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Hence, the theoretical problem arises concerning how to discretize the continuous theory while either preserving or well approximating the desirable theoretical properties that lead to the choice of the Gaussian kernel (see the article on scale-space axioms).
Once established, the axioms narrow the possible scale-space representations to a smaller class, typically with only a few free parameters. A set of standard scale space axioms, discussed below, leads to the linear Gaussian scale-space, which is the most common type of scale space used in image processing and computer vision.
For temporal smoothing in real-time situations, one can instead use the temporal kernel referred to as the time-causal limit kernel, [71] which possesses similar properties in a time-causal situation (non-creation of new structures towards increasing scale and temporal scale covariance) as the Gaussian kernel obeys in the non-causal case. The ...
From this classification, it is apparent that we require a continuous semi-group structure, there are only three classes of scale-space kernels with a continuous scale parameter; the Gaussian kernel which forms the scale-space of continuous signals, the discrete Gaussian kernel which forms the scale-space of discrete signals and the time-causal ...
A simple answer is to sample the continuous Gaussian, yielding the sampled Gaussian kernel. However, this discrete function does not have the discrete analogs of the properties of the continuous function, and can lead to undesired effects, as described in the article scale space implementation.
Output after kernel PCA, with a Gaussian kernel. Note in particular that the first principal component is enough to distinguish the three different groups, which is impossible using only linear PCA, because linear PCA operates only in the given (in this case two-dimensional) space, in which these concentric point clouds are not linearly separable.
The Gaussian kernel is continuous. Most commonly, the discrete equivalent is the sampled Gaussian kernel that is produced by sampling points from the continuous Gaussian. An alternate method is to use the discrete Gaussian kernel [10] which has superior characteristics for some purposes.
A kernel smoother is a statistical technique to estimate a real valued function: as the weighted average of neighboring observed data. The weight is defined by the kernel, such that closer points are given higher weights. The estimated function is smooth, and the level of smoothness is set by a single parameter.