## 1 Introduction  
Liver cancer caused 830,200 deaths globally in 2020, ranking among the top three causes of cancer mortality in 46 countries, and is projected to increase by more than 55% by 2040, highlighting its significant global health threat [1]. One of the most common types of malignant liver tumors is hepatocellular carcinoma, the leading cause of liver cancer-related deaths. These tumors often appear as irregular nodules with hard surfaces and uneven edges, providing essential information for medical diagnosis and treatment planning [2]. Medical professionals widely use radiological imaging techniques such as CT, MRI, and ultrasound to detect liver abnormalities and guide therapeutic decisions. Among these methods, computed tomography (CT) is preferred due to its higher spatial resolution, rapid acquisition process, wide availability, and durability, making it the primary imaging modality for liver analysis [3]. Segmenting the liver and liver tumors from CT images accurately is crucial for therapeutic applications. However, the variability in tumor size, shape, and location makes the accurate, semi-automatic segmentation of hepatic tumors a highly challenging task [4]. Computer-aided image interpretation powered by artificial intelligence offers greater objectivity, speed, and accuracy compared to human analysis [5; 6]. Manual segmentation methods remain widely used by clinicians; however, they are time-consuming and susceptible to errors due to human interpretation and differences in radiologists' expertise. This highlights the urgent need for automated and highly efficient segmentation techniques [7]. Extensive research has led to the development of various machine learning algorithms for liver and tumor segmentation, which can be broadly classified into traditional and deep learning-based approaches [8]. Traditional methods include region-growing [9; 10], thresholding [11], active contour [12; 13], and graph cutting [14; 15]. However, these methods have certain limitations. For instance, the region-growing model requires the human selection of an initial growth point, which can lead to uneven pixel distribution and over-segmentation. Additionally, traditional methods rely on manually designed features, often limited in representation capability and heavily influenced by prior knowledge. Methods like thresholding struggle with blurry lesions, resulting in low segmentation efficiency. Consequently, traditional approaches exhibit low accuracy and perform poorly on irregular targets, such as liver tumors. In recent years, deep learning-based segmentation methods, particularly those utilizing deep neural networks, have emerged as dominant approaches in medical image analysis. However, these approaches depend on accurate pixel-wise annotation, which is difficult to acquire due to the high cost of annotation and the need for deep expertise [16]. Accurate tumor segmentation is crucial for generating volume-based quantitative data, such as textural features essential for hepatic tumor classification, treatment response evaluation, and patient survival prediction [17]. Despite the benefits of automatic liver and tumor segmentation methods in delivering consistent and objective outcomes, their practical implementation remains hindered by many challenges. One major issue is the low-intensity contrast between the liver and surrounding organs, such as the lungs, pancreas, or stomach, which complicates accurate segmentation [18]. Additionally, contrast injection protocols, commonly used by radiologists to enhance tumor visibility, often introduce noise and blur liver boundaries, further complicating the process. Also, the segmentation of the tumor is more challenging, given that the number, location, size, and shape of liver tumors differ considerably between patients [19]. Moreover, the indistinct boundaries of tumors and the poor contrast between tumors and surrounding tissues present additional hurdles for computer-assisted segmentation methods [20].