New Fundamental Formulas of Image Restoration in Spatial and Frequency Domains

Circular convolutions and the corresponding frequency domain formula are fundamentally important in image restoration; however, in this paper, we'll prove that the usual computing method of circular convolutions violates the physical meaning of blur producing. Especially for the image restoration algorithms in frequency domain, this violation will affect the restoration result. Relevant problems are proved rigorously and modified formulas are given in both spatial and frequency domains. Experiments are done to show the effects of new formulas. For clarity of proving, the one-dimensional case is dealt with first, which may be useful in one-dimensional signal processing.