A common practice in low bit-rate image/video compression is uniform spatial down-sampling at the encoder and upsampling at the decoder. The down-sampling is performed in conjunction with deterministic low-pass filtering (e.g., Gaussian or the alike) to prevent aliasing. The down-sampled image is compressed and decompressed as usual; the upsampling is treated as an image restoration problem. In this paper, we show that the rate-distortion performance of the above low bit-rate image coding system can be improved, if the deterministic low-pass down-sampling filter is replaced by a random convolution kernel. The resulting down-sampled image is a two-dimensional array of local random measurements; this smaller image is still compressible in most cases. Accordingly, the decoder recovers the image from these local random measurements in the framework of compressive sensing. Theoretical analysis is conducted to support the superior performance of the proposed new method over its predecessors, and it is corroborated by our simulation results. At low to medium bit rates, the new method outperforms not only JPEG 2000 but also our earlier low bit-rate image codec CADU, with clear advantages over the competing methods in the reconstruction of high frequency features. In addition, the new method retains the system advantages of low encoder complexity and standard compliance as in CADU.