Optimal transform coding in the presence of quantization noise

The optimal linear Karhunen-Loeve transform (KLT) attains the minimum recon struction error for a fixed number of transform coefficients assuming that these coefficients do not contain noise. In any real coding system, however , the representation of the coefficients using finite number of bits requir es the presence of quantizers. In this paper, we formulate the optimal line ar transform using a data model that incorporates quantization noise, Our s olution does not correspond to an orthogonal transform and in fact, it achi eves smaller mean squared error (MSE) compared to the KLT, in the noisy cas e. Like the KLT our solution depends on the statistics of the input signal, but it also depends on the bit-rate used for each coefficient. Especially for images, based on our optimality theory, we propose a simple modificatio n of the discrete cosine transform (DCT). Our coding experiments show. peak signal-to-noise ratio (SNR) performance improvement over JPEG of the order of 0.2 dB with overhead less than 0.01 b/pixel.