectrum of ϕ ∈ L 2 (SE(2, N )) are the collections of matrices for any λ, λ 1 , λ 2 ∈ S,

PS ϕ (λ) := ϕ(T λ ) • ϕ(T λ ) * BS ϕ (λ 1 , λ 2 ) := ϕ(T λ1 ) ⊗ ϕ(T λ2 ) • ϕ(T λ1 ⊗ T λ2 ) * .
The next result generalizes, with a simplified proof, the result presented in [START_REF] Smach | Generalized Fourier descriptors with applications to objects recognition in SVM context[END_REF]. Let us mention that this result is indeed true in a more general setting, as it will be shown in a forthcoming paper by Prandi and Gauthier.

Theorem 1 Let K ⊂ R 2 be a compact. The generalized power-spectrum and bispectrum are weakly complete on L 2 (Z N × K). In particular, they discriminate on the open and dense set G ⊂ L 2 (Z N ×K) of functions ϕ supported in Z N × K and whose Fourier transform φ(T λ ) is invertible for an open and dense set of λ's.

That is, ϕ 1 , ϕ 2 ∈ G are such that PS ϕ1 = PS ϕ2 and BS ϕ1 = BS ϕ2 if and only if ϕ 1 = Λ(x, k)ϕ 2 for some (x, k) ∈ SE(2, N ).
Proof The fact that G is open and dense is proved in Lemma 1 in Appendix B. Let ϕ, η ∈ G be such that P S ϕ1 = P S ϕ2 and BS ϕ = BS η . The equality of the generalized bispectrum implies that the set of λ's where φ(T λ ) and η(T λ ) fail to be invertible is the same. We will denote it by I and let

U (T λ ) := φ(T λ ) -1 η(T λ ) ∀λ ∈ I.
In order to complete the proof of the statement, we will prove that U (T λ ) can be defined for all λ's in R 2 and, moreover, that U (T λ ) = T λ (a) for some a ∈ SE(2, N ). Indeed, by [START_REF] Choksuriwong | Comparative study of global invariant descriptors for object recognition[END_REF] this will readily implies that ϕ = Λ(a)η as announced.

We claim that U (T λ ) is unitary for all λ ∈ I. Indeed, by the equality of the generalized power-spectrum we have

U (T λ ) * U (T λ ) = η(T λ ) * PS ϕ (λ)η(T λ ) = I.
Observe that the equality of the generalized bispectrum and the definition of U , imply that for all

λ 1 , λ 2 ∈ I it holds BS ϕ (λ 1 , λ 2 ) = φ(T λ1 ) ⊗ φ(T λ2 ) • U (T λ1 ) ⊗ U (T λ2 ) • η(T λ1 ⊗ T λ2 ) * .
By the invertibility of φ(T λ1 )⊗ φ(T λ2 ) and the unitarity of U , this yields

φ(T λ1 ⊗ T λ2 ) • U (T λ1 ) ⊗ U (T λ2 ) = η(T λ1 ⊗ T λ2 ). (11)
The announced result is then a consequence of the following three facts, which are proved in Appendix B.

U (T λ ) = T λ (a). ⊓ ⊔
An immediate corollary is the following.

Corollary 1 Let L : L 2 (R 2 ) → L 2 (SE(2, N
)) be an injective lift operator (not necessarily satisfying (2)).

Assume that there exists a residual set R ⊂ L 2 (R 2 ) such that L(R) ∩ G is residual. Then, the generalized power-spectrum and bispectrum are weakly complete on

L 2 (R 2 ). Namely, for any f, g ∈ R it holds that BS Lf = BS Lg if and only if f = π(x, k)g for some (x, k) ∈ SE(2, N ).
Remark 3 In [START_REF] Smach | Generalized Fourier descriptors with applications to objects recognition in SVM context[END_REF], the authors applied their version of Theorem 1 to a non-left-invariant lift L, called cyclic lift. Indeed, for this cyclic lift, when N is odd, it is possible to prove that, for any compact K ⊂ R 2 , there exists a residual set R ⊂ L 2 (K) satisfying the assumptions of Corollary 1.

Unfortunately, Corollary 1 can never be applied to lifts of the form (2). In fact, as proved in Appendix C, letting ω f (λ

) := ( f (R -k λ)) N -1 k=0 ∈ C N , we have that Lf (T λ ) = ω Ψ (λ) * ⊗ ω f (λ) * , (12)
where for v, w ∈ C N , we let v * = (v k ) k and (v ⊗w) k,h = v k w h , so that (v ⊗ w)u = w, u v for all u ∈ C N . This immediately implies that rank Lf (T λ ) ≤ 1 and hence that range L ∩ G = ∅ whenever N > 1.

Rotational Fourier descriptors

To bypass the difficulty posed by the non-invertibility of the Fourier transform for lifted functions, we are led to consider the following stronger descriptors.

Definition 2

The rotational power-spectrum and bispectrum of ϕ ∈ L 2 (SE(2, N )) are the collections of matrices, for any λ, λ 1 , λ 2 ∈ S and

h ∈ Z N , RPS ϕ(λ, h) := ϕ(T R h λ ) • ϕ(T λ ) * RBS ϕ (λ 1 , λ 2 , h) := ϕ(T R h λ1 ) ⊗ ϕ(T λ2 ) • ϕ(T λ1 ⊗ T λ2 ) * .
As already mentioned in the introduction, the rotational descriptors are invariant only under the action of Z N ⊂ SE(2, N ) but not under translations. To avoid this problem, let us fix a compact K ⊂ R 2 and consider the set A ⊂ L 2 (R 2 ) of functions compactly supported in K, with non-zero average 5 . Observe that this is an open and dense subset of L 2 (K). We can then define the barycenter c f ∈ R 2 of f ∈ A as

c f = 1 avg f R 2 x 1 f (x) dx, R 2 x 2 f (x) dx ,
and the centering operator Φ : A → A as

Φf (x) := f (x -c f ). ( 13
)
Then, considering the centered lift L c = L • Φ, we have that L c f = L c g if and only if g is a translate of f . In particular,

L c f = Λ(0, k)L c g ⇐⇒ f = π(x, k)g for some x ∈ R 2 .
Let us consider the following set of functions.

