In this paper, we examine new motion compensation methods based on the affine or bilinear transformation and derive fast algorithms for affine and bilinear transformation using vector relationship. We also develop a more effective motion estimation method than the conventional image warping method in terms of computational complexity, reconstructed image quality, and the number of coding bits. The performance of the proposed motion compensation method, which combines the affine or the bilinear transformation with the proposed adaptive partial matching, is evaluated experimentally. We simulate our proposed motion compensation method in a DCT- based coder by encoding CIF (Common Intermediate Format) images at bitrates of below 64 kb/s. The proposed adaptive partial matching method can reduce the computational complexity below about 50% of the hexagonal matching method, while maintaining the image quality comparable to the hexagonal method.