Soizic Terrien
email: s.terrien@auckland.ac.nz

Bernd Krauskopf

Neil G R Broderick

Rémy Braive

Gregoire Beaudoin

Isabelle Sagnes

Sylvain Barbay

Grégoire Beaudoin

Pulse train interaction and control in a microcavity laser with delayed optical feedback

INTRODUCTION

Controllable sources of short optical pulse trains are at the heart of many recent developments in physics, with potential applications to optical telecommunications and signal processing. Trains of short pulses can arise from cw, coherently-driven optical fibre cavities [1] or nonlinear microcavities [2,3], forming dissipative temporal solitons. In the laser regime, mode-locked dissipative solitons have been observed in fibre laser cavities [4] and in a face-to-face VCSEL configuration [5]. Pulse trains can also be produced by using the regenerative self-pulsing configuration of an excitable laser with time-delayed feedback. This configuration was implemented in a coherently driven VCSEL [6], an opto-electronic system [7] and a micropillar laser with integrated saturable absorber [8,9]. Excitability [START_REF] Izhikevich | Dynamical systems in neuroscience[END_REF][START_REF] Barbay | [END_REF] is associated with an all-or-none response to external perturbations, depending on whether or not the amplitude of the perturbation exceeds the excitability threshold. In the excitable regime, a laser with saturable absorber (LSA) subject to a perturbation either emits an excitable response in the form of a short, high-amplitude pulse, or relaxes back to its off-state [START_REF] Barbay | [END_REF]. In the presence of an optical feedback loop with delay τ, the reinjection of an initial excitable pulse can trigger another excitable response. As the process repeats, this results in a train of pulses with a repetition rate close to the feedback delay [9]. This picture is however oversimplified, and a theoretical analysis predicts more complicated dynamics, including the coexistence of several selfpulsing modes with the stable off-state [START_REF] Krauskopf | Bifurcation Study of a Semiconductor Laser with Saturable Absorber and Delayed Optical Feedback[END_REF][START_REF] Terrien | [END_REF]. In particular, bistability between a pulsing periodic solution and the off-state is a condition for the on and off switchability of the regenerative pulse trains [9]. The pulsing dynamics is also driven by the nonlinear latency between the reinjection and the regeneration of a pulse [14][15][START_REF] Erneux | Analytical studies of an excitable micropillar laser with a saturable absorber[END_REF], which is strongly related to the net gain dynamics of the laser active medium accounting for the saturable gain and saturable loss. Despite similar features of the pulsing dynamics, it is worth noting that the physics involved here differs considerably from the one of temporal dissipative solitons in e.g. mode-locked lasers [5]: the micropillar LSA with delayed feedback is intrinsically a single mode system and does not involve the locking dynamics of several external cavity modes.

In this Letter, we present experimental results on the optical control and interaction of pulse trains in an excitable micropillar laser with integrated saturable absorber and delayed optical feedback. We show that the Yamada model with incoherent delayed optical feedback describes the pulsing dynamics in very good agreement with the experiment [9,[START_REF] Barbay | [END_REF], and allows us to explain the pulse-to-pulse interactions by the carrier dynamics of the laser. We reconcile the bifurcation analysis of the model with the numerical and experimental observations showing that non-regularly timed pulse trains are in fact very long transients, which evolve to solutions with equidistant pulses in the external cavity as a consequence of the carrier dynamics. In constrast to temporal dissipative solitons in other systems [START_REF] Jang | [END_REF]18], we show that pulse trains can also show attractive interactions. Finally, we show that reliable all-optical pulse train control is possible experimentally with short optical control pulses.

EXPERIMENTAL SETUP AND MODEL

The experimental setup is similar to the one described in [9], but with improved feedback strength thanks to a considerable reduction of feedback losses. A micropillar laser with integrated saturable absorber [19,20] emitting around 980nm is optically pumped at 800nm. The laser emission is fed back after a delay τ through a mirror located several tens of centimeters away. Part of the emitted beam is selected by a beamsplitter and analyzed by a camera and a fast avalanche photodetector (> 5GHz bandwitdh). The solitary micropillar laser is excitable for a large range of pump powers below threshold [START_REF] Barbay | [END_REF]. The 80ps optical perturbations triggering an excitable response are produced by a mode-locked Ti:Sa laser emitting at ∼800nm.

This system is modelled by the Yamada rate equations with incoherent delayed optical feedback [START_REF] Krauskopf | Bifurcation Study of a Semiconductor Laser with Saturable Absorber and Delayed Optical Feedback[END_REF]21]:

Ġ = γ G (A -G -GI), Q = γ Q (B -Q -aQI), İ = (G -Q -1)I + κI(t -τ), ( 1
)
for the dimensionless gain G, absorption Q and intensity I. Here, A is the pump parameter, B is the non-saturable absorption, a is the saturation parameter and γ G and γ Q are the carrier recombination rates in the gain and absorption media, respectively. The delayed term in the intensity equation describes incoherent feedback with delay τ and strength κ. The parameter values used in simulation are, unless otherwise stated, A = 2.4, B = 2.2, γ G = 0.01, γ Q = 0.02, a = 5, κ = 0.05 and τ = 1100. These are chosen both to match the known physical parameters and the experimental observations. The results presented here are robust in the presence of reasonable amounts of both spontaneous emission and pump noise. Hence, we do not include noise terms in (1) and concentrate on the deterministic dynamics.

WRITING AND COEXISTENCE OF PULSE TRAINS

Fig. 1 shows a direct comparison between observed and simulated pulse trains. The temporal traces are folded at the delay time τ and stacked vertically in a pseudo-space representation [22]. In panel (a1), a single external perturbation leads, as expected, to a pulse train whose repetition rate is slightly larger than the delay time τ, due to the nonlinear latency time of the excitable response [14,15]. Pulse trains can be sustained for very long time, depending on the feedback strength and on the amount of noise in the system [9]. For very low feedback losses, the observed pulse train duration is limited by the acquisition memory of the oscilloscope.

In panel (b1), two successive perturbations are sent with a time difference of 12.5ns. This leads to the coexistence of two pulse trains in the cavity, separated by a time interval of approximately 0.4 × τ which seems to remain constant on the timescale of 250 round trips. This suggests that this pulse train does not correspond to a harmonic regime. Three coexisting pulse trains can also be observed, as in Fig. 1 For the same parameters, Fig. 2 represents the phase portrait of system (1) in the (G, I)-plane, calculated with the continuation toolbox DDE Biftool [23,24]. It shows the coexistence of seven stable solutions: a non-lasing equilibrium and six periodic pulsing solutions. These coexist with several unstable periodic solutions and equilibria, which are not represented here. The stable pulsing solutions have periods T close to submultiples of τ, and hence correspond to different numbers of equidistant pulses in the external cavity [START_REF] Terrien | [END_REF]. Apart from the one for which T is close to τ, they all are only weakly stable: the modulus of their leading Floquet multiplier, shown in Fig. 2

(c1). As shown in

LONG-TERM BEHAVIOUR

This raises the question of the long-term behaviour of pulse trains with non-equidistant pulses (Fig. 1(b)-(c)). Fig. 3(a-c) represents the relative timing of two coexisting pulses of (1) over several hundreds of round trips, showing that the two pulses become equidistant in the long term. In fact, regimes with non equidistant pulses are very long transients toward one of the (weakly) attracting periodic solutions of Fig. 2(a). The bifurcation analysis of equations (1) shows that the amplitudes of the periodic solutions with largest periods are very close to each other (see the two largest orbits in Fig. 2) [START_REF] Terrien | [END_REF]: as such, no significant difference is observed in the amplitudes when one or several pulses exist in the external cavity.

The slow convergence towards the stable periodic solution is explained entirely by the dynamics of the net gain G = G -Q -1. In the solitary laser, each excitable pulse is followed by an absolute and a relative refractory periods, during which no other pulse can be triggered, or the conditions to trigger another pulse are more stringent, respectively [8]. These time windows are related to the recovery time of the carriers in the gain and absorber sections of the microlaser, described by γ G and γ Q . In Fig. 3(d-e), the absorption recovers faster than the gain (γ Q > γ G ): after a pulse, the low net gain G increases back to its saturated value as the gain G recovers. The second perturbation is introduced in the relative refractory period of the first pulse, where the absorber recovered entirely, and the gain has recovered sufficiently for the second pulse to be sustained. However, it did not recover entirely, so that the net gain experienced by the second pulse train is slightly smaller than for the first pulse train. Because the net gain determines the latency of the excitable response to a perturbation [15], this results in a slightly larger response time, and hence repetition period, for the second pulse train compared to the first pulse train. Round trip after round trip, the second pulse is reinjected further away from the first one, until both pulse trains experience an identical net gain, and their repetition periods become equal. This convergence to the solution with equidistant pulses gives the impression of a repulsion between pulses.

Because the system converges in the long term to a stable periodic orbit of the phase portrait, the maximum number of pulses that can be sustained simultaneously is related to the number of stable periodic solutions. In Fig. 2, the periodic orbit with the smallest period corresponds to the coexistence of six pulses in the external cavity. In Fig. 4(a), six pulse trains are triggered and indeed sustained in simulation, and they become equidistant after several round trips. Three different scenarios can result from a seventh perturbation, depending on its timing. When sent in the absolute refractory period of an existing pulse, the low gain prevents a pulse from being triggered. If sent just before an existing pulse (case A), a new pulse is triggered but the absolute refractory period of the newly created pulse prevents the next pre-existing pulse to be regenerated. This results in the retiming of the pulse train while keeping the number of coexisting pulses unchanged; see also the experimental plot in Fig. 4(b). When the seventh perturbation is sent in the relative refractory period of a pre-existing pulse (case B), the gain G did not recover entirely, and the newly created pulse thus has a small amplitude, which prevents its regeneration after a delay τ. Moreover, the absolute refractory period of this single pulse results in a switch off by preventing the regeneration of the next pulse. The final state has only five coexisting pulses, which then converge towards an equidistant configuration. Overall, there is no suitable perturbation timing to trigger a sustained seventh pulse. From a mathematical point of view, these different scenarios are related to the basins of attraction of the different stable periodic solutions. The ability of a perturbation to bring the system outside of the basin of a periodic orbit obviously depends on both amplitude and timing along the orbit. A detailed study of these basins is beyond the scope of this Letter, and will be discussed elsewhere.