Definition 3 Let R ⊂ L 2 (R 2
) be the set of real-valued functions f supported in K, such that f (λ) = 0 for a.e. λ ∈ R 2 and the family

Ω f = {S k ω f (λ)} N -1 k=0 is a basis for C N , if N is odd, or, if N is even, for X = {v ∈ C N | v(h) = v(h + N/2) ∀h ∈ Z N }.
The dependence of this definition on the parity of N comes from the well-known fact that f (λ) = f (-λ). Indeed, for N even, this implies that S k ω f (λ) ∈ X for any k ∈ Z N . As such, there is no hope for the family Ω f to generate the whole C N .

Finally, we have the following Theorem.

Theorem 2 For any compact K ⊂ R 2 , if the mother wavelet Ψ ∈ R, the rotational power-spectrum and bispectrum are weakly complete on L 2 (K) ∩ A. Namely, the set R is open and dense in L 2 (K) and for any f, g ∈ R ∩ A it holds that RPS Lcf = RPS Lcg and

RBS Lcf = RBS Lcg if and only if f = π(x, k)g for some (x, k) ∈ SE(2, N ).
Here, we content ourselves to present only a sketch of the proof of this result for the case N odd. The parity of N does not introduce essential problems, up to exploit the fact that range Lf (T λ ) ⊂ X for all f ∈ R and λ ∈ S and that the equivalence A of the Induction-Reduction Theorem quotients nicely to an equivalence between X × X and k∈ZN X . However, in order to prove the key technical point [START_REF] Dubnov | Polyspectra as measures of sound texture and timbre[END_REF] we need a much finer study of the properties of circulant operators, which is outside the scope of this work and we defer to a forthcoming paper by Prandi and Gauthier. 5 Recall that the average of f : R 2 → R is avg f = R 2 f (x) dx, which is always well-defined for L 2 (R 2 ) functions with compact support.

Proof (Sketch in the case N odd) The fact that R is open and dense in L 2 (K) follows from the same arguments in Lemma 1.

Let Circ v be the circulant matrix associated with v, that is, Circ v = [v, Sv, . . . , S N -1 v]. Then the condition on Ω f for f ∈ R is equivalent to the invertibility of Circ ω f (λ) for an open and dense set of λ's. By the properties of the Fourier transform on R 2 w.r.t. translations it follows that

ω Φf (λ) = diag k e -i λ,R k c f ω f (λ).
This entails that Circ ω f (λ) is invertible if and only if Circ ω Φf (λ) is. Hence, the statement is equivalent to the fact that for any couple f, g ∈ R we have

RBS Lf = RBS Lg if and only if f = R k g for some k ∈ Z N .
The proof is similar to the one of Theorem 1, but with additional technical difficulties. Let I be the set where Circ ω f (λ) and Circ ω g (λ) are invertible. By assumption I is open and dense. To overcome the noninvertibility of Lf in the definition the candidate intertwining representation U , we exploit the invertibility of the circulant matrices Circ ω f (λ) and Circ ω g (λ) on an open and dense set. Namely, for any λ ∈ I we let

U (T λ ) * := Circ ω g (λ) (Circ ω f (λ)) -1 .
By definition, U (T λ ) is circulant and U (T λ ) * S k ω f (λ) = S k ω g (λ) for any k ∈ Z N . Moreover, by [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF], this is equivalent to

Lf (T R k λ ) U (T λ ) = Lg(T R k λ ), ∀k ∈ Z N .
In particular, λ → U (T λ ) is constant on orbits {R k λ} k∈ZN . Finally, U (T λ ) is unitary as a consequence, e.g., of Theorem 3.

The main difficulty in the proof is now to derive the equivalent of identity [START_REF] Citti | A cortical based model of perceptual completion in the roto-translation space[END_REF], that is, that for an open and dense set of couples (λ 1 , λ 2 ) we have

Lf (T R k λ1 ⊗ T R k λ2 ) U (T λ1 ) ⊗ U (T λ2 ) = Lg(T R k λ1 ⊗ T R k λ2 ), ∀k ∈ Z N . ( 14
)
As already mentioned, the proof of this identity requires a deep use of properties of circulant operators, which is outside the scope of this paper. We thus defer it to a forthcoming paper.

Once ( 14) is known, the statement follows applying the same arguments as those in Theorem 1. Namely, 1. The function λ → U (T λ ) is continuous on I. This can be done via the same arguments as in Lemma 2. 2. The function λ → U (T λ ) can be extended to a continuous function on S still satisfying [START_REF] Dubnov | Polyspectra as measures of sound texture and timbre[END_REF]. This can be done exactly as in Lemma 3.

3. There exists k ∈ Z N such that U (T λ ) = T λ (0, k). This is proved following Lemma 4. Indeed, the fact that now λ → U (T λ ) is constant on the orbits {R k λ} k∈ZN implies that the ϕ k 's obtained there have to be independent of k. Since ϕ k (λ) = e i R k x0,λ for some x 0 ∈ R 2 , this implies that x 0 = 0 and hence ϕ k ≡ 0.

Obviously, this proves that U (T λ ) = S k = T λ (0, k), for some k ∈ Z N . ⊓ ⊔

Practical computation of the Fourier descriptors

Here, we present some explicit formulae for the computation of the Fourier descriptors presented in this section.

