We propose two new image compression-decompression methods that reproduce images with better visual fidelity, less blocking artifacts, and better PSNR, particularly in low bit rates, than those processed by the JPEG Baseline method at the same bit rates. The additional computational cost is small, i.e., linearly proportional to the number of pixels in an input image. The first method, the "full mode" polyharmonic local cosine transform (PHLCT), modifies the encoder and decoder parts of the JPEG Baseline method. The goal of the full mode PHLCT is to reduce the code size in the encoding part and reduce the blocking artifacts in the decoder part. The second one, the "partial mode" PHLCT (or PPHLCT for short), modifies only the decoder part, and consequently, accepts the JPEG files, yet decompresses them with higher quality with less blocking artifacts. The key idea behind these algorithms is a decomposition of each image block into a polyharmonic component and a residual. The polyharmonic component in this paper is an approximate solution to Poisson's equation with the Neumann boundary condition, which means that it is a smooth predictor of the original image block only using the image gradient information across the block boundary. Thus, the residual--obtained by removing the polyharmonic component from the original image block--has approximately zero gradient across the block boundary, which gives rise to the fast-decaying DCT coefficients, which, in turn, lead to more efficient compression-decompression algorithms for the same bit rates. We show that the polyharmonic component of each block can be estimated solely by the first column and row of the DCT coefficient matrix of that block and those of its adjacent blocks and can predict an original image data better than some of the other AC prediction methods previously proposed. Our numerical experiments objectively and subjectively demonstrate the superiority of PHLCT over the JPEG Baseline method and the improvement of the JPEG-compressed images when decompressed by PPHLCT.