PULSE TRAINS INTERACTION

We showed the existence of a weak "repulsion" of a second pulse in Fig. 3. More surprisingly, "attractive" interaction is also observed, as shown in Fig. 5(a)-(b) for experimental and simulated data. The explanation of this phenomenon is more subtle and now requires consideration of the dynamics of both the net gain G and the gain G. Here, γ G must be larger than γ Q , so that the net gain G decreases in between two pulses, as shown in Fig. 5(c). The second perturbation is introduced in the relative refractory period of the first pulse, where it benefits from a higher net gain than the first pulse train. The response time of the laser is thus shorter, which results in a slightly smaller repetition rate for the second pulse train compared to the first one. Round trip after round trip, the second pulse is reinjected closer and closer to the first pulse and benefits from a higher and higher net gain, which shortens further the repetition rate and explains the impression of an attraction between the pulses. Eventually, the second pulse is reinjected when insufficient gain G is available, which prevents its regeneration and stops the second pulse train. The pseudo-space representation in Fig. 3-5 may suggest that the pulses interact in the external cavity. However, we stress that the only mechanism for interaction is through the gain and absorption in the microlaser itself. Although the systems are fundamentally different, a similar mechanism was suggested in [25] for a model for mode-locked lasers, that explains the stability properties of mode-locked solutions with one and several pulses in the laser cavity. However, to our best knowledge, attractive interaction has never been observed.

OPTICAL CONTROL OF PULSE TRAIN DURATION

Based on these mechanisms, we show that a short optical external perturbation can reliably switch on and off a pulse train. This is much faster than existing techniques; in particular, it does not involve a modulation of the pumping current or a precise control of the phase of the holding beam [START_REF] Jang | [END_REF]18]. Fig. 6(a) shows two successive perturbations, and the resulting experimental temporal trace is Fig. 6(b). The pulse train is reliably switched off by the second perturbation, which has been checked for one thousand experimental realisations. There is a definite window in the amplitude and timing of the second perturbation to switch off the pulse train. In particular, the two perturbations are sent 12.5 ns apart for technical reasons, but the feedback delay (4.2ns) is chosen such that the second perturbation arrives just before the regeneration of the fourth pulse. This phenomenon can be reproduced in the model, which requires a careful choice of parameter values, so that a second perturbation brings the system into the basin of attraction of the off-state.

CONCLUSION

We have shown that an excitable micropillar laser with delayed optical feedback can sustain several pulse trains in the external cavity, which can be reliably and independently controlled all-optically by short pulses. By using a suitable mathematical model, we show that all pulsing dynamics are long transients towards a periodic solution with equidistant pulses in the external cavity. The impression of attractive and repulsive pulse interactions is explained entirely by latencies generated by the gain and net gain dynamics in the microlaser itself. As such, this interaction mechanism does not involve the notion of force. Rather, the only ingredients needed are excitability and self-feedback. These are encountered in many systems, including neurons and cardiac cells [START_REF] Izhikevich | Dynamical systems in neuroscience[END_REF]. As such, our results might be applicable to systems beyond the particular device considered here.

Fig. 1 (

 1 Fig.1shows a direct comparison between observed and simulated pulse trains. The temporal traces are folded at the delay time τ and stacked vertically in a pseudo-space representation[22]. In panel (a1), a single external perturbation leads, as expected, to a pulse train whose repetition rate is slightly larger than the delay time τ, due to the nonlinear latency time of the excitable response[14,15]. Pulse trains can be sustained for very long time, depending on the feedback strength and on the amount of noise in the system[9]. For very low feedback losses, the observed pulse train duration is limited by the acquisition memory of the oscilloscope.In panel (b1), two successive perturbations are sent with a time difference of 12.5ns. This leads to the coexistence of two pulse trains in the cavity, separated by a time interval of approximately 0.4 × τ which seems to remain constant on the timescale of 250 round trips. This suggests that this pulse train does not correspond to a harmonic regime. Three coexisting pulse trains can also be observed, as in Fig.1(c1). As shown inFig. 1(a2)-(c2), these experimental results can be matched almost perfectly by the Yamada model (1).For the same parameters, Fig.2represents the phase portrait of system (1) in the (G, I)-plane, calculated with the continuation toolbox DDE Biftool[23,24]. It shows the coexistence of seven stable solutions: a non-lasing equilibrium and six periodic pulsing solutions. These coexist with several unstable periodic solutions and equilibria, which are not represented here. The stable pulsing solutions have periods T close to submultiples of τ, and hence correspond to different numbers of equidistant pulses in the external cavity[START_REF] Terrien | [END_REF]. Apart from the one for which T is close to τ, they all are only weakly stable: the modulus of their leading Floquet multiplier, shown in Fig.2(b), is very close

Fig. 1 .

 1 Fig.1shows a direct comparison between observed and simulated pulse trains. The temporal traces are folded at the delay time τ and stacked vertically in a pseudo-space representation[22]. In panel (a1), a single external perturbation leads, as expected, to a pulse train whose repetition rate is slightly larger than the delay time τ, due to the nonlinear latency time of the excitable response[14,15]. Pulse trains can be sustained for very long time, depending on the feedback strength and on the amount of noise in the system[9]. For very low feedback losses, the observed pulse train duration is limited by the acquisition memory of the oscilloscope.In panel (b1), two successive perturbations are sent with a time difference of 12.5ns. This leads to the coexistence of two pulse trains in the cavity, separated by a time interval of approximately 0.4 × τ which seems to remain constant on the timescale of 250 round trips. This suggests that this pulse train does not correspond to a harmonic regime. Three coexisting pulse trains can also be observed, as in Fig.1(c1). As shown inFig. 1(a2)-(c2), these experimental results can be matched almost perfectly by the Yamada model (1).For the same parameters, Fig.2represents the phase portrait of system (1) in the (G, I)-plane, calculated with the continuation toolbox DDE Biftool[23,24]. It shows the coexistence of seven stable solutions: a non-lasing equilibrium and six periodic pulsing solutions. These coexist with several unstable periodic solutions and equilibria, which are not represented here. The stable pulsing solutions have periods T close to submultiples of τ, and hence correspond to different numbers of equidistant pulses in the external cavity[START_REF] Terrien | [END_REF]. Apart from the one for which T is close to τ, they all are only weakly stable: the modulus of their leading Floquet multiplier, shown in Fig.2(b), is very close

Fig. 2 .Fig. 3 .

 23 Fig. 2. (a): Phase portrait of (1) in the (G, I)-plane, showing one stable equilibrium (dot) and six stable periodic solutions (curves). (b) Modulus of the leading Floquet multiplier of each periodic solution, with respect to its period.

Fig. 4 .

 4 Fig. 4. (a) Simulated pulse trains following six perturbations, and effect of a seventh perturbation for two different timing (A and B). (b) Experimental retiming of a pulse train.

Fig. 5 .

 5 Fig. 5. (a) and (b) Attractive interaction between two experimental and simulated pulse trains. (c) and (d) Temporal evolution of G and G in the shaded area of panel (b). In simulation, the parameters are A = 2, B = 2, γ G = 0.02, γ Q = 0.01, a = 10, κ = 0.1 and τ = 1000.

Fig. 6 .

 6 Fig. 6. (a) External perturbations and (b) response of the experimental device.


Thomine Simon
email: simon.thomine@utt.fr

Snoussi Hichem
email: hichem.snoussi@utt.fr

Distillation-based fabric anomaly detection

Keywords: unsupervised, anomaly, texture, pattern, domain-generalized, fabric, knowledge distillation 1

Unsupervised texture anomaly detection has been a concerning topic in a vast amount of industrial processes. Patterned textures inspection, particularly in the context of fabric defect detection, is indeed a widely encountered use case. This task involves handling a diverse spectrum of colors and textile types, encompassing a wide range of fabrics. Given the extensive variability in colors, textures, and defect types, fabric defect detection poses a complex and challenging problem in the field of patterned textures inspection. In this article, we propose a knowledge distillation-based approach tailored specifically for addressing the challenge of unsupervised anomaly detection in textures resembling fabrics. Our method aims to redefine the recently introduced reverse distillation approach, which advocates for an encoder-decoder design to mitigate classifier bias and to prevent the student from reconstructing anomalies. In this study, we present a new reverse distillation technique for the specific task of fabric defect detection. Our approach involves a meticulous design selection that strategically highlights high-level features. To demonstrate the capabilities of our approach both in terms of performance and inference speed, we conducted a series of experiments on multiple texture datasets, including MVTEC AD, AITEX, and TILDA, alongside conducting experiments on a dataset acquired from a textile manufacturing facility. The main contributions of this paper are the following: a robust texture anomaly detector utilizing a reverse knowledge-distillation technique suitable for both anomaly detection and domain generalization and a novel dataset encompassing a diverse range of fabrics and defects.

