Optimal transform coding in the presence of quantization noise

The optimal linear Karhunen-Loeve transform (KLT) attains the minimum reconstruction 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 a finite number of bits requires the presence of quantizers. We formulate the optimal linear transform using a data model that incorporates the quantization noise. Our solution does not correspond to an orthogonal transform and in fact, it achieves a smaller mean squared error (MSE) compared to the KLT, in the noisy case. 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 modification of the discrete cosine transform (DCT). Our coding experiments show a peak signal-to noise ratio (SNR) performance improvement over JPEG of the order of 0.2 dB with an overhead less than 0.01 b/pixel.