In the following, we show that, under some assumptions on the mother wavelet Ψ , the concrete computation of the generalized power-spectrum and bispectrum and of their rotational counterparts, depend only on the 2D Fourier transform of f . Theorem 3 Assume that the mother wavelet Ψ ∈ R. Then:

-For any f ∈ R, the generalized power-spectrum and bispectrum of Lf are respectively determined by the quantities, for a.e. λ, λ 1 , λ 2 ∈ S,

I λ 1 (f ) = ω f (λ) 2 = N -1 k=0 | f (R -k λ)| 2 I λ1,λ2 1 (f ) = ω f (λ 1 ) ⊙ ω f (λ 2 ), ω f (λ 1 + λ 2 ) = N -1 k=0 f (R -k λ 1 ) f (R -k λ 2 ) f (R -k (λ 1 + λ 2 )).
-For any f ∈ A ∩ R, the rotational power-spectrum and bispectrum of L c f are respectively determined by the quantities, for a.e. λ, λ 1 , λ 2 ∈ S and h ∈ Z N ,

I λ,h 2 (f ) = ω Φf (R h λ), ω Φf (λ) = N -1 k=0 f (R -k+h λ) f (R -k λ), I λ1,λ2,h 2 (f ) = ω Φf (R h λ 1 ) ⊙ ω Φf (λ 2 ), ω Φf (λ 1 + λ 2 ) = N -1 k=0 f (R -k+h λ 1 ) f (R -k λ 2 ) f (R -k (λ 1 + λ 2 )).
Here, Φ : A → A is the centering operator defined in [START_REF] Derrode | Robust and efficient fouriermellin transform approximations for invariant grey-level image description and reconstruction[END_REF].

Remark 4 Theorem 3 shows in particular that the result of Theorem 2 is indeed stronger than the completeness result for the generalized bispectrum of the cyclic lift obtained in [START_REF] Smach | Generalized Fourier descriptors with applications to objects recognition in SVM context[END_REF]. Indeed, in that work is proved that the latter (for odd N ) is determined exactly by the quantities, for a.e. λ 1 , λ 2 ∈ S and h, k ∈ Z N , Ĩλ1,λ2,k,h

2 = ω φf (R h λ 1 ) ⊙ ω φf (R k λ 2 ), ω φf (λ 1 + R h+k λ 2 ) .
In particular, for each λ 1 , λ 2 ∈ S one has to compute N times more quantities than those for the rotational bispectrum.

As a corollary of Theorem 3 we show that, in order to compare the power-spectra and bispectra, it is usually enough to compare only the latter.

Corollary 2 Let Ψ ∈ R and f, g ∈ R ∩ A. Then, if
Lf and Lg have the same generalized (resp. rotational) bispectrum, they have also the same generalized (resp. rotational) power-spectrum.

Proof We only prove the result for the rotational descriptors. In order to prove the one for the generalized descriptors, it will be enough to fix h = 0 in the following. By Theorem 3 it is enough to show that whenever I λ1,λ2,h 2 (f ) = I λ1,λ2,h h (g) for a.e. λ 1 , λ 2 ∈ S and any h ∈ Z N , then I λ,h 2 (f ) = I λ,h 1 (g) for a.e. λ ∈ S and any h ∈ Z N . We start by observing that by the Paley-Wiener Theorem all these quantities are analytic, since f and g are compactly supported. Moreover, lim λ1,λ2↓0

I λ1,λ2,h 2 (f ) = N f (0)| f (0)| 2 = N avg(f ) 3 ,
and the same is true for g. Thus, avg(f ) = avg(g). Finally, the result follows observing that

lim λ2↓0 I λ1,λ2,h 2 (f ) = avg(f ) I λ1,h 2 (f ).⊓ ⊔

Experimental results

The goal of this section is to evaluate the performance of the invariant Fourier descriptors defined in the previous section on a large image database for object recognition. In addition to the generalized power-spectrum (PS) and bispectrum (BS) and the rotational powerspectrum (RPS) and bispectrum (RBS), we also consider the combination of the RPS and BS descriptors. Indeed, combining these two descriptors seems to be a good compromise between the theoretical result of completeness given by Theorem 2, which only holds for the RBS, and computational demands, as the results on the COIL-100 database will show.

After showing how to efficiently compute these descriptors and presenting the image data set, we analyze some experimental results. In order to estimate the features capabilities, we use a support vector machine (SVM) [START_REF] Vapnik | Statistical learning theory[END_REF] as supervised classification method. The recognition performances of the different descriptors regarding invariance to rotation, discrimination capability and robustness against noise are compared.

Implementation

As proved in Theorem 3, the equality of the Fourier descriptors we introduced does not depend on the choice of the mother wavelet Ψ . Accordingly, in our implementation we only computed the quantities introduced in Theorem 3, whose complexity is reduced to the efficient computation of the vector ω f (λ), for a given λ ∈ S. We recall that this vector is obtained by evaluating the Fourier transform of f on the orbit of λ under the action of discrete rotations R -k for k ∈ Z N .

Let us remark that, although in our implementation we chose this approach, in principle fixing a specific mother wavelet could be useful to appropriately weight descriptors depending on the associated frequencies. Indeed, preliminary tests with a Gabor mother wavelet (which can be easily shown to be in R) showed slightly better results at a bigger computational cost.

For the implementation we chose to consider N = 6 and to work with images composed of hexagonal pixels. There are two reasons for this choice:

-It is well-known that retinal cells are distributed in a hexagonal grid, and thus it is reasonable to assume that cortical activations reflect this fact. -Hexagonal grids are invariant under the action of Z 6 and discretized translations, which is the most we can get in the line of the invariance w.r.t. SE [START_REF] Bay | Surf: Speeded up robust features[END_REF][START_REF] Bressloff | Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex[END_REF]. Indeed, apart from the hexagonal lattice, the only other lattices on R 2 which are invariant by some Z N and appropriate discrete translations are obtained with N = 2, 3, 4.

