Finsler geometry approaches in image processing

The main purpose of this doctoral thesis was to develop a new image processing application on images called compelling images. Our main motivation has been to apply the newly developed technology from image processing operations to noise removal, edge detection and advanced segmentation operations. Because the success of many image processing applications basically depends on a good preprocessing step. At the basis of these preprocessing steps are partial differential structures that emerge from an excellent mathematical language. Many inventions in image processing in the world have emerged by reading and developing differential structures in an ingenious mathematical order. Calculations made with Euclidean metrics long ago gave way to calculations made with Riemann metrics. Today, all experiments with stability and precision in the field of technology are carried out with Finsler metrics. Because Finsler gives the best results in non-linner and anisotropic environments. In the first part of the thesis, segmentation studies in the literature are mentioned, since it is necessary to understand the Riemann structures first in order to understand Finsler, the segmentation process with the Riemann model is examined in the second part. The segmentation process in the Riemann model is based on an energy integral. Therefore, light transitions or energy transitions in the structure of the image play a major role. In this model, since the evolution of a given initial curve is guided by partial differentials, there is actually a gradient flow. For this reason, good results cannot be obtained in difficult images where light transitions are not evident. If a definition is made with the direction information that will guide the curve while defining the model, the evolution of the initial curve can be achieved using direction information even in difficult images. Such a model is possible with the Finsler model. Naturally, in the third part of the thesis, Finsler curve evolution is discussed and its mathematical structures are examined. From this, it is understood that the cost function should be determined in two ways for the curve evolution in Finsler's sense. The first is to choose as the cost function the direction that includes any point on the initial curve and gives the largest change in the image. The second would be to define a cost function that measures distance with level curves. Since both are costly, in the fourth part of the thesis, the Polyakov energy integral, which allows energy minimizations in images and allows the use of a flexible metric, is examined. This model is the most advanced energy model in existing technology. In this part of the thesis, image filtering results and performance comparisons for a new class of anisotropic filters are presented. The new filters are derived from a special selection of edge-preserving anisotropic Finsler metrics that generate anisotropic Laplace-Beltrami flows to minimize the Polyakov energy integral. This new anisotropic Filters were applied to noisy images and compared according to their noise reduction properties. It has been experimentally observed that the new anisotropic filters give superior noise removal performances compared to conventional isotropic filters according to various image quality criteria such as MSE, PSNR, SSIM, etc. It is important to note that anisotropic filters successfully remove noise from smooth regions while perfectly preserving edge information, thanks to their space-changing nature. In summary, in this doctoral thesis, the approaches of Finsler geometry to image processing applications are investigated and mathematical models are presented. Based on these models, a new family of anisotropic filters has been proposed. Quantitative results were obtained comparing the proposed new filters with the classical filters and it was seen that they performed significantly better than the classical isotropic filters. It is thought that this filter family, which is effective in processing difficult images in image processing, will be a guide for future studies by bringing this filter family to the literature.