We extend recent work on mathematical morphology for signal processing on weighted graphs, based ... more We extend recent work on mathematical morphology for signal processing on weighted graphs, based on discrete tropical algebra. The framework is general and can be applied to any scalar function defined on a graph. We show applications in structure tensors analysis and the regularisation of greyscale images.
Equivariance of neural networks to transformations helps to improve their performance and reduce ... more Equivariance of neural networks to transformations helps to improve their performance and reduce generalization error in computer vision tasks, as they apply to datasets presenting symmetries (e.g. scalings, rotations, translations). The method of moving frames is classical for deriving operators invariant to the action of a Lie group in a manifold. Recently, a rotation and translation equivariant neural network for image data was proposed based on the moving frames approach. In this paper we significantly improve that approach by reducing the computation of moving frames to only one, at the input stage, instead of repeated computations at each layer. The equivariance of the resulting architecture is proved theoretically and we build a rotation and translation equivariant neural network to process volumes, i.e. signals on the 3D space. Our trained model overperforms the benchmarks in the medical volume classification of most of the tested datasets from MedMNIST3D.
In discrete signal and image processing, many dilations and erosions can be written as the max-pl... more In discrete signal and image processing, many dilations and erosions can be written as the max-plus and min-plus product of a matrix on a vector. Previous studies considered operators on symmetrical, unbounded complete lattices, such as Cartesian powers of the completed real line. This paper focuses on adjunctions on closed hypercubes, which are the complete lattices used in practice to represent digital signals and images. We show that this constrains the representing matrices to be doubly-0-astic and we characterise the adjunctions that can be represented by them. A graph interpretation of the defined operators naturally arises from the adjacency relationship encoded by the matrices, as well as a max-plus spectral interpretation.
The present dissertation compares the human visual perception to computer vision algorithms based... more The present dissertation compares the human visual perception to computer vision algorithms based on a mathematical model called a contrario theory. To this aim, it focuses on two visual tasks that are at the same time easy to model and convenient to test in psychophysical experiments. Both tasks consist in the perceptual grouping of oriented elements, namely Gabor patches. The first one is the detection of alignments and the second one extends to curves, that is to say to more general arrangements of elements in good continuation. In both cases, alignments and curves, psychophysical experiments were set up to collect data on the human visual perception in a masking context. The non-accidentalness principle states that spatial relations are perceptually relevant when their accidental occurrence is unlikely. The a contrario theory is a formalization of this principle, and is used in computer vision to set detection thresholds accordingly. In this thesis, the a contrario framework is applied in two practical algorithms designed to detect non-accidental alignments and curves respectively. These algorithms play the part of artificial subjects for our experiments. The experimental data of human subjects is then compared to the detection algorithms on the very same tasks, yielding two main results. First, this procedure shows that the Number of False Alarms (NFA), which is the scalar measure of non-accidentalness in the a contrario theory, strongly correlates with the detection rates achieved by human subjects on a large variety of stimuli. Secondly, the algorithms' responses match very well the average behavior of human observers. The contribution of this thesis is therefore two-sided. On the one hand, it provides a rigorous validation of the a contrario theory's relevance to estimate visual thresholds and implement visual tasks in computer vision. On the other hand, it reinforces the importance of the non-accidentalness principle in human vision. Aiming at reproducible research, all the methods are submitted to IPOL journal, including detailed descriptions of the algorithms, commented reference source codes, and online demonstrations for each one. Les années de thèse ont été pour moi une période privilégiée, durant laquelle il m'a été offert d'apprendre en me consacrant pleinement à un projet unique, et ce dans les meilleures conditions. Je suis sincèrement reconnaissant envers tous ceux qui ont participé à me donner cette chance. J'ai trouvé au CMLA un cadre idéal pour travailler dans la sérénité et la bonne humeur. Je le dois en grande partie à Carine, Christophe, Micheline, Nicolas P., Sandra, Véronique et Virginie, dont la bienveillance et la compétence ont assuré le bon déroulement de mon séjour au laboratoire, tant sur les plans logistique et administratif qu'humain. Je remercie également tous les professeurs du CMLA pour leur accessibilité et l'image positive qu'ils m'ont donnée du métier d'enseignant-chercheur. Mes remerciements vont tout particulièrement à mes directeurs Jean-Michel Morel et Rafael Grompone von Gioi, qui m'ont laissé la plus grande liberté dans mon travail tout en me prévenant d'éventuels égarements par leurs idées et leurs conseils. Ils ont fait preuve d'une disponibilité exceptionnelle que je souhaite à tous les doctorants. Par ailleurs, je garderai un souvenir enthousiaste du groupe "images" du CMLA, où la qualité scientifique va de paire avec un esprit de partage et d'entraide. Merci également aux chercheurs du CibPSI, en particulier Alejandra Carboni et Alejandro Maiche, et à Gregory Randall qui a encouragé cette collaboration avec l'Universidad de la República de Montevideo. Les journées au laboratoire n'auraient pas été aussi plaisantes sans mes camarades du CMLA: entre risotti, débats et plaisanteries, la cuisine a été le lieu des meilleures pauses grâce à vous. Enfin, merci à toutes celles et tous ceux que je devrais remercier chaque jour-ils se reconnaîtront.