Introduction

Fabric defect detection has been a really worrisome problem for the textile industry as the quality of the garment is a priority for the customer. Due to the absence of superior alternatives, inspection is currently carried out mainly by humans, which represents a significant labor cost and does not guarantee flawless detection due to visual fatigue and distraction. Currently, the accuracy of humans in detecting defects is around 60% and decreases for visually difficult fabrics such as striped fabrics. Several constraints are relevant to consider when dealing with this problem: industrial processes are well oiled and cannot suffer from stoppage or slowing of any type, the variety of defects/fabric type is colossal and evolving over time with new collections, defects can be tiny and therefore require an industrial camera with high resolution. As a result, we should aim for a method that does not require defective samples in the training, while offering generalization capability and a fast inference time. Recently, a predominant trend in unsupervised anomaly detection methods has emerged. It involves the utilization of features extracted from pre-trained networks to learn descriptive representations of the analyzed objects or textures [START_REF] He | Deep Residual Learning for Image Recognition[END_REF][START_REF] Krizhevsky | ImageNet classification with deep convolutional neural networks[END_REF][START_REF] Tan | EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks[END_REF][START_REF] Yu | FastFlow: Unsupervised anomaly detection and localization via 2d normalizing flows[END_REF][START_REF] Rudolph | Fully convolutional crossscale-flows for image-based defect detection[END_REF][START_REF] Roth | Towards total recall in industrial anomaly detection[END_REF][START_REF] Bae | PNI : Industrial Anomaly Detection using Position and Neighborhood Information[END_REF][START_REF] Salehi | Multiresolution knowledge distillation for anomaly detection[END_REF][START_REF] Salehi | Multiresolution knowledge distillation for anomaly detection[END_REF][START_REF] Wang | Student-teacher feature pyramid matching for anomaly detection[END_REF][START_REF] Deng | Anomaly detection via reverse distillation from one-class embedding[END_REF][START_REF] Lee | CFA: Coupled-hypersphere-based Feature Adaptation for Target-Oriented Anomaly Localization[END_REF][START_REF] Thomine | MixedTeacher: Knowledge distillation for fast inference textural anomaly detection[END_REF] with several approaches such as normalizing flow [START_REF] Rezende | Variational Inference with Normalizing Flows[END_REF], memory bank and knowledge distillation [START_REF] Hinton | Distilling the Knowledge in a Neural Network[END_REF]. Our approach is based on a knowledge distillation approach, primarily due to its advantages in terms of inference speed and its well-established capability in detecting defects on various types of textures [START_REF] Thomine | MixedTeacher: Knowledge distillation for fast inference textural anomaly detection[END_REF]. Within the framework of knowledge distillation, knowledge transfer occurs between a teacher-student pair during the training. In the context of unsupervised anomaly detection, where the student model is trained solely on normal samples, the student model generates disparate representations compared to the teacher model when confronted with an anomalous sample. Our approach draws inspiration from the reverse distillation architecture [START_REF] Deng | Anomaly detection via reverse distillation from one-class embedding[END_REF]. This architecture follows a student-teacher paradigm, where the teacher model serves as the encoder and the student model as the decoder. By employing this architecture, we aim to leverage the knowledge and guidance provided by the teacher model to enhance the reconstruction capabilities of the student model in the context of anomaly detection. By design, this architecture emphasizes the deeper selected layers and therefore neglects the shallower reconstructed layers. In the domain of texture defect detection, the utmost significance lies in the high-level attributes, as the contextual information has less significance [START_REF] Thomine | MixedTeacher: Knowledge distillation for fast inference textural anomaly detection[END_REF]. Since we aim at detecting defects on fabrics-like textures, we want to keep shallower information and to that extent we introduced residual connections between the teacher and the student along with custom attention layers. Another crucial component of the reverse distillation approach is the embedding bottleneck, which is responsible for gathering features from specific layers of the teacher model and creating a feature map that serves as input for the student model. In our bottleneck embedding architecture, we strive to design a texture-specific approach that selects the most representative features from the teacher model without altering the descriptive capabilities. To that extent, we carefully designed a bottleneck based mainly on 1×1 convolutions along with a custom attention block and an SSPCAB layer [START_REF] Ristea | Self-supervised predictive convolutional attentive block for anomaly detection[END_REF]. In industrial deployment, interrupting the production chain for training a model on a specific type of texture is not feasible. Hence, we propose a slight modification of our model to introduce a domain-generalized model capable of detecting defects in any plain fabric. The primary idea behind this approach is to enable defect detection while a more precise and specific model is being trained. By adopting this strategy, we can ensure that the production chain remains operational, albeit with a concession in terms of precision. Our approach offers outstanding results for defect detection on plain textures, along with a fast inference time and domain generalization capabilities compared to the other state-of-the-art methods.

The primary contributions of this paper are outlined as follows:

• A robust texture anomaly detector utilizing a reverse knowledge-distillation technique. This approach enhances the model ability to detect anomalies in fabric-like textures with improved accuracy and reliability. A slight modification of this approach is also proposed for domain generalization purposes. • Introduction of a novel dataset encompassing a diverse range of fabrics and defects.

This dataset provides a valuable resource for training and evaluating fabric anomaly detection models, facilitating research and advancements in the textile domain.

Following the introduction, the subsequent section of this paper is dedicated to reviewing existing literature concerning both feature extraction and deep learning approaches for unsupervised anomaly detection. Subsequently, we present our novel reverse distillation approach with a precise description of the model. The following two sections focus on introducing a new dataset and conducting experiments to evaluate the effectiveness of our method. The final section provides a conclusion for the paper.

Related work

To handle the problem of texture defect detection, we first have to extract the main characteristics that describe the texture. As mentioned in [START_REF] Humeau-Heurtier | Texture feature extraction methods: A survey[END_REF], we can distinguish seven categories of approaches to handle this problem namely statistical approaches, structural approaches, transform-based approaches, model-based approaches, graphbased approaches, learning-based approaches and entropy-based approaches. Each of these approaches offers different advantages and drawbacks and some of them offer competitive results while being really fast with recent computing ressources such as GLCM [START_REF] Partio | Rock texture retrieval using gray level co-occurrence matrix[END_REF] and local binary pattern [START_REF] Ojala | Multiresolution gray-scale and rotation invariant texture classification with local binary patterns[END_REF] [START_REF] Mäenpää | TEXTURE ANALYSIS WITH LOCAL BINARY PATTERNS[END_REF]. More recently, deep learning based feature extraction brought outstanding results for extracting the most important features characterizing a texture. An example of supervised training approaches consists of training a deep texture classifier and then extracting the last layer to obtain informative features. [START_REF] Simon | Deep learning based feature extraction for texture classification[END_REF] Autoencoder-based approaches are the most common unsupervised approaches to extract features [START_REF] Kramer | Nonlinear principal component analysis using autoassociative neural networks[END_REF] but still suffer from high generalization capabilities which can prevent accurate defect detection.

To tackle the problem of defect detection in the industrial case, gathering every defect of a given object or texture is a laborious task and can lead to poor performances if every type of defect was not considered [START_REF] Han | ADBench: Anomaly Detection Benchmark[END_REF]. Certain methods still utilize supervised training methods like mobile-unet [START_REF] Jing | Mobile-unet: An efficient convolutional neural network for fabric defect detection 92[END_REF] which integrates classification and segmentation techniques, or the utilization of SSD-based approaches [START_REF] Xie | An improved fabric defect detection method based on[END_REF]. Unsupervised anomaly detection deals with the problem of detecting defects on an object or a texture without any prior on the type of possible defects. Consequently, many methods emerged proposing different types of algorithms such as autoencoders [START_REF] Mei | Automatic fabric defect detection with a multi-scale convolutional denoising autoencoder network model[END_REF] and variational autoencoder variants [START_REF] Nguyen | GEE: A gradient-based explainable variational autoencoder for network anomaly detection[END_REF] [START_REF] Zavrtanik | DRAEM -A discriminatively trained reconstruction embedding for surface anomaly detection[END_REF]. Another common way of detecting anomalies is Generative Adversarial Networks (GAN),introduced by [START_REF] Goodfellow | Generative adversarial networks[END_REF], adapted to unsupervised anomaly detection such as Ano-GAN [START_REF] Schlegl | f-AnoGAN: Fast unsupervised anomaly detection with generative adversarial networks[END_REF], G2D [START_REF] Pourreza | G2d: Generate to detect anomaly[END_REF] and OCR-GAN [START_REF] Liang | Omni-frequency channel-selection representations for unsupervised anomaly detection[END_REF]. More recently, approaches using a classifier pretrained on natural images as feature extractor made their way in industrial anomaly detection, offering outstanding performance. These methods, such as normalizing flow, knowledge distillation and memory banks, use selected layers output features to obtain descriptive feature maps. Then, a training is conducted based on the prior that this feature maps describe the most important characteristics of the given object. The normalizing flow approach consists of a flow training based on relevant features of good samples from a pretrained network such as AlexNet [START_REF] He | Deep Residual Learning for Image Recognition[END_REF], Resnet [START_REF] Krizhevsky | ImageNet classification with deep convolutional neural networks[END_REF], efficient-net [START_REF] Tan | EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks[END_REF] and even vision transformers [START_REF] Vaswani | Attention Is All You Need[END_REF] [33] trained on imageNet. Different strategies were used to enhance performances, such as a 2D flow [START_REF] Yu | FastFlow: Unsupervised anomaly detection and localization via 2d normalizing flows[END_REF] or a cross-scale flow [START_REF] Rudolph | Fully convolutional crossscale-flows for image-based defect detection[END_REF]. The use of a memory bank has the advantage of offering a description over the whole class to compare and detect if there is an anomaly [6] [7]. Distilling the knowledge from a teacher to a student [START_REF] Hinton | Distilling the Knowledge in a Neural Network[END_REF] was recently adapted for unsupervised anomaly detection [8] [9]. This concept involves training a student network on good samples using the output features of a teacher network that has already been pre-trained for classification purposes. In the test phase, the student network will be capable of replicating the output features of the teacher network when provided with good samples, but when confronted with anomalous samples, the student will not be able to extract descriptive features and therefore a relevant anomaly score can be computed.

Several methods used this principle with different strategies such as a multi-layer feature selection [START_REF] Wang | Student-teacher feature pyramid matching for anomaly detection[END_REF], a reverse distillation approach [START_REF] Deng | Anomaly detection via reverse distillation from one-class embedding[END_REF], a coupled-hypersphere-based feature adaptation [START_REF] Lee | CFA: Coupled-hypersphere-based Feature Adaptation for Target-Oriented Anomaly Localization[END_REF] and a mixed-teacher approach [START_REF] Thomine | MixedTeacher: Knowledge distillation for fast inference textural anomaly detection[END_REF].

Residual Reverse distillation using specifically selected embeddings

This part is dedicated to the use of reverse distillation to detect defects in textures.

To this extent, we divided this chapter in four subsections: the reverse distillation approach along with the residual connections, the description of the bottleneck embedding that transforms the teacher's output into input for the student model, the loss and score calculation strategy and the adaptation for domain generalization.

Residual reverse distillation

Approach: The idea of reverse distillation was introduced in [START_REF] Deng | Anomaly detection via reverse distillation from one-class embedding[END_REF] with the purpose of proposing an encoder-decoder version of the knowledge distillation paradigm applied to unsupervised anomaly detection. An important problem of unsupervised anomaly detection is the generalization capability, which is problematic since our purpose is to be specific to a texture for detecting any kind of defect. To tackle this specific problem, we propose a residual reverse distillation with a reconstructed embedding emphasizing high-level features. The main idea is to extract the most relevant features describing the texture using a limited encoding and residual connections, allowing a selective and descriptive embedding for input to the decoder student.

Problem definition: Given a training dataset of images without anomaly D = [I 1 , I 2 , ..., I n ], our goal is to extract the information of the L bottom layers. For an image I k ∈ R w * h * c where w is the width, h the height, c the number of channels and l the lth bottom layer, the teacher outputs features F l t (I k ) ∈ R w l * h l * c l . The decoder output features are defined as F L-l s (I k ) ∈ R w l * h l * c l . During the training phase, the student model is trained to reproduce the teacher features on good samples. In testing, when inferring an anomaly, the teacher and the student will output significantly different features and thus will give us an anomaly score. In the anomaly detection setting, normal samples follow the same distribution in both F t and F s . Out-of-distribution samples are considered anomalies.

