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Linear continuous filters and integral representations are standard for theory and for many applications. In comparison with discrete filters, continuous ones are simpler in many aspects and have more transparent properties.
An integral version of LPA filters is introduced in two different ways. An integral form of the estimates is derived first directly from the LPA approach. To do this, the sum in the criterion in Eq. (2.24) is replaced by the corresponding integral. Calculations similar to the ones produced in Sec. 2.2 give the continuous estimates and their integral kernel representations.
Another approach starts from the regular-grid discrete filters with the sampling interval Î approaching zero. This limit gives the LPA kernels and estimates in the integral forms identical to the integral LPA. In this way the integral kernels and the estimates are considered as approximations of the discrete ones. The term "equivalent kernels" is used for the integral kernels obtained in this asymptotic consideration with a reference to the original discrete ones. We start from the 2D signals and then pass to the general d-dimensional case.
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