The different steps of computation of the descriptors 6 are described in Figure 1 and given as follow:

1. The input image is converted to grayscale mode, the Fourier transform is computed via FFT, and the zero-frequency component is shifted to the center of the spectrum (Fig 1.

Test protocol

We use the Fourier descriptors to feed an SVM classifier, via the MATLAB Statistics and Machine Learning Toolbox, applying it on a database of 7200 objects extracted from the Columbia Object Image Library (COIL-100) and a database of 400 faces extracted from ORL face database. Finally, we compare the results obtained with those obtained using traditional descriptors.

The result of the training step consists of the set of support vectors determined by the SVM based method. During the decision step, the classifier computes the Fourier descriptors and the model determined during the training step is used to perform the SVM decision. The output is the image class.

For COIL-100 database, two cases are studied: a case without noise and another with noise. In the first one, tests have been performed using 75% of the COIL-100 database images for training and 25% for testing.In the second one, we have used a learning data-set composed of all the 7200 images (100 objects with 72 views) without noise and a testing data-set composed of 15 randomly selected views per object to which an additive Gaussian noise with S d of 5, 10 and 20 was added. (See Fig. 4).

We evaluate separately the recognition rate obtained using the four previous invariant descriptors and the combination of the RPS & BS invariants to test their complementarity. Then, we compare their performance with the Hu's moments (HM), the Zernike's moments (ZM), the Fourier-Mellin transform (FM), described in Appendix A, and the local SIFT and HOG descriptors [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] whose performance under the same conditions has been tested in [START_REF] Choksuriwong | Comparative study of global invariant descriptors for object recognition[END_REF], Since we use the RBF kernel in the SVM classification process, this depends on the kernel size σ. The results presented here are obtained by choosing empirically the value σ opt that provided maximum recognition rate.

Experiments

The performances of the different invariant descriptors are analyzed with respect to the recognition rate given a learning set. Hence, for a given ratio, the learning and testing sets have been built by splitting randomly all examples. Then, due to randomness of this procedure, multiple trials have been performed with different random draws of the learning and testing set. In the case of an added noise, since as mentioned before the learning set is comprised of all images, this procedure is applied only to the testing set.

The parameters of our experiments are the following:

1. The learning set c i corresponding to the values of an descriptor computed on an image from the database; 2. The classes ĉi ∈ {1, 100} corresponding to the object class. 3. Algorithm performance: the efficiency is given through a percentage of the well recognized objects composing the testing set. 4. Number of random trials: fixed to 5. 5. Kernel K: a Gaussian kernel of bandwidth σ is chosen

K(x, y) = e -x-y 2 2σ 2
x and y correspond to the descriptors vectors of objects.

For solving a multi-class problem, the two most popular approaches are the one-against-all (OAA) method and the one-against-one (OAO) method [START_REF] Milgram | One Against One" or "One Against All": Which One is Better for Handwriting Recognition with SVMs?[END_REF]. For our purpose, we chose an OAO SVM because it is substantially faster to train and seems preferable for problems with a very large number of classes.

COIL-100 databases

The Columbia Object Image Library (COIL-100, Fig. 2) is a database of color images of 100 different objects, where 72 images of each object were taken at pose intervals of 5 • .

Classification performance

Table 2 presents results obtained testing our object recognition method with the COIL-100 database. The best results were achieved using the local SIFT descriptor. The RBS comes in the second place and the local HOG features come third. Indeed it has been demonstrated in the literature, these local methods currently give the best results. However, if noise is added on the image, the use of global approach is better than the use of local ones. The main reason is that the key-points detector used in the local method produce in these cases many key-points that are nor relevant for object recognition. This will be shown in the next subsection.

In Figure 3 we present the recognition rate as a function of the size of the training set. As expect, this is an increasing function and we remark that the RBS and Fig. 3 Classification rate for different size of the training database. The test results for ZM, HM, FM, and SIFT are taken from [START_REF] Choksuriwong | Comparative study of global invariant descriptors for object recognition[END_REF].

the combination of the RPS and the BS give better results than the other global invariant descriptors.

Robustness against noise

Also in this case, test results for ZM, HM, FM, and SIFT are taken from [START_REF] Choksuriwong | Comparative study of global invariant descriptors for object recognition[END_REF].

Results presented in Table 3 show that noise has little influence on classification performance when we use

The ORL database

The Cambridge University ORL face database (Fig. 5) is composed of 400 grey level images of ten different patterns for each of 40 persons. The variations of the images are across time, size, pose and facial expression (open/closed eyes, smiling/not smiling), and facial details (glasses/no glasses).

Face samples from the ORL database

In the literature, the protocol used for training and testing is different from one paper to another. In [START_REF] Samaria | Parameterisation of a stochastic model for human face identification[END_REF], a hidden Markov model (HMM) based approach is used, and the best model resulted in recognition rate of 95%, with high computational cost. In [START_REF] Hjelmas | Face Detection: A Survey[END_REF], Hjelmas reached a 85% recognition rate using the ORL database and feature vector consisting of Gabor coefficients.

We perform experiments on the ORL database using the RBS, BS, PS, RPS, ZM, HU, FM, and the combination of the RPS & BS descriptors. Since the local descriptors SIFT and HOG obtained, predictably, almost perfect scores, we do not present them. The results are shown in Table 4, where we clearly see that the RBS invariant descriptor gives the best recognition rate c = 89.8%, faring far better than before w.r.t. the combination of RPS and BS descriptors.