Our reverse distillation backbone draws inspiration from the one outlined in reverse distillation [START_REF] Deng | Anomaly detection via reverse distillation from one-class embedding[END_REF], differing in terms of layer selection, selective embedding, and residual connections. Given the texture specific architecture described in [START_REF] Thomine | MixedTeacher: Knowledge distillation for fast inference textural anomaly detection[END_REF], we have decided to put an emphasis on the first layers of the network, since they describe high-level features that are primordial in patterned textures. To this purpose, we employed adaptive loss functions to assign varying significance to individual layers, not to mention that with identical loss factors for every layer, we determine that the loss is more impacted by deeper layers with random weight initialization in the student. Also, we include the initial ResNet layer before the first residual block to guarantee a high-level representation of the texture. The goal of the teacher encoder T in the reverse distillation paradigm is to extract thorough representations so that the decoder student S can reconstruct the different selected teacher's features. As in previous work [START_REF] Deng | Anomaly detection via reverse distillation from one-class embedding[END_REF] [START_REF] Lee | CFA: Coupled-hypersphere-based Feature Adaptation for Target-Oriented Anomaly Localization[END_REF], we used ResNet pretrained on ImageNet as our backbone for T. As in any knowledge distillation training, the teacher T is frozen to avoid converging to a trivial solution for T and S. In [START_REF] Deng | Anomaly detection via reverse distillation from one-class embedding[END_REF], they used the same exact number of layer for the student and the teacher to match the intermediate representation. We demonstrate that a shallower student with less residual blocks between each dimension is capable of offering a good representation, which also prevents the student to generalize in case of defective sample and therefore increase our defect detection capability in textures.

As part of its inherent design, the traditional reverse distillation approach highlights the deeper selected layers, owing to their dimensional proximity with the embedding representation. As mentioned earlier, for textures anomaly detection, shallower layers are more descriptive and less subject to misinterpretation. In this respect, we had some residual connections between the teacher and the student to try to compensate of the loss of information caused by the embedding. However, adding a direct connection between the equivalent layers of the student and the teacher would most likely result in identity or zero-weights depending on the selected method of connection. To counteract this issue, we choose to connect the teacher layer l to the student layer L -l -1 with a convolution layer and a pooling layer as well as a self-attention layer, shown in figure 1.

Fig. 1: Attention module used for residual connection

With the purpose of enriching the feature representation and modulating the feature importance, we introduce a custom attention block, described in figure 1, which is similar to a spatial attention but uniquely relies on 1×1 convolutions. The inclusion of this layer enables the introduction of non-linearity, leading to improved model performance with minimal additional computational expense.

The complete architecture is shown in figure 2 Fig. 2: Reverse residual distillation architecture.

Bottleneck Embedding

One of the main contributions of the reverse distillation paradigm is the embedding representation. In other knowledge-distillation anomaly detector models, the main drawback was the student capacity which could eventually generalize to anomalous defects. MixedTeacher [START_REF] Thomine | MixedTeacher: Knowledge distillation for fast inference textural anomaly detection[END_REF] proposed a smaller student to counteract this problem and manage to increase anomaly detection on textures by a huge factor compared to the original STPM model [START_REF] Wang | Student-teacher feature pyramid matching for anomaly detection[END_REF]. Our approach goes even further with the selective embedding encoding approach that prevents the student from any unnecessary parameters. This approach is specifically outlined for texture defect detection; however, for object defect detection, the size of the embedded minimal representation could potentially vary. This issue revolves around striking a balance between the model's capacity and its capability to accurately reconstruct correct elements, all the while ensuring it remains incapable of reconstructing defective elements.

First, to aggregate the features from the selected teacher's layers, we used convolution of size 1 kernel as described in figure 3 to get all the features at the size of the last layer divided by 2 to prevent the prevalence of the last layer over the others. The obtained feature map is our bottleneck input and the bottleneck embedding will then learn to reconstruct this representation before giving to the student decoder.

To increase model performance, we have decided to include a SSPCAB layer [START_REF] Ristea | Self-supervised predictive convolutional attentive block for anomaly detection[END_REF] as the first layer of our bottleneck network since this specific type of layer shows increased performance for a wide variety of unsupervised anomaly detector. To make sure the local information was preserved, we decided to keep a relatively shallow encoding embedding. Each layer is constituted of a convolutionnal layer with a kernel size of 1 and a stride of 1 followed by a ReLu and a BatchNorm layer [START_REF] Ioffe | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift[END_REF] except for the last which have a stride of 2 to fit the decoder input. Employing convolutional operations with a filter size of 1 confers the benefit of computational efficiency, alongside the capacity to introduce non-linearities and trainable parameters.

Loss and Anomaly Score Calculation

Mathematically, given the problem definition parameter, the pixel-loss function along the channel axis is defined as:

M l (I k ) ij = 1 2 ∥norm(F l t (I k ) ij ) -norm(F L-l s (I k ) ij )∥ (1)
with M l ∈ R H l * W l ,the layer l loss function as :

loss l (I k ) = 1 w l h l w l i=1 h l j=1 M l (I k ) ij (2)
and the global loss is written as:

loss(I k ) = l α l loss l (I k ) (3)
Fig. 3: Feature fusion and bottleneck architecture with α l the loss factor for the layer l.

At the testing stage, we want an anomaly score and an anomaly localization. To begin with, let's address the anomaly localization problem. The paradigm of knowledge distillation for anomaly detection and localization stipulates that, in case of anomalous sample, the teacher is able to represent the defect while the student lacks generalization capability as it has learnt to generate features only for defect-free samples. This means that the student and the teacher features will be different and this will reflect on the anomaly map calculated in equation 1. To account for the anomaly maps of all the features, we regroup all these features maps with upsampling in a single localization map denoted L map :

L map = l i=1 U pSample(M l ) (4)
A Gaussian filter is then used on L map to smooth the defect localization.

For the image anomaly score or defect detection, to avoid the prevalence of huge defects over small defects, we decided not to use the mean of the anomaly map but the max value instead based on the paradigm that a non-defective anomaly map will have no significant out-of-distribution value especially after the application of the Gaussian filter.

Reverse distillation for domain generalization

Domain generalization in deep learning refers to the task of training a model on data from multiple source domains, and then generalizing this model to perform well on a new target domain. The goal of domain generalization is to improve the generalization performance of deep learning models, especially in situations where the test data comes from a different distribution than the training data. In the specific domain of fabric defect detection, our objective is to develop an anomaly detection system capable of identifying defects in all types of textiles. However, it is important to note that our design and testing procedures are specifically tailored for homogeneous textures and may not be directly applicable to fabric designs that exhibit more intricate patterns or variations. As demonstrated in FABLE [START_REF] Thomine | FABLE : Fabric anomaly detection automation process[END_REF], the knowledge distillation architecture has outstanding domain-generalized performance for homogeneous textures. The domain generalization approach for fabric anomaly detection can be viewed as a strategy that provides additional training samples to the model. By incorporating a diverse range of fabric samples during training, the model becomes more adept at recognizing fabric-specific characteristics, such as thread patterns and lattice structures. This broader exposure helps the model generalize better across different fabric types, enhancing its ability to detect anomalies across a variety of textile materials. Indeed, while the idea of learning inherent textile patterns is appealing, achieving effective fabric anomaly detection is a complex task. There are various challenges associated with the intricacies of fabric textures, including variations in weave patterns, colors, textures, and defect types. Additionally, factors such as lighting conditions, scale, and resolution can impact the detection accuracy. Developing a robust and accurate fabric anomaly detection model requires careful consideration of these complexities and the implementation of sophisticated techniques to account for them.

As previously discussed, simpler patterns like thread and lattices are often captured by the shallower layers of the model architecture. This observation motivated the inclusion of a residual connection, which places emphasis on these particular layers. By doing so, the model can effectively capture and learn these fundamental fabric patterns, thereby enhancing its ability to detect anomalies associated with thread and lattice structures.

On the contrary, in the case of domain generalization, we believe that all deeper layers are a key to understanding the intricacies of such a descriptive problem. Consequently, we have made the decision to eliminate the residual connection mentioned in Section 3.1.1, as we consider it to be less relevant in this context.

Industrial textile dataset

In this part, we introduce the industrial fabrics dataset containing 2150 images with 6 types/colors of textile. We constitute two nomenclature for both domain-generalized and classic unsupervised anomaly detection training. We decide to use the same folder architecture as MVTEC AD dataset [START_REF] Bergmann | MVTec AD -a comprehensive real-world dataset for unsupervised anomaly detection[END_REF].

To constitute the dataset, we placed 2 cameras from different brand in front of a visiting machine with strong and nearly uniform light source.

Industry constraints

Industrial deployments are subject to certain constraints pertaining to the positioning of cameras and the intensity of light sources to prevent any disruption to the machine operator's activities. Our aim was to carefully choose an ideal pairing of light intensity and shutter duration to guarantee the capture of images that are sharp and clearly defined. However, due to variations in fabric color and the velocity of the visiting machine, our chosen light source did not consistently provide adequate brightness. As a result, certain images within the dataset exhibit a slight degree of blurriness, contributing to the creation of a more demanding and authentic dataset. This reflects the fact that achieving flawless conditions, even in an industrial setting, is frequently unachievable.

Variety of defects

As previously stated, defects within the textile industry exhibit a wide range of variations. Furthermore, our dataset will continue to expand as we accumulate additional instances defects through the ongoing use of our cameras. In addition, it is worth noting that certain anomalies present in our database may consist of dust particles that have settled on the textile, which may not typically be classified as defects in most textile industries. However, due to our inability to definitively determine whether these anomalies are genuine defects or simply dust, we have chosen to include them in the dataset. Figure 4 showcases examples of such anomalies.

