The problem of interactive foreground/background segmentation in still images is of great practical importance in image editing. They avoid the boundary-length bias of graph-cut methods and results in increased sensitivity to seed placement. A new proposed method of fully automatic processing frameworks is given based on Graph-cut and Geodesic Graph cut algorithms. This paper addresses the problem of segmenting liver and tumor regions from the abdominal CT images. The lack of edge modelling in geodesic or similar approaches limits their ability to precisely localize object boundaries, something at which graph-cut methods generally excel. A predicate is defined for measuring the evidence for a boundary between two regions using Geodesic Graph-based representation of the image. The algorithm is applied to image segmentation using two different kinds of local neighborhoods in constructing the graph. Liver and hepatic tumor segmentation can be automatically processed by the Geodesic graph-cut based method. This system has concentrated on finding a fast and interactive segmentation method for liver and tumor segmentation. In the pre-processing stage, Mean shift filter is applied to CT image process and statistical thresholding method is applied for reducing processing area with improving detections rate. In the Second stage, the liver region has been segmented using the algorithm of the proposed method. Next, the tumor region has been segmented using Geodesic Graph cut method. Results show that the proposed method is less prone to shortcutting than typical graph cut methods while being less sensitive to seed placement and better at edge localization than geodesic methods. This leads to increased segmentation accuracy and reduced effort on the part of the user. Finally Segmented Liver and Tumor Regions were shown from the abdominal Computed Tomographic image.