Conclusion and perspectives

In this paper we presented four Fourier descriptors over the semidiscrete roto-translation group SE(2, N ). Then, we proved that the generalized power-spectrum (PS) and bispectrum (BS) -and thus the rotational powerspectrum (RPS) and bispectrum (RBS) -are weakly complete, in the sense that they allow to distinguish over an open and dense set of compactly supported functions ϕ ∈ L 2 (SE(2, N )) up to the SE(2, N ) action. This generalizes a result of [START_REF] Smach | Generalized Fourier descriptors with applications to objects recognition in SVM context[END_REF]. We then considered a framework for the application of these Fourier descriptors to roto-translation invariant object recognition, inspired by some neurophysiological facts on the human primary visual cortex. In this framework, we showed that the rotational bispectrum is indeed a weakly complete roto-translation invariant for planar images. Moreover, although the proposed Fourier descriptors are given in terms of complex mathematical objects, we showed that they can be implemented in a straightforward way as linear combinations of the values of the 2D Fourier transform of the image.

In the second part of the paper, we proposed an evaluation of the performances of these Fourier descriptors in object recognition and we presented the results obtained on different databases: the COIL-100 database, composed of several objects undergoing 3D rotation and scales changes, and the ORL-database, on which different human faces are subject to several kind of variations. For both these databases, the global Fourier descriptors introduced in this paper are the most efficient global descriptors tested, equalled only, for noisy images, by the Zernike Moments. Although for unperturbed images the local SIFT descriptor gives better recognition rate, the addition of noise leads to the global descriptors outperforming the local ones. These results thus show the rotational bispectrum (RBS) to be a very good Fourier descriptor for object recognition, consistently with the theoretical weak completeness result. When the dimension of the feature vector is an issue, the RBS can be replaced by a combination of the generalized bispectrum (BS) and the rotational power-spectrum (RPS), which yields slightly worse results with a feature vector of length almost one third.

An extension of the object recognition method presented in this paper to an AdaBoost framework for the problem of object detection is currently undergoing.

A Moment invariants and Fourier-Mellin transform

In this section, following [START_REF] Choksuriwong | Comparative study of global invariant descriptors for object recognition[END_REF], we review the two most used classes of moment invariants, Hu and Zernike, and Fourier-Mellin descriptors, that we use as a comparison for our generalized Fourier descriptors.

Moment invariants were first introduced to the pattern recognition and image processing community in 1962 by Hu [START_REF] Hu | Visual pattern recognition by moment invariants[END_REF], with the introduction of the seven Hu moments which are invariants under translation, rotation and scaling. These are derived from a scaling and translation invariant modification of the standard moments of an image I : R 2 → R. Namely, v p,q = u p,q u (1+ p+q 2 ) 0,0 , where u p,q = R 2 (x -x 0 ) p (y -y 0 ) q I(x, y)dxdy, and x 0 = m 1,0 m 0,0 and y 0 = m 0,1 m 0,0 are the coordinates of the barycenter computed via the standard (p + q)-th order moments of I:

m p,q = R 2
x p y q I(x, y)dxdy.

Another important class of moments are the Zernike ones, introduced in [START_REF] Chong | Translation invariants of zernike moments[END_REF] and computed via orthogonal Zernike polynomials. The Zernike moment of order (m, n) is:

Z mn = m + 1 n x y I(x, y) [V mn (x, y)]
where x 2 + y 2 < 1 and V mn (x, y) are the Zernike polynomials defined in polar coordinates as V mn (r, θ) = R mn (r)e jnθ , where

R mn (r) = m-|n| 2 s=0 (-1) s (m -s)!r m-2s s!( m+|n| 2 -s)!( m-|n| 2 -s)!
Title Suppressed Due to Excessive Length 13 These moments present several advantages. Indeed, beside a rotation and translation invariance they have nice orthogonality properties and are considered to be robust against image noise. In particular, the orthogonality property helps in achieving a near zero value of redundancy measure in a set of moments functions [START_REF] Chong | A comparative analysis of algorithms for fast computation of Zernike moments[END_REF]. Finally, strictly related to Fourier descriptors are the descriptors obtained via Fourier-Mellin transform (FMT), presented in [START_REF] Derrode | Robust and efficient fouriermellin transform approximations for invariant grey-level image description and reconstruction[END_REF]. The FMT of an image I, that we assume to be given in polar coordinates, is defined as:

M I (u, v) = 1 2π 2π 0 ∞ 0 I(r, θ)r
-iv e -iuθ dr r dθ.

Following [START_REF] Derrode | Robust and efficient fouriermellin transform approximations for invariant grey-level image description and reconstruction[END_REF], we will indeed compute the analytical Fourier-Mellin transform (AFMT). That is, we replace I in the above definition with its regularized version I σ (r, θ) = r σ I(r, θ), where σ > 0. Finally, each feature M I σ (u, v) is modified in order to compensate for the rotation, translation and size changes of the object.

B Auxiliary lemmata for the proof of Theorem 1

Lemma 1 The set G introduced in Theorem 1 is open and dense in L 2 (K × Z N ).
Proof We start by showing that G = ∅. To this aim, it suffices to consider ϕ such that ϕ(

•, k) ≡ 0 for all k ∈ Z N \ {0} and ϕ(•, 0) = 0 such that supp F (ϕ(•, 0)) = R 2 . By (6), we then have ϕ ∈ G, since det φ(T λ ) = k∈Z N F (ϕ(•, 0))(R -k λ) = 0 ∀λ ∈ S.
For any ϕ ∈ G and k ∈ Z N , the Paley-Wiener Theorem implies that F (ϕ(•, k)) is analytic. In particular, by [START_REF] Bressloff | Geometric visual hallucinations, euclidean symmetry and the functional architecture of striate cortex[END_REF], λ → det φ(T λ ) is analytic. Thus, ϕ ∈ G if and only if ϕ(T λ 0 ) is invertible for some λ 0 ∈ S.

