Influence of Unknown Exterior Samples on Interpolated Values for Band-Limited Images

The growing size of digital images and their increasing information content in terms of bits per pixel (or SNR) leads to ask to which extent the known samples permit to restore the underlying continuous image. In the context of band-limited data the Shannon-Whittaker theory gives an adequate theoretical answer provided that infinitely many samples are measured. Yet, we show that the current accuracy of digital images will be limited in the future by the truncation error, which is no-more negligible with respect to other perturbations such as quantization or aliasing. To do so, we propose a method to estimate the truncation error. All of our results are expressed in terms of Root Mean Squared Error (RMSE) under the common hypothesis of band-limited weakly stationary random processes. As a first contribution, we present a general expression of the truncation RMSE involving the spectral content of the image. We then derive a simple and generic scheme to evaluate bounds on the truncation error. The actual computation of error bounds is conducted for two standard interpolation schemes, namely the Shannon-Whittaker and the DFT interpolators. These theoretical bounds reveal a specific decay of the truncation error as a function of the distance from the sample to the image boundary. The tight estimates obtained and validated on a set of experiments confirm that the truncation error can become the main error term in HDR images. In classic 8 bit images it is bound by the quantization error at a moderate distance from the image boundary, but still requires large images to become manageable.