Fig. 4: Images extracted from our proposed dataset with defect that could be dust

Dataset description

The dataset is composed of patches of size 256x256 extracted from 2 industrial cameras of resolution 3088x2076 and 2464x2056 respectively. The defects included in the dataset are not artificially generated but rather originate from real-world instances within the textile industry. However, we added some non-defect from the industry like writing on the fabric or the sewing attaching two pieces of fabric. These added "defects" aim to diversify the dataset and provide a broader range of defect types for training and analysis purposes. The dataset contains 6 classes with between 199 and 588 images for training and 30 and 114 for testing. The precise distribution is presented in table 1. It is important to acknowledge that the dataset poses a potential long-tail distribution challenge, especially for domain generalization training, but remains suitable for "classic" unsupervised anomaly detection where distinct models are trained for each class. An overview of the dataset can be seen in figure 5.

Existing datasets

In order to provide contextualization and to position our dataset within the broader landscape of existing datasets, it is essential to present other relevant datasets that specifically address the problem of texture anomaly detection. One of the widely used industrial unsupervised anomaly detection datasets is the MVTEC AD dataset [START_REF] Bergmann | MVTec AD -a comprehensive real-world dataset for unsupervised anomaly detection[END_REF], which comprises 10 objects and 5 textures. While 4 of these textures align well with the problem of textile anomaly detection, one texture is significantly different. Other datasets, such as TILDA [START_REF]TILDA Textile Texture-Database[END_REF], Aitex [START_REF] Silvestre-Blanes | A public fabric database for defect detection methods and results[END_REF] and DAGM [START_REF] Wieler | Weakly Supervised Learning for Industrial Optical Inspection[END_REF] , offer grayscale fabrics-like datasets for anomaly detection purposes. In addition, datasets like BTech [START_REF] Mishra | VT-ADL: A vision transformer network for image anomaly detection and localization[END_REF], NEU-CLS [START_REF] Bao | Triplet-graph reasoning network for few-shot metal generic surface defect segmentation[END_REF], KolektorSDD [START_REF] Tabernik | Segmentation-based deeplearning approach for surface-defect detection[END_REF], Magnetic Tile Dataset [START_REF] Huang | Surface Defect Saliency of Magnetic Tile[END_REF] and RSDDs [START_REF] Lu | SCueUnet: Efficient damage detection method for railway rail[END_REF] provide supplementary datasets for surface defect detection, which are not specific to textiles. Our dataset strives to offer color images of fabrics captured under authentic conditions, showcasing a wide range of real defects. With an emphasis on diversity, our dataset encompasses an extensive array of defect types, ensuring its applicability in addressing various anomaly detection challenges within the textile industry.

Category

CFA PatchCore [START_REF] Roth | Towards total recall in industrial anomaly detection[END_REF] FastFlow [START_REF] Yu | FastFlow: Unsupervised anomaly detection and localization via 2d normalizing flows[END_REF] Reverse distillation [START_REF] Deng | Anomaly detection via reverse distillation from one-class embedding[END_REF] MixedTeacher [START_REF] Thomine | MixedTeacher: Knowledge distillation for fast inference textural anomaly detection[END_REF]

Experiments

This part showcases performances of our knowledge distillation approach both in terms of AUROC score and inference time since these are the most concerning application requirements in anomaly detection.

We have also added a coverage measure to determine the actual benefit of this approach in real case scenario.

Training and inference were done on an RTX 3080ti. To ensure efficient inference for industrial applications, we employ ResNet18 and ResNet34 as the backbone architectures for our anomaly detection system. The teacher model, which serves as the reference model, is a ResNet34 trained on the ImageNet dataset. In contrast, the student model consists of a ResNet18 backbone. It is worth noting that the student model is structured in a backward manner, as explained in the part 3. We used ADAM optimizer [START_REF] Kingma | Adam: A Method for Stochastic Optimization[END_REF], a learning rate of 0.005 with scheduler to reduce the learning rate when hitting a plateau during the training. The training was done for 100 epochs with a batch size of 4. We resized all the images to a size of 256x256 keeping respectively 70% and 30% for training and validation. We kept the checkpoint with the lowest validation loss and introduce early stopping.

State of the art AUROC comparison

To evaluate our architecture, we use several datasets including MVTEC AD textures [START_REF] Bergmann | MVTec AD -a comprehensive real-world dataset for unsupervised anomaly detection[END_REF], TILDA [START_REF]TILDA Textile Texture-Database[END_REF] dataset, AITEX dataset [START_REF] Silvestre-Blanes | A public fabric database for defect detection methods and results[END_REF] and our dataset presented in section 4.

For MVTEC AD, we selected the four fabric-like textures and compared our results to the other state-of-the-art methods. The results are presented in table 2 for anomaly detection and table 3 for anomaly localization.

For the TILDA dataset, we restructured the data to conform to the MVTEC nomenclature. To ensure a fair comparison with state-of-the-art methods, the anomalib library [START_REF] Akcay | Anomalib: A Deep Learning Library for Anomaly Detection[END_REF] For the aitex dataset, we separate each 4096x256 images into 16 patches of size 256x256 and used the MVTEC AD nomenclature. The results of our experiment are presented in table 5. Examples of input images and their corresponding results are visually depicted in figure 8 Subsequently, we conduct assessments on the dataset for which our model has been meticulously tailored. This dataset is derived from an authentic industrial process, aligning with our model's primary objective of optimizing performance specifically for this designated task. Results are depicted in table 6 and images and anomaly maps are shown in figure 9. Our results on the dataset are exceptionally accurate as our model was specifically designed and optimized for it. However, we continue to record images using the camera and actively strive to gather a larger and more diverse dataset. By expanding the dataset, we aim to introduce greater challenges and complexities, which will further enhance the robustness and generalization capabilities of our model.

Fig. 9: Input image and anomaly map in industrial textile database. We showcase speed-blurry image on purpose to demonstrate our capability to deal with such quality differences

The domain generalization results are presented in table 7 Results interpretation : Results on the TILDA dataset demonstrates that our approach offers outstanding results on plain fabrics and simple patterns such as checkered or striped. The approach is therefore limited for more complex patterns that Table 8: Inference speed require more contextual information and a higher receptive field. That consideration establishes the limitation of such a restrictive architecture that emphasizes mostly high-level features along with 1×1 convolutions that do not increase the receptive field. Nonetheless, the results on MVTEC AD, AITEX and our dataset show that our model outperforms state-of-the-art approaches for "simple" textures.

Inference speed comparison

When designing the architecture, one of the primary concerns was the inference speed. Our approach of knowledge distillation offers the advantage of fast inference due to the utilization of "smaller" networks such as ResNet18 and ResNet34. Despite their reduced size, these networks can still deliver state-of-the-art performance, as demonstrated in the previous part. The efficacy of both the bottleneck embedding component and the residual connection component is significantly reliant upon the utilization of 1×1 convolutions, which exert minimal influence on the overall computational velocity. This combination allows for efficient and accelerated anomaly detection inference while maintaining high detection accuracy. The added embedding and residual connections are even lighter and therefore our method offer fast inference time.

In our specific scenario, we process high resolution images of size 3000x2000 and therefore, we will have to divide this image in smaller patches of size 256x256. The corresponding number of patches is 96 patches.

Coverage Measure

Applications of this nature are primarily designed to assist humans in the task of detecting defects in the visiting machine. By leveraging automated anomaly detection methods, the goal is to provide support and assistance to human operators in their inspection and quality control tasks. These applications aim to improve the efficiency and accuracy of defect detection processes, enabling operators to identify and address anomalies more effectively. Consequently, we have made the decision to incorporate a metric that accurately reflects the practical impact of our method on assisting industrial processes. This metric will serve as an objective measure to assess the effectiveness and efficiency of our approach in real-world scenarios.

The coverage metric quantifies the level of confidence exhibited by the model while maintaining a fixed precision and recall. For example, setting a target of 80 percent coverage with a precision of 95 percent would imply that the model will classify 80 percent of defects with a precision of 95 percent, while external assistance would be required for the remaining 20 percent.

In an industrial scenario, the tolerance for false negatives is typically extremely low, indicating that the detection of all defects is essential. Therefore, a recall of 100 percent, ensuring that no defects are missed, is a critical requirement for fabric defect detection. Additionally, to prioritize precision and avoid false positives, we have decided to set the precision at 100 percent as well. This means that all identified defects are guaranteed to be genuine, providing a high level of confidence in the accuracy of the defect detection system. In conjunction with the outcomes related to inference speed and domain generalization performance, these results collectively support the potential applicability of our approach within a real-world industrial setting, potentially involving collaboration with a human operator. The table 9 showcases coverage value for the tested MVTEC textures.

Conclusion

In summary, we have introduced a novel unsupervised anomaly detection approach utilizing knowledge distillation principles, with a specific emphasis on reverse distillation. By using thoughtful layer selection and incorporating residual connections, we have achieved exceptional outcomes on the MVTEC anomaly detection dataset, surpassing the performance of previous state-of-the-art methods specifically for these targeted classes. In order to address the challenge of fabric anomaly detection, we conducted extensive testing of our model using a diverse fabric defect database. Our findings validate the superior performance of our unsupervised fabric defect detection approach. However, the architectural design of our approach is specifically tailored for the problem of plain fabric defect detection and is limited when dealing with big patterns such as drawings. Due to the lack of existing databases tailored to our specific fabric anomaly detection problem, which involves high-resolution color images captured by industrial cameras in real-world scenarios, we have taken the initiative to create a novel dataset. It comprises 2150 images of diverse fabric types, encompassing several commonly encountered defect types prevalent in the fabric industry. Presently, this dataset is limited in terms of the pool of defects gathered since in a fabric factory some defects are rarer than others. It will, therefore, undergo regular updates to incorporate additional classes and to introduce new types of defects. This continuous expansion aims to enhance the dataset diversity and ensure its relevance to evolving fabric anomaly detection research and industry requirements. This dataset will be freely available for the scientific community.(https://github.com/SimonThomine/IndustrialTextileDataset)

Fig. 5 :

 5 Fig. 5: Images of our proposed datasets, (upper) defect-free images, (lower) defective images. Some images are a bit blurry due to motion blur. Other kind of defective samples are presented in the experiment part.