We claim that the set G is dense. Indeed, let ϕ / ∈ G and fix some η ∈ G and λ 0 ∈ S such that η(T λ 0 ) is invertible. By analyticity of ε → det( φ(T λ 0 ) + εη(T λ 0 )) follows that ϕ + εη ∈ G for sufficiently small ε > 0, which entails that ϕ ∈ Ḡ, proving the claim.

Let us prove that G is open in L 2 (K×Z N ). To this aim, fix ϕ ∈ G and ϕ n → ϕ in L 2 (K ×Z N ). This implies that φn → φ in L 2 ( SE(2, N )), and in particular that φn → φ in measure. By definition of convergence in measure, this implies that for sufficiently big n it has to hold det φn (T λ 0 ) = 0. Hence ϕ n ∈ G for n sufficiently big and G is open.

⊓ ⊔

Before diving into the proofs of the other auxiliary lemmata, we make the following observation. Let λ 1 , λ 2 ∈ I be such that λ 1 + R k λ 2 ∈ I for all k ∈ Z N . Applying the Induction-Reduction theorem (9) to [START_REF] Citti | A cortical based model of perceptual completion in the roto-translation space[END_REF] yields

A • U (T λ 1 ) ⊗ U (T λ 2 ) • A -1 = k∈Z N ϕ(T λ 1 +R k λ 2 ) -1 η(T λ 1 +R k λ 2 ) = k∈Z N U (T λ 1 +R k λ 2 ). ( 15
) Lemma 2 The function λ → U (T λ ) is continuous on I. Proof Fix λ 0 ∈ I and an open set V ⊂ R 2 such that V U (T λ 2 ) * i,j dλ 2 > 0 ∀i, j ∈ Z N .
This is possible since U ≡ 0. Since the set I is open dense, up to reducing V we can assume that there exists a neighborhood W of λ 0 such that V +λ ⊂ I for any λ ∈ W . Then, (15) holds for λ 1 ∈ W and λ 2 ∈ V . Explicitly computing the 0, 0 block of (15), we have

U (T λ 1 ) i,j U (T λ 2 ) * i,j = U (T λ 1 +λ 2 ) i,j ∀i, j ∈ Z N .
Then, integrating it over V w.r.t. λ 2 yields

U (T λ 1 ) i,j = V +λ U (T λ 2 ) i,j dλ 2 V U (T λ 2 ) * i,j dλ 2 ∀λ 1 ∈ W, ∀i, j ∈ Z N
Since the function on the r.h.s. is clearly continuous on W this proves the continuity at λ 0 of λ → U (T λ ), completing the proof. ⊓ ⊔ Lemma 3 The function λ → U (T λ ) can be extended to a continuous function on R 2 for which (11) is still true.

Proof Let λ 0 / ∈ I. Since I is an open and dense set, this implies that λ 0 is in its closure and that we can choose λ 1 , λ 2 ∈ I such that λ 0 = λ 1 +R k 0 λ 2 for some k 0 ∈ Z N and λ 1 +R k λ 2 ∈ I for any k = k 0 . We then let

U (T λ 0 ) := A • U (T λ 1 ) ⊗ U (T λ 2 ) • A * k 0 ,k 0 . (16)
We now prove that the above definition does not depend on the choice of λ 1 , λ 2 and k 0 . By openness of I, there exists a neighborhood V of λ 2 entirely contained in I. Then, up to taking a smaller V , it holds that λ 1 + R k 0 λ ′ 2 ∈ I for any λ ′ 2 ∈ V \ {λ 2 }. By [START_REF] Duits | Left-invariant parabolic Evolutions on SE(2) and Contour Enhancement via Invertible Orientation Scores. Part I: linear left-invariant diffusion equations on SE(2)[END_REF], this implies that for any µ

1 + R ℓ µ 2 = λ 0 it holds (A • U (T λ 1 ) ⊗ U (T λ ′ 2 ) • A * ) k 0 ,k 0 = (A • U (T µ 1 ) ⊗ U (T µ ′ 2 ) • A * ) ℓ,ℓ .
for λ ′ 2 and µ ′ 2 sufficiently near, but different, to λ 2 and µ 2 , respectively. By the continuity of U on I, proved in Lemma 2, this implies that this equation has to hold also for λ ′ 2 = λ 2 and µ ′ 2 = µ 2 . Hence, (16) does not depend on the choice of λ 1 , λ 2 and k 0 .

Finally, the fact that f (T 16) and [START_REF] Duits | Left-invariant parabolic Evolutions on SE(2) and Contour Enhancement via Invertible Orientation Scores. Part I: linear left-invariant diffusion equations on SE(2)[END_REF].

λ 1 ⊗ T λ 2 ) • U (T λ 1 ) ⊗ U (T λ 2 ) = ĝ(T λ 1 ⊗ T λ 2 ) for any λ 1 , λ 2 follows from (
⊓ ⊔ Lemma 4 There exists a ∈ SE(2, N ) such that U (T λ ) = T λ (a).
Proof By definition of U it holds that

k∈Z N U (T λ 1 +R k λ 2 )•A = A•U (T λ 1 )⊗U (T λ 2 ) ∀λ 1 , λ 2 = 0.
Then, for any i, j, ℓ, k,

U (T λ 1 ) ℓ,i U (T λ 2 ) ℓ-k,j = U (T λ 1 +R k λ 2 ) ℓ,i if j = i -k, 0 otherwise. ( 17
)
By invertibility of U (T λ 1 ), there exists i 0 ∈ Z N such that U (T λ 1 ) 0,i 0 = 0. Using [START_REF] Führ | Continuous wavelet transforms from semidirect products : Cyclic representations and Plancherel measure[END_REF] this implies that U (T λ 2 ) -k,j = 0 for any j = i 0 -k. Namely, we have proved that there exists a family of functions ϕ

