Partition of Unity

How can we assess the inherent quality of a given synthesis function? We answer this question gradually, developing it more in the next section, and we proceed at first more by intuition than by a rigorous analysis. Let us consider that the discrete sequence of data we want to interpolate is made of samples that all take exactly the same value fk = f0 for any k e Zq. In this particular case, we intuitively expect that the interpolated continuous function f (x) should also take a constant value (preferably the same f0) for all arguments x e Rq. This desirable property is called the reproduction of the constant. Its relevance is particularly high in image processing because the spectrum of images is very often concentrated toward low frequencies. From (1), we derive

This last equation is also known as the partition of unity. It is equivalent to impose that its Fourier transform satisfies some sort of interpolation property in the Fourier domain (see Appendix). The reproduction of the constant is also desirable for a noninterpolating synthesis function which is equivalent to asking that it satisfy the partition of unity condition, too. To see why, we remember that the set of coefficients Ck used in the reconstruction equation (4) is obtained by digital filtering of the sequence of samples { ..., fo, fo, fo,...}. Since the frequency representation of this sequence is exclusively concentrated at the origin, filtering will simply result in another sequence of coefficients { ..., co, c0, co,...} that also have a constant value. We have that

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