Fig. 7 :

 7 Fig. 7: Input image and anomaly map in all 8 TILDA textile. As expected, the model is too restrictive to handle textures with patterns like flowers

Fig. 8 :

 8 Fig. 8: Input image and anomaly map in all 4 Aitex textile

Table 1 :

 1 Industrial textile dataset distribution

	Category	total train testGood testDefective
	type1cam1	386	272	28	86
	type2cam2	257	199	19	39
	type3cam1	689	588	54	47
	type4cam2	229	199	19	11
	type5cam2	298	199	19	80
	type6cam2	291	199	19	73
	total	2150	1656	158	336

Table 2 :

 2 Anomaly detection results with AUROC on MVTEC fabric-like textures

	Ours

Table 3 :

 3 Anomaly localization results with AUROC on MVTEC fabric-like textures

  was employed. The results are shown in table 4. Examples of input Fig. 6: Input image and anomaly map in the 4 fabric-like MVTEC AD textures

	Tilda	fabric1	fabric2 fabric3 fabric4 fabric5 fabric6 fabric7 fabric8 mean
	reverseDistillation[10]	94.8	88.2	91.4	59.6	67.4	78.7	78.6	84.5	80.4
	ours	96.9	95.8	92.5	75.0	87.2	88.6	70.7	61.9	83.6

Table 4 :

 4 Anomaly detection results with AUROC on Tilda textures images and results can be seen in figure7

Table 5 :

 5 Anomaly detection results with AUROC on Aitex textures

Table 6 :

 6 Anomaly detection results with AUROC on Industrial textile dataset

	ITD	type1cam1 type2cam2 type3cam1 type4cam2 type5cam2	type6cam2 mean
	reverseDistillation [10]	100	100	98.3	100	100	100	99.7
	ours		100	100	100	100	100	100	100
	Category	Epi-FRC+ [47] EISNet+[48] DGTSAD [49] Fable [35] Ours
	carpet	91.6		98.2	94.3	98.5	99.3
	tile	95.1		85.1	99.4	96.5	98.8
	wood	94.1		97.9	96.2	99.9	99.7
	leather	100		100	100	99.1	99.9
	Mean	95.2		95.3	97.5	98.5	99.4

Table 7 :

 7 Domain generalization comparison

	Batch size	Inference time FPS
	16 : time per patch	0.7ms	1450
	32 : time per patch	0.5ms	2000
	whole image (96 patches)	45ms	22

  To report industrial plausible inference time, we processed in batch of size 32 and reported the inference time for the whole image processing (3 batches) as shown in table 8.

	Category	Coverage Coverage +/-0.02 Coverage +/-0.05
	carpet	100	97	91
	tile	91	81	71
	wood	99	96	91
	leather	100	100	100
	Mean	97.5	93.5	88.3

Table 9 :

 9 Coverage measure with fixed precision and recall at 100 percent. The +/-indicates the score tolerance to consider that 2 samples are ambiguous. Base coverage is with score tolerance 0


domain: spi info math

text:
Valentin Noël Satie
email: valentin.noel1@ens-paris-saclay.fr

Faster Simulation Approach Dedicated to Infrared Thermographic Inspection of Anisotropic Delaminated Planar Pieces

Keywords: destructive testing, infrared thermography, anisotropic materials, isotropic materials

In this work, we propose an analytical model based on an approximation of the heat transfer within a part by successive and simultaneous convolution operations for the simulation of thermographic inspection (single or double-sided). Infrared thermography has already proven to be an important method for non-destructive evaluation, due to the fact that it provides information in an immediate, fast and low cost manner. The temperature distribution is approximated by its expansion on a tensor. The configurations treated by this model are laminated flat parts affected by thin delamination defects. The sources considered are lamps providing thermal excitation to the surface of the inspected part. The description of the delamination defects as thin air gaps between the layers of the part is equivalent to the introduction of a surface resistance to the thermal flow, thus allowing their treatment by the modal approach applied without additional discretization. The proposed model, although developed particularly for fibrous polymeric materials, has been generalized to be applicable to all types of materials (i.e. (an-)isotropic).

I. INTRODUCTION

Thermal/infrared non-destructive testing (T/INDT) techniques have received increasing attention in recent years due to their advantages of fast, high-resolution, non-contact testing and technical advances in infrared cameras and data acquisition equipment. The detection of irregularities is based on the principle that all bodies emit infrared radiation when their temperature is above 0 K. The emitted infrared radiation can be measured by infrared cameras, and the images are then analyzed for defect detection and characterization. Irregularities present within the part under study affect the rate of heat diffusion, resulting in thermal contrast on the surface(s) of the part. By analyzing the alterations or contrast in the thermal pattern of the material surface, information about subsurface defects can be obtained. The simulation of TNDT procedures involves, in a first step, the solution of the thermal conduction problem in the considered part, with and without defects, in order to obtain the temperature distribution at the interfaces of the part, which constitutes the measurement. The complete solution of this problem can be obtained by using a numerical technique such as the finite element method (FEM) or the finite integration technique (FIT) [START_REF] Wen | Finite integration method for partial differential equations[END_REF]. Nevertheless, in practical situations, it is often useful to forgo the detailed information of the full numerical solution in favor of fast analytical or semi-analytical approximations, like the TREE approach proposed in [START_REF] Ratsakou | Fast simulation approach dedicated to infrared thermographic inspection of delaminated planar pieces[END_REF], which retain the essence of the heat flow behavior. In other words, the popularity of FEM and FIT methods is only matched by their robustness and large computational cost. This paper proposes an analytical model based on an approximation of the heat transfer equation by successive convolution operations. This singular approach, stemming from an intuition and from the approach used by Comsol for the modelling of convective exchanges as put forward in [START_REF] Fang | A method of defect depth estimation for simulated infrared thermography data with deep learning[END_REF], is based on a homogenization of the constitutive parameters of the fibered polymeric materials (CFRP) with the final objective of being a reliable method while remaining from a computational point of view very light, which would help to tend towards a fast use and thus perfectly adapted to the generation of data sets for inverse problems.

II. CONDUCTION HEAT TRANSFER IN (IN-)HOMOGENEOUS

MATERIALS

To calculate the temperature field in homogeneous materials, the classical heat conduction solutions have been summarized by Carslaw and Jaeger [START_REF] Carlslaw | Conduction of heat in solids[END_REF] 1 a (i)

∂T (i) (x, y, z, t) ∂t = ∂ 2 T (i) (x, y, z, t) ∂x 2 + ∂ 2 T (i) (x, y, z, t) ∂y 2 + ∂ 2 T (i) (x, y, z, t) ∂z 2 , i = 1, ..., M (1)
T (i) | t0 = T in (2)
T (i) | zj = T (i+1) | zj (3) -σ (i) ∂T (i) ∂z zj = -σ (i+1) ∂T (i+1) ∂z zj (4) • T (i) temperature in the i-region • M number of layers • T in initial piece temperature • a (i) = σ (i) /(ρ (i) C (i)
p ) thermal diffusivity in the i-region These equations are therefore applicable in the case where the material of interest is homogeneous. In the case where we are interested in anisotropic materials, these equations are no longer applicable. Indeed, in the case of isotropic materials, the observed thermal conductivity is a scalar (varying or invariant depending on the assumptions). In other words, whatever the direction from which we observe the heat flow, the conductivity will be identical in the whole material, independently of the direction and the layer studied. On the other hand, in the case of polymeric materials reinforced by fibers (carbon, glass, etc.), the conductivity sigma is no longer representable as a scalar [START_REF] Macias | Thermal characterization of carbon fiber-reinforced carbon composites[END_REF] and [START_REF] Bard | Influence of fiber volume content on thermal conductivity in transverse and fiber direction of carbon fiber-reinforced epoxy laminates[END_REF]. Indeed, in a CFRP material, each layer is composed of an alignment of carbon fibers located within a matrix (considered here as expoxy), each layer having its own fiber orientation. The thermal conductivity within the expoy matrix is of the order of σ m = 0.25W/mK, within the carbon fiber σ f = 129 W/mK.

Therefore, the use of a scalar conductivity is impossible. Let us observe a square cell of surface S, composed of an epoxy matrix representing a surface S 1 and a carbon fiber of surface S 2 . The fraction of surface of the fiber is thus equal to S f = S2 S1+S2 . If we consider a square piece of length L, V f = S f * L. Therefore, to determine the thermal conductivity σ // parallel to the fiber

σ // = σ f V f + (1 -V f )σ m (5)
Following the same logic, we can determine the transverse conductivity k ⊥

1 σ ⊥ = V f σ f + (1 -V f ) σ m (6)
It is important to note that this value is the lower limit of the transverse conductivity. This homogenization allows us to define the parameters σ // and σ ⊥ as a function of the conductivities of the epoxy matrix σ m and of the fiber σ f , with the knowledge a priori of a ratio of 5 between the parallel and transverse conductivities. One of the major points of contention in the egstion of thermal conductivities is the necessity to consider a conductivity through fold σ CP such that σ CP ̸ = σ CP or not. In this paper, we consider a conductivity σ CP ̸ = σ CP . From these newly defined parameters, it is now possible to define the different conductivities in any direction of the part as a function of the orientation θ of the carbon fibers of the studied layer, as first introduced in [START_REF] Mityushev | Analytical methods for heat conduction in composites[END_REF]

] σ =   σ // cos 2 θ + σ ⊥ sin 2 θ σ // -σ ⊥ 2 sin2θ 0 σ // -σ ⊥ 2 sin2θ σ // cos 2 θ + σ ⊥ sin 2 θ 0 0 0 σ CP   (7
) with σ the conductivity tensor, which will be used for the Comsol implementation when declaring the conductivities in the layer.

This approach, called homogenization, leads to several remarks. Such an approach allows us to simplify the architecture of our part by allowing us to no longer consider the epoxy matrix and the carbon fiber separately. This has the effect of facilitating the implementation of finite element approaches such as Comsol and simultaneously decreasing the required computational costs. However, the disadvantage of homogenization is that it brings some additional uncertainty to the model of the part, which was estimated at about 4% dans [START_REF] Li | On homogenization of fiber-reinforced polymers[END_REF].

III. CONVOLUTION APPROACH

The development of convolution kernels for this application consists of three steps. The heat transfer in a CFRP material composed of several plies is defined as follows: for each ply, different influences are to be considered: the heat transfer from the layer above and the layer below, as well as the heat transfer in the layer itself. In the case of the presence of defect(s) at the interfaces surrounding the studied layer, the influence of these defects must also be taken into account.

Therefore, we can divide the different influences into five distinct parts:

• influence layer n -1

• influence layer n + 1

• influence layer n • influence of defects on interfaces n and n + 1 These influences are characterized by the layer from which the incoming heat transfer originates. That is, the influence of the n -1 layer is characterized by the n -1 layer and thus the convolution kernel corresponding to this layer. Similar reasoning is applied for the n + 1 layer. As for the other influences, they are all characterized by the studied layer, the n layer. Thus, in the case where we study the n layer such that the n -1 and n + 1 layers exist, five influences and three convolution kernels will be used. In the case where the n -1 or n + 1 layer does not exist (edges of the part), only four influences take place and two convolution kernels are used.