HAL (Le Centre pour la Communication Scientifique Directe), Aug 29, 2020
Mathematical Morphology and Logics: In propositional logics, considering the lattice of formulas,... more Mathematical Morphology and Logics: In propositional logics, considering the lattice of formulas, morphological operators will act on formulas (and on their models). In modal logics, dilation and erosion can define modal operators. Such examples will be described in several logics. Then we will show how they can be used to define revision operators satisfying the AGM postulates, merging operators, or explanatory relations.
The algorithm presented in this paper is an application of a general framework for morphological ... more The algorithm presented in this paper is an application of a general framework for morphological processing of signals on weighted graphs. Here we apply it to images by defining what we call a co-circularity graph. In this graph, the vertices are the pixels and the weighted edges depend on a consistency criterion (co-circularity) between local orientations estimated from the structure tensors. This graph induces anisotropic adaptive morphological operators which are related both to anisotropic diffusion in images and path optimality in graphs. We present several applications such as the enhancement of fiber-like structures, completion of interrupted edges and the regularization of grayscale images. We also discuss the parameters setting depending on the application. Source Code The reviewed source code and documentation for this algorithm are available from the web page of this article 1. Compilation and usage instructions are included in the README.txt file of the archive.
In neural networks, the property of being equivariant to transformations improves generalization ... more In neural networks, the property of being equivariant to transformations improves generalization when the corresponding symmetry is present in the data. In particular, scale-equivariant networks are suited to computer vision tasks where the same classes of objects appear at different scales, like in most semantic segmentation tasks. Recently, convolutional layers equivariant to a semigroup of scalings and translations have been proposed. However, the equivariance of subsampling and upsampling has never been explicitly studied even though they are necessary building blocks in some segmentation architectures. The U-Net is a representative example of such architectures, which includes the basic elements used for state-of-the-art semantic segmentation. Therefore, this paper introduces the Scale Equivariant U-Net (SEU-Net), a U-Net that is made approximately equivariant to a semigroup of scales and translations through careful application of subsampling and upsampling layers and the use of aforementioned scale-equivariant layers. Moreover, a scale-dropout is proposed in order to improve generalization to different scales in approximately scale-equivariant architectures. The proposed SEU-Net is trained for semantic segmentation of the Oxford Pet IIIT and the DIC-C2DH-HeLa dataset for cell segmentation. The generalization metric to unseen scales is dramatically improved in comparison to the U-Net, even when the U-Net is trained with scale jittering, and to a scale-equivariant architecture that does not perform upsampling operators inside the equivariant pipeline. The scale-dropout induces better generalization on the scale-equivariant models in the Pet experiment, but not on the cell segmentation experiment.
Quantitative approaches are part of the understanding of contour integration and the Gestalt law ... more Quantitative approaches are part of the understanding of contour integration and the Gestalt law of good continuation. The present study introduces a new quantitative approach based on the a contrario theory, which formalizes the non-accidentalness principle for good continuation. This model yields an ideal observer algorithm, able to detect non-accidental alignments in Gabor patterns. More precisely, this parameterless algorithm associates with each candidate percept a measure, the Number of False Alarms (NFA), quantifying its degree of masking. To evaluate the approach, we compared this ideal observer with the human attentive performance on three experiments of straight contours detection in arrays of Gabor patches. The experiments showed a strong correlation between the detectability of the target stimuli and their degree of non-accidentalness, as measured by our model. What is more, the algorithm's detection curves were very similar to the ones of human subjects. This fact seems to validate our proposed measurement method as a convenient way to predict the visibility of alignments. This framework could be generalized to other Gestalts.
