We present a new variational-based method for automatic liver vessels segmentation from abdominal CTA images. The segmentation task is formulated as a functional minimization problem within a variational framework. We introduce a new functional that incorporates both geometrical vesselness measure and vessels surface properties. The functional describes the distance between the desired segmentation and the original image. To minimize the functional, we derive the Euler-Lagrange equation from it and solve it using the conjugate gradients algorithm. Our approach is automatic and improves upon other Hessian-based methods in the detection of bifurcations and complex vessels structures by incorporating a surface term into the functional. To assess our method, we conducted with an expert radiologist two comparative studies on 8 abdominal CTA clinical datasets. In the first study, the radiologist assessed the presence of 11 vascular bifurcations on each dataset, totaling of 73 bifurcations. The radiologist qualitatively compared the bifurcations segmentation of our method and that of a Hessian-based threshold method. Our method correctly segmented 88% of the bifurcations with a higher visibility score of 82%, as compared to only 55% in the Hessian-based method with a visibility score of 33%. In the second study, the radiologist assessed the individual vessels visibility on the 3D segmentation images and on the original CTA slices. Ten main liver vessels were examined in each dataset The overall visibility score was 93%. These results indicate that our method is suitable for the automatic segmentation and visualization of the liver vessels.