-k : S → C such that U (T λ 1 ) -k,• = ϕ -k (λ 1 ) δ i 0 -k or, equivalently, that U (T λ ) = diag k ϕ k (λ) S i 0 .
By the explicit expression (4) of T λ , in order to complete the proof it suffices to prove that ϕ k (λ) = e i x 0 ,R -k λ for some

x 0 ∈ R 2 .
By continuity and unitarity of U , the ϕ k 's are continuous and satisfy |ϕ h (λ)| = 1. Using again [START_REF] Führ | Continuous wavelet transforms from semidirect products : Cyclic representations and Plancherel measure[END_REF] with j = i 0 -k, we obtain

ϕ ℓ (λ 1 + R k λ 2 ) = ϕ ℓ (λ 1 )ϕ ℓ-k (λ 2 ), (18)
for any λ 1 , λ 2 = 0 and ℓ, k ∈ Z N . We claim that the ϕ ℓ 's are characters of R 2 . Indeed, let us fix k = 0 in [START_REF] Galerne | Random phase textures: Theory and synthesis[END_REF]:

ϕ ℓ (λ 1 + λ 2 ) = ϕ ℓ (λ 1 )ϕ ℓ (λ 2 ). ( 19
)
Choosing λ 2 = -λ 1 in the above shows that ϕ ℓ can be extended at 0. Moreover, letting λ 1 = 0 and taking the limit λ 2 → 0 shows that this extension is continuous. Since characters of R 2 are exactly the continuous functions satisfying [START_REF] Granlund | Fourier Preprocessing for Hand Print Character Recognition[END_REF], the claim is proved. By Pontryiagin duality, there exists x ℓ ∈ R 2 such that ϕ ℓ (λ) = e i λ,x ℓ . Finally, by [START_REF] Galerne | Random phase textures: Theory and synthesis[END_REF] with k ∈ Z N one obtains that R -k x ℓ = x ℓ-k , which proves that there exists x 0 ∈ R 2 such that ϕ ℓ (λ) = e i x 0 ,R -k λ . This completes the proof of the statement.

⊓ ⊔

C Proofs

Proof (Formula (12)) Let λ ∈ S and consider v ∈ C N . Observe that (x, k) -1 = (-R -k x, -k). Then, by ( 5), [START_REF] Bay | Surf: Speeded up robust features[END_REF], and (4), for any h ∈ Z N we have

( L(T λ ).v) h = N-1 k=0 R 2 L(x, k)e -i λ,R h-k x v h-k dx = N-1 k=0 v h-k R 2 R 2 f (y) Ψ (R -k (y -x))e -i λ,R h-k x dy dx = N-1 k=0 v h-k R 2 R 2
Ψ (z)f (y)e -i R k-h λ,y-z dy dz

= Ψ (R h λ) N-1 k=0 v h-k f(R k-h λ) = ω Ψ (λ) h ω f (λ), v .
By definition of ω Ψ (λ) * ⊗ ω f (λ) * this completes the proof.

⊓ ⊔

In order to prove Theorem 3 we need the following explicit description of the equivalence in the Induction-Reduction theorem of Section 3.1. M i,j N i-k,ℓ v(j, ℓ).

Since is straightforward to check that A -1 :

k∈Z N C N → C N×N is given by [A -1 (w ℓ ) ℓ∈Z N ](k, h) = w k-h (k), we then have

[A • (M ⊗ N ) • A -1 .(w h ) h∈Z N ] k (i) = N-1 j,ℓ=0 M i,j N i-k,ℓ w j-ℓ (j) = N-1 j,h=0
M i,j N i-k,j-h w h (j).

By [START_REF] Chong | A comparative analysis of algorithms for fast computation of Zernike moments[END_REF], this completes the proof.

⊓ ⊔

Proof (Proof of Theorem 3) Without loss of generality we can restrict ourselves to consider functions such that φf = f and L c f = Lf . We start by the trivial remark that the result on the rotational descriptors contains the one on the generalized ones. Let us consider

I λ,h 2 (f ) := ω f (R h λ), ω f (λ) , I λ 1 ,λ 2 ,h 2 (f ) := ω f (R h λ 1 ) ⊙ ω f (λ 2 ), ω f (λ 1 + λ 2 ) .
Since f is assumed to be compactly supported, its Fourier transform is analytic, and so are the functions λ → I λ,h Let us recall the following properties of the tensor product, valid for all v, v 1 , v 2 , w, w 1 , w 2 ∈ C N :

1. (v ⊗ w) * = v ⊗ w, 2. (v 1 ⊗ w 1 ) • (v 2 ⊗ w 2 ) = w 1 , v 2 v 1 ⊗ w 2 ,
By these and (12), we immediately have

RPS L c f (λ, h) = ω φf (R h λ) * , ω φf (λ) * ω Ψ (R h λ) * ⊗ ω Ψ (λ) * = I λ,h 2 (f ) ω Ψ (R h λ) * ⊗ ω Ψ (λ) * .
Hence, whenever ω Ψ (R h λ) * ⊗ ω Ψ (λ) * = 0, RPS L c f (λ, h) = RPS L c g (λ, h) if and only if I λ,h 2 (f ) = I λ,h 2 (g). Since ω Ψ (R h λ) * ⊗ ω Ψ (λ) * = 0 if and only if ω Ψ (λ) = 0, by the fact that Ψ ∈ R this is true for a.e. λ ∈ S. This completes the proof of the part of the statement regarding the rotational power-spectrum.

