The existing lensless compressive camera (L2C2) [1] suffers from low capture rates, resulting in low resolution images when acquired over a short time. In this work, we propose a new regime to mitigate these drawbacks. We replace the global-based compressive sensing used in the existing L2C2 by the local block (patch) based compressive sensing. We use a single sensor for each block, rather than for the entire image, thus forming a multiple but spatially parallel sensor L2C2. This new camera retains the advantages of existing L2C2 while leading to the additional benefits of fast acquisition, real time reconstruction, and the flexibility to image size. We develop multiple geometries of this block-wise L2C2 in this paper. We have built prototypes of the proposed blockwise L2C2 and demonstrated excellent results of real data.