# On Alpha-Expansion-Based Graph-Cut Optimization for Decoder-Side Depth Estimation
## 1 Introduction  
The evolution of immersive video has dramatically enhanced how users interact with digital content, as new immersive experiences characterized by high levels of realism and interactivity are presented [1]. In complex immersive video systems, it is necessary to provide 3D representation of a scene. This is a critical component in achieving realism, as it is required to render virtual views requested by the viewer [2]. It can be acquired, for instance, using the depth estimation process or depth cameras [3]. When a larger number of depth maps is sent, the bitrate required to preserve proper geometry becomes too high, especially for low bitrates (as it can constitute more than 50% of the whole bitstream [4]). The use of a dedicated codec for depth maps provides satisfactory results, but such codecs are not standardized and are not sufficiently widespread to be used in practical cases. To address this, the current state-of-the-art ISO/IEC 23090-12 Moving Picture Experts Group (MPEG) Immersive Video (MIV) coding standard [2] is codec-agnostic; i.e., it assumes the use of the same internal video compression to encode texture and geometry. Compressing depth maps with traditional video encoding methods is also challenging due to the unique statistical properties of depth maps, which differ significantly from natural video content [4]. Such traditional video encoding methods struggle to preserve sharp edges and precise depth information critical for rendering accurate 3D scenes. It can lead to potential inaccuracies and a diminished user experience in immersive applications [5]. To overcome these challenges, decoder-side depth estimation has been proposed [6] as one of the types of immersive video coding in which depth information must be reconstructed from compressed video data. In such an approach, the primary challenge lies in dealing with compression artifacts and ensuring that depth estimation algorithms operate effectively within the constraints of minimizing the complexity of the decoding process. Compression-induced artifacts can impact the quality of the video and, as a result, depthestimation accuracy. These errors include the displacement of object edges (which can lead to inaccuracies in spatial relationships between views) and minimized levels of detail (which reduces the texture and nuances within the scene, making it challenging to achieve precise depth estimation) [7]. In the incoming second edition of MIV [8], the depth estimation step is assumed to simultaneously enhance the quality of a subset of transmitted depth maps as well as the quality of depth maps estimated for remaining views [9]. This further increases the importance of providing faster and more reliable depth estimation methods for immersive video. Unfortunately, conventional depth estimation methods often struggle to provide high quality in real time, while learning-based methods can be used only for content captured with a linear arrangement of cameras [4]. Even for such content, studies show that the quality of depth estimated with learning-based methods for immersive video is lower than for conventional depth estimators [10]. Graph-cut optimization techniques, particularly the alpha-expansion algorithm, have shown promise in various computer vision tasks because of their ability to solve pixel labeling problems efficiently [11]. Graph-cut-based techniques constitute the basis for the two last reference depth estimation methods of MPEG video coding (i.e., Depth Estimation Reference Software--DERS [12] and Immersive Video Depth Estimation--IVDE [4]). Unfortunately, this step has not been fully optimized for decoder-side depth estimation, where specific challenges such as compressed video artifacts and limited computational resources must be addressed. Adapting these techniques to handle the distortions introduced by compression is crucial for improving depth estimation accuracy in immersive video. The abovementioned challenges motivated our research presented in this paper.