Morphological Network: How Far Can We Go with Morphological Neurons?

TL;DR

This paper provides a theoretical analysis of morphological neurons, demonstrating their greater power than previously thought, and shows that networks with these neurons can approximate any continuous function, with promising experimental results.

Contribution

The paper offers the first comprehensive theoretical analysis of morphological neurons, revealing their capacity to approximate any continuous function and their potential as powerful nonlinear operators.

Findings

Morphological neurons are more powerful than previously anticipated.

Networks with morphological neurons can approximate any continuous function.

Experimental results show their effectiveness over similar neural networks.

Abstract

Morphological neurons, that is morphological operators such as dilation and erosion with learnable structuring elements, have intrigued researchers for quite some time because of the power these operators bring to the table despite their simplicity. These operators are known to be powerful nonlinear tools, but for a given problem coming up with a sequence of operations and their structuring element is a non-trivial task. So, the existing works have mainly focused on this part of the problem without delving deep into their applicability as generic operators. A few works have tried to utilize morphological neurons as a part of classification (and regression) networks when the input is a feature vector. However, these methods mainly focus on a specific problem, without going into generic theoretical analysis. In this work, we have theoretically analyzed morphological neurons and have shown that these are far more powerful than previously anticipated. Our proposed morphological block, containing dilation and erosion followed by their linear combination, represents a sum of hinge functions. Existing works show that hinge functions perform quite well in classification and regression problems. Two morphological blocks can even approximate any continuous function. However, to facilitate the theoretical analysis that we have done in this paper, we have restricted ourselves to the 1D version of the operators, where the structuring element operates on the whole input. Experimental evaluations also indicate the effectiveness of networks built with morphological neurons, over similarly structured neural networks.