Implementing Continuous Convolutions
نویسنده
چکیده
The convolution operator plays an important role in image processing. In many cases, convolutions are used in continuous domain models, for example, Gaussian convolutions used in scale-space (Koenderink, 1984). Given that image data is inherently discrete, a common approach to approximate the continuous model is to convert the continuous convolutions to their discrete counterparts by replacing the the integrals with sums. This however may lead to inaccuracies (Hummel & Lowe, 1989; van den Boomgaard, 2001). Alternatively, Hummel and Lowe (Hummel & Lowe, 1989) and later van den Boomgaard and van der Weij (van den Boomgaard, 2001) propose the following scheme: (1) reconstruct the continuous form of the given discrete signal, (2) apply the continuous convolution, and (3) sample the result. Equivalently, this scheme can be reduced to a discrete convolution of the input signal in the discrete domain by the sampled version of the result of convolving the reconstruction kernel (i.e., interpolation) with the filter (computed offline). The remainder of this note formalizes this scheme.
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