Jesús Angulo

Tin Barisin, Jesus Angulo, Katja Schladitz et al.

Scattering networks yield powerful and robust hierarchical image descriptors which do not require lengthy training and which work well with very few training data. However, they rely on sampling the scale dimension. Hence, they become sensitive to scale variations and are unable to generalize to unseen scales. In this work, we define an alternative feature representation based on the Riesz transform. We detail and analyze the mathematical foundations behind this representation. In particular, it inherits scale equivariance from the Riesz transform and completely avoids sampling of the scale dimension. Additionally, the number of features in the representation is reduced by a factor four compared to scattering networks. Nevertheless, our representation performs comparably well for texture classification with an interesting addition: scale equivariance. Our method yields superior performance when dealing with scales outside of those covered by the training dataset. The usefulness of the equivariance property is demonstrated on the digit classification task, where accuracy remains stable even for scales four times larger than the one chosen for training. As a second example, we consider classification of textures.

Santiago Velasco-Forero, Jesús Angulo

This paper analyses both nonlinear activation functions and spatial max-pooling for Deep Convolutional Neural Networks (DCNNs) by means of the algebraic basis of mathematical morphology. Additionally, a general family of activation functions is proposed by considering both max-pooling and nonlinear operators in the context of morphological representations. Experimental section validates the goodness of our approach on classical benchmarks for supervised learning by DCNN.

Joao P. C. Bertoldo, Santiago Velasco-Forero, Jesus Angulo et al.

We propose an incremental improvement to Fully Convolutional Data Description (FCDD), an adaptation of the one-class classification approach from anomaly detection to image anomaly segmentation (a.k.a. anomaly localization). We analyze its original loss function and propose a substitute that better resembles its predecessor, the Hypersphere Classifier (HSC). Both are compared on the MVTec Anomaly Detection Dataset (MVTec-AD) -- training images are flawless objects/textures and the goal is to segment unseen defects -- showing that consistent improvement is achieved by better designing the pixel-wise supervision.

Yufei Hu, Nacim Belkhir, Jesus Angulo et al.

Deep Neural Networks (DNNs) are generated by sequentially performing linear and non-linear processes. Using a combination of linear and non-linear procedures is critical for generating a sufficiently deep feature space. The majority of non-linear operators are derivations of activation functions or pooling functions. Mathematical morphology is a branch of mathematics that provides non-linear operators for a variety of image processing problems. We investigate the utility of integrating these operations in an end-to-end deep learning framework in this paper. DNNs are designed to acquire a realistic representation for a particular job. Morphological operators give topological descriptors that convey salient information about the shapes of objects depicted in images. We propose a method based on meta-learning to incorporate morphological operators into DNNs. The learned architecture demonstrates how our novel morphological operations significantly increase DNN performance on various tasks, including picture classification and edge detection.

Alexandre Kirszenberg, Guillaume Tochon, Elodie Puybareau et al.

Integrating mathematical morphology operations within deep neural networks has been subject to increasing attention lately. However, replacing standard convolution layers with erosions or dilations is particularly challenging because the min and max operations are not differentiable. Relying on the asymptotic behavior of the counter-harmonic mean, p-convolutional layers were proposed as a possible workaround to this issue since they can perform pseudo-dilation or pseudo-erosion operations (depending on the value of their inner parameter p), and very promising results were reported. In this work, we present two new morphological layers based on the same principle as the p-convolutional layer while circumventing its principal drawbacks, and demonstrate their potential interest in further implementations within deep convolutional neural network architectures.

Bastien Ponchon, Santiago Velasco-Forero, Samy Blusseau et al.

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 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.

Jean Serra, Jesus Angulo, B Ravi Kiran

Consider a family $Z=\{\boldsymbol{x_{i}},y_{i}$,$1\leq i\leq N\}$ of $N$ pairs of vectors $\boldsymbol{x_{i}} \in \mathbb{R}^d$ and scalars $y_{i}$ that we aim to predict for a new sample vector $\mathbf{x}_0$. Kriging models $y$ as a sum of a deterministic function $m$, a drift which depends on the point $\boldsymbol{x}$, and a random function $z$ with zero mean. The zonality hypothesis interprets $y$ as a weighted sum of $d$ random functions of a single independent variables, each of which is a kriging, with a quadratic form for the variograms drift. We can therefore construct an unbiased estimator $y^{*}(\boldsymbol{x_{0}})=\sum_{i}λ^{i}z(\boldsymbol{x_{i}})$ de $y(\boldsymbol{x_{0}})$ with minimal variance $E[y^{*}(\boldsymbol{x_{0}})-y(\boldsymbol{x_{0}})]^{2}$, with the help of the known training set points. We give the explicitly closed form for $λ^{i}$ without having calculated the inverse of the matrices.

Gianni Franchi, Jesus Angulo, Dino Sejdinovic

We propose a novel approach for pixel classification in hyperspectral images, leveraging on both the spatial and spectral information in the data. The introduced method relies on a recently proposed framework for learning on distributions -- by representing them with mean elements in reproducing kernel Hilbert spaces (RKHS) and formulating a classification algorithm therein. In particular, we associate each pixel to an empirical distribution of its neighbouring pixels, a judicious representation of which in an RKHS, in conjunction with the spectral information contained in the pixel itself, give a new explicit set of features that can be fed into a suite of standard classification techniques -- we opt for a well-established framework of support vector machines (SVM). Furthermore, the computational complexity is reduced via random Fourier features formalism. We study the consistency and the convergence rates of the proposed method and the experiments demonstrate strong performance on hyperspectral data with gains in comparison to the state-of-the-art results.

Guillaume Noyel, Jesus Angulo, Dominique Jeulin

The present paper introduces the $η$ and η connections in order to add regional information on $λ$-flat zones, which only take into account a local information. A top-down approach is considered. First $λ$-flat zones are built in a way leading to a sub-segmentation. Then a finer segmentation is obtained by computing $η$-bounded regions and $μ$-geodesic balls inside the $λ$-flat zones. The proposed algorithms for the construction of new partitions are based on queues with an ordered selection of seeds using the cumulative distance. $η$-bounded regions offers a control on the variations of amplitude in the class from a point, called center, and $μ$-geodesic balls controls the "size" of the class. These results are applied to hyperspectral images.