This paper introduces a highly efficient algorithm capable of jointly estimating scale and rotation between two images with sub-pixel precision. Image alignment serves as a critical process for spatially registering images captured from different viewpoints, and finds extensive use in domains such as medical imaging and computer vision. Traditional phase-correlation techniques are effective in determining translational shifts; however, they are inadequate when addressing scale and rotation changes, which often arise due to camera zooming or rotational movements. In this paper, we propose a novel algorithm that integrates scale and rotation estimation based on the Fourier transform in log-polar coordinates with a cross-correlation maximization strategy, leveraging the auxiliary function method. By incorporating sub-pixel-level cross-correlation our method enables precise estimation of both scale and rotation. Experimental results demonstrate that the proposed method achieves lower mean estimation errors for scale and rotation than conventional Fourier transform-based techniques that rely on discrete cross-correlation.