High-resolution imaging with near-field millimeter-wave radar typically relies on extensive data acquisition. Existing time-domain and frequency-domain imaging algorithms process signals under the condition of satisfying the Nyquist sampling rate, which imposes significant burdens on data collection and hardware costs. This paper proposes a millimeter-wave radar sparse imaging algorithm based on compressive sensing theory, leveraging the sparsity of the measured target echo signals to effectively reduce data requirements. The algorithm constructs a sparse model based on the sparsity exhibited by undersampled data in the wavenumber domain, and optimizes it to reconstruct the signal. A matched filtering algorithm is applied in the azimuth direction to achieve two-dimensional imaging of the target. Experimental results demonstrate that under conditions of data undersampling, the proposed algorithm can achieve high-resolution imaging of the target, significantly reducing data requirements. Moreover, the image quality outperforms other compressed sensing optimization algorithms in all metrics. Even under conditions where the target object is partially occluded, the algorithm can effectively restore the occluded portions of the image, demonstrating strong interference resistance and robustness.