This paper presents a method to be used in psychophysical experiments to compare directly visual ... more This paper presents a method to be used in psychophysical experiments to compare directly visual perception to an a contrario algorithm, on a straight patterns detection task. The method is composed of two parts. The first part consists in building a stimulus, namely an array of oriented elements, in which an alignment is present with variable salience. The second part focuses on a detection algorithm, based on the a contrario theory, which is designed to predict which alignment will be considered as the most salient in a given stimulus.
This paper addresses the issue of building a part-based representation of a dataset of images. Mo... more This paper addresses the issue of building a part-based representation of a dataset of images. More precisely, we look for a non-negative, sparse decomposition of the images on a reduced set of atoms, in order to unveil a morphological and explainable structure of the data. Additionally, we want this decomposition to be computed online for any new sample that is not part of the initial dataset. Therefore, our solution relies on a sparse, non-negative auto-encoder, where the encoder is deep (for accuracy) and the decoder shallow (for explainability). This method compares favorably to the state-of-the-art online methods on two benchmark datasets (MNIST and Fashion MNIST) and on a hyperspectral image, according to classical evaluation measures and to a new one we introduce, based on the equivariance of the representation to morphological operators.
This paper addresses the issue of building a part-based representation of a dataset of images. Mo... more This paper addresses the issue of building a part-based representation of a dataset of images. More precisely, we look for a nonnegative, sparse decomposition of the images on a reduced set of atoms, in order to unveil a morphological and interpretable structure of the data. Additionally, we want this decomposition to be computed online for any new sample that is not part of the initial dataset. Therefore, our solution relies on a sparse, non-negative auto-encoder where the encoder is deep (for accuracy) and the decoder shallow (for interpretability). This method compares favorably to the state-of-the-art online methods on two datasets (MNIST and Fashion MNIST), according to classical metrics and to a new one we introduce, based on the invariance of the representation to morphological dilation.
Following recent advances in morphological neural networks, we propose to study in more depth how... more Following recent advances in morphological neural networks, we propose to study in more depth how Max-plus operators can be exploited to define morphological units and how they behave when incorporated in layers of conventional neural networks. Besides showing that they can be easily implemented with modern machine learning frameworks, we confirm and extend the observation that a Max-plus layer can be used to select important filters and reduce redundancy in its previous layer, without incurring performance loss. Experimental results demonstrate that the filter selection strategy enabled by a Max-plus layer is highly efficient and robust, through which we successfully performed model pruning on two neural network architectures. We also point out that there is a close connection between Maxout networks and our pruned Max-plus networks by comparing their respective characteristics. The code for reproducing our experiments is available online 4 .
HAL (Le Centre pour la Communication Scientifique Directe), 2019
Motivated by recent advances in morphological neural networks, we further study the properties of... more Motivated by recent advances in morphological neural networks, we further study the properties of morphological units when incorporated in layers of conventional neural networks. We confirm and extend the observation that a Max-plus layer can be used to select relevant filters and reduce redundancy in its previous layer, without incurring performance loss. We present several experiments in image processing, showing that this filter selection property seems efficient and robust. We also point out the close connection between Maxout networks and our pruned Max-plus networks. The code related to our experiments is available online (https://github.com/yunxiangzhang).
HAL (Le Centre pour la Communication Scientifique Directe), Jun 23, 2023
Dans ce chapitre nous présentons la Morphologie Mathématique comme une approche non linéaire du t... more Dans ce chapitre nous présentons la Morphologie Mathématique comme une approche non linéaire du traitement d'images, basée sur des critères de forme et de taille. Nous essaierons de montrer son attrait en mettant en avant l'élégance de sa théorie ainsi que la puissance des outils qu'elle permet de construire : fonction distance, filtres, algorithme du Watershed pour la segmentation et autres représentations hiérarchiques, pour les principaux.
We develop a machine-learning image segmentation pipeline that detects ductile (as opposed to bri... more We develop a machine-learning image segmentation pipeline that detects ductile (as opposed to brittle) fracture in fractography images. To demonstrate the validity of our approach, use is made of a set of fractography images representing fracture surfaces from cold-spray deposits. The coatings have been subjected to varying heat treatments in an effort
We extend recent work on mathematical morphology for signal processing on weighted graphs, based ... more We extend recent work on mathematical morphology for signal processing on weighted graphs, based on discrete tropical algebra. The framework is general and can be applied to any scalar function defined on a graph. We show applications in structure tensors analysis and the regularisation of greyscale images.
