Interval + Image = Wavelet: For Image Processing under Interval Uncertainty, Wavelets Are Optimal

In computer and electronic manufacturing, it is very important to be able to automatically check whether the surface mounted devices (SMD) are correctly placed on the printed circuit boards. The inspection of these boards has to be done on a shop floor, where statistical characteristics of the noise vary so much that, in essence, we only have interval estimates for this noise. We show that under this interval uncertainty, the optimal image processing technique consists of using Haar wavelets. Wavelets indeed lead to much better results than previously used Fourier transform techniques. On a more fundamental level, our result is a step towards solving an important problem related to wavelets: that wavelet transforms often empirically work much better than other methods, but there are very few theoretical explanations of this efficiency. Our results shows that, probably, such a theoretical explanation can be obtained if we take interval uncertainty into consideration.