To prove the statement regarding the rotational bispectrum, let B f = A • RBS Lf (λ 1 , λ 2 , h) • A -1 , where A is the equivalence given by the Induction-Reduction Theorem and defined in [START_REF] Chou | Pavement distress classification using neural networks[END_REF]. Since A is invertible, determining RBS Lf (λ 1 , λ 2 , h) is equivalent to determining B f . Exploiting the fact that the r.h.s. of ( 9) is a diagonal matrix we have

B k,ℓ f = A • Lf (T R h λ 1 ) ⊗ Lf(T λ 1 ) • A -1 k,ℓ
• Lf(T λ 1 +R ℓ λ 2 ) * . By Lemma 5, formula [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF], and explicit computations, we then get

B k,ℓ f = ω f (R h λ 1 ) ⊙ ω f (R ℓ λ 2 ), ω f (λ 1 + R ℓ λ 2 ) × (ω Ψ (R h λ 1 ) * ⊙ ω Ψ (R h+k λ 2 ) * ) ⊗ ω Ψ (λ 1 + R ℓ λ 2 ).
Similarly to before, (ω Ψ (R h λ 1 ) * ⊙ ω Ψ (R h+k λ 2 ) * )⊗ω Ψ (λ 1 + R ℓ λ 2 ) = 0 for a.e. λ 1 , λ 2 ∈ S since Ψ ∈ R. For these couples, B f = B g if and only if

ω f (R h λ 1 ) ⊙ ω f (R ℓ λ 2 ), ω f (λ 1 + R ℓ λ 2 ) = ω g (R h λ 1 ) ⊙ ω g (R ℓ λ 2 ), ω g (λ 1 + R ℓ λ 2 ) .
Finally, making the change of variables R ℓ λ 2 → λ 2 completes the proof of the theorem.

⊓ ⊔

Fig. 1

 1 Fig. 1 Steps of computing the invariant descriptors. (S1) computation of the shifted F F T of the the image f , (S2) generation of the hexagonal grid, (S3) extraction of different hexagons, (S4) evaluation of the F F T of f on each extracted hexagon, (S5) generation of the vector ω f (λ) and (S6) computation of the four invariants

Fig. 2

 2 Fig. 2 Sample objects of COIL-100 database

Fig. 4

 4 Fig. 4 Sample of COIL-100 noisy object

Lemma 5

 5 For any M, N ∈ C N×N we haveA • (M ⊗ N ) • A -1 k,h = (M i,j N i-k,j-h ) i,j∈Z N . Proof Observe that for any v ∈ C N ⊗ C N it holds (M ⊗ N ).v = M • v • N T . Thus, [A • (M ⊗ N ).v] k (i) =

  and (λ 1 , λ 2 ) → I λ 1 ,λ 2 ,h 2 (f ) for any h ∈ Z N . Thus, the statement of the proposition reduces to show that RPS Lf = RPS Lg (resp. RB Lf = RB Lg ) if and only ifI λ,h 2 (f ) = I λ,h 2 (g) (resp. I λ 1 ,λ 2 ,h 2 (f ) = I λ 1 ,λ2 ,h 2 (g)) for a.e. λ, λ 1 , λ 2 ∈ S and all h ∈ Z N

  1. Lemma 2: The function λ → U (T λ ) is continuous on I. 2. Lemma 3: The function λ → U (T λ ) can be extended to a continuous function on R 2 for which (11) is still true. 3. Lemma 4: There exists a ∈ SE(2, N ) such that

Table 1

 1 Dimension of the feature vectors for the Fourier descriptors under consideration

				1. S1).
	2. For cost computational reasons and since we are
	dealing with natural images, for which the relevant
	frequencies are the low ones, we extract a grid of
	16 × 16 pixels around the origin (Fig 1. S2).
	6 MATLAB	sample	code	for	the	implementa-
	tion of the rotational bispectral invariants can be
	found	at	https://nbviewer.jupyter.org/github/
	dprn/bispectral-invariant-svm/blob/master/
	Invariant_computation_matlab.ipynb

Table 2

 2 Recognition rate for each descriptor using the COIL-100 database. The test results for ZM, HM, FM, and SIFT are taken from[START_REF] Choksuriwong | Comparative study of global invariant descriptors for object recognition[END_REF].

	Descriptors Recognition rates
	RBS	95.5%
	BS	88%
	PS	84.3%
	RPS	89.8%
	RPS+BS	92.8%
	ZM	91.9%
	HM	80.2%
	FM	89.6%
	HOG	95.3%
	SIFT	100%

Table 3

 3 Classification rate on COIL-100 noisy database. The test results for ZM, HM, FM, and SIFT are taken from[START_REF] Choksuriwong | Comparative study of global invariant descriptors for object recognition[END_REF].

	S d	RBS	BS	PS	RPS	RPS+BS	ZM	HM	FM	SIFT	HOG
	5	100% 100% 71.5% 99.8%	100%	100% 95.2% 98.6% 89.27%	4%
	10	100% 100% 71.2% 99.8%	100%	100% 95.2% 95.2% 88.89%	1.2%
	20	100% 100% 67.8% 99.8%	100%	100% 91.4% 90.2% 85.46%	1%

Table 4

 4 Recognition rate for each descriptor using the ORL database

	Descriptors Recognition rates
	RBS	89.8%
	BS	67.9%
	PS	49.2%
	RPS	76.9%
	RPS+BS	79.8%
	ZM	75%
	HM	43.5%
	FM	47.6%

http://www.cs.columbia.edu/CAVE/software/softlib/ coil-100.php

http://www.cl.cam.ac.uk/research/dtg/attarchive/ facedatabase.html

Acknowledgements This research has been supported by the European Research Council, ERC StG 2009 "GeCoMethods", contract n. 239748. The second and last authors were partially supported by the Grant ANR-15-CE40-0018 of the ANR.