We present an up-to-date survey on the topic of adaptive mathematical morphology. A broad review of research performed within the field is provided, as well as an in-depth summary of the theoretical advances within the field. Adaptivity can come in many different ways, based on different attributes, measures, and parameters. Similarities and differences between a few selected methods for adaptive structuring elements are considered, providing perspective on the consequences of different types of adaptivity. We also provide a brief analysis of perspectives and trends within the field, discussing possible directions for future studies.

Image analysis methods for processing 3D profile data have been investigated and developed. These methods include; Image reconstruction by prioritized incremental normalized convolution, morphology-based crack detection for steel slabs, and adaptive morphology based on the local structure tensor. The methods have been applied to a number of industrial applications.An issue with 3D profile data captured by laser triangulation is occlusion, which occurs when the line-of-sight between the projected laser light and the camera sensor is obstructed. To overcome this problem, interpolation of missing surface in rock piles has been investigated and a novel interpolation method for filling in missing pixel values iteratively from the edges of the reliable data, using normalized convolution, has been developed.3D profile data of the steel surface has been used to detect longitudinal cracks in casted steel slabs. Segmentation of the data is done using mathematical morphology, and the resulting connected regions are assigned a crack probability estimate based on a statistic logistic regression model. More specifically, the morphological filtering locates trenches in the data, excludes scale regions for further analysis, and finally links crack segments together in order to obtain a segmented region which receives a crack probability based on its depth and length.Also suggested is a novel method for adaptive mathematical morphology intended to improve crack segment linking, i.e. for bridging gaps in the crack signature in order to increase the length of potential crack segments. Standard morphology operations rely on a predefined structuring element which is repeatedly used for each pixel in the image. The outline of a crack, however, can range from a straight line to a zig-zag pattern. A more adaptive method for linking regions with a large enough estimated crack depth would therefore be beneficial. More advanced morphological approaches, such as morphological amoebas and path openings, adapt better to curvature in the image. For our purpose, however, we investigate how the local structure tensor can be used to adaptively assign to each pixel an elliptical structuring element based on the local orientation within the image. The information from the local structure tensor directly defines the shape of the elliptical structuring element, and the resulting morphological filtering successfully enhances crack signatures in the data.

Classical morphological image processing, where the same structuring element is used to process the whole image, has its limitations. Consequently, adaptive mathematical morphology is attracting more and more attention.So far, however, the use of non-flat adaptive structuring functions is very limited. This work presents a method for defining quadratic structuring functions from the well known local structure tensor, building on previous work for flat adaptive morphology. The result is a novel approach to adaptive mathematical morphology, suitable for enhancement and linking of directional features in images. Moreover, the presented strategy can be quite efficiently implemented and is easy to use as it relies on just two user-set parameters which are directly related to image measures.

As technological advances drives the evolution of sensors as well as the systems using them, processing and analysis of multi-dimensional signals such as images becomes more and more common in a wide range of applications ranging from consumer products to automated systems in process industry. Image processing is often needed to enhance or suppress features in the acquired data, enabling better analysis of the signals and thereby better use of the system in question. Since imaging applications can be very different, image processing covers a wide range of methods and sub-fields.Mathematical morphology constitutes a well defined framework for non-linear image processing based on set relations. It relies on minimum and maximum values over neighborhoods (i.e. regions surrounding the individual points) defined by shapes or functions known as structuring elements. Classical morphological operations use a predefined structuring element which is used repeatedly for each point in the image. This is often not ideal, however, which has motivated the evolution of adaptive morphological filtering where the structuring element changes from point to point. The field of adaptive mathematical morphology includes many different concepts with different strengths and weaknesses, and the specific choice of method should be made with the specific application in mind.The main contribution of this thesis is a novel method for adaptive morphological filtering using Elliptical Adaptive Structuring Elements (EASE). The method enhances directional structures in images by orienting the structuring elements along the existing structure, and can be efficiently used to close gaps in such structures. The method is introduced by summarizing underlying theory as well as presenting a practical application motivating it:~crack detection in casted steel. Furthermore, it is demonstrated how the method can be extended to allow for filtering of incomplete (i.e. partially missing) image data without need for pre-filtering. The EASE concept is also put in relation to other related work by presenting a survey of the field of adaptive mathematical morphology.In conclusion, EASE allows for fast structure-based adaptive morphological filtering of images based on solid mathematical theory, successfully enhancing directional structures such as lines, borders, etc. in the data. The method is user-friendly, as it does not require more than a few user-defined parameters, and can also be adapted for direct filtering of incomplete data.

