## 1 Introduction  
Partial differential Equation (PDE)-based image compression techniques use PDEs to model the image data and convert it into a set of parameters that can be efficiently encoded. There are several PDE-based image compression techniques that differ in the way they model and transform the image data. PDE-based image enhancement methods were invented to restore missing image regions by uniformly transferring information from the known surrounding regions. The filling effect has also become the main feature of PDE-based inpainting methods such as [1; 2; 3; 4]. The main idea is to consider the image data as Dirichlet boundary conditions and interpolate the data in the inpainting regions by solving the appropriate boundary value problems. **Why use PDE-based image compression? ** PDE-based image compression methods offer several advantages over traditional compression techniques such as JPEG and MPEG. Firstly, they offer higher compression ratios while maintaining better image quality. This is achieved by modeling the image as a set of partial differential equations and transforming it into a set of parameters that can be efficiently encoded. The resulting compressed image contains much less data, causing it to be easier to store and transmit [5]. Secondly, PDE-based compression methods are more robust to noise and distortion. Unlike traditional compression techniques, PDE-based methods rely on modeling the image using differential equations, which can adapt to changes in the image data, causing them to be more tolerant to noise and distortion. This causes them to be particularly useful in scenarios where the image quality needs to be maintained despite noise and other distortions [6; 7]. Thirdly, PDE-based compression methods are versatile and can be adapted to various applications. For instance, they can be used for compressing medical images, satellite images, and video streams, among others. This causes them to be a valuable tool in various domains, including medical imaging, remote sensing, and multimedia [8; 9]. Overall, PDE-based image compression methods offer a compelling alternative to traditional compression techniques. They offer higher compression ratios, better image quality, and are more robust to noise and distortion. Moreover, they are versatile and can be adapted to different applications. **Existing PDE-based compression algorithms. ** The Embedded Zero-Tree Wavelet (EZW) algorithm is a compression technique that uses the wavelet transform to decompose an image into sub-bands, and then applies a PDE-based thresholding to produce a sparse representation of the image. The resulting sparse representation is then encoded using entropy coding. EZW is widely used in image and video compression [10]. The SPIHT (Set Partitioning In Hierarchical Trees) algorithm is a wavelet-based compression algorithm that uses a PDE-based algorithm to sort and partition the wavelet coefficients in a hierarchical tree structure. This tree structure is then used to encode the wavelet coefficients using binary arithmetic coding. SPIHT has been used in various applications, such as satellite imagery and medical imaging [11]. The Geometric Image Compression (GIC) algorithm is a PDE-based compression algorithm that uses a geometric representation of images based on a set of partial differential equations. GIC models the image data as a series of curves, areas, and volumes, and then applies PDE-based compression techniques to the curves, areas, and volumes to create a compact representation of the image. GIC has proven to be very powerful in compressing geometric data such as 3D meshes and point clouds [12]. The Variational Image Compression (VIC) algorithm is a PDE-based compression algorithm that uses a variational approach to model the image data. VIC solves an optimization problem that minimizes an energy function containing both fidelity to the original image and a regularization term that promotes the smoothness of the compressed image. The resulting compressed image is then encoded using entropy coding.