Equivariance of neural networks to transformations helps to improve their performance and reduce ... more Equivariance of neural networks to transformations helps to improve their performance and reduce generalization error in computer vision tasks, as they apply to datasets presenting symmetries (e.g. scalings, rotations, translations). The method of moving frames is classical for deriving operators invariant to the action of a Lie group in a manifold. Recently, a rotation and translation equivariant neural network for image data was proposed based on the moving frames approach. In this paper we significantly improve that approach by reducing the computation of moving frames to only one, at the input stage, instead of repeated computations at each layer. The equivariance of the resulting architecture is proved theoretically and we build a rotation and translation equivariant neural network to process volumes, i.e. signals on the 3D space. Our trained model overperforms the benchmarks in the medical volume classification of most of the tested datasets from MedMNIST3D.
In discrete signal and image processing, many dilations and erosions can be written as the max-pl... more In discrete signal and image processing, many dilations and erosions can be written as the max-plus and min-plus product of a matrix on a vector. Previous studies considered operators on symmetrical, unbounded complete lattices, such as Cartesian powers of the completed real line. This paper focuses on adjunctions on closed hypercubes, which are the complete lattices used in practice to represent digital signals and images. We show that this constrains the representing matrices to be doubly-0-astic and we characterise the adjunctions that can be represented by them. A graph interpretation of the defined operators naturally arises from the adjacency relationship encoded by the matrices, as well as a max-plus spectral interpretation.
The present dissertation compares the human visual perception to computer vision algorithms based... more The present dissertation compares the human visual perception to computer vision algorithms based on a mathematical model called a contrario theory. To this aim, it focuses on two visual tasks that are at the same time easy to model and convenient to test in psychophysical experiments. Both tasks consist in the perceptual grouping of oriented elements, namely Gabor patches. The first one is the detection of alignments and the second one extends to curves, that is to say to more general arrangements of elements in good continuation. In both cases, alignments and curves, psychophysical experiments were set up to collect data on the human visual perception in a masking context. The non-accidentalness principle states that spatial relations are perceptually relevant when their accidental occurrence is unlikely. The a contrario theory is a formalization of this principle, and is used in computer vision to set detection thresholds accordingly. In this thesis, the a contrario framework is applied in two practical algorithms designed to detect non-accidental alignments and curves respectively. These algorithms play the part of artificial subjects for our experiments. The experimental data of human subjects is then compared to the detection algorithms on the very same tasks, yielding two main results. First, this procedure shows that the Number of False Alarms (NFA), which is the scalar measure of non-accidentalness in the a contrario theory, strongly correlates with the detection rates achieved by human subjects on a large variety of stimuli. Secondly, the algorithms' responses match very well the average behavior of human observers. The contribution of this thesis is therefore two-sided. On the one hand, it provides a rigorous validation of the a contrario theory's relevance to estimate visual thresholds and implement visual tasks in computer vision. On the other hand, it reinforces the importance of the non-accidentalness principle in human vision. Aiming at reproducible research, all the methods are submitted to IPOL journal, including detailed descriptions of the algorithms, commented reference source codes, and online demonstrations for each one. Les années de thèse ont été pour moi une période privilégiée, durant laquelle il m'a été offert d'apprendre en me consacrant pleinement à un projet unique, et ce dans les meilleures conditions. Je suis sincèrement reconnaissant envers tous ceux qui ont participé à me donner cette chance. J'ai trouvé au CMLA un cadre idéal pour travailler dans la sérénité et la bonne humeur. Je le dois en grande partie à Carine, Christophe, Micheline, Nicolas P., Sandra, Véronique et Virginie, dont la bienveillance et la compétence ont assuré le bon déroulement de mon séjour au laboratoire, tant sur les plans logistique et administratif qu'humain. Je remercie également tous les professeurs du CMLA pour leur accessibilité et l'image positive qu'ils m'ont donnée du métier d'enseignant-chercheur. Mes remerciements vont tout particulièrement à mes directeurs Jean-Michel Morel et Rafael Grompone von Gioi, qui m'ont laissé la plus grande liberté dans mon travail tout en me prévenant d'éventuels égarements par leurs idées et leurs conseils. Ils ont fait preuve d'une disponibilité exceptionnelle que je souhaite à tous les doctorants. Par ailleurs, je garderai un souvenir enthousiaste du groupe "images" du CMLA, où la qualité scientifique va de paire avec un esprit de partage et d'entraide. Merci également aux chercheurs du CibPSI, en particulier Alejandra Carboni et Alejandro Maiche, et à Gregory Randall qui a encouragé cette collaboration avec l'Universidad de la República de Montevideo. Les journées au laboratoire n'auraient pas été aussi plaisantes sans mes camarades du CMLA: entre risotti, débats et plaisanteries, la cuisine a été le lieu des meilleures pauses grâce à vous. Enfin, merci à toutes celles et tous ceux que je devrais remercier chaque jour-ils se reconnaîtront.
