The kernel design on the sector is done using a modified version of the Local Polynomial Approximation procedure.
This technique allows to design estimation kernels for smoothing and for differentiation.
1) Unlike many other transforms which start from the continuous domain and then discretized, this technique works directly in the multidimensional discrete domain;
Directional LPA kernels have a number of valuable benefits:
2) The designed kernel are truly multivariable, non-separable and anisotropic with arbitrary orientation, width and length;
3) The desirable smoothness of the kernel along and across the main direction is enabled by the corresponding vanishing moment conditions;
4) The kernel support can be flexibly shaped to any desirable geometry in order to capture geometrical structural and pictorial information. In this way a special design can be done for complex form objects and specific applications;
5) The smoothing and corresponding differentiating directional kernels can be designed;
6) These kernels are by definition asymmetric, allowing efficient edge adaptation. Traditional symmetric or nearly-symmetric supports tend to produce either so-called ringing artifacts or oversmoothing in the vicinity of the edges.
The Directional LPA allows to consider several different problems within a unified framework.