The application of complex measurement matrices in image-compressed sensing (CS) has attracted significant attention; however, effectively combining their structural, random, and orthogonal characteristics remains a substantial challenge. We propose a complex measurement matrix that integrates randomness and orthogonality into a structured discrete Fourier transform (DFT) matrix, termed ROS-DFT. We first introduce randomness to the DFT matrix through random phase encoding (RPE). Subsequently, sequences generated by the Logistic map are employed to shuffle index arrays of the matrix, which further augments matrix randomness. Finally, the Gram–Schmidt orthogonalization is applied to the columns of the measurement matrix, recovering column orthogonality, which is lost during the randomization operation. In addition, sparsification of the measurement matrix is achieved by randomly preserving a fixed number of non-zero elements in each column to reduce the storage space and minimize the computational complexity. Extensive qualitative and quantitative experiments demonstrate the effectiveness of the proposed method in significantly enhancing the quality of reconstructed images across various CS ratios compared with state-of-the-art methods.