The Euler number is an important topological property in a binary image, and it can be computed by counting certain bit-quads in the binary image. This paper proposes a further improved bit-quad-based algorithm for computing the Euler number. By scanning image rows two by two and utilizing the information obtained while processing the previous pixels, the number of pixels to be checked for processing a bit-quad can be decreased from 2 to 1.5. Experimental results demonstrated that our proposed algorithm significantly outperforms conventional Euler number computing algorithms.