Deep morphological networks

Abstract

Mathematical morphology provides powerful nonlinear operators for a variety of image processing tasks such as filtering, segmentation, and edge detection. In this paper, we propose a way to use these nonlinear operators in an end-toend deep learning framework and illustrate them on different applications. We demonstrate on various examples that new layers making use of the morphological non-linearities are complementary to convolution layers. These new layers can be used to integrate the non-linear operations and pooling into a joint operation. We finally enhance results obtained in boundary detection using this new family of layers with just 0.01% of the parameters of competing state-of-the-art methods.

Image Restoration by Learning Morphological Opening-Closing Network

“…Similarly, the gradient for the erosion layer can be derived. A worked-out example of gradient calculation for the erosion layer is shown in [13].…”

Approximating morphological operators with part-based representations learned by asymmetric auto-encoders

“…This activation function has shown the best performances in similar architectures [16]. The sparsity of the encoding is achieved using the same approach as in [2,10], that consists in adding to the previous loss function the regularization term described in Equations (8) and (9). We only enforced the non-negativity of the weights of the decoder, as they define the dictionary of images of our learned representation and as enforcing the non-negativity of the encoder weights would bring nothing but more constraints to the network and lower its capacity.…”

“…Hence, we take advantage of the recent advances in deep, sparse and non-negative auto-encoders to design a new framework able to learn part-based representations of an image database, compatible with morphological processing. To that extent, this work is part of the resurgent research line investigating interactions between deep learning and mathematical morphology [9,22,23,27,32]. However with respect to these studies, focusing mainly on introducing morphological operators in neural networks, the present paper addresses a different question.…”

Tropical Geometry and Machine Learning

“…The name "tropical semiring" initially referred to the min-plus semiring and was used in finite automata [57], [99], speech recognition using graphical models [82], and tropical geometry [68], [80]. However, nowadays, the term, tropical semiring, may refer to both the max-plus and its dual min-plus arithmetic, whose combinations with corresponding nonlinear matrix algebra and nonlinear signal convolutions have been used in operations research and scheduling [25]; discrete event systems, max-plus control, and optimization [1], [2], [6], [15], [22], [37], [39], [48], [78], [110]; convex analysis [65], [85], [94]; morphological image analysis [49], [73], [79], [95], [96]; nonlinear difference equations for distance transforms [11], [71]; nonlinear PDEs of the Hamilton-Jacobi type for vision scale spaces [14], [50]; speech recognition and natural language processing [56], [82]; neural networks [18], [19], [34], [40], [83], [89], [93], [103], [114], [115]; and idempotent mathematics (nonlinear functional analysis) [63], [64].…”