HAL (Le Centre pour la Communication Scientifique Directe), Aug 29, 2020
Mathematical Morphology and Logics: In propositional logics, considering the lattice of formulas,... more Mathematical Morphology and Logics: In propositional logics, considering the lattice of formulas, morphological operators will act on formulas (and on their models). In modal logics, dilation and erosion can define modal operators. Such examples will be described in several logics. Then we will show how they can be used to define revision operators satisfying the AGM postulates, merging operators, or explanatory relations.
The algorithm presented in this paper is an application of a general framework for morphological ... more The algorithm presented in this paper is an application of a general framework for morphological processing of signals on weighted graphs. Here we apply it to images by defining what we call a co-circularity graph. In this graph, the vertices are the pixels and the weighted edges depend on a consistency criterion (co-circularity) between local orientations estimated from the structure tensors. This graph induces anisotropic adaptive morphological operators which are related both to anisotropic diffusion in images and path optimality in graphs. We present several applications such as the enhancement of fiber-like structures, completion of interrupted edges and the regularization of grayscale images. We also discuss the parameters setting depending on the application. Source Code The reviewed source code and documentation for this algorithm are available from the web page of this article 1. Compilation and usage instructions are included in the README.txt file of the archive.
In neural networks, the property of being equivariant to transformations improves generalization ... more In neural networks, the property of being equivariant to transformations improves generalization when the corresponding symmetry is present in the data. In particular, scale-equivariant networks are suited to computer vision tasks where the same classes of objects appear at different scales, like in most semantic segmentation tasks. Recently, convolutional layers equivariant to a semigroup of scalings and translations have been proposed. However, the equivariance of subsampling and upsampling has never been explicitly studied even though they are necessary building blocks in some segmentation architectures. The U-Net is a representative example of such architectures, which includes the basic elements used for state-of-the-art semantic segmentation. Therefore, this paper introduces the Scale Equivariant U-Net (SEU-Net), a U-Net that is made approximately equivariant to a semigroup of scales and translations through careful application of subsampling and upsampling layers and the use of aforementioned scale-equivariant layers. Moreover, a scale-dropout is proposed in order to improve generalization to different scales in approximately scale-equivariant architectures. The proposed SEU-Net is trained for semantic segmentation of the Oxford Pet IIIT and the DIC-C2DH-HeLa dataset for cell segmentation. The generalization metric to unseen scales is dramatically improved in comparison to the U-Net, even when the U-Net is trained with scale jittering, and to a scale-equivariant architecture that does not perform upsampling operators inside the equivariant pipeline. The scale-dropout induces better generalization on the scale-equivariant models in the Pet experiment, but not on the cell segmentation experiment.
Quantitative approaches are part of the understanding of contour integration and the Gestalt law ... more Quantitative approaches are part of the understanding of contour integration and the Gestalt law of good continuation. The present study introduces a new quantitative approach based on the a contrario theory, which formalizes the non-accidentalness principle for good continuation. This model yields an ideal observer algorithm, able to detect non-accidental alignments in Gabor patterns. More precisely, this parameterless algorithm associates with each candidate percept a measure, the Number of False Alarms (NFA), quantifying its degree of masking. To evaluate the approach, we compared this ideal observer with the human attentive performance on three experiments of straight contours detection in arrays of Gabor patches. The experiments showed a strong correlation between the detectability of the target stimuli and their degree of non-accidentalness, as measured by our model. What is more, the algorithm's detection curves were very similar to the ones of human subjects. This fact seems to validate our proposed measurement method as a convenient way to predict the visibility of alignments. This framework could be generalized to other Gestalts.
