Low bit-rate image coding using adaptive geometric piecewise polynomial approximation

We present a new image coding algorithm, the geometric piecewise polynomials (GPP) method, that draws on recent developments in the theory of adaptive multivariate piecewise polynomials approximation. The algorithm relies on a segmentation stage whose goal is to minimize a functional that is conceptually similar to the Mumford-Shah functional except that it measures the smoothness of the segmentation instead of the length. The initial segmentation is "pruned" and the remaining curve portions are lossy encoded. The image is then further partitioned and approximated by low order polynomials on the subdomains. We show examples where our algorithm outperforms state-of-the-art wavelet coding in the low bit-rate range. The GPP algorithm significantly outperforms wavelet based coding methods on graphic and cartoon images. Also, at the bit rate 0.05 bits per pixel, the GPP algorithm achieves on the test image Cameraman, which has a geometric structure, a PSNR of 21.5 dB, while the JPEG2000 Kakadu software obtains PSNR of 20 dB. For the test image Lena, the GPP algorithm obtains the same PSNR as JPEG2000, but with better visual quality at 0.03 bpp.