Morphological Network: Network With Morphological Neurons

Abstract

Image processing with traditional approaches mainly use the tools of linear systems. However, linear approaches are not well suited and may even fail to solve problems involving geometrical aspects of the image. Thus, nonlinear geometric approaches like morphological operations are very popular in those cases. Morphological operations are nonlinear operations based on a set and lattice-theoretic methodology for image analysis that are capable of describing the geometrical structure of image objects quantitatively. It is suitable for various problems in image processing, computer vision, and pattern recognition. While solving problems with morphology, a particular structuring element is defined. Structuring elements have particular shape and size which are applied spatially in the images. Finding such structuring elements for each task are very difficult and hand engineered. In this thesis, we develop networks with trainable morphological structuring elements for solving several problems. Our main idea is to learn appropriate structuring element(s) given an objective. The elementary operations of morphology are dilation and erosion. Similar to convolutional neural networks, a network is built with dilation and erosion operators with trainable structuring elements. For example, we have considered a gray scale rainy dataset. Since the rain streak has a particular shape and is considered as white noise, the network is able to remove rain in grayscale images using learned structuring elements. Dilation and Erosion in particular order constitute opening and closing operations. Opening and closing are popular in removing bright and dark noise from images. We have relied more on the training of structuring elements and built a network with dilation and erosion so that it may perform opening or closing operations based on the necessity. We have empirically proved that opening and closing is happening in the network. Further the network is applied for image restoration tasks and evaluated on colour image de-raining and image dehazing. Dilation and Erosion are composed with max and min operation. To make it more generic like a neural network, we have theoretically analyzed the morphological network and have built a dense morphological network to process 1-dimensional feature vectors. Morphological block has been defined by a dilation-erosion layer followed by a linear combination layer. We have shown that a morphological block represents a sum of hinge functions. With this morphological block our network is able to perform many classification tasks. Further, we have proved that two sequential morphological blocks can approximate any continuous function. We have also analyzed the network with deep multilayer configuration and shown many properties of the network. Next, We have extended the dense morphological concept and built a 2D network so that it can be applied in general image processing tasks. We build a network with a basic 2D morphological block i.e dilation erosion followed by linear combination of feature map. We have repeated this block and built a network for general image processing tasks such as classification of pixels. We have also evaluated the performance of the network on image processing tasks like segmentation of blood vessels from fundus images, segmentation of lungs from chest x-ray and image dehazing.