This paper presents a method to be used in psychophysical experiments to compare directly visual ... more This paper presents a method to be used in psychophysical experiments to compare directly visual perception to an a contrario algorithm, on a straight patterns detection task. The method is composed of two parts. The first part consists in building a stimulus, namely an array of oriented elements, in which an alignment is present with variable salience. The second part focuses on a detection algorithm, based on the a contrario theory, which is designed to predict which alignment will be considered as the most salient in a given stimulus.
This paper addresses the issue of building a part-based representation of a dataset of images. Mo... more This paper addresses the issue of building a part-based representation of a dataset of images. More precisely, we look for a non-negative, sparse decomposition of the images on a reduced set of atoms, in order to unveil a morphological and explainable structure of the data. Additionally, we want this decomposition to be computed online for any new sample that is not part of the initial dataset. Therefore, our solution relies on a sparse, non-negative auto-encoder, where the encoder is deep (for accuracy) and the decoder shallow (for explainability). This method compares favorably to the state-of-the-art online methods on two benchmark datasets (MNIST and Fashion MNIST) and on a hyperspectral image, according to classical evaluation measures and to a new one we introduce, based on the equivariance of the representation to morphological operators.
This paper addresses the issue of building a part-based representation of a dataset of images. Mo... more This paper addresses the issue of building a part-based representation of a dataset of images. More precisely, we look for a nonnegative, sparse decomposition of the images on a reduced set of atoms, in order to unveil a morphological and interpretable structure of the data. Additionally, we want this decomposition to be computed online for any new sample that is not part of the initial dataset. Therefore, our solution relies on a sparse, non-negative auto-encoder where the encoder is deep (for accuracy) and the decoder shallow (for interpretability). This method compares favorably to the state-of-the-art online methods on two datasets (MNIST and Fashion MNIST), according to classical metrics and to a new one we introduce, based on the invariance of the representation to morphological dilation.
Following recent advances in morphological neural networks, we propose to study in more depth how... more Following recent advances in morphological neural networks, we propose to study in more depth how Max-plus operators can be exploited to define morphological units and how they behave when incorporated in layers of conventional neural networks. Besides showing that they can be easily implemented with modern machine learning frameworks, we confirm and extend the observation that a Max-plus layer can be used to select important filters and reduce redundancy in its previous layer, without incurring performance loss. Experimental results demonstrate that the filter selection strategy enabled by a Max-plus layer is highly efficient and robust, through which we successfully performed model pruning on two neural network architectures. We also point out that there is a close connection between Maxout networks and our pruned Max-plus networks by comparing their respective characteristics. The code for reproducing our experiments is available online 4 .
HAL (Le Centre pour la Communication Scientifique Directe), 2019
Motivated by recent advances in morphological neural networks, we further study the properties of... more Motivated by recent advances in morphological neural networks, we further study the properties of morphological units when incorporated in layers of conventional neural networks. We confirm and extend the observation that a Max-plus layer can be used to select relevant filters and reduce redundancy in its previous layer, without incurring performance loss. We present several experiments in image processing, showing that this filter selection property seems efficient and robust. We also point out the close connection between Maxout networks and our pruned Max-plus networks. The code related to our experiments is available online (https://github.com/yunxiangzhang).
HAL (Le Centre pour la Communication Scientifique Directe), Jun 23, 2023
Dans ce chapitre nous présentons la Morphologie Mathématique comme une approche non linéaire du t... more Dans ce chapitre nous présentons la Morphologie Mathématique comme une approche non linéaire du traitement d'images, basée sur des critères de forme et de taille. Nous essaierons de montrer son attrait en mettant en avant l'élégance de sa théorie ainsi que la puissance des outils qu'elle permet de construire : fonction distance, filtres, algorithme du Watershed pour la segmentation et autres représentations hiérarchiques, pour les principaux.
We develop a machine-learning image segmentation pipeline that detects ductile (as opposed to bri... more We develop a machine-learning image segmentation pipeline that detects ductile (as opposed to brittle) fracture in fractography images. To demonstrate the validity of our approach, use is made of a set of fractography images representing fracture surfaces from cold-spray deposits. The coatings have been subjected to varying heat treatments in an effort
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