These convolution kernels are developed as follows: each layer has the same conductivity components in the direction of the fiber and orthogonal to it. However, each layer has its own fiber orientation. With this information, it is possible to define the convolution kernels corresponding to each of these layers. Indeed, with the expression of the Gaussian kernel being:

G 2D (x, y; σ sd ) = 1 2πσ 2 sd e -x 2 +y 2 2σ 2 sd ( 8
)
with σ sd the standard deviation, σ 2 sd the variance, we can determine the Gaussian kernel for zero orientation with the parallel and transverse conductivities σ / and σ / :

G 2D (x, y; σ sd ) = 1 2πσ 2 sd e - σ ⊥ ×x 2 +σ // ×y 2 2σ 2 sd (9)
To obtain the convolution kernels oriented according to the fiber orientation, it is sufficient to define the fiber orientation for each layer and to perform a rotation of the same angle to the convolution kernel.

In order to visualize the implementation more easily, pseudo-code algorithms are available in Alg.1, 2 and 3. The kernels have been named as follows in our implementation:

• h 1 : convolution kernel of the layer n -1 • h 2 : convolution kernel of the layer n • h 3 : convolution kernel of the layer n + 1 It is necessary to note that these kernels are normalized and that the sum of the three kernels (thus the three influences) is unitary

3 i=1 l l=1 W w=1 h i (l, w) = 1 ( 10
)
where L is the length and W is the width of the top surface of the part under study.

Algorithm 1 Pseudo-code for heat transfer model for n ̸ = (1, 10)

1: procedure HEAT TRANSFER(C, I, h 1 , h 2 , h 3 , τ obs , ∆τ ) 2:
while j < τ obs ∆τ do 3:

C τ +1 n ← C τ n * h 2 + (C τ n .× ∼ I n ) * h 2 + (C τ n .× ∼ I n+1 ) * h 2 + (C τ n-1 . × I n ) * h 1 + (C τ n+1 . × I n+1 ) * h 3 4: j ← j + 1 5:
end while

6:

return C n 7: end procedure Algorithm 2 Pseudo-code for heat transfer model for n = 1

procedure HEAT TRANSFER(C, h 1 , h 2 , h 3 , τ obs , ∆τ ) 2: while j < τ obs ∆τ do C τ +1 n ← C τ n * h 2 + (C τ n .× ∼ I n ) * h 2 + (C τ n .× ∼ I n+1 ) * h 2 + (C τ n+1 . × I n+1 ) * h 3 4: j ← j + 1 end while 6: return C n end procedure
One of the problems we have encountered is the setting of the standard deviation σ. Empirically, we observe that the value leading to the lowest estimation error is σ = 2.1, or σ = k // -k ⊥ .

A. Convolution approach: hypothesis

The presented approach, called convolution, calculates at each time step the temperature evolution within each layer of the studied part. The constraint of this approach is that the temperatures given in such a context are the temperatures at Algorithm 3 Pseudo-code for heat transfer model for n = 10 procedure HEAT TRANSFER(C, h 1 , h 2 , h 3 , τ obs , ∆τ ) while j < τ obs ∆τ do 3:

C τ +1 n ← C τ n * h 1 + (C τ n .× ∼ I n ) * h 2 + (C τ n .× ∼ I n+1 ) * h 2 + (C τ n-1 . × I n ) * h 1 j ← j + 1 end while 6:
return C n end procedure the center of each layer. Thus, in order to perform the different convolution operations simulating the heat transfer within the studied material, it is necessary to initialize the temperature within the first layer. Similar to the implementation in Comsol, both the ambient temperature and the initial temperature of the part are equal to T 0 = 25 • C.

To do this, we will first determine the evolution of the temperature on the upper and lower surfaces of the top layer of our model, of thickness L 1 , using the equations in [START_REF] Wen | Finite integration method for partial differential equations[END_REF]. If we call the initial temperature distribution on the top surface of an isotropic material of thickness L 1 , T (x, 0), the temperature distribution on this same surface at any other time t is given by Carslaw and Jaeger in [START_REF] Carlslaw | Conduction of heat in solids[END_REF] as:

T (x, t) = 1 L L 0 T (x, 0)dx + 2 L ∞ n=1 exp -n 2 π 2 αt L 2 × cos nπx L L 0 T (x, 0) cos nπx L dx, (11)
where α is the thermal diffusivity in cm 2 /s. The power density Q of the energy source is here expressed in cal/cm 2 . The assumption made here is that if Q is instantaneously and uniformly absorbed by a small thickness g at the top surface (x = 0) of the part under study, the temperature distribution at that instant is given by:

T (x, 0) = Q/ρCg for 0 < x < g T (x, 0) = 0 for g < x < L
With this initial condition, the equation 11 can be written as:

T (x, t) = Q ρCL 1 + 2 ∞ n=1 cos nπx L sin(nπg/L) (nπg/L) × exp -n 2 π 2 L 2 (12)
where rho is the density in g/cm 3 and C is the specific heat in cal/g • C. Because g is very small for opaque materials, the simplification sin(nπg) ≃ nπg. Moreover, with thermal conductivity K = αρC and a pulse duration τ , giving a Dirac pulse energy W = Q × τ , the equation 12 can be written:

T (x, t) = W α KL 1 + 2 ∞ n=1 exp -n 2 π 2 L 2 (13)
The knowledge of these different parameters and that of the thermal diffusivity of the material (a = 3.15times10 -7 m 2 /s) allows us to determine the evolution of the temperature at the point of impact of the flash lamp beam on the upper surface of the part via the equation Eq.14 and the temperature of the lower surface via the equation Eq.15 in [START_REF] Vavilov | A novel approach for one-sided thermal nondestructive testing of composites by using infrared thermography[END_REF]:

T = W a σ CP L (1 + 2 ∞ n=1 e -n 2 π 2 F0 ) (14)
T = W a σ CP L (1 + 2 ∞ n=1 (-1) n e -n 2 π 2 F0 ) (15)
with F 0 = aτ L 2 the Fourier number and sigma CP being the conductivity through plies.

From the equation 15 it is thus possible to determine the temperature evolution of the lower and upper surface of the first layer of our part. This information will allow us to define two key parameters for our convolution approach.

Indeed, the fact that we have the temperature evolution of the lower surface of the first layer of the part allows us to determine when is observable the first temperature rise on the lower surface. This first temperature rise being observed at a time t = ∆τ , we can thus consider that the necessary time step between each operation of calculation of heat transfer from layer to layer is carried out in a periodic way with a period equal to ∆τ . The knowledge of the time step allows us to define another parameter. Our convolution approach thus carries out the convolution operations in a periodic way with a period ∆τ , the initialization of the temperature of the upper layer of our part must correspond to the temperature at the center of this same layer at t = ∆τ . Thus, by taking again the equation Eq.13 and by replacing the length of a layer L by the value of the half of its length (L/2), we obtain the evolution of the temperature at the center of the first layer, with as particular interest the temperature at t = ∆τ , which will be the initial temperature of the first layer of our model.

IV. NUMERICAL SIMULATIONS

In this section, the different implementations and the characteristics related to the numerical simulations performed will be described. It is important to note that for all the parametric values given, these are common to both implementations: the first one with our convolution approach and the second one with Comsol Multiphysics.

A. Material characteristics

The purpose of this sub-section is to define the thermal constants (conductivities, densities, specific heats) and the geometries of the parts, their organizations and the characteristics of the defects. For the different numerical simulations, we will use three different solids: steel, aluminum and CFRP. The number of layers for each of the studied parts has been determined in an arbitrary way, as the proposed approach can deal with a theoretically unlimited number of layers, with for example a material having 22 layers in [START_REF] Saeed | Automatic defects detection in cfrp thermograms, using convolutional neural networks and transfer learning[END_REF].

B. Flash lamp characteristics

Several types of illumination can be considered in a thermal inspection framework. In the proposed case, we use a flash lamp, a frequently used approach, because of its simple mathematical modeling [START_REF] Chiwu | Inspection on cfrp sheet with subsurface defects using pulsed thermographic technique[END_REF] and [START_REF] Waugh | Modelling and evaluation of pulsed and pulse phase thermography through application of composite and metallic case studies[END_REF].

The source considered for the thermal inspection is a flash lamp located at 40 cm from the studied part. The surface illuminated by this lamp is a spatial support of slightly smaller dimension than the surface of the studied part. As described, the studied parts having been arbitrarily chosen of surface 50 × 50 mm, the corresponding spatial support is here of 46 × 46 mm. The illumination on this spatial support, which is assumed and described as uniform, corresponds to a pulse of duration ∆τ with a power density Q. All the quoted parameters are grouped in the table Tab.VI.

C. Comsol hypothesis

In addition to the parameters shared and described by Tab.I-V, the Comsol Multiphysics implementation has other subtleties.

The numerical solution proposed by COMSOL Multiphysics is based on a finite element method (FEM). In the case of our study, we have performed a heat transfer study in solids (module "Heat Transfer in Solids"). The boundary conditions chosen are the so-called Dirichlet conditions, as expressed in [START_REF] Moës | Imposing dirichlet boundary conditions in the extended finite element method[END_REF]

T (i) | x=0 = T (i) | x=L = 0 (16)
T (i) | y=0 = T (i) | y=W = 0 (17)
where L and W are the height and width of the room respectively.

Like the Comsol Multiphysics implementation, the same boundary conditions are applied in the implementation of the convolution approach.

The equation observed by COMSOL Multiphysics is:

ρC p ∂T ∂t + ∇ • q = Q (18)
q = -σ∇T , σ tenseur des conductivités. First, we have defined the architecture of our part. Our part is a layered material, each layer having a thickness of 0.2mm. Each delamination is implemented as a thin layer (Thin Layer mode) of resistance R s = d s /σ s , d s its thickness and σ s its thermal conductivity, modeled as a surface condition

-n d • q d = - T u -T d R s (19) -n u • q u = - T d -T u R s (20)
Subsequently, we define a spatial support of (46 × 46 mm) using the plane creation tool. In the analysis, it is assumed that the spatial support is uniformly heated. This is an idealized situation. On this plane is defined a surface illumination by a flash lamp of power density Q = 10 6 W/m 2 for a duration of τ = 1 ms. To do this, a step function of amplitude Q and duration τ have been implemented. The corresponding equation is:

-n • q = Q b (21) avec Q b = Qδ(t -τ ), [W/m 2 ].
In order to compare the heat transfer results of Comsol vs. our method, the study was given the same time step as the one used by our method.