Inter-Site Distance (ISD) is a common measure for characterizing the site density in a mobile network However, obtaining a good estimation of the ISD for a real world network is not trivial since the physical layout is usually quite more complex than a perfect theoretical hexagonal grid, due to a number of unavoidable factors such as site availability and traffic density. Voronoi diagrams have been suggested for approximating cells from network layouts, providing a method for partitioning the covered area into cells defined by the proximity to the given set of sites. This yields a framework for site coverage approximation based on the actual site distribution, rather than an underlying theoretical model.We present a novel measure, based on Voronoi diagrams, for characterizing the site density of a cellular network and provide a comparison to the more traditional ISD measure. This measure improves capacity assessments and modeling of real networks.

A priority-based method for pixel reconstruction and incrementalhole filling in incomplete images and 3D surface data is presented.The method is primarily intended for reconstruction of occluded areasin 3D surfaces and makes use of a novel prioritizing scheme, based on apixelwise defined confidence measure, that determines the order in whichpixels are iteratively reconstructed. The actual reconstruction of individualpixels is performed by interpolation using normalized convolution.The presented approach has been applied to the problem of reconstructing3D surface data of a rock pile as well as randomly sampled imagedata. It is concluded that the method is not optimal in the latter case,but the results show an improvement to ordinary normalized convolutionwhen applied to the rock data and are in this case comparable to thoseobtained from normalized convolution using adaptive neighborhood sizes.

Mathematical Morphology is a common strategy for non-linear filtering of image data. In its traditional form the filters used, known as structuring elements, have constant shape once set. Such rigid structuring elements are excellent for detecting patterns of a specific shape, but risk destroying valuable information in the data as they do not adapt in any way to its structure.We present a novel method for adaptive morphological filtering where the local structure tensor, a well-known method for estimation of structure within image data, is used to construct adaptive elliptical structuring elements which vary from pixel to pixel depending on the local image structure. More specifically, their shape varies from lines in regions of strong single-directional characteristics to disks at locations where the data has no prevalent direction.

Continuous casting is a highly efficient process used to produce most of the world steel production tonnage, but can cause cracks in the semi-finished steel product output. These cracks may cause problems further down the production chain, and detecting them early in the process would avoid unnecessary and costly processing of the defective goods. In order for a crack detection system to be accepted in industry, however, false detection of cracks in non-defective goods must be avoided. This is further complicated by the presence of scales; a brittle, often cracked, top layer originating from the casting process. We present an approach for an automated on-line crack detection system, based on 3D profile data of steel slab surfaces, utilizing morphological image processing and statistical classification by logistic regression.The initial segmentation successfully extracts 80\% of the crack length present in the data, while discarding most potential pseudo-defects (non-defect surface features similar to defects). The subsequent statistical classification individually has a crack detection accuracy of over 80\% (with respect to total segmented crack length), while discarding all remaining manually identified pseudo-defects. Taking more ambiguous regions into account gives a worst-case false classification of 131~mm within the 30~600~mm long sequence of 150~mm wide regions used as validation data. The combined system successfully identifies over 70\% of the manually identified (unambiguous) crack length, while missing only a few crack regions containing short crack segments.The results provide proof-of-concept for a fully automated crack detection system based on the presented method.

We demonstrate how known convolution techniques for uncertain data can be used to set the shapes of structuring elements in adaptive mathematical morphology, enabling robust morphological processing of partially occluded or otherwise incomplete data. Results are presented for filtering of both gray-scale images containing missing data and 3D profile data where information is missing due to occlusion effects. The latter demonstrates the intended use of the method: enhancement of crack signatures in a surface inspection system for casted steel.The presented method is able to disregard unreliable data in a systematic and robust way, enabling adaptive morphological processing of the available information while avoiding any false edges or other unwanted features introduced by the values of faulty pixels.

A novel method for automated inspection of small corner cracks in casted steel is presented, using a photometric stereo setup consisting of two light sources of different colors in conjunction with a line-scan camera. The resulting image is separated into two different reflection patterns which are used to cancel shadow effects and estimate the surface gradient. Statistical methods are used to first segment the image and then provide an estimated crack probability for each segmented region. Results show that true cracks are successfully assigned a high crack probability, while only a minor proportion of other regions cause similar probability values. About 80% of the cracks present in the segmented regions are given a crack probability higher than 70%, while the corresponding number for other non-crack regions is only 5%. The segmented regions contain over 70% of the manually identified crack pixels. We thereby provide proof-of-concept for the presented method.

Future mobile networks will need new tools to deal with the challenges of emerging technologies. In particular, more flexible networks will require localization of transmitters in the networks. In this work we present a novel method for transmitter localization, suitable for rich multipath mmWave 5G scenarios such as dense urban environments. Our work combines stochastic estimation of Radio Environmental Maps (REMs) with the well known concept of triangulation, generalizing the latter into a method for localization in anisotropic propagation environments. It can be considered a conceptual bridge from classical distance-based triangulation into a generalized version where the propagation environment is taken into account. The result is a highly flexible tool for network planning in general and transmitter localization in particular.