Intravoxel incoherent motion (IVIM) imaging of diffusion and perfusion parameters in the brain using parallel imaging suffers from local noise amplification. To address the issue, signal correlations in space and along the diffusion encoding dimension are exploited jointly using a constrained image reconstruction approach. IVIM imaging was performed on a clinical 3 T MR system with diffusion weighting along six gradient directions and 16 b‐values encoded per direction across a range of 0–900 s/mm2. Data were collected in 11 subjects, retrospectively undersampled in k‐space with net factors ranging from 2 to 6 and reconstructed using CG‐SENSE and the proposed k‐b PCA approach. Results of k‐b PCA and CG‐SENSE from retrospectively undersampled data were compared with those from the fully sampled reference. In addition, prospective single‐shot k‐b undersampling was implemented and data were acquired in five additional volunteers. IVIM parameter maps were derived using a segmented least‐squares method. The proposed k‐b PCA method outperformed CG‐SENSE in terms of reconstruction errors for effective undersampling factors of 3 and beyond. Undersampling artifacts were effectively removed with k‐b PCA up to sixfold undersampling. At net sixfold undersampling, relative errors (compared with the fully sampled reference) of image magnitude and IVIM parameters (D, f and D*) were (median ± interquartile range): 3.5 ± 3.7 versus 25.3 ± 25.8%, 2.7 ± 3.6 versus 14.2 ± 20.4%, 15.1 ± 26.1 versus 96.6 ± 67.4% and 14.8 ± 26.6 versus 100 ± 195.1% for k‐b PCA versus CG‐SENSE, respectively. Acquisition with sixfold prospective undersampling yielded average IVIM parameters in the brain of 0.79 ± 0.18 × 10−3 mm2/s for D, 7.35 ± 7.27% for f and 7.11 ± 2.39 × 10−3 mm2/s for D*. Constrained reconstruction using k‐b PCA improves IVIM parameter mapping from undersampled data when compared with CG‐SENSE reconstruction. Prospectively undersampled single‐shot echo planar imaging acquisition was successfully employed using k‐b PCA, demonstrating a reduction of image artifacts and noise relative to parallel imaging.