D. Comsol: comparisons

This sub-section is reserved for the comparison of the temperature evolution results at the bottom and top layers of the studied part. We consider here that both the top and bottom surface of the part are accessible. In the case where the bottom surface of the tested sample is not accessible, the simulations must be performed in reflection mode. In this configuration, the heat source must be moved, so that the camera has a view of the surface under test during excitation. This will affect the spatial distribution of the heating power density.

For the tests, three configurations will be used, shown in Fig. 1. The part consists of two layers, each with a thickness of 2.5 : mm. The first layer is composed of aluminum, while the second layer is composed of steel, with the parameters listed in Tab.I and Tab.III. The evolution of the temperatures within the lower and upper layers for the n • 1 configuration is illustrated in Fig. 2 and in Fig. 3 for the n • 2 configuration.

It is noticed that in the case of the configuration n irc 1, the evolution of the temperature given by our method is very close (mean square error M SE = 0.01. In the case of the n irc 2 configuration, the similarity of the curves is less (M SE = 0.14). This is related to the fact that in the second configuration the defect is located in the center of the part, below the point at which the measurement of the temperature evolution is taken, coupled with the fact that the delamination is located at the interface between two different materials, thus having particularly different thermal conductivities.

2) Case 2: inhomogeneous CFRP part: The part consists of four layers, each with a thickness of 0.2 mm with the parameters listed in the tables Tab.I, Tab.II and Tab.III. The evolution of the temperatures within the lower and upper layers for the n • 2 configuration is illustrated in Fig. 4 and in Fig. 5 for the n • 3 configuration. The observations made for the bi-material part for the second configuration are not applicable to the part composed of CFRP. Indeed, the similarity of the curves (M SE = 0.01) for both the n • 2 and n • 3 configurations. These results confirm that the problems encountered in the previous case were related to the difference in thermal conductivity between the two materials. The deeper the delamination, the lower the equilibrium temperature reached at the end of the observation. This can be explained by the fact that, the closer a defect is to the surface, the greater the reflected heat flux will be, which will cause a greater increase in the surface temperature of the part.The differences between these curves are minimal, for example between the one obtained for a defect at z = 0.2 mm and the one for a defect at z = 0.6 mm, the mean square error M SE = 0.06. From this observation it is possible to note that, in an inverse problem approach, the determination of the exact depth of a defect within a part will require a very particular precision.

2) Case 4: two inhomogeneous CFRP parts, one oriented and the other uniaxial with a central defect: The part consists of four layers, each with a thickness of 0.2 mm with the parameters listed in Tab.I, Tab.II and Tab.III. The evolution of the temperatures within the lower and upper layers for the n • 2 configuration as a function of the orientation of the studied part (orientation defined in Tab.V and otherwise unidirectional) is illustrated in Fig. 7.

The difference in the temperature evolution in the upper layer between an oreinted part and a unidirectional part is moderate, with a M SE = 0.26. The difference is indeed significant, linked to the difference between a unidirectional material, thus quasi-isotropic, and an anisotropic material.

3) Case 5: inhomogeneous CFRP part, with and without central defect: The part consists of four layers, each with a thickness of 0.2 mm with the parameters listed in the tables Tab.I, Tab.II and Tab.III. The evolution of the temperatures within the lower and upper layers for the n • 2 configuration as a function of the presence (or not) of the central defect is illustrated in Fig. 8.

The presence of a defect in a part will have a moderate impact on the temperature evolution curves in the upper and Fig. 8: Temperature evolution of the top surface depending on the presence of a defect lower layers. Indeed, in the case where a defect is present, a slight increase in temperature is observed after the reflection of the chalking flux on the defect, and in parallel a decrease in temperature in the lower layer. The M SE between these curves (average lower and upper layer) is M SE = 0.18, slightly lower than when comparing an oriented part with a unidirectional part. It is important to specify that the defect is in this case very thin, with a thickness of 10 -3 mm. This parameter has an influence, which will be studied in case n°6. The figure Fig. 8 allows us to observe that with the presence (or not) of defect, the equilibrium temperature remains the same. 4) Case n°6: inhomogeneous CFRP part, with central defect with different thicknesses: The part consists of four layers, each with a thickness of 0.2 mm with the parameters listed in Tab.I, Tab.II and Tab.III. The evolution of the temperatures within the lower and upper layers for the n • 2 configuration as a function of the thickness of the central defect is illustrated in Fig. 9. It is observed that the temperature evolution is impacted by the thickness of the defect. Indeed, the thicker the defect observed, the greater the heat increase observed in the upper layer. This knowledge, merged with those obtained by the study of the other cases presented in this article, allow us to have a priori information, useful for an inverse problem solving approach of determination and characterization of defects within an inhomogeneous part by thermographic inspection.

It is necessary to note that the numerically calculated temperatures have a high accuracy. In order to simulate the raw thermograms obtained experimentally, the temperature values have to be rounded up to one or two decimal places. In this way, a minimum temperature resolution is assumed and the sensitivity of an infrared camera is simulated. Once the temperature values resulting from the calculations are rounded according to the assumed sensitivity, they can be processed by the proposed techniques. Indeed, if the sensitivity of an infrared camera is not simulated assuming a minimum temperature resolution, the final results are more favorable but less realistic.

F. Computational costs

The main interest of this convolution approach was to propose an alternative with lower computational costs. In order to provide an objective comparison with the performance of the Comsol Multiphysics FEM approach, the same observation period of the heat transfer evolution and the same time step were used. In order to obtain a reliable representation of the computational costs, each approach was run with different heat transfer configurations in inhomogeneous media. It is noted that the Comsol Multiphysics simulations were run with a normal mesh. The resulting average computational cost results on a computer with an Intel Core i7 10875H CPU (2.30 From the point of view of the computational costs involved, the convolution approach is, as expected, much lighter than the FEM approach. Such a decrease in computational costs while using a CPU implementation opens doors for the fast generation of heat transfer datasets.

V. CONCLUSION

A convolution approach to heat transfer in an anisotropic material has been described in this paper. The results obtained and their quasi-coincidence with the results obtained by Comsol Multiphysics have been demonstrated for two different configurations. Tests were then carried out to determine the influence of the different parameters inherent to the implementation of the heat transfer problem in a fibered polymeric part. The impact of these parameters could be quantified, allowing to gather a priori information that can be used to define an inverse problem solving for the determination and characterization of defects in a CFRP.

At the same time, the computational efficiency of our convolution approach has been highlighted, underlining the possibility of developing dynamic heat transfer data sets in CFRPs in a low-cost manner. Such datasets can be exploited to determine the presence of defects in a CFRP and its parameters, for example through a neural network approach [START_REF] Duan | Automated defect classification in infrared thermography based on a neural network[END_REF]. In future work, improvements in the management and duration of lighting, which is defined here as uniform, can be considered, as well as the development of solutions to the underlying inverse problem.

VI. ACKNOWLEDGEMENTS

Fig. 1 : 3 1)

 13 Fig. 1: Sketches of the diferent configurations: a) Configuration n • 1, b) Configuration n • 2, c) Configuration n • 3

Fig. 2 :Fig. 3 :

 23 Fig. 2: Comparison plots of the temperatures' evolution at the central point of the top and bottom layers for configuration n • 1

Fig. 4 :Fig. 5 :

 45 Fig. 4: Comparison plots of the temperatures' evolution at the central point of the top and bottom layers for configuration n • 2

E. Comsol: other tests 1 )Fig. 6 :

 16 Fig. 6: Temperature evolution of the top surface depending of the depth of the central defect

Fig. 7 :

 7 Fig. 7: Temperature evolution of the top surface for an unilateral piece and an oriented piece

2 Fig. 9 :

 29 Fig. 9: Temperature evolution of the top surface depending of the thckness of the central defect at z = 0.4 mm

  The delaminations are considered as exclusively composed of air. The characteristics of these different solids and gases are available in the table Tab.I for homogeneous materials and in the table Tab.II.

		Isotropic Materials Characteristics
	Material	Thermal Conductivity σ	Density ρ	Heat Capacity Cp
		[W/mK]
	Steel	44.5	7850	475
	Aluminium	237	2707	897
	Air	0.02454	1.1843	1005

TABLE I :

 I Isotropic Materials Characteristics

		Anisotropic Material Characteristics
	Material Thermal Conductivity σ	Density ρ	Heat Capacity Cp
		[W/mK]
	CFRP	σ // = 2.71, σ ⊥ =	1530	917
		0.61, σ CP = 0.53

TABLE II :

 II Anisotropic Materials CharacteristicsThe choice was made to give all test pieces the same surface size. Depending on the materials used, the thickness of each part differs, as shown in TableTab.III.

		Studied Layers Geometries
	Material	Width [mm]	Height [mm]	Thickness [mm]
	Steel	50	50	2.5
	Aluminium	50	50	2.5
	CFRP	50	50	0.2

TABLE III :

 III Studied Layers GeometriesDuring the different numerical simulations, different configurations and positions of defects in the studied parts will be tested. We will therefore refer to the names given in the table Tab.IV for each delamination.

		Studied Defect Characteristics [mm]
	Name	Width	Height	Depth	Thickness
	Defect 1	10	10	0.4	10 -3
	Defect 2	6	10	0.4	10 -3
	Defect 3	10	10	0.2	10 -2
	Defect 4	6	10	0.6	10 -2

TABLE IV

 IV

	: Studied Defect Characteristics
	More specifically, during the CFRP study, different fiber
	stacking sequences will be implemented, with their complete
	description according to the corresponding numerical simula-
	tion in the table Tab.V.
	Carbon-Fiber Reinforced Polymer Piece
	Number of layers	Stacking Sequence
	4	[0 • , 45 • ,90 • ,-45 • ]

TABLE V :

 V Carbon-Fiber Reinforced Polymer Piece, Characteristics

TABLE VI :

 VI Flash lamp characteristics

  GHz) are available in Table Tab.VII.

	Computational costs [s]
	Convolution approach	2.36
	Comsol Multiphysics approach	139.44

TABLE VII :

 VII Carbon-Fiber Reinforced Polymer Heat transfer, Computational costs