Hypercomplex Fourier transforms based on quaternions have been used for gray and color image processing. In this paper, we present the relations between the quaternion Fourier spectral coefficients. Using these relations, we can separate the scalar and vector part of quaternion image for frequency domain and find the constraints of the Fourier spectral coefficients which the quaternion Fourier transform of these spectral coefficients will have zero scalar part. In addition, we can calculate the DQFT of four real images, or the DQFT of two complex images by only one DQFT and we can design a color cosine image from frequency domain. Finally, we discuss the property of the DQFT of a causal image.