## 1 Introduction
VIC has shown good performance in compressing natural and medical images [13]. Variational and PDE-based interpolation and inpainting techniques have been used to interpolate the scattered data. Lower order PDEs often result in singularities at isolated interpolation points on images, higher-order PDEs provide smoother solutions, but the violation of an extremum principle can result in undesirable overshoots and undershoots [14]. Recently, PDE-based image compression techniques have been developed using PDE-based interpolation of scattering data [15; 16; 17; 18; 19; 20]. The idea is to keep only a small number of pixels and reconstruct the remaining data using PDE-based interpolation. There are many subdivision strategies, in particular methods based on quadtree decompositions [21; 22]) and adaptive triangulation ideas can be found in [23; 24; 25]. In R-EED [18], the adaptive triangulation from [17] is replaced by a subdivision into a rectangular structure with various modifications. **Reliability and efficiency. ** PDE-based image compression techniques have been proposed as a means of reducing the size of image data while preserving image quality. However, these methods have both strengths and limitations that affect their reliability and efficiency. One of the limitations of PDE-based compression techniques is their high computational complexity, which can cause them to be computationally intensive and time-consuming, especially for large images or real-time applications [26]. Another limitation of PDE-based compression techniques is their sensitivity to image content. PDE-based methods rely on modeling the image data using PDEs, which may not always accurately capture the structure and content of the image. In some cases, PDE-based methods may not perform well for certain types of images, such as those with sharp edges or high-frequency content [27]. Most PDE-based compression techniques are lossy, which means that they discard some information from the original image in order to achieve compression; while lossy compression can be effective in reducing the size of the image data, it can also result in perceptual distortions or artifacts in the compressed image, which can affect its quality and reliability [28]. In addition, PDE-based compression techniques may not always achieve the same level of compression performance as other state-of-the-art image compression methods, such as those based on deep learning or neural networks. In some cases, PDEbased methods may not be able to achieve the same level of compression while preserving the same level of image quality [7; 29]. In conclusion, while PDE-based image compression techniques have shown promise in reducing the size of image data, their reliability and efficiency depend on various factors, such as the specific method used, the nature of the image data being compressed, and the compression requirements of the application. Further research is needed to improve the reliability and efficiency of PDE-based compression methods, and to address their limitations and challenges. **Our contribution. ** The aim of this paper is to investigate the compression capabilities of unaided PDE-based compression methods and PDE-based compression methods using data transformations and symbol prediction. The compression methods and ratios for grayscale compression and binary tree structure compression are evaluated in terms of the extent to which the quality of a reconstructed image depends on the choice of compression methods. The improvement of the compression algorithms is performed by applying data transforms to the input data stream. The data transformations used are delta coding, context tree weighting, Burrows-Wheeler transform, prediction by partial fitting, and dynamic Markov coding. **Related work. ** PDE-based compression was evaluated with range coding and Burrows-Wheeler transform for grey values only [30] and concluded that it performed better than EED with this combination.