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.